Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6256,2,Mod(1,6256)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6256, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6256.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6256 = 2^{4} \cdot 17 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6256.a (trivial) |
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: | yes |
| Analytic conductor: | \(49.9544115045\) |
| Analytic rank: | \(0\) |
| Dimension: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} - 3x^{8} - 11x^{7} + 35x^{6} + 30x^{5} - 112x^{4} - 24x^{3} + 116x^{2} + 8x - 24 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 3128) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - 3x^{8} - 11x^{7} + 35x^{6} + 30x^{5} - 112x^{4} - 24x^{3} + 116x^{2} + 8x - 24 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{8} - 2\nu^{7} - 12\nu^{6} + 20\nu^{5} + 41\nu^{4} - 42\nu^{3} - 54\nu^{2} + 8\nu + 20 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{8} - \nu^{7} - 14\nu^{6} + 9\nu^{5} + 59\nu^{4} - 12\nu^{3} - 82\nu^{2} - 16\nu + 20 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{8} - \nu^{7} - 14\nu^{6} + 9\nu^{5} + 61\nu^{4} - 14\nu^{3} - 98\nu^{2} - 4\nu + 36 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{8} + 2\nu^{7} + 13\nu^{6} - 22\nu^{5} - 50\nu^{4} + 60\nu^{3} + 66\nu^{2} - 36\nu - 16 ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -2\nu^{8} + 3\nu^{7} + 26\nu^{6} - 29\nu^{5} - 100\nu^{4} + 56\nu^{3} + 134\nu^{2} - 4\nu - 32 ) / 4 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{8} - \nu^{7} - 14\nu^{6} + 9\nu^{5} + 60\nu^{4} - 13\nu^{3} - 88\nu^{2} - 10\nu + 20 ) / 2 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 3\nu^{8} - 5\nu^{7} - 38\nu^{6} + 51\nu^{5} + 139\nu^{4} - 118\nu^{3} - 172\nu^{2} + 48\nu + 36 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} - \beta_{4} - \beta_{3} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + 2\beta_{6} - \beta_{4} + \beta_{3} + 2\beta_{2} + 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 9\beta_{7} + 2\beta_{6} - 7\beta_{4} - 9\beta_{3} + 2\beta_{2} + 24 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{8} + 11\beta_{7} + 24\beta_{6} - 9\beta_{4} + 9\beta_{3} + 20\beta_{2} + 42\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 4\beta_{8} + 73\beta_{7} + 30\beta_{6} + 4\beta_{5} - 51\beta_{4} - 69\beta_{3} + 26\beta_{2} + 4\beta _1 + 164 \)
|
| \(\nu^{7}\) | \(=\) |
\( 30\beta_{8} + 103\beta_{7} + 228\beta_{6} + 8\beta_{5} - 73\beta_{4} + 69\beta_{3} + 172\beta_{2} + 314\beta _1 + 8 \)
|
| \(\nu^{8}\) | \(=\) |
\( 68 \beta_{8} + 589 \beta_{7} + 338 \beta_{6} + 64 \beta_{5} - 387 \beta_{4} - 513 \beta_{3} + \cdots + 1196 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | −2.69728 | 0 | −2.98041 | 0 | −0.264746 | 0 | 4.27530 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.62454 | 0 | 0.645241 | 0 | 3.74850 | 0 | −0.360874 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.12474 | 0 | −1.27669 | 0 | −0.203887 | 0 | −1.73496 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −0.549526 | 0 | 3.39050 | 0 | −0.410400 | 0 | −2.69802 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0.493996 | 0 | −3.17846 | 0 | 0.764783 | 0 | −2.75597 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 1.57363 | 0 | 3.35376 | 0 | 3.49762 | 0 | −0.523675 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 1.68051 | 0 | −2.18065 | 0 | −1.20246 | 0 | −0.175877 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 2.30487 | 0 | 1.81717 | 0 | −1.60240 | 0 | 2.31242 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 2.94307 | 0 | 0.409530 | 0 | 4.67299 | 0 | 5.66166 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(17\) | \( -1 \) |
| \(23\) | \( +1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 6256.2.a.bd | 9 | |
| 4.b | odd | 2 | 1 | 3128.2.a.f | ✓ | 9 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3128.2.a.f | ✓ | 9 | 4.b | odd | 2 | 1 | |
| 6256.2.a.bd | 9 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6256))\):
|
\( T_{3}^{9} - 3T_{3}^{8} - 11T_{3}^{7} + 35T_{3}^{6} + 30T_{3}^{5} - 112T_{3}^{4} - 24T_{3}^{3} + 116T_{3}^{2} + 8T_{3} - 24 \)
|
|
\( T_{5}^{9} - 26T_{5}^{7} - 4T_{5}^{6} + 219T_{5}^{5} + 48T_{5}^{4} - 631T_{5}^{3} - 46T_{5}^{2} + 467T_{5} - 144 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} \)
|
| $3$ |
\( T^{9} - 3 T^{8} + \cdots - 24 \)
|
| $5$ |
\( T^{9} - 26 T^{7} + \cdots - 144 \)
|
| $7$ |
\( T^{9} - 9 T^{8} + \cdots + 2 \)
|
| $11$ |
\( T^{9} - 11 T^{8} + \cdots + 408 \)
|
| $13$ |
\( T^{9} - T^{8} + \cdots - 1097 \)
|
| $17$ |
\( (T - 1)^{9} \)
|
| $19$ |
\( T^{9} - 4 T^{8} + \cdots - 22032 \)
|
| $23$ |
\( (T + 1)^{9} \)
|
| $29$ |
\( T^{9} - 13 T^{8} + \cdots - 88584 \)
|
| $31$ |
\( T^{9} - 6 T^{8} + \cdots - 324984 \)
|
| $37$ |
\( T^{9} - T^{8} + \cdots + 752 \)
|
| $41$ |
\( T^{9} + 7 T^{8} + \cdots + 72 \)
|
| $43$ |
\( T^{9} - 14 T^{8} + \cdots - 6016 \)
|
| $47$ |
\( T^{9} - 30 T^{8} + \cdots + 361939 \)
|
| $53$ |
\( T^{9} - 14 T^{8} + \cdots - 195136 \)
|
| $59$ |
\( T^{9} - 41 T^{8} + \cdots - 40487012 \)
|
| $61$ |
\( T^{9} + 5 T^{8} + \cdots - 423602 \)
|
| $67$ |
\( T^{9} - 23 T^{8} + \cdots - 19297216 \)
|
| $71$ |
\( T^{9} - 24 T^{8} + \cdots - 147208 \)
|
| $73$ |
\( T^{9} + 10 T^{8} + \cdots + 69998808 \)
|
| $79$ |
\( T^{9} - 30 T^{8} + \cdots + 101022 \)
|
| $83$ |
\( T^{9} - 25 T^{8} + \cdots - 322461584 \)
|
| $89$ |
\( T^{9} + 34 T^{8} + \cdots - 39817088 \)
|
| $97$ |
\( T^{9} + 14 T^{8} + \cdots + 147101736 \)
|
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