Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8304,2,Mod(1,8304)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8304, 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("8304.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8304 = 2^{4} \cdot 3 \cdot 173 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8304.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: | \(66.3077738385\) |
| Analytic rank: | \(0\) |
| Dimension: | \(13\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{13} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{13} - 6 x^{12} - 25 x^{11} + 169 x^{10} + 246 x^{9} - 1791 x^{8} - 1116 x^{7} + 8787 x^{6} + \cdots + 506 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 4152) |
| 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 \(1,\beta_1,\ldots,\beta_{12}\) 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
\( x^{13} - 6 x^{12} - 25 x^{11} + 169 x^{10} + 246 x^{9} - 1791 x^{8} - 1116 x^{7} + 8787 x^{6} + \cdots + 506 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 855529776 \nu^{12} - 3250177047 \nu^{11} - 31703776068 \nu^{10} + 92583602422 \nu^{9} + \cdots - 531854709109 ) / 267576184165 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 1641177529 \nu^{12} + 12675255338 \nu^{11} + 20987650282 \nu^{10} - 326113212193 \nu^{9} + \cdots - 47696517269 ) / 267576184165 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 1734972322 \nu^{12} - 11442447784 \nu^{11} - 33837239246 \nu^{10} + 297337905034 \nu^{9} + \cdots - 2136061877618 ) / 267576184165 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 4824347277 \nu^{12} - 26031008799 \nu^{11} - 138746410976 \nu^{10} + 745591139699 \nu^{9} + \cdots - 5661758613228 ) / 535152368330 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 3298002587 \nu^{12} - 19069717994 \nu^{11} - 79081100906 \nu^{10} + 485426106914 \nu^{9} + \cdots - 354139975993 ) / 267576184165 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 3338659631 \nu^{12} - 16070881777 \nu^{11} - 107365683288 \nu^{10} + 473196715817 \nu^{9} + \cdots - 3407521504429 ) / 267576184165 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 4722074022 \nu^{12} + 36380203534 \nu^{11} + 55113380471 \nu^{10} - 875954520499 \nu^{9} + \cdots + 210756204288 ) / 267576184165 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 7125452397 \nu^{12} + 32892866614 \nu^{11} + 232303772036 \nu^{10} - 935993062229 \nu^{9} + \cdots + 5549368400783 ) / 267576184165 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 8766629926 \nu^{12} - 45568121952 \nu^{11} - 253291422318 \nu^{10} + 1262106274422 \nu^{9} + \cdots - 4163790962689 ) / 267576184165 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 17824085667 \nu^{12} - 98251435679 \nu^{11} - 474361045696 \nu^{10} + 2658102921079 \nu^{9} + \cdots - 9311130058728 ) / 535152368330 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 26212464481 \nu^{12} - 136893553327 \nu^{11} - 752499964408 \nu^{10} + 3789663557557 \nu^{9} + \cdots - 10768054065844 ) / 535152368330 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{10} - \beta_{9} - \beta_{3} + 2\beta _1 + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{12} - \beta_{11} - 3\beta_{10} - 3\beta_{9} - \beta_{7} - 3\beta_{3} - \beta_{2} + 15\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4 \beta_{12} - 2 \beta_{11} - 22 \beta_{10} - 18 \beta_{9} + 2 \beta_{6} - 3 \beta_{4} - 19 \beta_{3} + \cdots + 46 \)
|
| \(\nu^{5}\) | \(=\) |
\( 23 \beta_{12} - 21 \beta_{11} - 81 \beta_{10} - 72 \beta_{9} + 2 \beta_{8} - 13 \beta_{7} + 12 \beta_{6} + \cdots + 85 \)
|
| \(\nu^{6}\) | \(=\) |
\( 100 \beta_{12} - 70 \beta_{11} - 434 \beta_{10} - 347 \beta_{9} + 8 \beta_{8} - 2 \beta_{7} + 78 \beta_{6} + \cdots + 580 \)
|
| \(\nu^{7}\) | \(=\) |
\( 484 \beta_{12} - 436 \beta_{11} - 1788 \beta_{10} - 1497 \beta_{9} + 66 \beta_{8} - 141 \beta_{7} + \cdots + 1725 \)
|
| \(\nu^{8}\) | \(=\) |
\( 2191 \beta_{12} - 1790 \beta_{11} - 8610 \beta_{10} - 6827 \beta_{9} + 280 \beta_{8} - 58 \beta_{7} + \cdots + 8866 \)
|
| \(\nu^{9}\) | \(=\) |
\( 10245 \beta_{12} - 9260 \beta_{11} - 37346 \beta_{10} - 30057 \beta_{9} + 1606 \beta_{8} - 1441 \beta_{7} + \cdots + 32681 \)
|
| \(\nu^{10}\) | \(=\) |
\( 47048 \beta_{12} - 40998 \beta_{11} - 172578 \beta_{10} - 134718 \beta_{9} + 7051 \beta_{8} + \cdots + 151295 \)
|
| \(\nu^{11}\) | \(=\) |
\( 218133 \beta_{12} - 197543 \beta_{11} - 764867 \beta_{10} - 596415 \beta_{9} + 35358 \beta_{8} + \cdots + 614326 \)
|
| \(\nu^{12}\) | \(=\) |
\( 1005734 \beta_{12} - 897054 \beta_{11} - 3477007 \beta_{10} - 2656608 \beta_{9} + 157734 \beta_{8} + \cdots + 2741221 \)
|
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 | −1.00000 | 0 | −3.51269 | 0 | 0.725875 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.00000 | 0 | −3.41659 | 0 | −2.97811 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.00000 | 0 | −2.32292 | 0 | −2.42295 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.00000 | 0 | −0.876733 | 0 | 4.02900 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −1.00000 | 0 | −0.533740 | 0 | −1.76592 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −1.00000 | 0 | 0.0509003 | 0 | −1.95995 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | −1.00000 | 0 | 0.116677 | 0 | 3.06598 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | −1.00000 | 0 | 0.947597 | 0 | −5.10937 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | −1.00000 | 0 | 2.79734 | 0 | −0.527488 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | −1.00000 | 0 | 2.85220 | 0 | −0.655017 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0 | −1.00000 | 0 | 3.46711 | 0 | −4.68098 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0 | −1.00000 | 0 | 3.58405 | 0 | −0.125632 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 0 | −1.00000 | 0 | 3.84680 | 0 | 4.40455 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(1\) |
| \(173\) | \(-1\) |
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 | 8304.2.a.bm | 13 | |
| 4.b | odd | 2 | 1 | 4152.2.a.q | ✓ | 13 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4152.2.a.q | ✓ | 13 | 4.b | odd | 2 | 1 | |
| 8304.2.a.bm | 13 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8304))\):
|
\( T_{5}^{13} - 7 T_{5}^{12} - 19 T_{5}^{11} + 216 T_{5}^{10} - 44 T_{5}^{9} - 2220 T_{5}^{8} + 2406 T_{5}^{7} + \cdots + 28 \)
|
|
\( T_{7}^{13} + 8 T_{7}^{12} - 26 T_{7}^{11} - 342 T_{7}^{10} - 217 T_{7}^{9} + 4391 T_{7}^{8} + \cdots + 1024 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{13} \)
|
| $3$ |
\( (T + 1)^{13} \)
|
| $5$ |
\( T^{13} - 7 T^{12} + \cdots + 28 \)
|
| $7$ |
\( T^{13} + 8 T^{12} + \cdots + 1024 \)
|
| $11$ |
\( T^{13} + 2 T^{12} + \cdots + 23968 \)
|
| $13$ |
\( T^{13} - 16 T^{12} + \cdots - 975904 \)
|
| $17$ |
\( T^{13} - 6 T^{12} + \cdots - 1901776 \)
|
| $19$ |
\( T^{13} + 10 T^{12} + \cdots - 259072 \)
|
| $23$ |
\( T^{13} + 21 T^{12} + \cdots + 2146304 \)
|
| $29$ |
\( T^{13} - 11 T^{12} + \cdots - 96091904 \)
|
| $31$ |
\( T^{13} + 5 T^{12} + \cdots + 20536736 \)
|
| $37$ |
\( T^{13} - 28 T^{12} + \cdots + 3437504 \)
|
| $41$ |
\( T^{13} - 8 T^{12} + \cdots - 49694912 \)
|
| $43$ |
\( T^{13} + \cdots + 2864094016 \)
|
| $47$ |
\( T^{13} + 8 T^{12} + \cdots + 7426048 \)
|
| $53$ |
\( T^{13} - 15 T^{12} + \cdots - 5298048 \)
|
| $59$ |
\( T^{13} + \cdots + 858024448 \)
|
| $61$ |
\( T^{13} + \cdots - 420804928 \)
|
| $67$ |
\( T^{13} + \cdots - 9290412736 \)
|
| $71$ |
\( T^{13} + \cdots - 6253479184 \)
|
| $73$ |
\( T^{13} + \cdots + 110403926 \)
|
| $79$ |
\( T^{13} + 22 T^{12} + \cdots - 2866432 \)
|
| $83$ |
\( T^{13} + \cdots + 24223886656 \)
|
| $89$ |
\( T^{13} + \cdots + 144218336192 \)
|
| $97$ |
\( T^{13} + \cdots + 110567256064 \)
|
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