Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [256,10,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, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("256.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 256 = 2^{8} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| 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: | \(131.849174058\) |
| 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: |
\( x^{8} - 1131x^{6} + 409780x^{4} - 50526912x^{2} + 1648955392 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{52}\cdot 3^{2} \) |
| Twist minimal: | no (minimal twist has level 8) |
| 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
\( x^{8} - 1131x^{6} + 409780x^{4} - 50526912x^{2} + 1648955392 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -69\nu^{7} - 45449\nu^{5} + 45862044\nu^{3} - 4708400320\nu ) / 177809664 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 103\nu^{6} - 58477\nu^{4} + 3049676\nu^{2} + 502572640 ) / 81714 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3509\nu^{7} - 3701575\nu^{5} + 1231606564\nu^{3} - 120396053312\nu ) / 177809664 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -55\nu^{6} + 14301\nu^{4} - 1983884\nu^{2} + 1776768672 ) / 81714 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 77603\nu^{7} - 74725505\nu^{5} + 19756287292\nu^{3} - 723641639104\nu ) / 88904832 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -179513\nu^{7} + 163504883\nu^{5} - 41176127092\nu^{3} + 2316148697152\nu ) / 177809664 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 3175\nu^{6} - 2885741\nu^{4} + 740661452\nu^{2} - 42892741024 ) / 81714 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{6} + \beta_{5} + 5\beta_{3} - 98\beta_1 ) / 4096 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{7} - 64\beta_{4} - 65\beta_{2} + 2316288 ) / 8192 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 465\beta_{6} + 353\beta_{5} + 7013\beta_{3} - 59090\beta_1 ) / 4096 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -21\beta_{7} - 38208\beta_{4} - 19755\beta_{2} + 941262848 ) / 8192 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 215767\beta_{6} + 149383\beta_{5} + 3736355\beta_{3} - 35318462\beta_1 ) / 4096 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -41531\beta_{7} - 19797184\beta_{4} - 2792069\beta_{2} + 425837131776 ) / 8192 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 98710953\beta_{6} + 67994105\beta_{5} + 1859056733\beta_{3} - 19879486274\beta_1 ) / 4096 \)
|
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 | −247.414 | 0 | −1417.55 | 0 | 5087.57 | 0 | 41530.8 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −150.106 | 0 | 292.339 | 0 | −9955.46 | 0 | 2848.66 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −100.481 | 0 | 2583.09 | 0 | 6967.65 | 0 | −9586.48 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −67.6316 | 0 | −506.862 | 0 | 300.249 | 0 | −15109.0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 67.6316 | 0 | 506.862 | 0 | 300.249 | 0 | −15109.0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 100.481 | 0 | −2583.09 | 0 | 6967.65 | 0 | −9586.48 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 150.106 | 0 | −292.339 | 0 | −9955.46 | 0 | 2848.66 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 247.414 | 0 | 1417.55 | 0 | 5087.57 | 0 | 41530.8 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
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 | 256.10.a.s | 8 | |
| 4.b | odd | 2 | 1 | 256.10.a.p | 8 | ||
| 8.b | even | 2 | 1 | inner | 256.10.a.s | 8 | |
| 8.d | odd | 2 | 1 | 256.10.a.p | 8 | ||
| 16.e | even | 4 | 2 | 32.10.b.a | 8 | ||
| 16.f | odd | 4 | 2 | 8.10.b.a | ✓ | 8 | |
| 48.i | odd | 4 | 2 | 288.10.d.b | 8 | ||
| 48.k | even | 4 | 2 | 72.10.d.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8.10.b.a | ✓ | 8 | 16.f | odd | 4 | 2 | |
| 32.10.b.a | 8 | 16.e | even | 4 | 2 | ||
| 72.10.d.b | 8 | 48.k | even | 4 | 2 | ||
| 256.10.a.p | 8 | 4.b | odd | 2 | 1 | ||
| 256.10.a.p | 8 | 8.d | odd | 2 | 1 | ||
| 256.10.a.s | 8 | 1.a | even | 1 | 1 | trivial | |
| 256.10.a.s | 8 | 8.b | even | 2 | 1 | inner | |
| 288.10.d.b | 8 | 48.i | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(256))\):
|
\( T_{3}^{8} - 98416T_{3}^{6} + 2654022240T_{3}^{4} - 24101856384768T_{3}^{2} + 63696237642309888 \)
|
|
\( T_{5}^{8} - 9024192T_{5}^{6} + 16402176226816T_{5}^{4} - 4781063725538918400T_{5}^{2} + 294381261264939950080000 \)
|
|
\( T_{7}^{4} - 2400T_{7}^{3} - 83936384T_{7}^{2} + 378295830528T_{7} - 105959154151424 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + \cdots + 63\!\cdots\!88 \)
|
| $5$ |
\( T^{8} + \cdots + 29\!\cdots\!00 \)
|
| $7$ |
\( (T^{4} + \cdots - 105959154151424)^{2} \)
|
| $11$ |
\( T^{8} + \cdots + 18\!\cdots\!00 \)
|
| $13$ |
\( T^{8} + \cdots + 18\!\cdots\!00 \)
|
| $17$ |
\( (T^{4} + \cdots + 49\!\cdots\!48)^{2} \)
|
| $19$ |
\( T^{8} + \cdots + 48\!\cdots\!72 \)
|
| $23$ |
\( (T^{4} + \cdots - 42\!\cdots\!76)^{2} \)
|
| $29$ |
\( T^{8} + \cdots + 46\!\cdots\!00 \)
|
| $31$ |
\( (T^{4} + \cdots + 67\!\cdots\!56)^{2} \)
|
| $37$ |
\( T^{8} + \cdots + 17\!\cdots\!48 \)
|
| $41$ |
\( (T^{4} + \cdots + 74\!\cdots\!80)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 94\!\cdots\!08 \)
|
| $47$ |
\( (T^{4} + \cdots + 54\!\cdots\!12)^{2} \)
|
| $53$ |
\( T^{8} + \cdots + 11\!\cdots\!88 \)
|
| $59$ |
\( T^{8} + \cdots + 77\!\cdots\!08 \)
|
| $61$ |
\( T^{8} + \cdots + 10\!\cdots\!00 \)
|
| $67$ |
\( T^{8} + \cdots + 13\!\cdots\!68 \)
|
| $71$ |
\( (T^{4} + \cdots + 60\!\cdots\!52)^{2} \)
|
| $73$ |
\( (T^{4} + \cdots - 27\!\cdots\!12)^{2} \)
|
| $79$ |
\( (T^{4} + \cdots + 14\!\cdots\!40)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 37\!\cdots\!68 \)
|
| $89$ |
\( (T^{4} + \cdots + 17\!\cdots\!20)^{2} \)
|
| $97$ |
\( (T^{4} + \cdots + 41\!\cdots\!72)^{2} \)
|
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