Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [96,5,Mod(65,96)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(96, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("96.65");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 96 = 2^{5} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 96.e (of order \(2\), degree \(1\), 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: | \(9.92351645605\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 60x^{6} + 1368x^{4} - 13864x^{2} + 54756 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{24}\cdot 3^{2} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 60x^{6} + 1368x^{4} - 13864x^{2} + 54756 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{7} + 8\nu^{5} + 530\nu^{3} - 7404\nu ) / 936 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} - 26\nu^{4} + 214\nu^{2} - 432 ) / 54 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -5\nu^{7} + 222\nu^{5} - 3330\nu^{3} + 22832\nu ) / 468 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -4\nu^{7} + 188\nu^{5} - 3184\nu^{3} + 18120\nu ) / 351 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 11\nu^{7} - 556\nu^{5} + 8834\nu^{3} - 41796\nu ) / 936 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} - 98\nu^{4} + 2662\nu^{2} - 21600 ) / 54 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -3\nu^{6} + 142\nu^{4} - 2242\nu^{2} + 11472 ) / 6 \)
|
| \(\nu\) | \(=\) |
\( ( 2\beta_{5} - 3\beta_{4} + 6\beta_{3} - 6\beta_1 ) / 96 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{7} + 8\beta_{6} + 19\beta_{2} + 1440 ) / 96 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -40\beta_{5} - 129\beta_{4} + 96\beta_{3} - 24\beta_1 ) / 96 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 17\beta_{7} + 100\beta_{6} + 359\beta_{2} + 10368 ) / 48 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -493\beta_{5} - 813\beta_{4} + 357\beta_{3} - 321\beta_1 ) / 24 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 335\beta_{7} + 1744\beta_{6} + 9893\beta_{2} + 136224 ) / 48 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -25892\beta_{5} - 36087\beta_{4} + 8940\beta_{3} - 34212\beta_1 ) / 48 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/96\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(37\) | \(65\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 65.1 |
|
0 | −7.33223 | − | 5.21904i | 0 | 2.14012i | 0 | −37.9402 | 0 | 26.5233 | + | 76.5344i | 0 | ||||||||||||||||||||||||||||||||||||||
| 65.2 | 0 | −7.33223 | + | 5.21904i | 0 | − | 2.14012i | 0 | −37.9402 | 0 | 26.5233 | − | 76.5344i | 0 | ||||||||||||||||||||||||||||||||||||||
| 65.3 | 0 | −4.02968 | − | 8.04746i | 0 | − | 36.8160i | 0 | 79.9534 | 0 | −48.5233 | + | 64.8574i | 0 | ||||||||||||||||||||||||||||||||||||||
| 65.4 | 0 | −4.02968 | + | 8.04746i | 0 | 36.8160i | 0 | 79.9534 | 0 | −48.5233 | − | 64.8574i | 0 | |||||||||||||||||||||||||||||||||||||||
| 65.5 | 0 | 4.02968 | − | 8.04746i | 0 | 36.8160i | 0 | −79.9534 | 0 | −48.5233 | − | 64.8574i | 0 | |||||||||||||||||||||||||||||||||||||||
| 65.6 | 0 | 4.02968 | + | 8.04746i | 0 | − | 36.8160i | 0 | −79.9534 | 0 | −48.5233 | + | 64.8574i | 0 | ||||||||||||||||||||||||||||||||||||||
| 65.7 | 0 | 7.33223 | − | 5.21904i | 0 | − | 2.14012i | 0 | 37.9402 | 0 | 26.5233 | − | 76.5344i | 0 | ||||||||||||||||||||||||||||||||||||||
| 65.8 | 0 | 7.33223 | + | 5.21904i | 0 | 2.14012i | 0 | 37.9402 | 0 | 26.5233 | + | 76.5344i | 0 | |||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 4.b | odd | 2 | 1 | inner |
| 12.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 96.5.e.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 96.5.e.a | ✓ | 8 |
| 4.b | odd | 2 | 1 | inner | 96.5.e.a | ✓ | 8 |
| 8.b | even | 2 | 1 | 192.5.e.g | 8 | ||
| 8.d | odd | 2 | 1 | 192.5.e.g | 8 | ||
| 12.b | even | 2 | 1 | inner | 96.5.e.a | ✓ | 8 |
| 24.f | even | 2 | 1 | 192.5.e.g | 8 | ||
| 24.h | odd | 2 | 1 | 192.5.e.g | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 96.5.e.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 96.5.e.a | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 96.5.e.a | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
| 96.5.e.a | ✓ | 8 | 12.b | even | 2 | 1 | inner |
| 192.5.e.g | 8 | 8.b | even | 2 | 1 | ||
| 192.5.e.g | 8 | 8.d | odd | 2 | 1 | ||
| 192.5.e.g | 8 | 24.f | even | 2 | 1 | ||
| 192.5.e.g | 8 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 1360T_{5}^{2} + 6208 \)
acting on \(S_{5}^{\mathrm{new}}(96, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + 44 T^{6} + \cdots + 43046721 \)
|
| $5$ |
\( (T^{4} + 1360 T^{2} + 6208)^{2} \)
|
| $7$ |
\( (T^{4} - 7832 T^{2} + 9201808)^{2} \)
|
| $11$ |
\( (T^{4} + 5104 T^{2} + 379456)^{2} \)
|
| $13$ |
\( (T^{2} + 124 T - 46844)^{4} \)
|
| $17$ |
\( (T^{4} + 322560 T^{2} + 18536988672)^{2} \)
|
| $19$ |
\( (T^{4} - 147416 T^{2} + 12852112)^{2} \)
|
| $23$ |
\( (T^{4} + 386496 T^{2} + 15265096704)^{2} \)
|
| $29$ |
\( (T^{4} + 1142224 T^{2} + 236101966912)^{2} \)
|
| $31$ |
\( (T^{4} + \cdots + 1065598868368)^{2} \)
|
| $37$ |
\( (T^{2} + 604 T + 40516)^{4} \)
|
| $41$ |
\( (T^{4} + \cdots + 7454324933632)^{2} \)
|
| $43$ |
\( (T^{4} + \cdots + 2169366617488)^{2} \)
|
| $47$ |
\( (T^{4} + \cdots + 7749943312384)^{2} \)
|
| $53$ |
\( (T^{4} + 15020752 T^{2} + 841263965248)^{2} \)
|
| $59$ |
\( (T^{4} + \cdots + 108088626178624)^{2} \)
|
| $61$ |
\( (T^{2} + 3036 T - 6261948)^{4} \)
|
| $67$ |
\( (T^{4} + \cdots + 64084344692368)^{2} \)
|
| $71$ |
\( (T^{4} + \cdots + 450666025378816)^{2} \)
|
| $73$ |
\( (T^{2} - 7524 T + 13341636)^{4} \)
|
| $79$ |
\( (T^{4} + \cdots + 937983749427088)^{2} \)
|
| $83$ |
\( (T^{4} + \cdots + 365660980485696)^{2} \)
|
| $89$ |
\( (T^{4} + \cdots + 10\!\cdots\!48)^{2} \)
|
| $97$ |
\( (T^{2} - 7796 T - 70012124)^{4} \)
|
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