Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [392,6,Mod(177,392)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(392, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 2]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("392.177");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 392 = 2^{3} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 392.i (of order \(3\), degree \(2\), not minimal) |
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: | no |
| Analytic conductor: | \(62.8704573667\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-3}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 8) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/392\mathbb{Z}\right)^\times\).
| \(n\) | \(197\) | \(295\) | \(297\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-\zeta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 177.1 |
|
0 | −10.0000 | − | 17.3205i | 0 | 37.0000 | − | 64.0859i | 0 | 0 | 0 | −78.5000 | + | 135.966i | 0 | ||||||||||||||||||
| 361.1 | 0 | −10.0000 | + | 17.3205i | 0 | 37.0000 | + | 64.0859i | 0 | 0 | 0 | −78.5000 | − | 135.966i | 0 | |||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 392.6.i.b | 2 | |
| 7.b | odd | 2 | 1 | 392.6.i.e | 2 | ||
| 7.c | even | 3 | 1 | 8.6.a.a | ✓ | 1 | |
| 7.c | even | 3 | 1 | inner | 392.6.i.b | 2 | |
| 7.d | odd | 6 | 1 | 392.6.a.b | 1 | ||
| 7.d | odd | 6 | 1 | 392.6.i.e | 2 | ||
| 21.h | odd | 6 | 1 | 72.6.a.f | 1 | ||
| 28.f | even | 6 | 1 | 784.6.a.l | 1 | ||
| 28.g | odd | 6 | 1 | 16.6.a.a | 1 | ||
| 35.j | even | 6 | 1 | 200.6.a.a | 1 | ||
| 35.l | odd | 12 | 2 | 200.6.c.a | 2 | ||
| 56.k | odd | 6 | 1 | 64.6.a.g | 1 | ||
| 56.p | even | 6 | 1 | 64.6.a.a | 1 | ||
| 77.h | odd | 6 | 1 | 968.6.a.a | 1 | ||
| 84.n | even | 6 | 1 | 144.6.a.k | 1 | ||
| 112.u | odd | 12 | 2 | 256.6.b.d | 2 | ||
| 112.w | even | 12 | 2 | 256.6.b.f | 2 | ||
| 140.p | odd | 6 | 1 | 400.6.a.l | 1 | ||
| 140.w | even | 12 | 2 | 400.6.c.d | 2 | ||
| 168.s | odd | 6 | 1 | 576.6.a.g | 1 | ||
| 168.v | even | 6 | 1 | 576.6.a.h | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8.6.a.a | ✓ | 1 | 7.c | even | 3 | 1 | |
| 16.6.a.a | 1 | 28.g | odd | 6 | 1 | ||
| 64.6.a.a | 1 | 56.p | even | 6 | 1 | ||
| 64.6.a.g | 1 | 56.k | odd | 6 | 1 | ||
| 72.6.a.f | 1 | 21.h | odd | 6 | 1 | ||
| 144.6.a.k | 1 | 84.n | even | 6 | 1 | ||
| 200.6.a.a | 1 | 35.j | even | 6 | 1 | ||
| 200.6.c.a | 2 | 35.l | odd | 12 | 2 | ||
| 256.6.b.d | 2 | 112.u | odd | 12 | 2 | ||
| 256.6.b.f | 2 | 112.w | even | 12 | 2 | ||
| 392.6.a.b | 1 | 7.d | odd | 6 | 1 | ||
| 392.6.i.b | 2 | 1.a | even | 1 | 1 | trivial | |
| 392.6.i.b | 2 | 7.c | even | 3 | 1 | inner | |
| 392.6.i.e | 2 | 7.b | odd | 2 | 1 | ||
| 392.6.i.e | 2 | 7.d | odd | 6 | 1 | ||
| 400.6.a.l | 1 | 140.p | odd | 6 | 1 | ||
| 400.6.c.d | 2 | 140.w | even | 12 | 2 | ||
| 576.6.a.g | 1 | 168.s | odd | 6 | 1 | ||
| 576.6.a.h | 1 | 168.v | even | 6 | 1 | ||
| 784.6.a.l | 1 | 28.f | even | 6 | 1 | ||
| 968.6.a.a | 1 | 77.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 20T_{3} + 400 \)
acting on \(S_{6}^{\mathrm{new}}(392, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} + 20T + 400 \)
|
| $5$ |
\( T^{2} - 74T + 5476 \)
|
| $7$ |
\( T^{2} \)
|
| $11$ |
\( T^{2} + 124T + 15376 \)
|
| $13$ |
\( (T - 478)^{2} \)
|
| $17$ |
\( T^{2} - 1198 T + 1435204 \)
|
| $19$ |
\( T^{2} + 3044 T + 9265936 \)
|
| $23$ |
\( T^{2} + 184T + 33856 \)
|
| $29$ |
\( (T + 3282)^{2} \)
|
| $31$ |
\( T^{2} - 5728 T + 32809984 \)
|
| $37$ |
\( T^{2} + 10326 T + 106626276 \)
|
| $41$ |
\( (T + 8886)^{2} \)
|
| $43$ |
\( (T + 9188)^{2} \)
|
| $47$ |
\( T^{2} + 23664 T + 559984896 \)
|
| $53$ |
\( T^{2} + 11686 T + 136562596 \)
|
| $59$ |
\( T^{2} + 16876 T + 284799376 \)
|
| $61$ |
\( T^{2} - 18482 T + 341584324 \)
|
| $67$ |
\( T^{2} - 15532 T + 241243024 \)
|
| $71$ |
\( (T + 31960)^{2} \)
|
| $73$ |
\( T^{2} - 4886 T + 23872996 \)
|
| $79$ |
\( T^{2} + \cdots + 1985593600 \)
|
| $83$ |
\( (T - 67364)^{2} \)
|
| $89$ |
\( T^{2} + \cdots + 5183136036 \)
|
| $97$ |
\( (T - 48866)^{2} \)
|
| show more | |
| show less |