Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [384,8,Mod(193,384)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(384, 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("384.193");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 384 = 2^{7} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 384.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: | \(119.955849786\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 277x^{4} + 19236x^{2} + 9216 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{16}\cdot 3^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 277x^{4} + 19236x^{2} + 9216 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} + 181\nu^{3} + 5892\nu ) / 4032 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -13\nu^{5} - 18481\nu^{3} - 1963572\nu ) / 4032 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -19\nu^{5} + 12689\nu^{3} + 2162100\nu ) / 4032 \)
|
| \(\beta_{4}\) | \(=\) |
\( 2\nu^{4} + 274\nu^{2} - 205 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2\nu^{4} + 306\nu^{2} + 2747 ) / 7 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 32\beta_1 ) / 96 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 7\beta_{5} - \beta_{4} - 2952 ) / 32 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -39\beta_{3} - 47\beta_{2} - 1352\beta_1 ) / 32 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -959\beta_{5} + 153\beta_{4} + 407704 ) / 32 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 5095\beta_{3} + 6543\beta_{2} + 310888\beta_1 ) / 32 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(133\) | \(257\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 193.1 |
|
0 | − | 27.0000i | 0 | − | 334.405i | 0 | −359.725 | 0 | −729.000 | 0 | ||||||||||||||||||||||||||||||||||
| 193.2 | 0 | − | 27.0000i | 0 | 96.4778i | 0 | 1147.84 | 0 | −729.000 | 0 | ||||||||||||||||||||||||||||||||||||
| 193.3 | 0 | − | 27.0000i | 0 | 349.927i | 0 | −856.119 | 0 | −729.000 | 0 | ||||||||||||||||||||||||||||||||||||
| 193.4 | 0 | 27.0000i | 0 | − | 349.927i | 0 | −856.119 | 0 | −729.000 | 0 | ||||||||||||||||||||||||||||||||||||
| 193.5 | 0 | 27.0000i | 0 | − | 96.4778i | 0 | 1147.84 | 0 | −729.000 | 0 | ||||||||||||||||||||||||||||||||||||
| 193.6 | 0 | 27.0000i | 0 | 334.405i | 0 | −359.725 | 0 | −729.000 | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 384.8.d.a | ✓ | 6 |
| 4.b | odd | 2 | 1 | 384.8.d.b | yes | 6 | |
| 8.b | even | 2 | 1 | inner | 384.8.d.a | ✓ | 6 |
| 8.d | odd | 2 | 1 | 384.8.d.b | yes | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 384.8.d.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 384.8.d.a | ✓ | 6 | 8.b | even | 2 | 1 | inner |
| 384.8.d.b | yes | 6 | 4.b | odd | 2 | 1 | |
| 384.8.d.b | yes | 6 | 8.d | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(384, [\chi])\):
|
\( T_{5}^{6} + 243584T_{5}^{4} + 15873740800T_{5}^{2} + 127455068160000 \)
|
|
\( T_{7}^{3} + 68T_{7}^{2} - 1087632T_{7} - 353498688 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( (T^{2} + 729)^{3} \)
|
| $5$ |
\( T^{6} + \cdots + 127455068160000 \)
|
| $7$ |
\( (T^{3} + 68 T^{2} + \cdots - 353498688)^{2} \)
|
| $11$ |
\( T^{6} + \cdots + 21\!\cdots\!56 \)
|
| $13$ |
\( T^{6} + \cdots + 67\!\cdots\!00 \)
|
| $17$ |
\( (T^{3} + \cdots - 3423209334392)^{2} \)
|
| $19$ |
\( T^{6} + \cdots + 19\!\cdots\!64 \)
|
| $23$ |
\( (T^{3} + \cdots + 5673212960256)^{2} \)
|
| $29$ |
\( T^{6} + \cdots + 15\!\cdots\!76 \)
|
| $31$ |
\( (T^{3} + \cdots - 26\!\cdots\!72)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 19\!\cdots\!44 \)
|
| $41$ |
\( (T^{3} + \cdots - 36\!\cdots\!12)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 17\!\cdots\!44 \)
|
| $47$ |
\( (T^{3} + \cdots - 12\!\cdots\!60)^{2} \)
|
| $53$ |
\( T^{6} + \cdots + 45\!\cdots\!76 \)
|
| $59$ |
\( T^{6} + \cdots + 72\!\cdots\!64 \)
|
| $61$ |
\( T^{6} + \cdots + 49\!\cdots\!96 \)
|
| $67$ |
\( T^{6} + \cdots + 10\!\cdots\!16 \)
|
| $71$ |
\( (T^{3} + \cdots + 15\!\cdots\!84)^{2} \)
|
| $73$ |
\( (T^{3} + \cdots - 97\!\cdots\!56)^{2} \)
|
| $79$ |
\( (T^{3} + \cdots - 20\!\cdots\!56)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 13\!\cdots\!56 \)
|
| $89$ |
\( (T^{3} + \cdots + 66\!\cdots\!52)^{2} \)
|
| $97$ |
\( (T^{3} + \cdots + 56\!\cdots\!96)^{2} \)
|
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