Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [392,4,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, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("392.177");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 392 = 2^{3} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| 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: | \(23.1287487223\) |
| 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 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 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 | 3.00000 | + | 5.19615i | 0 | 4.00000 | − | 6.92820i | 0 | 0 | 0 | −4.50000 | + | 7.79423i | 0 | ||||||||||||||||||
| 361.1 | 0 | 3.00000 | − | 5.19615i | 0 | 4.00000 | + | 6.92820i | 0 | 0 | 0 | −4.50000 | − | 7.79423i | 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 | 392.4.i.h | 2 | |
| 7.b | odd | 2 | 1 | 392.4.i.a | 2 | ||
| 7.c | even | 3 | 1 | 392.4.a.a | 1 | ||
| 7.c | even | 3 | 1 | inner | 392.4.i.h | 2 | |
| 7.d | odd | 6 | 1 | 56.4.a.b | ✓ | 1 | |
| 7.d | odd | 6 | 1 | 392.4.i.a | 2 | ||
| 21.g | even | 6 | 1 | 504.4.a.a | 1 | ||
| 28.f | even | 6 | 1 | 112.4.a.b | 1 | ||
| 28.g | odd | 6 | 1 | 784.4.a.q | 1 | ||
| 35.i | odd | 6 | 1 | 1400.4.a.b | 1 | ||
| 35.k | even | 12 | 2 | 1400.4.g.b | 2 | ||
| 56.j | odd | 6 | 1 | 448.4.a.c | 1 | ||
| 56.m | even | 6 | 1 | 448.4.a.n | 1 | ||
| 84.j | odd | 6 | 1 | 1008.4.a.e | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 56.4.a.b | ✓ | 1 | 7.d | odd | 6 | 1 | |
| 112.4.a.b | 1 | 28.f | even | 6 | 1 | ||
| 392.4.a.a | 1 | 7.c | even | 3 | 1 | ||
| 392.4.i.a | 2 | 7.b | odd | 2 | 1 | ||
| 392.4.i.a | 2 | 7.d | odd | 6 | 1 | ||
| 392.4.i.h | 2 | 1.a | even | 1 | 1 | trivial | |
| 392.4.i.h | 2 | 7.c | even | 3 | 1 | inner | |
| 448.4.a.c | 1 | 56.j | odd | 6 | 1 | ||
| 448.4.a.n | 1 | 56.m | even | 6 | 1 | ||
| 504.4.a.a | 1 | 21.g | even | 6 | 1 | ||
| 784.4.a.q | 1 | 28.g | odd | 6 | 1 | ||
| 1008.4.a.e | 1 | 84.j | odd | 6 | 1 | ||
| 1400.4.a.b | 1 | 35.i | odd | 6 | 1 | ||
| 1400.4.g.b | 2 | 35.k | even | 12 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(392, [\chi])\):
|
\( T_{3}^{2} - 6T_{3} + 36 \)
|
|
\( T_{5}^{2} - 8T_{5} + 64 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} - 6T + 36 \)
|
| $5$ |
\( T^{2} - 8T + 64 \)
|
| $7$ |
\( T^{2} \)
|
| $11$ |
\( T^{2} + 56T + 3136 \)
|
| $13$ |
\( (T - 28)^{2} \)
|
| $17$ |
\( T^{2} + 90T + 8100 \)
|
| $19$ |
\( T^{2} - 74T + 5476 \)
|
| $23$ |
\( T^{2} - 96T + 9216 \)
|
| $29$ |
\( (T + 222)^{2} \)
|
| $31$ |
\( T^{2} + 100T + 10000 \)
|
| $37$ |
\( T^{2} + 58T + 3364 \)
|
| $41$ |
\( (T + 422)^{2} \)
|
| $43$ |
\( (T - 512)^{2} \)
|
| $47$ |
\( T^{2} - 148T + 21904 \)
|
| $53$ |
\( T^{2} - 642T + 412164 \)
|
| $59$ |
\( T^{2} + 318T + 101124 \)
|
| $61$ |
\( T^{2} - 720T + 518400 \)
|
| $67$ |
\( T^{2} - 412T + 169744 \)
|
| $71$ |
\( (T - 448)^{2} \)
|
| $73$ |
\( T^{2} - 994T + 988036 \)
|
| $79$ |
\( T^{2} - 296T + 87616 \)
|
| $83$ |
\( (T + 386)^{2} \)
|
| $89$ |
\( T^{2} + 6T + 36 \)
|
| $97$ |
\( (T - 138)^{2} \)
|
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