Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [49,22,Mod(1,49)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(49, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 22, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("49.1");
S:= CuspForms(chi, 22);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 49 = 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 22 \) |
| Character orbit: | \([\chi]\) | \(=\) | 49.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: | \(136.943898701\) |
| Analytic rank: | \(0\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 1) |
| 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 |
|
−288.000 | 128844. | −2.01421e6 | −2.16410e7 | −3.71071e7 | 0 | 1.18407e9 | 6.14042e9 | 6.23259e9 | |||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \(-1\) |
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 | 49.22.a.a | 1 | |
| 7.b | odd | 2 | 1 | 1.22.a.a | ✓ | 1 | |
| 21.c | even | 2 | 1 | 9.22.a.c | 1 | ||
| 28.d | even | 2 | 1 | 16.22.a.c | 1 | ||
| 35.c | odd | 2 | 1 | 25.22.a.a | 1 | ||
| 35.f | even | 4 | 2 | 25.22.b.a | 2 | ||
| 56.e | even | 2 | 1 | 64.22.a.a | 1 | ||
| 56.h | odd | 2 | 1 | 64.22.a.g | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1.22.a.a | ✓ | 1 | 7.b | odd | 2 | 1 | |
| 9.22.a.c | 1 | 21.c | even | 2 | 1 | ||
| 16.22.a.c | 1 | 28.d | even | 2 | 1 | ||
| 25.22.a.a | 1 | 35.c | odd | 2 | 1 | ||
| 25.22.b.a | 2 | 35.f | even | 4 | 2 | ||
| 49.22.a.a | 1 | 1.a | even | 1 | 1 | trivial | |
| 64.22.a.a | 1 | 56.e | even | 2 | 1 | ||
| 64.22.a.g | 1 | 56.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(49))\):
|
\( T_{2} + 288 \)
|
|
\( T_{3} - 128844 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T + 288 \)
|
| $3$ |
\( T - 128844 \)
|
| $5$ |
\( T + 21640950 \)
|
| $7$ |
\( T \)
|
| $11$ |
\( T + 94724929188 \)
|
| $13$ |
\( T - 80621789794 \)
|
| $17$ |
\( T + 3052282930002 \)
|
| $19$ |
\( T - 7920788351740 \)
|
| $23$ |
\( T + 73845437470344 \)
|
| $29$ |
\( T + 4253031736469010 \)
|
| $31$ |
\( T + 1900541176310432 \)
|
| $37$ |
\( T - 22\!\cdots\!22 \)
|
| $41$ |
\( T - 20\!\cdots\!58 \)
|
| $43$ |
\( T + 19\!\cdots\!44 \)
|
| $47$ |
\( T + 14\!\cdots\!32 \)
|
| $53$ |
\( T - 20\!\cdots\!06 \)
|
| $59$ |
\( T - 59\!\cdots\!20 \)
|
| $61$ |
\( T + 61\!\cdots\!62 \)
|
| $67$ |
\( T - 16\!\cdots\!52 \)
|
| $71$ |
\( T + 56\!\cdots\!28 \)
|
| $73$ |
\( T - 43\!\cdots\!94 \)
|
| $79$ |
\( T + 51\!\cdots\!60 \)
|
| $83$ |
\( T + 48\!\cdots\!56 \)
|
| $89$ |
\( T - 50\!\cdots\!30 \)
|
| $97$ |
\( T + 80\!\cdots\!82 \)
|
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