Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8001,2,Mod(1,8001)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8001, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("8001.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8001 = 3^{2} \cdot 7 \cdot 127 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8001.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: | \(63.8883066572\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 2 x^{15} - 20 x^{14} + 38 x^{13} + 155 x^{12} - 275 x^{11} - 593 x^{10} + 957 x^{9} + 1177 x^{8} + \cdots + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 889) |
| 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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 2 x^{15} - 20 x^{14} + 38 x^{13} + 155 x^{12} - 275 x^{11} - 593 x^{10} + 957 x^{9} + 1177 x^{8} + \cdots + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 116093 \nu^{15} + 228893 \nu^{14} + 2347523 \nu^{13} - 4412026 \nu^{12} - 18385016 \nu^{11} + \cdots - 128712 ) / 239255 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 118542 \nu^{15} - 282272 \nu^{14} - 2211727 \nu^{13} + 5170999 \nu^{12} + 15614719 \nu^{11} + \cdots + 299938 ) / 239255 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 246643 \nu^{15} - 556793 \nu^{14} - 4734663 \nu^{13} + 10433381 \nu^{12} + 34632721 \nu^{11} + \cdots + 1167877 ) / 239255 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 416133 \nu^{15} + 848713 \nu^{14} + 8205093 \nu^{13} - 16039026 \nu^{12} - 62183706 \nu^{11} + \cdots - 1601912 ) / 239255 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 420868 \nu^{15} + 831558 \nu^{14} + 8486798 \nu^{13} - 16040641 \nu^{12} - 66156781 \nu^{11} + \cdots - 2488852 ) / 239255 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 86516 \nu^{15} - 193256 \nu^{14} - 1686213 \nu^{13} + 3671187 \nu^{12} + 12597346 \nu^{11} + \cdots + 403609 ) / 47851 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 558146 \nu^{15} + 1242406 \nu^{14} + 10825416 \nu^{13} - 23508057 \nu^{12} - 80263942 \nu^{11} + \cdots - 2398759 ) / 239255 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 609731 \nu^{15} - 1339991 \nu^{14} - 11975331 \nu^{13} + 25605932 \nu^{12} + 90323612 \nu^{11} + \cdots + 2939939 ) / 239255 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 614167 \nu^{15} + 1330387 \nu^{14} + 12063962 \nu^{13} - 25378599 \nu^{12} - 90934424 \nu^{11} + \cdots - 2942448 ) / 239255 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 673264 \nu^{15} + 1527129 \nu^{14} + 13105079 \nu^{13} - 29129163 \nu^{12} - 97629743 \nu^{11} + \cdots - 5229541 ) / 239255 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 907437 \nu^{15} - 1921932 \nu^{14} - 18024552 \nu^{13} + 36966909 \nu^{12} + 137780884 \nu^{11} + \cdots + 5318248 ) / 239255 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 183898 \nu^{15} + 398886 \nu^{14} + 3621479 \nu^{13} - 7620756 \nu^{12} - 27413174 \nu^{11} + \cdots - 1000302 ) / 47851 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 929511 \nu^{15} - 2103041 \nu^{14} - 18058381 \nu^{13} + 39940442 \nu^{12} + 134317747 \nu^{11} + \cdots + 5430944 ) / 239255 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 5\beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{15} + 2\beta_{14} - \beta_{11} + \beta_{9} + \beta_{8} - \beta_{7} + 5\beta_{2} + 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( 9 \beta_{15} + 9 \beta_{14} + 10 \beta_{13} + 10 \beta_{12} + \beta_{11} + \beta_{10} - 9 \beta_{5} + \cdots - 9 \)
|
| \(\nu^{6}\) | \(=\) |
\( 10 \beta_{15} + 22 \beta_{14} + 3 \beta_{13} + 3 \beta_{12} - 7 \beta_{11} + 3 \beta_{10} + 9 \beta_{9} + \cdots + 63 \)
|
| \(\nu^{7}\) | \(=\) |
\( 68 \beta_{15} + 72 \beta_{14} + 81 \beta_{13} + 81 \beta_{12} + 15 \beta_{11} + 16 \beta_{10} + \cdots - 71 \)
|
| \(\nu^{8}\) | \(=\) |
\( 81 \beta_{15} + 193 \beta_{14} + 49 \beta_{13} + 45 \beta_{12} - 30 \beta_{11} + 46 \beta_{10} + \cdots + 320 \)
|
| \(\nu^{9}\) | \(=\) |
\( 492 \beta_{15} + 555 \beta_{14} + 618 \beta_{13} + 614 \beta_{12} + 161 \beta_{11} + 171 \beta_{10} + \cdots - 537 \)
|
| \(\nu^{10}\) | \(=\) |
\( 619 \beta_{15} + 1562 \beta_{14} + 537 \beta_{13} + 469 \beta_{12} - 45 \beta_{11} + 490 \beta_{10} + \cdots + 1658 \)
|
| \(\nu^{11}\) | \(=\) |
\( 3511 \beta_{15} + 4202 \beta_{14} + 4602 \beta_{13} + 4524 \beta_{12} + 1495 \beta_{11} + 1567 \beta_{10} + \cdots - 3961 \)
|
| \(\nu^{12}\) | \(=\) |
\( 4639 \beta_{15} + 12147 \beta_{14} + 5003 \beta_{13} + 4231 \beta_{12} + 822 \beta_{11} + 4497 \beta_{10} + \cdots + 8626 \)
|
| \(\nu^{13}\) | \(=\) |
\( 24939 \beta_{15} + 31494 \beta_{14} + 33856 \beta_{13} + 32868 \beta_{12} + 12804 \beta_{11} + \cdots - 28730 \)
|
| \(\nu^{14}\) | \(=\) |
\( 34545 \beta_{15} + 92328 \beta_{14} + 42883 \beta_{13} + 35508 \beta_{12} + 13085 \beta_{11} + \cdots + 44462 \)
|
| \(\nu^{15}\) | \(=\) |
\( 176958 \beta_{15} + 234588 \beta_{14} + 247371 \beta_{13} + 237044 \beta_{12} + 104286 \beta_{11} + \cdots - 206072 \)
|
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 |
|
−2.57541 | 0 | 4.63271 | 3.61540 | 0 | −1.00000 | −6.78030 | 0 | −9.31112 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.24781 | 0 | 3.05263 | 0.411504 | 0 | −1.00000 | −2.36611 | 0 | −0.924982 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.72082 | 0 | 0.961212 | 4.11468 | 0 | −1.00000 | 1.78756 | 0 | −7.08062 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.48078 | 0 | 0.192706 | −1.52223 | 0 | −1.00000 | 2.67620 | 0 | 2.25409 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.45188 | 0 | 0.107956 | −0.584615 | 0 | −1.00000 | 2.74702 | 0 | 0.848791 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.532475 | 0 | −1.71647 | 0.118172 | 0 | −1.00000 | 1.97893 | 0 | −0.0629236 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.191617 | 0 | −1.96328 | −3.92161 | 0 | −1.00000 | 0.759431 | 0 | 0.751445 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −0.170601 | 0 | −1.97090 | 0.206353 | 0 | −1.00000 | 0.677439 | 0 | −0.0352041 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0.102239 | 0 | −1.98955 | −0.280585 | 0 | −1.00000 | −0.407889 | 0 | −0.0286868 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0.625678 | 0 | −1.60853 | 0.920707 | 0 | −1.00000 | −2.25778 | 0 | 0.576066 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 1.24549 | 0 | −0.448744 | 2.43368 | 0 | −1.00000 | −3.04990 | 0 | 3.03114 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 1.33134 | 0 | −0.227522 | 2.92411 | 0 | −1.00000 | −2.96560 | 0 | 3.89299 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 1.79993 | 0 | 1.23975 | −3.74624 | 0 | −1.00000 | −1.36840 | 0 | −6.74297 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 2.18919 | 0 | 2.79257 | −0.868150 | 0 | −1.00000 | 1.73509 | 0 | −1.90055 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 2.37299 | 0 | 3.63108 | 3.84175 | 0 | −1.00000 | 3.87053 | 0 | 9.11644 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 2.70451 | 0 | 5.31438 | 1.33706 | 0 | −1.00000 | 8.96376 | 0 | 3.61609 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(1\) |
| \(127\) | \(1\) |
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 | 8001.2.a.t | 16 | |
| 3.b | odd | 2 | 1 | 889.2.a.c | ✓ | 16 | |
| 21.c | even | 2 | 1 | 6223.2.a.k | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 889.2.a.c | ✓ | 16 | 3.b | odd | 2 | 1 | |
| 6223.2.a.k | 16 | 21.c | even | 2 | 1 | ||
| 8001.2.a.t | 16 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):
|
\( T_{2}^{16} - 2 T_{2}^{15} - 20 T_{2}^{14} + 38 T_{2}^{13} + 155 T_{2}^{12} - 275 T_{2}^{11} - 593 T_{2}^{10} + \cdots + 1 \)
|
|
\( T_{5}^{16} - 9 T_{5}^{15} - 7 T_{5}^{14} + 273 T_{5}^{13} - 532 T_{5}^{12} - 2135 T_{5}^{11} + 7523 T_{5}^{10} + \cdots + 16 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} - 2 T^{15} + \cdots + 1 \)
|
| $3$ |
\( T^{16} \)
|
| $5$ |
\( T^{16} - 9 T^{15} + \cdots + 16 \)
|
| $7$ |
\( (T + 1)^{16} \)
|
| $11$ |
\( T^{16} - 22 T^{15} + \cdots - 417463 \)
|
| $13$ |
\( T^{16} + 4 T^{15} + \cdots + 9972315 \)
|
| $17$ |
\( T^{16} - 18 T^{15} + \cdots + 22881193 \)
|
| $19$ |
\( T^{16} + 15 T^{15} + \cdots + 906809 \)
|
| $23$ |
\( T^{16} + \cdots + 1853701360 \)
|
| $29$ |
\( T^{16} + \cdots + 8925976496 \)
|
| $31$ |
\( T^{16} + 32 T^{15} + \cdots + 713813 \)
|
| $37$ |
\( T^{16} + \cdots - 5726959039 \)
|
| $41$ |
\( T^{16} + \cdots + 19825104281 \)
|
| $43$ |
\( T^{16} + \cdots - 219978965872 \)
|
| $47$ |
\( T^{16} + \cdots + 49063403511 \)
|
| $53$ |
\( T^{16} + \cdots + 114032072912 \)
|
| $59$ |
\( T^{16} + \cdots - 58448663216 \)
|
| $61$ |
\( T^{16} + \cdots - 123057095311 \)
|
| $67$ |
\( T^{16} + \cdots + 15509381648 \)
|
| $71$ |
\( T^{16} + \cdots + 120439326815165 \)
|
| $73$ |
\( T^{16} + \cdots - 145782520397 \)
|
| $79$ |
\( T^{16} + \cdots + 845588972425 \)
|
| $83$ |
\( T^{16} + \cdots - 3653038864 \)
|
| $89$ |
\( T^{16} + \cdots + 10200551550832 \)
|
| $97$ |
\( T^{16} + \cdots - 269740302224 \)
|
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