Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4008,2,Mod(1,4008)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4008, 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("4008.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4008 = 2^{3} \cdot 3 \cdot 167 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4008.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: | \(32.0040411301\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{12} - 4 x^{11} - 31 x^{10} + 131 x^{9} + 309 x^{8} - 1453 x^{7} - 1072 x^{6} + 6350 x^{5} + \cdots + 3008 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 4 x^{11} - 31 x^{10} + 131 x^{9} + 309 x^{8} - 1453 x^{7} - 1072 x^{6} + 6350 x^{5} + \cdots + 3008 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 2612041 \nu^{11} + 172266458 \nu^{10} - 431149861 \nu^{9} - 5604811029 \nu^{8} + \cdots - 506734859797 ) / 6887737669 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 48270257 \nu^{11} - 393971708 \nu^{10} - 803884765 \nu^{9} + 12731441603 \nu^{8} + \cdots + 623931970190 ) / 13775475338 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 433554181 \nu^{11} - 1027661208 \nu^{10} - 15151502491 \nu^{9} + 32370295155 \nu^{8} + \cdots - 810005934996 ) / 27550950676 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 924523625 \nu^{11} + 3335040372 \nu^{10} + 28727151823 \nu^{9} - 105506939067 \nu^{8} + \cdots - 1194286516536 ) / 55101901352 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 497566513 \nu^{11} - 583717328 \nu^{10} - 19421435387 \nu^{9} + 18043299379 \nu^{8} + \cdots - 2291097916892 ) / 27550950676 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 608147925 \nu^{11} + 2540267068 \nu^{10} + 17866911943 \nu^{9} - 80927081243 \nu^{8} + \cdots - 1570311572784 ) / 27550950676 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 1276718751 \nu^{11} - 4447291652 \nu^{10} - 40378997177 \nu^{9} + 141144188077 \nu^{8} + \cdots + 1003995641592 ) / 55101901352 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 672160257 \nu^{11} + 2096323188 \nu^{10} + 22136844839 \nu^{9} - 66600085467 \nu^{8} + \cdots + 103637063844 ) / 27550950676 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 825574067 \nu^{11} - 1232246264 \nu^{10} - 31068710789 \nu^{9} + 37574547953 \nu^{8} + \cdots - 3936944027888 ) / 27550950676 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 1494430261 \nu^{11} + 4167534940 \nu^{10} + 50046579515 \nu^{9} - 131337919583 \nu^{8} + \cdots + 1214478641340 ) / 27550950676 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{9} + \beta_{7} - \beta_{6} + \beta_{4} + 7 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{10} + \beta_{7} - 2\beta_{5} + \beta_{4} + \beta_{3} + 2\beta_{2} + 12\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{11} - 13 \beta_{9} + 2 \beta_{8} + 15 \beta_{7} - 15 \beta_{6} + 19 \beta_{4} - 2 \beta_{3} + \cdots + 78 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2 \beta_{11} - 15 \beta_{10} + \beta_{9} - 3 \beta_{8} + 11 \beta_{7} - \beta_{6} - 31 \beta_{5} + \cdots + 31 \)
|
| \(\nu^{6}\) | \(=\) |
\( 26 \beta_{11} + 6 \beta_{10} - 157 \beta_{9} + 34 \beta_{8} + 188 \beta_{7} - 202 \beta_{6} + 4 \beta_{5} + \cdots + 976 \)
|
| \(\nu^{7}\) | \(=\) |
\( 49 \beta_{11} - 199 \beta_{10} + 29 \beta_{9} - 57 \beta_{8} + 96 \beta_{7} - 32 \beta_{6} - 401 \beta_{5} + \cdots + 484 \)
|
| \(\nu^{8}\) | \(=\) |
\( 465 \beta_{11} + 136 \beta_{10} - 1897 \beta_{9} + 446 \beta_{8} + 2264 \beta_{7} - 2645 \beta_{6} + \cdots + 12688 \)
|
| \(\nu^{9}\) | \(=\) |
\( 810 \beta_{11} - 2629 \beta_{10} + 581 \beta_{9} - 879 \beta_{8} + 663 \beta_{7} - 715 \beta_{6} + \cdots + 7620 \)
|
| \(\nu^{10}\) | \(=\) |
\( 7141 \beta_{11} + 2177 \beta_{10} - 22913 \beta_{9} + 5288 \beta_{8} + 26926 \beta_{7} - 34360 \beta_{6} + \cdots + 167606 \)
|
| \(\nu^{11}\) | \(=\) |
\( 11158 \beta_{11} - 35307 \beta_{10} + 10456 \beta_{9} - 13224 \beta_{8} + 1478 \beta_{7} - 13396 \beta_{6} + \cdots + 119224 \)
|
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 | −3.64003 | 0 | −1.45249 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | 1.00000 | 0 | −3.49351 | 0 | 4.58171 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 1.00000 | 0 | −2.09997 | 0 | 2.97416 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.00000 | 0 | −1.14709 | 0 | −3.32442 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 1.00000 | 0 | −0.716151 | 0 | −3.33002 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 1.00000 | 0 | −0.619387 | 0 | 0.954126 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 1.00000 | 0 | 1.53322 | 0 | −0.915565 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 1.00000 | 0 | 1.56937 | 0 | 3.21458 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 1.00000 | 0 | 2.31305 | 0 | 4.58648 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 1.00000 | 0 | 2.93317 | 0 | 1.35675 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0 | 1.00000 | 0 | 3.59055 | 0 | 4.10828 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0 | 1.00000 | 0 | 3.77676 | 0 | −1.75359 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(167\) | \(-1\) |
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 | 4008.2.a.l | ✓ | 12 |
| 4.b | odd | 2 | 1 | 8016.2.a.bf | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4008.2.a.l | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 8016.2.a.bf | 12 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4008))\):
|
\( T_{5}^{12} - 4 T_{5}^{11} - 31 T_{5}^{10} + 131 T_{5}^{9} + 309 T_{5}^{8} - 1453 T_{5}^{7} - 1072 T_{5}^{6} + \cdots + 3008 \)
|
|
\( T_{7}^{12} - 11 T_{7}^{11} + 6 T_{7}^{10} + 297 T_{7}^{9} - 719 T_{7}^{8} - 2618 T_{7}^{7} + \cdots - 27584 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( (T - 1)^{12} \)
|
| $5$ |
\( T^{12} - 4 T^{11} + \cdots + 3008 \)
|
| $7$ |
\( T^{12} - 11 T^{11} + \cdots - 27584 \)
|
| $11$ |
\( T^{12} - T^{11} + \cdots + 128 \)
|
| $13$ |
\( T^{12} - 8 T^{11} + \cdots + 55424 \)
|
| $17$ |
\( T^{12} - 3 T^{11} + \cdots + 81616 \)
|
| $19$ |
\( T^{12} - 12 T^{11} + \cdots - 93568 \)
|
| $23$ |
\( T^{12} - 7 T^{11} + \cdots + 1079296 \)
|
| $29$ |
\( T^{12} - 5 T^{11} + \cdots - 2048 \)
|
| $31$ |
\( T^{12} + \cdots + 160663936 \)
|
| $37$ |
\( T^{12} + \cdots + 1249349168 \)
|
| $41$ |
\( T^{12} + 6 T^{11} + \cdots + 59055712 \)
|
| $43$ |
\( T^{12} + \cdots - 142411744 \)
|
| $47$ |
\( T^{12} + \cdots - 1855943024 \)
|
| $53$ |
\( T^{12} - 20 T^{11} + \cdots + 7459216 \)
|
| $59$ |
\( T^{12} + \cdots - 176553400 \)
|
| $61$ |
\( T^{12} + \cdots - 116056064 \)
|
| $67$ |
\( T^{12} + \cdots + 1073105416 \)
|
| $71$ |
\( T^{12} + \cdots + 23931793408 \)
|
| $73$ |
\( T^{12} - 22 T^{11} + \cdots + 63085376 \)
|
| $79$ |
\( T^{12} + \cdots - 3173513344 \)
|
| $83$ |
\( T^{12} - 26 T^{11} + \cdots - 56792 \)
|
| $89$ |
\( T^{12} + \cdots + 40701142672 \)
|
| $97$ |
\( T^{12} + \cdots - 1191476768 \)
|
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