Incorrect Hecke Operators in conda-forge Sage 9.7–10.6 (fixed in 10.7)

[MaximKontsevich]

Steps To Reproduce

Create a clean conda environment:

 conda create -n sage_test -c conda-forge sage=10.6
 conda activate sage_test

Run the following one-line test:

 CuspForms(Gamma1(2), 10).hecke_matrix(5)

This computes the Hecke operator T_5 on the 1-dimensional space S_10(Gamma1(2)).

Expected Behavior

Since S_10(Gamma1(2)) is 1-dimensional, the Hecke operator T_5 must be the 1×1 matrix

 [870]

This is the correct value and agrees with:

• conda-forge sage=10.7
• Debian/apt sage=9.5
• SageCell server
• Theoretical value of a_5 for the unique normalized weight-10 level-2 eigenform.

Actual Behavior

In conda-forge builds of Sage 9.7, 10.1, 10.4, 10.5, and 10.6, the same code produces:

[534154/3]

This value is mathematically incorrect:

• it is not equal to 870,
• it is not integral,
• it cannot arise from any normalization convention,
• and it does not match the true Hecke eigenvalue.

The error occurs even at a good prime p=5, for a 1-dimensional cusp form space,
so it is not related to basis choice, modular symbols, or sign issues.

In conda-forge sage=10.7, the correct value [870] is obtained.

Additional Information

This bug appears to affect all conda-forge builds of SageMath from 9.7 through 10.6.
It does NOT appear in:

• conda-forge sage=10.7 (correct)
• Debian/apt sage=9.5 (correct)
• SageCell (correct)

This suggests a conda-forge build-chain or ABI issue rather than a Sage
source code bug.

The example above is intentionally minimal: a single one-line computation
on a 1-dimensional space, so the Hecke eigenvalue is completely unambiguous.

A regression test such as:

CuspForms(Gamma1(2), 10).hecke_matrix(5) == matrix([[870]])

would detect this problem automatically.

Environment

For the conda-forge versions showing the bug:

  OS: Linux (Ubuntu 22.04 and ChromeOS Linux container; bug reproduced on both)
  Sage version: 9.7, 10.1, 10.4, 10.5, 10.6 (conda-forge)
  Python: default conda-forge Python (3.11 or 3.12 depending on environment)
  Installation method: conda create -n sageX -c conda-forge sage=X

For the correct versions:

  Sage version (working): 10.7 (conda-forge), 9.5 (Debian/apt), SageCell

Checklist

• I have searched the existing issues for a bug report that matches the one I want to file, without success.
• I have read the documentation and troubleshoot guide

[cxzhong]

In 10.8 rc0 sage-the-distro

zhongcx@LAPTOP-A6042VHQ:~/sage$ ./sage
┌────────────────────────────────────────────────────────────────────┐
│ SageMath version 10.8.rc0, Release Date: 2025-12-02                │
│ Using Python 3.14.1. Type "help()" for help.                       │
└────────────────────────────────────────────────────────────────────┘
┏━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓
┃ Warning: this is a prerelease version, and it may be unstable.     ┃
┗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛
sage: CuspForms(Gamma1(2), 10).hecke_matrix(5)
[870]
sage: CuspForms(Gamma1(2), 10).hecke_matrix(5) == matrix([[870]])
True

[MaximKontsevich]

Thank you for mentioning version 10.8.rc0.

Formally, the issue is already resolved since conda-forge 10.7 produces the correct value as well.

However, the underlying situation is still quite serious:

• Many users are on earlier conda-forge releases (9.7 → 10.6), and all of these appear to produce mathematically incorrect Hecke eigenvalues.
• It is unclear whether other installation methods (e.g. source builds, distro packages, Windows Subsystem builds) were affected or not.
• Most importantly, it would be good for the Sage developers to understand what exactly went wrong in the build chain, and how the issue remained undetected for several years.

[cxzhong]

@dimpase Can you investigate this? I have no ideas about it

[dimpase]

@tscrim - you might know?

[dimpase]

@MaximKontsevich - this appears to be a Conda-only bug. It not reproducible in Sage 10.5/6/7 on cocalc (which are more or less vanilla Sage releases). Perhaps @saraedum or @isuruf have an idea what went wrong with Conda versions.

~/sage$ sage-10.5
┌────────────────────────────────────────────────────────────────────┐
│ SageMath version 10.5, Release Date: 2024-12-04                    │
│ Create a "Sage Worksheet" file for the notebook interface.         │
│ Enhanced for CoCalc.                                               │
│ Using Python 3.12.5. Type "help()" for help.                       │
└────────────────────────────────────────────────────────────────────┘
WARNING: your terminal doesn't support cursor position requests (CPR).
sage: CuspForms(Gamma1(2), 10).hecke_matrix(5)
[870]
sage:                                                                                                         
~/sage$ sage-10.6
┌────────────────────────────────────────────────────────────────────┐
│ SageMath version 10.6, Release Date: 2025-03-31                    │
│ Create a "Sage Worksheet" file for the notebook interface.         │
│ Enhanced for CoCalc.                                               │
│ Using Python 3.12.5. Type "help()" for help.                       │
└────────────────────────────────────────────────────────────────────┘
WARNING: your terminal doesn't support cursor position requests (CPR).
sage: CuspForms(Gamma1(2), 10).hecke_matrix(5)
[870]
sage:                                                                                                         
~/sage$ sage-10.7
┌────────────────────────────────────────────────────────────────────┐
│ SageMath version 10.7, Release Date: 2025-08-09                    │
│ Create a "Sage Worksheet" file for the notebook interface.         │
│ Enhanced for CoCalc.                                               │
│ Using Python 3.12.5. Type "help()" for help.                       │
└────────────────────────────────────────────────────────────────────┘
/ext/sage/10.7/local/var/lib/sage/venv-python3.12.5/lib/python3.12/site-packages/line_profiler/toml_config.py:111: DeprecationWarning: path is deprecated. Use files() instead. Refer to https://importlib-resources.readthedocs.io/en/latest/using.html#migrating-from-legacy for migration advice.
  with importlib.resources.path(package + '.rc',
WARNING: your terminal doesn't support cursor position requests (CPR).
sage: CuspForms(Gamma1(2), 10).hecke_matrix(5)
[870]
sage:    

[dimpase]

It's also a bug in a specific Conda setup, not a uniform across any Conda install with Sage 10.6.
E.g. I just ran these (note that uname=Linux, uname -m = x86_64)

curl -L -O "https://github.com/conda-forge/miniforge/releases/latest/download/Miniforge3-$(uname)-$(uname -m).sh"
bash Miniforge3-$(uname)-$(uname -m).sh
...
conda create -n sage_test -c conda-forge sage=10.6
conda activate sage_test

on a Gentoo Lnux x86_64 machine, and the test returns [870].

---

no luck in reproducing this on Intel macOS Conda, either.

[dimpase]

@MaximKontsevich - there can be something funky in your conda install. Can you reproduce the issue on a fresh conda install (following e.g. https://doc.sagemath.org/html/en/installation/conda.html#sec-installation-conda) ?

[MaximKontsevich]

Besides the x86-64 Chromebook where I have Debian 13.0, I have tried the same test on two Linux x86-64 machines both with Ubuntu 24.04.3 LTS. I just typed

conda create -n sage_test -c conda-forge sage=10.6
conda activate sage_test

then ran Sage, and the test gave me the wrong result. When I changed the version to 10.7, everything worked correctly. What I am seeing is definitely observable and reproducible at least on Ubuntu and Debian, using miniforge which I installed following the same site https://doc.sagemath.org/html/en/installation/conda.html#sec-installation-conda

[dimpase]

I tried reproducing on in a WSL on a Windows x86_64 machine with Ubuntu 24.04, and, again, no luck reproducing this. (neither with Sage 10.6, nor 10.5).

I am afraid it might have to do with not only software, but hardware, as well.
Can you post CPU models of your machines? I tried on Intel(R) Core(TM) i5-7300U and on Intel(R) Core(TM) Ultra 7 165U. (look at the output of cat /proc/cpuinfo)

[dimpase]

another common source of trouble is user-set LD_LIBRARY_PATH. Conda environments aren't totally isolated from the rest of the OS, so if LD_LIBRARY_PATH is set the linker will try first to pick up suitable libraries from the list of directories it points at.
The code in question uses Flint and Pari-GP, so it might potentially happen that these are picked up from a wrong location.

Perhaps there are similar Conda-only. pitfalls I am not aware of, it's a question for Conda experts (I'm not one of them, again @saraedum or @isuruf, or @tobiasdiez might say more here ).
E.g. if you have libflint.so (installed as a OS package, say), which only fits Sage 10.7, but not 10.6 or earlier, and Conda's sage chooses to load it for some reason, it would explain that 10.7 is not broken, but 10.6 or earlier is.

[MaximKontsevich]

Chromebook with Debian:
Intel(R) Core(TM) Ultra 7 155U

Laptop #1 with Ubuntu:
Intel (R) Core(TM) i7-10750H CPU @ 2.60Ghz

Laptop #2 with Ubuntu:
Intel (R) Core(TM) Ultra 7 155H

I don't remember doing anything unusual about installing Linux and Sage, just followed the instructions. All installs are quite recent, about 6 months old.

[tobiasdiez]

Thanks for the report.

From the discussion above, it seems likely that the older Sage versions available through conda either bring in older dependency versions that contain a bug, or that those Sage versions are not fully compatible with newer versions of certain dependencies (for example, by interpreting their output differently). Unfortunately, issues of this kind are generally outside what we can reliably fix or prevent.

It would probably be possible to track down the precise cause by debugging the problematic configuration and comparing it with the behavior in 10.7. However, since the issue no longer occurs in the current release, it’s not entirely clear whether the effort required would be worthwhile.

[dimpase]

IMHO it makes sense to dig this clause up, if only to prevent future SNAFUs like this.

I don't remember doing anything unusual about installing Linux and Sage, just followed the instructions. All installs are quite recent, about 6 months old.

You might have other soft on your machines, which may interfere (even though it should not).

in activated sage_test with Sage 10.6, please run

sage -sh

and at its prompt, run

echo $LD_LIBRARY_PATH
echo $PATH

The first echo should output empty string, and the second one something like

/home/dima/miniforge3/envs/sage_test/bin:/home/dima/miniforge3/envs/sage_test/bin:/home/dima/miniforge3/condabin

followed by more stuff, possibly. Please post what you get here (or send this data by email to me, if you think it should not be posted here)

If all this looks fine, we need to narrow the scope of the trouble. The first step would be to understand whether CuspForms(Gamma1(2), 10) is already wrong, or everything used to to compute Hecke matrix looks as it should be. Below I step through the (working) Sage 10.6 Conda code on my machine - on your machine any step below might go elsewhere.

The outcome of this investigation is that [870] (or the wrong answer) is obtained via

cf=CuspForms(Gamma1(2), 10); cf
# Cuspidal subspace of dimension 1 of Modular Forms space of dimension 3 for Congruence Subgroup Gamma0(2) of weight 10 over Rational Field
cf.basis()
# [q + 16*q^2 - 156*q^3 + 256*q^4 + 870*q^5 + O(q^6)]
A=cf.modular_symbols(sign=1); A
# Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 3 for Gamma_0(2) of weight 10 with sign 1 over Rational Field
Ap=A.ambient_hecke_module()._compute_hecke_matrix_prime(5); Ap
# [1953126   62976  -62976]
# [      0  945510 1007616]
# [      0  944640 1008486]
Ap.restrict(A.free_module())
# [870]
A.free_module()
# Vector space of degree 3 and dimension 1 over Rational Field
# Basis matrix:
# [ 0  1 -1]

Once we get [ 0 1 -1] and Ap as above, we see that the latter is a left eigenvector of Ap with eigenvalue 870.

So either A.free_module() or Ap is wrong on your machine. Perhaps cf is already wrong?

---

sage: cf=CuspForms(Gamma1(2), 10)
sage: cf.hecke_matrix??

will print the code of the function computing the matrix:
```python
Signature: cf.hecke_matrix(n)
Docstring:
WARNING: the enclosing module is marked 'needs sage.libs.flint sage.libs.pari',
so doctests may not pass.

   Return the matrix of the n-th Hecke operator acting on given basis.

   EXAMPLES:

      sage: C = CuspForms(1, 16)
      sage: C.hecke_matrix(3)
      [-3348]
Source:   
    def hecke_matrix(self, n):
        """
        Return the matrix of the `n`-th Hecke operator acting on given basis.

        EXAMPLES::

            sage: C = CuspForms(1, 16)
            sage: C.hecke_matrix(3)
            [-3348]
        """
        n = int(n)
        if n <= 0:
            raise IndexError("n must be positive.")
        if n not in self._hecke_matrices:
            T = self._compute_hecke_matrix(n)
            T.set_immutable()
            self._hecke_matrices[n] = T
        return self._hecke_matrices[n]
File:      ~/miniforge3/envs/sage_test/lib/python3.12/site-packages/sage/modular/hecke/module.py
Type:      method

thus, the real work is done by cf._compute_hecke_matrix. Again, we can look at the output of cf._compute_hecke_matrix?? and see that it's doing

        if hasattr(self, '_compute_q_expansion_basis') and self.character() is not None:
            return hecke.HeckeModule_generic._compute_hecke_matrix(self, n)

        else:
            # Try to avoid doing unnecessary computations where possible.
            if self.is_cuspidal():
                M = self.ambient().cuspidal_submodule().hecke_matrix(n).block_sum(zero_matrix(self.base_ring(), self.ambient().eisenstein_submodule().rank()))
            elif self.is_eisenstein():
                M = zero_matrix(self.base_ring(), self.ambient().cuspidal_submodule().rank()).block_sum(self.ambient().eisenstein_submodule().hecke_matrix(n))
            else:
                M = self.ambient().hecke_matrix(n)
            return M.restrict(self.free_module(), check=(gcd(n, self.level()) > 1))
File:      ~/miniforge3/envs/sage_test/lib/python3.12/site-packages/sage/modular/modform/space.py
Type:      method

we see, as

sage: cf.character()
Dirichlet character modulo 2 of conductor 1
sage: hasattr(cf, '_compute_q_expansion_basis')
True

that it does

        if hasattr(self, '_compute_q_expansion_basis') and self.character() is not None:
            return hecke.HeckeModule_generic._compute_hecke_matrix(self, n)

now we need to demangle the last line:

sage: import_statements("HeckeModule_generic")
from sage.modular.hecke.module import HeckeModule_generic

so the console call is

sage: sage.modular.hecke.module.HeckeModule_generic._compute_hecke_matrix(cf, 5)
# indeed, I get [870]

ok, let's look at sage.modular.hecke.module.HeckeModule_generic._compute_hecke_matrix??
It will tell that the call in this case boils down to

sage: cf._compute_hecke_matrix_prime(5)
# [870]

now cf._compute_hecke_matrix_prime?? tells us to do

sage: A=cf.modular_symbols(sign=1)
sage: A
# Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 3 for Gamma_0(2) of weight 10 with sign 1 over Rational Field
sage: A.hecke_matrix(5)
# [870]

continuing, one gets

sage: A.ambient_hecke_module().hecke_matrix(5)
# [1953126   62976  -62976]
# [      0  945510 1007616]
# [      0  944640 1008486]

which should be restricted to A.free_module() to get [870]

sage: Ap=A.ambient_hecke_module().hecke_matrix(5)
sage: Ap.restrict(A.free_module())
# [870]

[MaximKontsevich]

I tried

 echo $LD_LIBRARY_PATH
 echo $PATH

on all 3 machines I have, and got
/home/maxim/miniforge3/envs/sage_test/bin:/home/maxim/.local/bin:/home/maxim/bin:/home/maxim/miniforge3/envs/sage_test/bin:/home/maxim/miniforge3/condabin:/usr/local/bin:/usr/bin:/bin:/usr/local/games:/usr/games
on Chromebook (Debian)

and identical output on two other machines (both Ubuntu)
/home/maxim/miniforge3/envs/sage_test/bin:/home/maxim/miniforge3/envs/sage_test/bin:/home/maxim/miniforge3/condabin:/home/maxim/.local/bin:/usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin:/usr/games:/usr/local/games:/snap/bin.

After that, in sage I typed cf=CuspForms(Gamma1(2), 10); cf and got
Cuspidal subspace of dimension 1 of Modular Forms space of dimension 3 for Congruence Subgroup Gamma0(2) of weight 10 over Rational Field exactly like you,
but the next command cf.basis() gave me some error message on all 3 machines:

# [<repr(<sage.modular.modform.cuspidal_submodule.CuspidalSubmodule_g0_Q_with_category.element_class at 0x7ffb3a457f10>) failed: ArithmeticError: subspace is not invariant under matrix>]

which is not what you get: # [q + 16*q^2 - 156*q^3 + 256*q^4 + 870*q^5 + O(q^6)] !

Presumably, it will be sufficient to stop there and explore what goes wrong.
Still, I continued further,

A=cf.modular_symbols(sign=1); A
# Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 3 for Gamma_0(2) of weight 10 with sign 1 over Rational Field
Ap=A.ambient_hecke_module()._compute_hecke_matrix_prime(5); Ap

and got wrong matrix Ap:

[  1953126  813208/3 -813208/3]
[        0    879014   1074112]
[        0 2102888/3 3756490/3]

Then cf.hecke_matrix?? produces exactly yours output, as well as cf._compute_hecke_matrix?? (at least I can see with naked eye, - if you want I really copy/paste my outputs).
Then I reproduced:

sage: cf.character()
Dirichlet character modulo 2 of conductor 1
sage: hasattr(cf, '_compute_q_expansion_basis')
True

and

sage: import_statements("HeckeModule_generic")
from sage.modular.hecke.module import HeckeModule_generic

Finally, sage.modular.hecke.module.HeckeModule_generic._compute_hecke_matrix(cf, 5) gave me wrong matrix
[534154/3].

The same wrong matrix comes from cf._compute_hecke_matrix_prime(5).

If I continue with

sage: A=cf.modular_symbols(sign=1)
sage: A
# Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 3 for Gamma_0(2) of weight 10 with sign 1 over Rational Field
sage: A.hecke_matrix(5)

I get again the same wrong 1 by 1 matrix.
A.ambient_hecke_module().hecke_matrix(5) gives the same wrong 3 by 3 matrix as I showed earlier.