Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [192,8,Mod(97,192)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(192, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("192.97");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 192 = 2^{6} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 192.d (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: | \(59.9779248930\) |
| 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: |
\( x^{8} - 4405x^{6} + 14555221x^{4} - 21358981620x^{2} + 23510900230416 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| 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
\( x^{8} - 4405x^{6} + 14555221x^{4} - 21358981620x^{2} + 23510900230416 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 17620\nu^{6} - 58220884\nu^{4} + 256462994020\nu^{2} - 235194428533032 ) / 17643853451421 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{7} - 4853209\nu^{5} + 10696470433\nu^{3} - 23553618193656\nu ) / 2940197162103 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -4405\nu^{7} + 14555221\nu^{5} - 32079700477\nu^{3} + 23510900230416\nu ) / 70543377757656 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{6} + 4405\nu^{4} - 9706417\nu^{2} + 10679490810 ) / 1212201 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 8809\nu^{7} - 43652449\nu^{5} + 160252989817\nu^{3} - 399813438411576\nu ) / 35271688878828 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -96\nu^{6} - 1027093692720 ) / 14555221 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 8810\nu^{7} - 29110442\nu^{5} + 106867666586\nu^{3} - 47021800460832\nu ) / 980065720701 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} - 24\beta_{5} + 48\beta_{3} + 6\beta_{2} ) / 192 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{6} + 24\beta_{4} + 26430\beta _1 + 211440 ) / 192 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2203\beta_{7} + 317136\beta_{3} ) / 96 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -4405\beta_{6} + 105720\beta_{4} + 58238502\beta _1 - 465908016 ) / 192 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 4855411\beta_{7} + 116529864\beta_{5} + 1164241488\beta_{3} - 145530186\beta_{2} ) / 192 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -14555221\beta_{6} - 1027093692720 ) / 96 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -10706170243\beta_{7} + 256948085832\beta_{5} - 3590753449296\beta_{3} - 448844181162\beta_{2} ) / 192 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/192\mathbb{Z}\right)^\times\).
| \(n\) | \(65\) | \(127\) | \(133\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 97.1 |
|
0 | − | 27.0000i | 0 | − | 435.979i | 0 | 34.1431 | 0 | −729.000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 97.2 | 0 | − | 27.0000i | 0 | − | 214.276i | 0 | 616.112 | 0 | −729.000 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 97.3 | 0 | − | 27.0000i | 0 | 214.276i | 0 | −616.112 | 0 | −729.000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 97.4 | 0 | − | 27.0000i | 0 | 435.979i | 0 | −34.1431 | 0 | −729.000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 97.5 | 0 | 27.0000i | 0 | − | 435.979i | 0 | −34.1431 | 0 | −729.000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 97.6 | 0 | 27.0000i | 0 | − | 214.276i | 0 | −616.112 | 0 | −729.000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 97.7 | 0 | 27.0000i | 0 | 214.276i | 0 | 616.112 | 0 | −729.000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 97.8 | 0 | 27.0000i | 0 | 435.979i | 0 | 34.1431 | 0 | −729.000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 8.b | even | 2 | 1 | inner |
| 8.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 192.8.d.c | ✓ | 8 |
| 3.b | odd | 2 | 1 | 576.8.d.h | 8 | ||
| 4.b | odd | 2 | 1 | inner | 192.8.d.c | ✓ | 8 |
| 8.b | even | 2 | 1 | inner | 192.8.d.c | ✓ | 8 |
| 8.d | odd | 2 | 1 | inner | 192.8.d.c | ✓ | 8 |
| 12.b | even | 2 | 1 | 576.8.d.h | 8 | ||
| 24.f | even | 2 | 1 | 576.8.d.h | 8 | ||
| 24.h | odd | 2 | 1 | 576.8.d.h | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 192.8.d.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 192.8.d.c | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
| 192.8.d.c | ✓ | 8 | 8.b | even | 2 | 1 | inner |
| 192.8.d.c | ✓ | 8 | 8.d | odd | 2 | 1 | inner |
| 576.8.d.h | 8 | 3.b | odd | 2 | 1 | ||
| 576.8.d.h | 8 | 12.b | even | 2 | 1 | ||
| 576.8.d.h | 8 | 24.f | even | 2 | 1 | ||
| 576.8.d.h | 8 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 235992T_{5}^{2} + 8727296400 \)
acting on \(S_{8}^{\mathrm{new}}(192, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T^{2} + 729)^{4} \)
|
| $5$ |
\( (T^{4} + 235992 T^{2} + 8727296400)^{2} \)
|
| $7$ |
\( (T^{4} - 380760 T^{2} + 442513296)^{2} \)
|
| $11$ |
\( (T^{4} + \cdots + 396271131033600)^{2} \)
|
| $13$ |
\( (T^{4} + 6562560 T^{2} + 10277093376)^{2} \)
|
| $17$ |
\( (T^{2} + 19476 T - 87834780)^{4} \)
|
| $19$ |
\( (T^{4} + \cdots + 25\!\cdots\!36)^{2} \)
|
| $23$ |
\( (T^{4} + \cdots + 21\!\cdots\!00)^{2} \)
|
| $29$ |
\( (T^{4} + \cdots + 41\!\cdots\!00)^{2} \)
|
| $31$ |
\( (T^{4} + \cdots + 22\!\cdots\!24)^{2} \)
|
| $37$ |
\( (T^{4} + \cdots + 50\!\cdots\!00)^{2} \)
|
| $41$ |
\( (T^{2} - 169332 T - 78581998044)^{4} \)
|
| $43$ |
\( (T^{4} + \cdots + 36\!\cdots\!00)^{2} \)
|
| $47$ |
\( (T^{4} + \cdots + 44\!\cdots\!24)^{2} \)
|
| $53$ |
\( (T^{4} + \cdots + 18\!\cdots\!00)^{2} \)
|
| $59$ |
\( (T^{4} + \cdots + 22\!\cdots\!44)^{2} \)
|
| $61$ |
\( (T^{4} + \cdots + 63\!\cdots\!64)^{2} \)
|
| $67$ |
\( (T^{4} + \cdots + 10\!\cdots\!44)^{2} \)
|
| $71$ |
\( (T^{4} + \cdots + 39\!\cdots\!04)^{2} \)
|
| $73$ |
\( (T^{2} + \cdots - 18394317198524)^{4} \)
|
| $79$ |
\( (T^{4} + \cdots + 73\!\cdots\!64)^{2} \)
|
| $83$ |
\( (T^{4} + \cdots + 74\!\cdots\!00)^{2} \)
|
| $89$ |
\( (T^{2} + \cdots - 4237589143836)^{4} \)
|
| $97$ |
\( (T^{2} + \cdots - 31779299973500)^{4} \)
|
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