Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [192,9,Mod(31,192)]
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([1, 1, 0]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("192.31");
S:= CuspForms(chi, 9);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 192 = 2^{6} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 192.b (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: | \(78.2166931317\) |
| 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: |
\( x^{8} - 2365x^{6} + 4196101x^{4} - 3304198260x^{2} + 1951955471376 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{20}\cdot 3^{2} \) |
| 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
\( x^{8} - 2365x^{6} + 4196101x^{4} - 3304198260x^{2} + 1951955471376 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{7} - 1399489\nu^{5} + 1656993793\nu^{3} - 1958563867896\nu ) / 1953606871944 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 4730\nu^{6} - 8392202\nu^{4} + 19847557730\nu^{2} - 9766384356276 ) / 1465618353381 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{6} + 2365\nu^{4} - 2798977\nu^{2} + 1652099130 ) / 349281 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2365\nu^{7} - 4196101\nu^{5} + 4968180037\nu^{3} - 1951955471376\nu ) / 1464379453674 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 48\nu^{6} + 79569176280 ) / 4196101 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -4729\nu^{7} + 12581209\nu^{5} - 24798960457\nu^{3} + 33203062614456\nu ) / 732189726837 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 2365\nu^{7} - 4196101\nu^{5} + 8270981173\nu^{3} - 1951955471376\nu ) / 325601145324 \)
|
| \(\nu\) | \(=\) |
\( ( 4\beta_{7} + 3\beta_{6} - 6\beta_{4} + 24\beta_1 ) / 96 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{5} + 12\beta_{3} + 14190\beta_{2} + 56760 ) / 96 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2366\beta_{7} - 10641\beta_{4} ) / 24 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 2365\beta_{5} + 28380\beta_{3} + 16793862\beta_{2} - 67175448 ) / 96 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 5602684\beta_{7} - 4202013\beta_{6} - 41949186\beta_{4} - 167796744\beta_1 ) / 96 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 4196101\beta_{5} - 79569176280 ) / 48 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -6639156892\beta_{7} - 4979367669\beta_{6} + 69476231922\beta_{4} - 277904927688\beta_1 ) / 96 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 31.1 |
|
0 | −46.7654 | 0 | − | 473.777i | 0 | 2524.24i | 0 | 2187.00 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 31.2 | 0 | −46.7654 | 0 | − | 2.65948i | 0 | 3252.24i | 0 | 2187.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 31.3 | 0 | −46.7654 | 0 | 2.65948i | 0 | − | 3252.24i | 0 | 2187.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 31.4 | 0 | −46.7654 | 0 | 473.777i | 0 | − | 2524.24i | 0 | 2187.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 31.5 | 0 | 46.7654 | 0 | − | 473.777i | 0 | − | 2524.24i | 0 | 2187.00 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 31.6 | 0 | 46.7654 | 0 | − | 2.65948i | 0 | − | 3252.24i | 0 | 2187.00 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 31.7 | 0 | 46.7654 | 0 | 2.65948i | 0 | 3252.24i | 0 | 2187.00 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 31.8 | 0 | 46.7654 | 0 | 473.777i | 0 | 2524.24i | 0 | 2187.00 | 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 | 192.9.b.a | ✓ | 8 |
| 4.b | odd | 2 | 1 | inner | 192.9.b.a | ✓ | 8 |
| 8.b | even | 2 | 1 | inner | 192.9.b.a | ✓ | 8 |
| 8.d | odd | 2 | 1 | inner | 192.9.b.a | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 192.9.b.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 192.9.b.a | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
| 192.9.b.a | ✓ | 8 | 8.b | even | 2 | 1 | inner |
| 192.9.b.a | ✓ | 8 | 8.d | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 224472T_{5}^{2} + 1587600 \)
acting on \(S_{9}^{\mathrm{new}}(192, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T^{2} - 2187)^{4} \)
|
| $5$ |
\( (T^{4} + 224472 T^{2} + 1587600)^{2} \)
|
| $7$ |
\( (T^{4} + \cdots + 67395233491600)^{2} \)
|
| $11$ |
\( (T^{4} + \cdots + 275666250240000)^{2} \)
|
| $13$ |
\( (T^{4} + \cdots + 31\!\cdots\!00)^{2} \)
|
| $17$ |
\( (T^{2} + 74388 T - 13132290780)^{4} \)
|
| $19$ |
\( (T^{4} + \cdots + 29\!\cdots\!16)^{2} \)
|
| $23$ |
\( (T^{4} + \cdots + 58\!\cdots\!00)^{2} \)
|
| $29$ |
\( (T^{4} + \cdots + 97\!\cdots\!00)^{2} \)
|
| $31$ |
\( (T^{4} + \cdots + 32\!\cdots\!00)^{2} \)
|
| $37$ |
\( (T^{4} + \cdots + 38\!\cdots\!76)^{2} \)
|
| $41$ |
\( (T^{2} + \cdots - 5251697952444)^{4} \)
|
| $43$ |
\( (T^{4} + \cdots + 82\!\cdots\!76)^{2} \)
|
| $47$ |
\( (T^{4} + \cdots + 31\!\cdots\!44)^{2} \)
|
| $53$ |
\( (T^{4} + \cdots + 95\!\cdots\!00)^{2} \)
|
| $59$ |
\( (T^{4} + \cdots + 26\!\cdots\!44)^{2} \)
|
| $61$ |
\( (T^{4} + \cdots + 19\!\cdots\!56)^{2} \)
|
| $67$ |
\( (T^{4} + \cdots + 77\!\cdots\!00)^{2} \)
|
| $71$ |
\( (T^{4} + \cdots + 47\!\cdots\!00)^{2} \)
|
| $73$ |
\( (T^{2} + \cdots + 861690230101732)^{4} \)
|
| $79$ |
\( (T^{4} + \cdots + 25\!\cdots\!76)^{2} \)
|
| $83$ |
\( (T^{4} + \cdots + 53\!\cdots\!96)^{2} \)
|
| $89$ |
\( (T^{2} + \cdots + 67\!\cdots\!52)^{4} \)
|
| $97$ |
\( (T^{2} + \cdots + 46\!\cdots\!40)^{4} \)
|
| show more | |
| show less |