Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [256,12,Mod(1,256)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(256, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("256.1");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 256 = 2^{8} \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 256.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: | \(196.695854223\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{8} - 4 x^{7} - 59846 x^{6} - 119616 x^{5} + 904265037 x^{4} + 7141379724 x^{3} + \cdots + 17099916310116 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{53}\cdot 3^{2} \) |
| Twist minimal: | no (minimal twist has level 64) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 4 x^{7} - 59846 x^{6} - 119616 x^{5} + 904265037 x^{4} + 7141379724 x^{3} + \cdots + 17099916310116 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 1161742031108 \nu^{7} + 7198449043742 \nu^{6} + \cdots + 15\!\cdots\!22 ) / 72\!\cdots\!05 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 59911076412152 \nu^{7} + \cdots - 17\!\cdots\!24 ) / 72\!\cdots\!05 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 11\!\cdots\!96 \nu^{7} + \cdots + 27\!\cdots\!16 ) / 11\!\cdots\!10 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 47\!\cdots\!68 \nu^{7} + \cdots - 12\!\cdots\!72 ) / 72\!\cdots\!05 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 12\!\cdots\!64 \nu^{7} + \cdots - 25\!\cdots\!52 ) / 11\!\cdots\!10 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 33\!\cdots\!56 \nu^{7} + \cdots + 53\!\cdots\!56 ) / 21\!\cdots\!15 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 53\!\cdots\!56 \nu^{7} + \cdots + 25\!\cdots\!12 ) / 19\!\cdots\!35 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + 4096\beta _1 + 8192 ) / 16384 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -16\beta_{6} + 16\beta_{5} - 7\beta_{4} + 112\beta_{3} + 512\beta_{2} + 17408\beta _1 + 122580992 ) / 8192 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 3456 \beta_{7} - 1272 \beta_{6} - 7944 \beta_{5} + 24249 \beta_{4} - 55608 \beta_{3} + \cdots + 1102913536 ) / 8192 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 864 \beta_{7} - 119248 \beta_{6} + 88528 \beta_{5} - 6373 \beta_{4} + 951472 \beta_{3} + \cdots + 454626757120 ) / 1024 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 12939696 \beta_{7} - 8826535 \beta_{6} - 48706265 \beta_{5} + 149568404 \beta_{4} + \cdots + 6868491732992 ) / 1024 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 206874432 \beta_{7} - 10773547152 \beta_{6} + 7187776656 \beta_{5} + 1619026311 \beta_{4} + \cdots + 27\!\cdots\!44 ) / 2048 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 1544209195968 \beta_{7} - 1623189223460 \beta_{6} - 8016105574876 \beta_{5} + \cdots + 11\!\cdots\!84 ) / 4096 \)
|
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 | −803.959 | 0 | −9050.16 | 0 | 71874.1 | 0 | 469202. | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −803.959 | 0 | 9050.16 | 0 | −71874.1 | 0 | 469202. | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −182.481 | 0 | −11689.5 | 0 | −26516.0 | 0 | −143848. | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −182.481 | 0 | 11689.5 | 0 | 26516.0 | 0 | −143848. | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −85.7231 | 0 | −3056.94 | 0 | −68336.8 | 0 | −169799. | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −85.7231 | 0 | 3056.94 | 0 | 68336.8 | 0 | −169799. | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 576.162 | 0 | −2270.57 | 0 | 37732.3 | 0 | 154816. | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 576.162 | 0 | 2270.57 | 0 | −37732.3 | 0 | 154816. | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 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 | 256.12.a.l | 8 | |
| 4.b | odd | 2 | 1 | 256.12.a.n | 8 | ||
| 8.b | even | 2 | 1 | 256.12.a.n | 8 | ||
| 8.d | odd | 2 | 1 | inner | 256.12.a.l | 8 | |
| 16.e | even | 4 | 2 | 64.12.b.c | ✓ | 16 | |
| 16.f | odd | 4 | 2 | 64.12.b.c | ✓ | 16 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 64.12.b.c | ✓ | 16 | 16.e | even | 4 | 2 | |
| 64.12.b.c | ✓ | 16 | 16.f | odd | 4 | 2 | |
| 256.12.a.l | 8 | 1.a | even | 1 | 1 | trivial | |
| 256.12.a.l | 8 | 8.d | odd | 2 | 1 | inner | |
| 256.12.a.n | 8 | 4.b | odd | 2 | 1 | ||
| 256.12.a.n | 8 | 8.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(256))\):
|
\( T_{3}^{4} + 496T_{3}^{3} - 386472T_{3}^{2} - 120671424T_{3} - 7245913968 \)
|
|
\( T_{5}^{8} - 233050752 T_{5}^{6} + \cdots + 53\!\cdots\!00 \)
|
|
\( T_{7}^{8} - 11962624512 T_{7}^{6} + \cdots + 24\!\cdots\!84 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T^{4} + 496 T^{3} + \cdots - 7245913968)^{2} \)
|
| $5$ |
\( T^{8} + \cdots + 53\!\cdots\!00 \)
|
| $7$ |
\( T^{8} + \cdots + 24\!\cdots\!84 \)
|
| $11$ |
\( (T^{4} + \cdots + 70\!\cdots\!52)^{2} \)
|
| $13$ |
\( T^{8} + \cdots + 16\!\cdots\!56 \)
|
| $17$ |
\( (T^{4} + \cdots + 10\!\cdots\!08)^{2} \)
|
| $19$ |
\( (T^{4} + \cdots + 10\!\cdots\!64)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 37\!\cdots\!00 \)
|
| $29$ |
\( T^{8} + \cdots + 13\!\cdots\!00 \)
|
| $31$ |
\( T^{8} + \cdots + 32\!\cdots\!04 \)
|
| $37$ |
\( T^{8} + \cdots + 55\!\cdots\!64 \)
|
| $41$ |
\( (T^{4} + \cdots + 14\!\cdots\!24)^{2} \)
|
| $43$ |
\( (T^{4} + \cdots + 12\!\cdots\!52)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 41\!\cdots\!96 \)
|
| $53$ |
\( T^{8} + \cdots + 15\!\cdots\!00 \)
|
| $59$ |
\( (T^{4} + \cdots + 24\!\cdots\!04)^{2} \)
|
| $61$ |
\( T^{8} + \cdots + 30\!\cdots\!16 \)
|
| $67$ |
\( (T^{4} + \cdots + 65\!\cdots\!32)^{2} \)
|
| $71$ |
\( T^{8} + \cdots + 55\!\cdots\!36 \)
|
| $73$ |
\( (T^{4} + \cdots + 96\!\cdots\!88)^{2} \)
|
| $79$ |
\( T^{8} + \cdots + 11\!\cdots\!04 \)
|
| $83$ |
\( (T^{4} + \cdots + 67\!\cdots\!00)^{2} \)
|
| $89$ |
\( (T^{4} + \cdots - 17\!\cdots\!68)^{2} \)
|
| $97$ |
\( (T^{4} + \cdots - 11\!\cdots\!28)^{2} \)
|
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