Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [128,12,Mod(1,128)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(128, 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("128.1");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 128 = 2^{7} \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 128.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: | \(98.3479271116\) |
| 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: |
\( x^{6} - 2x^{5} - 4442x^{4} + 153566x^{3} - 1333532x^{2} - 4433532x + 49754286 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{43}\cdot 3^{2} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 4442x^{4} + 153566x^{3} - 1333532x^{2} - 4433532x + 49754286 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 22274044 \nu^{5} - 418462072 \nu^{4} + 87404493092 \nu^{3} - 1674466085912 \nu^{2} + \cdots + 36980862273615 ) / 112429558305 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 124903048 \nu^{5} + 4134146512 \nu^{4} - 460640866040 \nu^{3} + 1939774343120 \nu^{2} + \cdots - 90301388773827 ) / 112429558305 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 1172219372 \nu^{5} + 17119693592 \nu^{4} - 4885995122548 \nu^{3} + 98676597031288 \nu^{2} + \cdots - 26\!\cdots\!09 ) / 37476519435 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 3310701220 \nu^{5} - 90663757192 \nu^{4} + 12727119015164 \nu^{3} - 120373678691624 \nu^{2} + \cdots + 38\!\cdots\!73 ) / 37476519435 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 39454874912 \nu^{5} + 547867641152 \nu^{4} - 165635503038688 \nu^{3} + \cdots - 20\!\cdots\!39 ) / 112429558305 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + \beta_{3} + 56\beta_{2} + 26\beta _1 + 4106 ) / 12288 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 37\beta_{5} - 93\beta_{4} - 551\beta_{3} - 4974\beta_{2} - 7876\beta _1 + 36406075 ) / 24576 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -3561\beta_{5} + 20459\beta_{4} + 55565\beta_{3} + 1049350\beta_{2} - 773528\beta _1 - 3555021155 ) / 49152 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 367135 \beta_{5} - 1449757 \beta_{4} - 5846283 \beta_{3} - 74555306 \beta_{2} - 44318904 \beta _1 + 357482673941 ) / 49152 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 26433905 \beta_{5} + 122446027 \beta_{4} + 411662141 \beta_{3} + 6319119158 \beta_{2} + \cdots - 26054324125227 ) / 49152 \)
|
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 | −795.850 | 0 | −3925.84 | 0 | 13521.5 | 0 | 456230. | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −219.634 | 0 | 4589.61 | 0 | 25262.2 | 0 | −128908. | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −181.938 | 0 | 4531.11 | 0 | −62513.8 | 0 | −144046. | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 28.4295 | 0 | −13262.1 | 0 | 35677.5 | 0 | −176339. | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 551.275 | 0 | −3654.34 | 0 | −40487.8 | 0 | 126757. | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 597.717 | 0 | 9917.53 | 0 | 77908.4 | 0 | 180119. | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
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 | 128.12.a.e | ✓ | 6 |
| 4.b | odd | 2 | 1 | 128.12.a.g | yes | 6 | |
| 8.b | even | 2 | 1 | 128.12.a.h | yes | 6 | |
| 8.d | odd | 2 | 1 | 128.12.a.f | yes | 6 | |
| 16.e | even | 4 | 2 | 256.12.b.p | 12 | ||
| 16.f | odd | 4 | 2 | 256.12.b.q | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 128.12.a.e | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 128.12.a.f | yes | 6 | 8.d | odd | 2 | 1 | |
| 128.12.a.g | yes | 6 | 4.b | odd | 2 | 1 | |
| 128.12.a.h | yes | 6 | 8.b | even | 2 | 1 | |
| 256.12.b.p | 12 | 16.e | even | 4 | 2 | ||
| 256.12.b.q | 12 | 16.f | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(128))\):
|
\( T_{3}^{6} + 20T_{3}^{5} - 688148T_{3}^{4} + 32764128T_{3}^{3} + 81557676144T_{3}^{2} + 8149590626112T_{3} - 297910794836160 \)
|
|
\( T_{5}^{6} + 1804 T_{5}^{5} - 170673668 T_{5}^{4} + 115715684000 T_{5}^{3} + \cdots - 39\!\cdots\!00 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + \cdots - 297910794836160 \)
|
| $5$ |
\( T^{6} + \cdots - 39\!\cdots\!00 \)
|
| $7$ |
\( T^{6} + \cdots + 24\!\cdots\!40 \)
|
| $11$ |
\( T^{6} + \cdots - 55\!\cdots\!28 \)
|
| $13$ |
\( T^{6} + \cdots - 31\!\cdots\!40 \)
|
| $17$ |
\( T^{6} + \cdots - 49\!\cdots\!40 \)
|
| $19$ |
\( T^{6} + \cdots + 21\!\cdots\!40 \)
|
| $23$ |
\( T^{6} + \cdots - 21\!\cdots\!96 \)
|
| $29$ |
\( T^{6} + \cdots - 11\!\cdots\!28 \)
|
| $31$ |
\( T^{6} + \cdots + 20\!\cdots\!48 \)
|
| $37$ |
\( T^{6} + \cdots + 15\!\cdots\!12 \)
|
| $41$ |
\( T^{6} + \cdots + 75\!\cdots\!92 \)
|
| $43$ |
\( T^{6} + \cdots - 91\!\cdots\!00 \)
|
| $47$ |
\( T^{6} + \cdots + 16\!\cdots\!00 \)
|
| $53$ |
\( T^{6} + \cdots - 19\!\cdots\!40 \)
|
| $59$ |
\( T^{6} + \cdots - 14\!\cdots\!80 \)
|
| $61$ |
\( T^{6} + \cdots + 71\!\cdots\!20 \)
|
| $67$ |
\( T^{6} + \cdots - 22\!\cdots\!76 \)
|
| $71$ |
\( T^{6} + \cdots + 10\!\cdots\!20 \)
|
| $73$ |
\( T^{6} + \cdots - 29\!\cdots\!80 \)
|
| $79$ |
\( T^{6} + \cdots + 13\!\cdots\!60 \)
|
| $83$ |
\( T^{6} + \cdots - 31\!\cdots\!72 \)
|
| $89$ |
\( T^{6} + \cdots - 60\!\cdots\!84 \)
|
| $97$ |
\( T^{6} + \cdots - 23\!\cdots\!84 \)
|
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