Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1288,4,Mod(1,1288)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1288, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1288.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1288 = 2^{3} \cdot 7 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1288.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: | \(75.9944600874\) |
| Analytic rank: | \(1\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 3 x^{11} - 197 x^{10} + 551 x^{9} + 13776 x^{8} - 35332 x^{7} - 433468 x^{6} + 942840 x^{5} + \cdots + 79691136 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{14} \) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 3 x^{11} - 197 x^{10} + 551 x^{9} + 13776 x^{8} - 35332 x^{7} - 433468 x^{6} + 942840 x^{5} + \cdots + 79691136 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 34 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 12789744192311 \nu^{11} + 24631759046189 \nu^{10} + \cdots - 16\!\cdots\!68 ) / 74\!\cdots\!80 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 9956710309439 \nu^{11} - 21470887680139 \nu^{10} + \cdots + 28\!\cdots\!68 ) / 49\!\cdots\!20 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 45916733872621 \nu^{11} - 356896223174839 \nu^{10} + \cdots + 28\!\cdots\!48 ) / 14\!\cdots\!60 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 9833405595791 \nu^{11} - 46978407500467 \nu^{10} + \cdots + 24\!\cdots\!28 ) / 29\!\cdots\!12 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 14570588171087 \nu^{11} + 117014655308153 \nu^{10} + \cdots - 96\!\cdots\!56 ) / 37\!\cdots\!40 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 62954753868131 \nu^{11} - 131993017336649 \nu^{10} + \cdots + 58\!\cdots\!48 ) / 14\!\cdots\!60 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 45758792580997 \nu^{11} + 89477290192303 \nu^{10} + \cdots - 29\!\cdots\!56 ) / 74\!\cdots\!80 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 33230376213181 \nu^{11} - 24256422060431 \nu^{10} + \cdots - 76\!\cdots\!28 ) / 37\!\cdots\!40 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 23337775077757 \nu^{11} - 33318552705077 \nu^{10} + \cdots + 19\!\cdots\!64 ) / 18\!\cdots\!20 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 34 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{10} - 2\beta_{8} + \beta_{7} - 2\beta_{6} + 3\beta_{5} - \beta_{4} - 2\beta_{3} + \beta_{2} + 58\beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 8 \beta_{11} - 7 \beta_{10} + 7 \beta_{9} + 3 \beta_{8} - 15 \beta_{6} + 13 \beta_{5} - 9 \beta_{4} + \cdots + 1957 \)
|
| \(\nu^{5}\) | \(=\) |
\( 30 \beta_{11} + 72 \beta_{10} + 18 \beta_{9} - 276 \beta_{8} + 96 \beta_{7} - 204 \beta_{6} + 336 \beta_{5} + \cdots + 72 \)
|
| \(\nu^{6}\) | \(=\) |
\( 900 \beta_{11} - 732 \beta_{10} + 801 \beta_{9} + 123 \beta_{8} - 105 \beta_{7} - 1812 \beta_{6} + \cdots + 140239 \)
|
| \(\nu^{7}\) | \(=\) |
\( 4104 \beta_{11} + 4936 \beta_{10} + 3864 \beta_{9} - 28208 \beta_{8} + 7897 \beta_{7} - 17813 \beta_{6} + \cdots + 52751 \)
|
| \(\nu^{8}\) | \(=\) |
\( 81866 \beta_{11} - 63337 \beta_{10} + 75802 \beta_{9} - 8079 \beta_{8} - 16293 \beta_{7} + \cdots + 11156530 \)
|
| \(\nu^{9}\) | \(=\) |
\( 436464 \beta_{11} + 347430 \beta_{10} + 494952 \beta_{9} - 2638326 \beta_{8} + 651009 \beta_{7} + \cdots + 9391725 \)
|
| \(\nu^{10}\) | \(=\) |
\( 7166142 \beta_{11} - 5318583 \beta_{10} + 6903738 \beta_{9} - 2254605 \beta_{8} - 1728561 \beta_{7} + \cdots + 933835612 \)
|
| \(\nu^{11}\) | \(=\) |
\( 42967836 \beta_{11} + 25254952 \beta_{10} + 52934928 \beta_{9} - 238676312 \beta_{8} + 54743029 \beta_{7} + \cdots + 1237400123 \)
|
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 | −9.54837 | 0 | 3.49009 | 0 | −7.00000 | 0 | 64.1714 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −6.50318 | 0 | −17.0723 | 0 | −7.00000 | 0 | 15.2913 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −5.75353 | 0 | −15.0501 | 0 | −7.00000 | 0 | 6.10308 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −4.94008 | 0 | 7.45554 | 0 | −7.00000 | 0 | −2.59564 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −4.22103 | 0 | 11.8806 | 0 | −7.00000 | 0 | −9.18289 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −1.91372 | 0 | −0.729835 | 0 | −7.00000 | 0 | −23.3377 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 1.67219 | 0 | −12.2658 | 0 | −7.00000 | 0 | −24.2038 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 2.21921 | 0 | 21.1410 | 0 | −7.00000 | 0 | −22.0751 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 4.25868 | 0 | 8.53407 | 0 | −7.00000 | 0 | −8.86367 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 5.34795 | 0 | −17.9873 | 0 | −7.00000 | 0 | 1.60054 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0 | 7.21386 | 0 | 2.11328 | 0 | −7.00000 | 0 | 25.0398 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0 | 9.16802 | 0 | −5.50910 | 0 | −7.00000 | 0 | 57.0526 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(7\) | \(1\) |
| \(23\) | \(-1\) |
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 | 1288.4.a.d | ✓ | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1288.4.a.d | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} + 3 T_{3}^{11} - 197 T_{3}^{10} - 551 T_{3}^{9} + 13776 T_{3}^{8} + 35332 T_{3}^{7} + \cdots + 79691136 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1288))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( T^{12} + 3 T^{11} + \cdots + 79691136 \)
|
| $5$ |
\( T^{12} + \cdots + 26865384448 \)
|
| $7$ |
\( (T + 7)^{12} \)
|
| $11$ |
\( T^{12} + \cdots + 67\!\cdots\!52 \)
|
| $13$ |
\( T^{12} + \cdots - 21\!\cdots\!32 \)
|
| $17$ |
\( T^{12} + \cdots + 13\!\cdots\!32 \)
|
| $19$ |
\( T^{12} + \cdots + 43\!\cdots\!12 \)
|
| $23$ |
\( (T - 23)^{12} \)
|
| $29$ |
\( T^{12} + \cdots - 25\!\cdots\!12 \)
|
| $31$ |
\( T^{12} + \cdots + 15\!\cdots\!92 \)
|
| $37$ |
\( T^{12} + \cdots + 50\!\cdots\!08 \)
|
| $41$ |
\( T^{12} + \cdots + 12\!\cdots\!80 \)
|
| $43$ |
\( T^{12} + \cdots + 63\!\cdots\!68 \)
|
| $47$ |
\( T^{12} + \cdots - 78\!\cdots\!40 \)
|
| $53$ |
\( T^{12} + \cdots + 35\!\cdots\!00 \)
|
| $59$ |
\( T^{12} + \cdots - 13\!\cdots\!36 \)
|
| $61$ |
\( T^{12} + \cdots + 18\!\cdots\!20 \)
|
| $67$ |
\( T^{12} + \cdots + 29\!\cdots\!28 \)
|
| $71$ |
\( T^{12} + \cdots + 65\!\cdots\!00 \)
|
| $73$ |
\( T^{12} + \cdots - 18\!\cdots\!64 \)
|
| $79$ |
\( T^{12} + \cdots - 98\!\cdots\!80 \)
|
| $83$ |
\( T^{12} + \cdots - 20\!\cdots\!80 \)
|
| $89$ |
\( T^{12} + \cdots - 82\!\cdots\!20 \)
|
| $97$ |
\( T^{12} + \cdots + 43\!\cdots\!40 \)
|
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