Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [432,3,Mod(161,432)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(432, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("432.161");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 432 = 2^{4} \cdot 3^{3} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 432.e (of order \(2\), degree \(1\), 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: | \(11.7711474204\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{8})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3^{2} \) |
| Twist minimal: | no (minimal twist has level 216) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 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
| \(\beta_{1}\) | \(=\) |
\( 3\zeta_{8}^{2} \)
|
| \(\beta_{2}\) | \(=\) |
\( 2\zeta_{8}^{3} + 2\zeta_{8} \)
|
| \(\beta_{3}\) | \(=\) |
\( -6\zeta_{8}^{3} + 6\zeta_{8} \)
|
| \(\zeta_{8}\) | \(=\) |
\( ( \beta_{3} + 3\beta_{2} ) / 12 \)
|
| \(\zeta_{8}^{2}\) | \(=\) |
\( ( \beta_1 ) / 3 \)
|
| \(\zeta_{8}^{3}\) | \(=\) |
\( ( -\beta_{3} + 3\beta_{2} ) / 12 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/432\mathbb{Z}\right)^\times\).
| \(n\) | \(271\) | \(325\) | \(353\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 161.1 |
|
0 | 0 | 0 | − | 5.82843i | 0 | 11.4853 | 0 | 0 | 0 | |||||||||||||||||||||||||||||
| 161.2 | 0 | 0 | 0 | − | 0.171573i | 0 | −5.48528 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 161.3 | 0 | 0 | 0 | 0.171573i | 0 | −5.48528 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 161.4 | 0 | 0 | 0 | 5.82843i | 0 | 11.4853 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 432.3.e.h | 4 | |
| 3.b | odd | 2 | 1 | inner | 432.3.e.h | 4 | |
| 4.b | odd | 2 | 1 | 216.3.e.c | ✓ | 4 | |
| 8.b | even | 2 | 1 | 1728.3.e.s | 4 | ||
| 8.d | odd | 2 | 1 | 1728.3.e.p | 4 | ||
| 9.c | even | 3 | 2 | 1296.3.q.l | 8 | ||
| 9.d | odd | 6 | 2 | 1296.3.q.l | 8 | ||
| 12.b | even | 2 | 1 | 216.3.e.c | ✓ | 4 | |
| 24.f | even | 2 | 1 | 1728.3.e.p | 4 | ||
| 24.h | odd | 2 | 1 | 1728.3.e.s | 4 | ||
| 36.f | odd | 6 | 2 | 648.3.m.f | 8 | ||
| 36.h | even | 6 | 2 | 648.3.m.f | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 216.3.e.c | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 216.3.e.c | ✓ | 4 | 12.b | even | 2 | 1 | |
| 432.3.e.h | 4 | 1.a | even | 1 | 1 | trivial | |
| 432.3.e.h | 4 | 3.b | odd | 2 | 1 | inner | |
| 648.3.m.f | 8 | 36.f | odd | 6 | 2 | ||
| 648.3.m.f | 8 | 36.h | even | 6 | 2 | ||
| 1296.3.q.l | 8 | 9.c | even | 3 | 2 | ||
| 1296.3.q.l | 8 | 9.d | odd | 6 | 2 | ||
| 1728.3.e.p | 4 | 8.d | odd | 2 | 1 | ||
| 1728.3.e.p | 4 | 24.f | even | 2 | 1 | ||
| 1728.3.e.s | 4 | 8.b | even | 2 | 1 | ||
| 1728.3.e.s | 4 | 24.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 \(S_{3}^{\mathrm{new}}(432, [\chi])\):
|
\( T_{5}^{4} + 34T_{5}^{2} + 1 \)
|
|
\( T_{7}^{2} - 6T_{7} - 63 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 34T^{2} + 1 \)
|
| $7$ |
\( (T^{2} - 6 T - 63)^{2} \)
|
| $11$ |
\( T^{4} + 226T^{2} + 2401 \)
|
| $13$ |
\( (T^{2} + 4 T - 284)^{2} \)
|
| $17$ |
\( T^{4} + 328T^{2} + 8464 \)
|
| $19$ |
\( (T^{2} + 16 T - 224)^{2} \)
|
| $23$ |
\( T^{4} + 1888 T^{2} + 430336 \)
|
| $29$ |
\( (T^{2} + 1764)^{2} \)
|
| $31$ |
\( (T^{2} - 2 T - 71)^{2} \)
|
| $37$ |
\( (T^{2} + 72 T + 1008)^{2} \)
|
| $41$ |
\( T^{4} + 5472 T^{2} + 5992704 \)
|
| $43$ |
\( (T^{2} - 28 T - 92)^{2} \)
|
| $47$ |
\( T^{4} + 5832 T^{2} + 374544 \)
|
| $53$ |
\( T^{4} + 9826 T^{2} + 83521 \)
|
| $59$ |
\( T^{4} + 13192 T^{2} + 35378704 \)
|
| $61$ |
\( (T^{2} - 104 T - 1904)^{2} \)
|
| $67$ |
\( (T^{2} - 116 T + 3076)^{2} \)
|
| $71$ |
\( T^{4} + 2248 T^{2} + 226576 \)
|
| $73$ |
\( (T^{2} + 142 T + 2449)^{2} \)
|
| $79$ |
\( (T^{2} - 148 T + 4324)^{2} \)
|
| $83$ |
\( T^{4} + 12658 T^{2} + 10195249 \)
|
| $89$ |
\( T^{4} + 16776 T^{2} + 61027344 \)
|
| $97$ |
\( (T^{2} - 22 T - 4487)^{2} \)
|
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