Hilbert newform lookup over field 4.4.1600.1

[Alexander Clouston]

Hello,

I am trying to identify the Hilbert modular newform attached to the elliptic curve

[
E/K:\quad y^2=x^3-(\sqrt2+\sqrt5)x+(\sqrt2+\sqrt5-1)
]

over

[
K=\mathbb Q(\sqrt2,\sqrt5),
]

LMFDB field label (4.4.1600.1).

The expected conductor is

[
\mathfrak N=
\mathfrak p_2^9(3,\sqrt2+\sqrt5)^2(41,\sqrt2+17,\sqrt5+13),
]

with norm

[
870580224.
]

The initial Frobenius/Hecke traces I have are:

[
N\mathfrak p=31: 0,-8,0,6,
]
[
N\mathfrak p=49: -2,8,
]
[
N\mathfrak p=71: 8,8,6,-4,
]
[
N\mathfrak p=79: -6,12,2,4,
]
[
N\mathfrak p=89: -2,14,6,6.
]

Could you tell me whether this Hilbert newform is in the database, and if so what its label is?

Thank you.

Alexander Clouston

[Edgar Costa]

https://www.lmfdb.org/ModularForm/GL2/TotallyReal/?field_label=4.4.1600.1&level_norm=870580224

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[Alexander Clouston]

Thank you, 

A quick follow-up: I ran Magma’s conductor computation for the elliptic curve

[

E/K:\quad y^2=x^3-(\sqrt2+\sqrt5)x+(\sqrt2+\sqrt5-1),

]

over (K=\mathbb Q(\sqrt2,\sqrt5)), field label (4.4.1600.1).

Magma gives conductor factorization:

[

\mathfrak N_E=\mathfrak p_2^9\mathfrak p_3^2\mathfrak p_{41},

]

with local information:

[

\mathfrak p_2:\quad v(\Delta)=10,\ f=9,\ \mathrm{III},

]

[

\mathfrak p_3:\quad v(\Delta)=3,\ f=2,\ \mathrm{III},

]

[

\mathfrak p_{41}:\quad v(\Delta)=1,\ f=1,\ I_1.

]

The conductor norm is

[

870580224.

]

I checked the public LMFDB search for field (4.4.1600.1) and level norm (870580224), and it returns no matches.

Would it be possible to identify or compute the Hilbert newform attached to this curve? The initial Frobenius traces are:

[

N\mathfrak p=31: 0,-8,0,6,

]

[

N\mathfrak p=49: -2,8,

]

[

N\mathfrak p=71: 8,8,6,-4,

]

[

N\mathfrak p=79: -6,12,2,4,

]

[

N\mathfrak p=89: -2,14,6,6.

]

Thank you, appreciate it! 

Alexander Clouston 

Sent from my iPhone

On 28 Apr 2026, at 20:57, Edgar Costa <edgardi...@gmail.com> wrote:



[Edgar Costa]

Hi Alexander,

You can try something like this https://gist.github.com/edgarcosta/e7eeedcf547e1ba79e4de7b531c557d5  but note that the space in question has dimension 72253439

I should also note that I did not write that, so take it all with a grain of salt, but it looks likely correct.

Cheers,
Edgar

[Alexander Clouston]

Thank you! 

Appreciate the response 

Sent from my iPhone

On 30 Apr 2026, at 17:54, Edgar Costa <edgardi...@gmail.com> wrote:



[Alexander Clouston]

Hi Edgar just a thought,

As that explains the timeout.

I’m in the UK and do not have university access to a full Magma installation. The public Magma calculator was enough to confirm the conductor, but not enough to run NewformsOfDegree1(M).

Would it be possible for you or someone on the LMFDB side to run the gist on a full Magma installation? I only need the degree-one newform/eigenpacket matching the elliptic curve traces, not a full decomposition of the 72{,}253{,}439-dimensional space.

If that is not possible, could you recommend someone who might be willing to run this computation?

Best,
Alexander

Sent from my iPhone

On 30 Apr 2026, at 17:54, Edgar Costa <edgardi...@gmail.com> wrote:

[Edgar Costa]

Hi Alexander,

I tried to run it, and found three independent walls:

1. I found a bug in Magma: any level divisible by P_2 * P_3^2 hits an assertion failure in proj1.m:343. Your conductor contains it, so NewformsOfDegree1 cannot run, regardless of compute. https://github.com/Magma-Maths/Magma/issues/46

2. Memory: ~200 KB per class on levels Magma can handle, so dim 72,253,439 needs ~14 TB.

3. Time: one T_p at the target dim is days-to-weeks, and identification needs several.

So the Magma form object is not producible. The eigenpacket itself, however, is just the system (a_p(f_E))_p, and by modularity (Freitas-Le Hung-Siksek) it equals (a_p(E))_p, which is cheap from the curve (~2 ms per prime). 

Cheers,

Edgar