Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [338,10,Mod(1,338)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(338, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("338.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 338 = 2 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 338.a (trivial) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
| Self dual: | yes |
| Analytic conductor: | \(174.082112623\) |
| Analytic rank: | \(0\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 26) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
16.0000 | −273.000 | 256.000 | −1015.00 | −4368.00 | −3955.00 | 4096.00 | 54846.0 | −16240.0 | |||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(13\) | \( +1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 338.10.a.c | 1 | |
| 13.b | even | 2 | 1 | 26.10.a.a | ✓ | 1 | |
| 39.d | odd | 2 | 1 | 234.10.a.b | 1 | ||
| 52.b | odd | 2 | 1 | 208.10.a.c | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 26.10.a.a | ✓ | 1 | 13.b | even | 2 | 1 | |
| 208.10.a.c | 1 | 52.b | odd | 2 | 1 | ||
| 234.10.a.b | 1 | 39.d | odd | 2 | 1 | ||
| 338.10.a.c | 1 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(338))\):
|
\( T_{3} + 273 \)
|
|
\( T_{5} + 1015 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T - 16 \)
|
| $3$ |
\( T + 273 \)
|
| $5$ |
\( T + 1015 \)
|
| $7$ |
\( T + 3955 \)
|
| $11$ |
\( T - 50998 \)
|
| $13$ |
\( T \)
|
| $17$ |
\( T - 509757 \)
|
| $19$ |
\( T - 626574 \)
|
| $23$ |
\( T - 653524 \)
|
| $29$ |
\( T + 4943006 \)
|
| $31$ |
\( T + 4071700 \)
|
| $37$ |
\( T + 2348883 \)
|
| $41$ |
\( T - 13350960 \)
|
| $43$ |
\( T + 7834847 \)
|
| $47$ |
\( T - 39637681 \)
|
| $53$ |
\( T - 73200924 \)
|
| $59$ |
\( T - 141141614 \)
|
| $61$ |
\( T + 132061256 \)
|
| $67$ |
\( T - 185673110 \)
|
| $71$ |
\( T + 224452625 \)
|
| $73$ |
\( T - 172523674 \)
|
| $79$ |
\( T + 643288156 \)
|
| $83$ |
\( T + 720077280 \)
|
| $89$ |
\( T - 73028106 \)
|
| $97$ |
\( T - 15879778 \)
|
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