Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [392,6,Mod(1,392)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(392, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("392.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 392 = 2^{3} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 392.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: | \(62.8704573667\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{177}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 44 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 56) |
| 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)
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{177}\). We also show the integral \(q\)-expansion of the trace form.
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 |
|
0 | −0.304135 | 0 | −35.5207 | 0 | 0 | 0 | −242.908 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 26.3041 | 0 | 97.5207 | 0 | 0 | 0 | 448.908 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(7\) | \(-1\) |
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 | 392.6.a.f | 2 | |
| 4.b | odd | 2 | 1 | 784.6.a.p | 2 | ||
| 7.b | odd | 2 | 1 | 56.6.a.c | ✓ | 2 | |
| 7.c | even | 3 | 2 | 392.6.i.g | 4 | ||
| 7.d | odd | 6 | 2 | 392.6.i.l | 4 | ||
| 21.c | even | 2 | 1 | 504.6.a.s | 2 | ||
| 28.d | even | 2 | 1 | 112.6.a.k | 2 | ||
| 56.e | even | 2 | 1 | 448.6.a.q | 2 | ||
| 56.h | odd | 2 | 1 | 448.6.a.z | 2 | ||
| 84.h | odd | 2 | 1 | 1008.6.a.bt | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 56.6.a.c | ✓ | 2 | 7.b | odd | 2 | 1 | |
| 112.6.a.k | 2 | 28.d | even | 2 | 1 | ||
| 392.6.a.f | 2 | 1.a | even | 1 | 1 | trivial | |
| 392.6.i.g | 4 | 7.c | even | 3 | 2 | ||
| 392.6.i.l | 4 | 7.d | odd | 6 | 2 | ||
| 448.6.a.q | 2 | 56.e | even | 2 | 1 | ||
| 448.6.a.z | 2 | 56.h | odd | 2 | 1 | ||
| 504.6.a.s | 2 | 21.c | even | 2 | 1 | ||
| 784.6.a.p | 2 | 4.b | odd | 2 | 1 | ||
| 1008.6.a.bt | 2 | 84.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 26T_{3} - 8 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(392))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} - 26T - 8 \)
|
| $5$ |
\( T^{2} - 62T - 3464 \)
|
| $7$ |
\( T^{2} \)
|
| $11$ |
\( T^{2} - 972T + 229824 \)
|
| $13$ |
\( T^{2} + 78T - 890736 \)
|
| $17$ |
\( T^{2} + 560T + 72028 \)
|
| $19$ |
\( T^{2} + 2642 T + 1502728 \)
|
| $23$ |
\( T^{2} - 2272 T - 9595712 \)
|
| $29$ |
\( T^{2} + 7808 T + 15206524 \)
|
| $31$ |
\( T^{2} + 5444 T + 4417984 \)
|
| $37$ |
\( T^{2} - 576 T - 74699556 \)
|
| $41$ |
\( T^{2} - 16888 T + 46010668 \)
|
| $43$ |
\( T^{2} + 8396 T - 2597984 \)
|
| $47$ |
\( T^{2} - 4532 T - 266143232 \)
|
| $53$ |
\( T^{2} - 1420 T - 234393308 \)
|
| $59$ |
\( T^{2} + 34146 T + 272791704 \)
|
| $61$ |
\( T^{2} + \cdots - 1891566584 \)
|
| $67$ |
\( T^{2} - 56952 T + 538727376 \)
|
| $71$ |
\( T^{2} + 7224 T - 73683456 \)
|
| $73$ |
\( T^{2} + \cdots + 3868158196 \)
|
| $79$ |
\( T^{2} - 52808 T + 273025408 \)
|
| $83$ |
\( T^{2} + \cdots - 3018190488 \)
|
| $89$ |
\( T^{2} + \cdots + 3342290308 \)
|
| $97$ |
\( T^{2} + \cdots + 9447942844 \)
|
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