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: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-59})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 14x^{2} - 15x + 225 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 56) |
| 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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{4} - x^{3} - 14x^{2} - 15x + 225 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 14\nu^{2} - 14\nu - 225 ) / 210 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{3} + 2\nu^{2} + 58\nu + 15 ) / 15 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 31\nu^{3} + 14\nu^{2} + 406\nu - 885 ) / 210 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} - 3\beta_1 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + 2\beta_{2} + 87\beta _1 + 87 ) / 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 14\beta_{3} - 7\beta_{2} + 66 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/392\mathbb{Z}\right)^\times\).
| \(n\) | \(197\) | \(295\) | \(297\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1 - \beta_{1}\) |
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 | −0.152067 | − | 0.263388i | 0 | −17.7603 | + | 30.7618i | 0 | 0 | 0 | 121.454 | − | 210.364i | 0 | ||||||||||||||||||||||||
| 177.2 | 0 | 13.1521 | + | 22.7800i | 0 | 48.7603 | − | 84.4554i | 0 | 0 | 0 | −224.454 | + | 388.765i | 0 | |||||||||||||||||||||||||
| 361.1 | 0 | −0.152067 | + | 0.263388i | 0 | −17.7603 | − | 30.7618i | 0 | 0 | 0 | 121.454 | + | 210.364i | 0 | |||||||||||||||||||||||||
| 361.2 | 0 | 13.1521 | − | 22.7800i | 0 | 48.7603 | + | 84.4554i | 0 | 0 | 0 | −224.454 | − | 388.765i | 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.l | 4 | |
| 7.b | odd | 2 | 1 | 392.6.i.g | 4 | ||
| 7.c | even | 3 | 1 | 56.6.a.c | ✓ | 2 | |
| 7.c | even | 3 | 1 | inner | 392.6.i.l | 4 | |
| 7.d | odd | 6 | 1 | 392.6.a.f | 2 | ||
| 7.d | odd | 6 | 1 | 392.6.i.g | 4 | ||
| 21.h | odd | 6 | 1 | 504.6.a.s | 2 | ||
| 28.f | even | 6 | 1 | 784.6.a.p | 2 | ||
| 28.g | odd | 6 | 1 | 112.6.a.k | 2 | ||
| 56.k | odd | 6 | 1 | 448.6.a.q | 2 | ||
| 56.p | even | 6 | 1 | 448.6.a.z | 2 | ||
| 84.n | even | 6 | 1 | 1008.6.a.bt | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 56.6.a.c | ✓ | 2 | 7.c | even | 3 | 1 | |
| 112.6.a.k | 2 | 28.g | odd | 6 | 1 | ||
| 392.6.a.f | 2 | 7.d | odd | 6 | 1 | ||
| 392.6.i.g | 4 | 7.b | odd | 2 | 1 | ||
| 392.6.i.g | 4 | 7.d | odd | 6 | 1 | ||
| 392.6.i.l | 4 | 1.a | even | 1 | 1 | trivial | |
| 392.6.i.l | 4 | 7.c | even | 3 | 1 | inner | |
| 448.6.a.q | 2 | 56.k | odd | 6 | 1 | ||
| 448.6.a.z | 2 | 56.p | even | 6 | 1 | ||
| 504.6.a.s | 2 | 21.h | odd | 6 | 1 | ||
| 784.6.a.p | 2 | 28.f | even | 6 | 1 | ||
| 1008.6.a.bt | 2 | 84.n | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 26T_{3}^{3} + 684T_{3}^{2} + 208T_{3} + 64 \)
acting on \(S_{6}^{\mathrm{new}}(392, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 26 T^{3} + \cdots + 64 \)
|
| $5$ |
\( T^{4} - 62 T^{3} + \cdots + 11999296 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + \cdots + 52819070976 \)
|
| $13$ |
\( (T^{2} - 78 T - 890736)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 5188032784 \)
|
| $19$ |
\( T^{4} + \cdots + 2258191441984 \)
|
| $23$ |
\( T^{4} + \cdots + 92077688786944 \)
|
| $29$ |
\( (T^{2} + 7808 T + 15206524)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 19518582624256 \)
|
| $37$ |
\( T^{4} + \cdots + 55\!\cdots\!36 \)
|
| $41$ |
\( (T^{2} + 16888 T + 46010668)^{2} \)
|
| $43$ |
\( (T^{2} + 8396 T - 2597984)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 70\!\cdots\!24 \)
|
| $53$ |
\( T^{4} + \cdots + 54\!\cdots\!64 \)
|
| $59$ |
\( T^{4} + \cdots + 74\!\cdots\!16 \)
|
| $61$ |
\( T^{4} + \cdots + 35\!\cdots\!56 \)
|
| $67$ |
\( T^{4} + \cdots + 29\!\cdots\!76 \)
|
| $71$ |
\( (T^{2} + 7224 T - 73683456)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 14\!\cdots\!16 \)
|
| $79$ |
\( T^{4} + \cdots + 74\!\cdots\!64 \)
|
| $83$ |
\( (T^{2} + 84486 T - 3018190488)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 11\!\cdots\!64 \)
|
| $97$ |
\( (T^{2} - 194624 T + 9447942844)^{2} \)
|
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