Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [624,2,Mod(305,624)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(624, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([0, 0, 6, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("624.305");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 624 = 2^{4} \cdot 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 624.cn (of order \(12\), degree \(4\), not minimal) |
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: | no |
| Analytic conductor: | \(4.98266508613\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{12})\) |
| Coefficient field: | 16.0.9349208943630483456.9 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + \cdots + 25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{2} \) |
| Twist minimal: | no (minimal twist has level 78) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + \cdots + 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 50 \nu^{14} - 350 \nu^{13} + 2019 \nu^{12} - 7564 \nu^{11} + 23298 \nu^{10} - 55495 \nu^{9} + \cdots + 2945 ) / 65 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 3456 \nu^{15} - 25920 \nu^{14} + 151876 \nu^{13} - 594074 \nu^{12} + 1879372 \nu^{11} + \cdots - 125460 ) / 17095 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 4446 \nu^{15} + 26244 \nu^{14} - 145319 \nu^{13} + 473557 \nu^{12} - 1320043 \nu^{11} + \cdots - 363138 ) / 3419 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 13659 \nu^{15} - 124929 \nu^{14} + 763986 \nu^{13} - 3301409 \nu^{12} + 11090331 \nu^{11} + \cdots - 2004525 ) / 17095 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 26630 \nu^{15} - 182630 \nu^{14} + 1056740 \nu^{13} - 3923413 \nu^{12} + 12097498 \nu^{11} + \cdots + 218885 ) / 17095 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 31151 \nu^{15} - 209568 \nu^{14} + 1205575 \nu^{13} - 4413594 \nu^{12} + 13470792 \nu^{11} + \cdots + 318955 ) / 17095 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 4446 \nu^{15} - 40446 \nu^{14} + 244733 \nu^{13} - 1050316 \nu^{12} + 3488215 \nu^{11} + \cdots - 506866 ) / 3419 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 26630 \nu^{15} - 216820 \nu^{14} + 1296070 \nu^{13} - 5311527 \nu^{12} + 17314892 \nu^{11} + \cdots - 2071845 ) / 17095 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 34456 \nu^{15} - 251845 \nu^{14} + 1471552 \nu^{13} - 5679382 \nu^{12} + 17866121 \nu^{11} + \cdots - 597160 ) / 17095 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 31151 \nu^{15} - 244547 \nu^{14} + 1450428 \nu^{13} - 5830901 \nu^{12} + 18791545 \nu^{11} + \cdots - 1570700 ) / 17095 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 31151 \nu^{15} - 267691 \nu^{14} + 1612436 \nu^{13} - 6769285 \nu^{12} + 22315745 \nu^{11} + \cdots - 2946190 ) / 17095 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 46358 \nu^{15} + 321385 \nu^{14} - 1862113 \nu^{13} + 6958330 \nu^{12} - 21520004 \nu^{11} + \cdots - 273965 ) / 17095 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 46358 \nu^{15} - 337691 \nu^{14} + 1976255 \nu^{13} - 7626613 \nu^{12} + 24045856 \nu^{11} + \cdots - 1289570 ) / 17095 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 70161 \nu^{15} - 497409 \nu^{14} + 2891923 \nu^{13} - 10958903 \nu^{12} + 34143086 \nu^{11} + \cdots - 701840 ) / 17095 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 70161 \nu^{15} - 555006 \nu^{14} + 3295102 \nu^{13} - 13295132 \nu^{12} + 42919133 \nu^{11} + \cdots - 3835485 ) / 17095 \)
|
| \(\nu\) | \(=\) |
\( ( - 2 \beta_{15} - 2 \beta_{14} + 3 \beta_{13} - 3 \beta_{12} + 3 \beta_{11} - 3 \beta_{10} + 6 \beta_{9} + \cdots + 3 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{15} - 3\beta_{12} + 3\beta_{11} - 3\beta_{10} + 3\beta_{9} - 3\beta_{5} + \beta_{3} - 6 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 7 \beta_{15} + 13 \beta_{14} - 21 \beta_{13} + 12 \beta_{12} - 9 \beta_{11} + 12 \beta_{10} - 21 \beta_{9} + \cdots - 24 ) / 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 20 \beta_{15} + 2 \beta_{14} - 6 \beta_{13} + 30 \beta_{12} - 18 \beta_{11} + 24 \beta_{10} - 24 \beta_{9} + \cdots + 18 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 7 \beta_{15} - 93 \beta_{14} + 132 \beta_{13} + 3 \beta_{12} + 18 \beta_{11} + 3 \beta_{10} + 57 \beta_{9} + \cdots + 183 ) / 6 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 135 \beta_{15} - 50 \beta_{14} + 78 \beta_{13} - 207 \beta_{12} + 93 \beta_{11} - 120 \beta_{10} + \cdots - 15 ) / 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 348 \beta_{15} + 604 \beta_{14} - 849 \beta_{13} - 537 \beta_{12} + 3 \beta_{11} - 327 \beta_{10} + \cdots - 1299 ) / 6 \)
|
| \(\nu^{8}\) | \(=\) |
\( 264 \beta_{15} + 212 \beta_{14} - 288 \beta_{13} + 392 \beta_{12} - 164 \beta_{11} + 152 \beta_{10} + \cdots - 202 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 4153 \beta_{15} - 3395 \beta_{14} + 5103 \beta_{13} + 6606 \beta_{12} - 369 \beta_{11} + 3078 \beta_{10} + \cdots + 8202 ) / 6 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 3833 \beta_{15} - 6390 \beta_{14} + 8640 \beta_{13} - 5082 \beta_{12} + 2922 \beta_{11} - 861 \beta_{10} + \cdots + 8814 ) / 3 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 38777 \beta_{15} + 13891 \beta_{14} - 24690 \beta_{13} - 60285 \beta_{12} + 4404 \beta_{11} + \cdots - 43491 ) / 6 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 10130 \beta_{15} + 55670 \beta_{14} - 77934 \beta_{13} + 6762 \beta_{12} - 19512 \beta_{11} + \cdots - 89109 ) / 3 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 316318 \beta_{15} + 444 \beta_{14} + 49461 \beta_{13} + 471657 \beta_{12} - 52155 \beta_{11} + \cdots + 151791 ) / 6 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 77259 \beta_{15} - 430001 \beta_{14} + 630756 \beta_{13} + 188841 \beta_{12} + 132069 \beta_{11} + \cdots + 760593 ) / 3 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 2288517 \beta_{15} - 846113 \beta_{14} + 818229 \beta_{13} - 3234570 \beta_{12} + 590439 \beta_{11} + \cdots + 356820 ) / 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/624\mathbb{Z}\right)^\times\).
| \(n\) | \(79\) | \(145\) | \(209\) | \(469\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{5}\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 305.1 |
|
0 | −1.73022 | − | 0.0795432i | 0 | −0.313444 | − | 0.313444i | 0 | 0.0745867 | − | 0.278362i | 0 | 2.98735 | + | 0.275255i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 305.2 | 0 | −0.933998 | − | 1.45865i | 0 | −2.76293 | − | 2.76293i | 0 | 0.657464 | − | 2.45369i | 0 | −1.25529 | + | 2.72474i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 305.3 | 0 | 0.933998 | + | 1.45865i | 0 | 0.313444 | + | 0.313444i | 0 | 0.0745867 | − | 0.278362i | 0 | −1.25529 | + | 2.72474i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 305.4 | 0 | 1.73022 | + | 0.0795432i | 0 | 2.76293 | + | 2.76293i | 0 | 0.657464 | − | 2.45369i | 0 | 2.98735 | + | 0.275255i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 353.1 | 0 | −1.45865 | − | 0.933998i | 0 | −0.428520 | − | 0.428520i | 0 | 0.735180 | − | 0.196991i | 0 | 1.25529 | + | 2.72474i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 353.2 | 0 | −0.0795432 | − | 1.73022i | 0 | 0.428520 | + | 0.428520i | 0 | 0.735180 | − | 0.196991i | 0 | −2.98735 | + | 0.275255i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 353.3 | 0 | 0.0795432 | + | 1.73022i | 0 | 2.02097 | + | 2.02097i | 0 | −3.46723 | + | 0.929042i | 0 | −2.98735 | + | 0.275255i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 353.4 | 0 | 1.45865 | + | 0.933998i | 0 | −2.02097 | − | 2.02097i | 0 | −3.46723 | + | 0.929042i | 0 | 1.25529 | + | 2.72474i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 401.1 | 0 | −1.73022 | + | 0.0795432i | 0 | −0.313444 | + | 0.313444i | 0 | 0.0745867 | + | 0.278362i | 0 | 2.98735 | − | 0.275255i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 401.2 | 0 | −0.933998 | + | 1.45865i | 0 | −2.76293 | + | 2.76293i | 0 | 0.657464 | + | 2.45369i | 0 | −1.25529 | − | 2.72474i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 401.3 | 0 | 0.933998 | − | 1.45865i | 0 | 0.313444 | − | 0.313444i | 0 | 0.0745867 | + | 0.278362i | 0 | −1.25529 | − | 2.72474i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 401.4 | 0 | 1.73022 | − | 0.0795432i | 0 | 2.76293 | − | 2.76293i | 0 | 0.657464 | + | 2.45369i | 0 | 2.98735 | − | 0.275255i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 449.1 | 0 | −1.45865 | + | 0.933998i | 0 | −0.428520 | + | 0.428520i | 0 | 0.735180 | + | 0.196991i | 0 | 1.25529 | − | 2.72474i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 449.2 | 0 | −0.0795432 | + | 1.73022i | 0 | 0.428520 | − | 0.428520i | 0 | 0.735180 | + | 0.196991i | 0 | −2.98735 | − | 0.275255i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 449.3 | 0 | 0.0795432 | − | 1.73022i | 0 | 2.02097 | − | 2.02097i | 0 | −3.46723 | − | 0.929042i | 0 | −2.98735 | − | 0.275255i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 449.4 | 0 | 1.45865 | − | 0.933998i | 0 | −2.02097 | + | 2.02097i | 0 | −3.46723 | − | 0.929042i | 0 | 1.25529 | − | 2.72474i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 13.f | odd | 12 | 1 | inner |
| 39.k | even | 12 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 624.2.cn.d | 16 | |
| 3.b | odd | 2 | 1 | inner | 624.2.cn.d | 16 | |
| 4.b | odd | 2 | 1 | 78.2.k.a | ✓ | 16 | |
| 12.b | even | 2 | 1 | 78.2.k.a | ✓ | 16 | |
| 13.f | odd | 12 | 1 | inner | 624.2.cn.d | 16 | |
| 39.k | even | 12 | 1 | inner | 624.2.cn.d | 16 | |
| 52.i | odd | 6 | 1 | 1014.2.g.d | 16 | ||
| 52.j | odd | 6 | 1 | 1014.2.g.c | 16 | ||
| 52.l | even | 12 | 1 | 78.2.k.a | ✓ | 16 | |
| 52.l | even | 12 | 1 | 1014.2.g.c | 16 | ||
| 52.l | even | 12 | 1 | 1014.2.g.d | 16 | ||
| 156.p | even | 6 | 1 | 1014.2.g.c | 16 | ||
| 156.r | even | 6 | 1 | 1014.2.g.d | 16 | ||
| 156.v | odd | 12 | 1 | 78.2.k.a | ✓ | 16 | |
| 156.v | odd | 12 | 1 | 1014.2.g.c | 16 | ||
| 156.v | odd | 12 | 1 | 1014.2.g.d | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 78.2.k.a | ✓ | 16 | 4.b | odd | 2 | 1 | |
| 78.2.k.a | ✓ | 16 | 12.b | even | 2 | 1 | |
| 78.2.k.a | ✓ | 16 | 52.l | even | 12 | 1 | |
| 78.2.k.a | ✓ | 16 | 156.v | odd | 12 | 1 | |
| 624.2.cn.d | 16 | 1.a | even | 1 | 1 | trivial | |
| 624.2.cn.d | 16 | 3.b | odd | 2 | 1 | inner | |
| 624.2.cn.d | 16 | 13.f | odd | 12 | 1 | inner | |
| 624.2.cn.d | 16 | 39.k | even | 12 | 1 | inner | |
| 1014.2.g.c | 16 | 52.j | odd | 6 | 1 | ||
| 1014.2.g.c | 16 | 52.l | even | 12 | 1 | ||
| 1014.2.g.c | 16 | 156.p | even | 6 | 1 | ||
| 1014.2.g.c | 16 | 156.v | odd | 12 | 1 | ||
| 1014.2.g.d | 16 | 52.i | odd | 6 | 1 | ||
| 1014.2.g.d | 16 | 52.l | even | 12 | 1 | ||
| 1014.2.g.d | 16 | 156.r | even | 6 | 1 | ||
| 1014.2.g.d | 16 | 156.v | odd | 12 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(624, [\chi])\):
|
\( T_{5}^{16} + 300T_{5}^{12} + 15606T_{5}^{8} + 2700T_{5}^{4} + 81 \)
|
|
\( T_{7}^{8} + 4T_{7}^{7} + 2T_{7}^{6} + 16T_{7}^{5} + 46T_{7}^{4} - 112T_{7}^{3} + 68T_{7}^{2} - 16T_{7} + 4 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} \)
|
| $3$ |
\( T^{16} - 6 T^{12} + \cdots + 6561 \)
|
| $5$ |
\( T^{16} + 300 T^{12} + \cdots + 81 \)
|
| $7$ |
\( (T^{8} + 4 T^{7} + 2 T^{6} + \cdots + 4)^{2} \)
|
| $11$ |
\( (T^{8} - 30 T^{6} + \cdots + 36)^{2} \)
|
| $13$ |
\( (T^{8} + 12 T^{7} + \cdots + 28561)^{2} \)
|
| $17$ |
\( T^{16} + \cdots + 151807041 \)
|
| $19$ |
\( (T^{8} - 8 T^{7} + \cdots + 5476)^{2} \)
|
| $23$ |
\( T^{16} + 60 T^{14} + \cdots + 18974736 \)
|
| $29$ |
\( T^{16} + \cdots + 33871089681 \)
|
| $31$ |
\( (T^{8} + 8 T^{7} + \cdots + 7744)^{2} \)
|
| $37$ |
\( (T^{8} - 8 T^{7} + \cdots + 375769)^{2} \)
|
| $41$ |
\( T^{16} + \cdots + 65697655057281 \)
|
| $43$ |
\( (T^{8} - 18 T^{6} + \cdots + 2916)^{2} \)
|
| $47$ |
\( T^{16} + \cdots + 41006250000 \)
|
| $53$ |
\( (T^{8} + 156 T^{6} + \cdots + 522729)^{2} \)
|
| $59$ |
\( T^{16} + \cdots + 43489065701376 \)
|
| $61$ |
\( (T^{8} + 12 T^{7} + \cdots + 47961)^{2} \)
|
| $67$ |
\( (T^{8} + 16 T^{7} + \cdots + 676)^{2} \)
|
| $71$ |
\( T^{16} + \cdots + 5143987297296 \)
|
| $73$ |
\( (T^{8} - 28 T^{7} + \cdots + 16834609)^{2} \)
|
| $79$ |
\( (T^{4} - 24 T^{3} + \cdots + 312)^{4} \)
|
| $83$ |
\( T^{16} + \cdots + 36804120336 \)
|
| $89$ |
\( T^{16} + \cdots + 59\!\cdots\!56 \)
|
| $97$ |
\( (T^{8} - 8 T^{7} + \cdots + 43264)^{2} \)
|
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