Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8967,2,Mod(1,8967)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8967, 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("8967.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8967 = 3 \cdot 7^{2} \cdot 61 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8967.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: | \(71.6018554925\) |
| Analytic rank: | \(0\) |
| Dimension: | \(19\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{19} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{19} - 4 x^{18} - 23 x^{17} + 104 x^{16} + 198 x^{15} - 1098 x^{14} - 729 x^{13} + 6066 x^{12} + \cdots + 350 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{18}\) 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
\( x^{19} - 4 x^{18} - 23 x^{17} + 104 x^{16} + 198 x^{15} - 1098 x^{14} - 729 x^{13} + 6066 x^{12} + \cdots + 350 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 7\nu^{2} - \nu + 6 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 1572924 \nu^{18} + 34306232 \nu^{17} - 78501240 \nu^{16} - 772233446 \nu^{15} + \cdots - 11127305929 ) / 147463873 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 3623208 \nu^{18} + 32573083 \nu^{17} + 7485634 \nu^{16} - 758336922 \nu^{15} + \cdots - 2060393174 ) / 294927746 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 6918642 \nu^{18} + 53354311 \nu^{17} + 54388112 \nu^{16} - 1271896460 \nu^{15} + \cdots - 9691114386 ) / 294927746 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 13815335 \nu^{18} - 58756055 \nu^{17} - 298190522 \nu^{16} + 1488599610 \nu^{15} + \cdots - 3888741532 ) / 294927746 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 9131134 \nu^{18} - 21388607 \nu^{17} - 261046748 \nu^{16} + 581002205 \nu^{15} + \cdots - 7848553972 ) / 147463873 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 9131134 \nu^{18} - 21388607 \nu^{17} - 261046748 \nu^{16} + 581002205 \nu^{15} + \cdots - 6078987496 ) / 147463873 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 22163555 \nu^{18} + 40529961 \nu^{17} + 689164952 \nu^{16} - 1183859354 \nu^{15} + \cdots + 30740651746 ) / 294927746 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 15921473 \nu^{18} + 114477341 \nu^{17} + 141335934 \nu^{16} - 2704279911 \nu^{15} + \cdots - 25295252865 ) / 147463873 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 39963316 \nu^{18} + 212222293 \nu^{17} + 676001912 \nu^{16} - 5186438520 \nu^{15} + \cdots - 22284733250 ) / 294927746 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 57406263 \nu^{18} + 242420676 \nu^{17} + 1224572230 \nu^{16} - 6070070148 \nu^{15} + \cdots - 6376579286 ) / 294927746 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 61816361 \nu^{18} - 272812290 \nu^{17} - 1268915794 \nu^{16} + 6782684210 \nu^{15} + \cdots + 12463997308 ) / 294927746 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 87363207 \nu^{18} + 425672799 \nu^{17} + 1622698460 \nu^{16} - 10465615152 \nu^{15} + \cdots - 34984506896 ) / 294927746 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 56224582 \nu^{18} + 237589087 \nu^{17} + 1200055354 \nu^{16} - 5956627724 \nu^{15} + \cdots - 3860296965 ) / 147463873 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 56771841 \nu^{18} - 265723817 \nu^{17} - 1111661158 \nu^{16} + 6594745704 \nu^{15} + \cdots + 12414662740 ) / 147463873 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 7\beta_{2} + \beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{17} - \beta_{16} - \beta_{14} + \beta_{12} + \beta_{6} - \beta_{5} - \beta_{4} + 11 \beta_{3} + \cdots + 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{17} - \beta_{16} - \beta_{14} + \beta_{12} - \beta_{10} + \beta_{9} + \beta_{6} - \beta_{5} + \cdots + 88 \)
|
| \(\nu^{7}\) | \(=\) |
\( 11 \beta_{17} - 12 \beta_{16} - \beta_{15} - 12 \beta_{14} + \beta_{13} + 12 \beta_{12} - \beta_{10} + \cdots + 18 \)
|
| \(\nu^{8}\) | \(=\) |
\( - \beta_{18} + 13 \beta_{17} - 16 \beta_{16} - 13 \beta_{14} + \beta_{13} + 16 \beta_{12} + \beta_{11} + \cdots + 552 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 2 \beta_{18} + 90 \beta_{17} - 108 \beta_{16} - 16 \beta_{15} - 106 \beta_{14} + 15 \beta_{13} + \cdots + 213 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 21 \beta_{18} + 120 \beta_{17} - 176 \beta_{16} - \beta_{15} - 121 \beta_{14} + 14 \beta_{13} + \cdots + 3590 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 46 \beta_{18} + 662 \beta_{17} - 887 \beta_{16} - 175 \beta_{15} - 839 \beta_{14} + 151 \beta_{13} + \cdots + 2118 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 285 \beta_{18} + 966 \beta_{17} - 1665 \beta_{16} - 31 \beta_{15} - 1001 \beta_{14} + 129 \beta_{13} + \cdots + 23900 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 662 \beta_{18} + 4631 \beta_{17} - 7027 \beta_{16} - 1638 \beta_{15} - 6314 \beta_{14} + 1280 \beta_{13} + \cdots + 19181 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 3177 \beta_{18} + 7242 \beta_{17} - 14584 \beta_{16} - 539 \beta_{15} - 7873 \beta_{14} + \cdots + 161854 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 7684 \beta_{18} + 31510 \beta_{17} - 54787 \beta_{16} - 14165 \beta_{15} - 46285 \beta_{14} + \cdots + 164054 \)
|
| \(\nu^{16}\) | \(=\) |
\( - 31682 \beta_{18} + 52018 \beta_{17} - 122196 \beta_{16} - 7190 \beta_{15} - 60467 \beta_{14} + \cdots + 1111030 \)
|
| \(\nu^{17}\) | \(=\) |
\( - 78910 \beta_{18} + 210682 \beta_{17} - 423817 \beta_{16} - 117235 \beta_{15} - 334351 \beta_{14} + \cdots + 1351190 \)
|
| \(\nu^{18}\) | \(=\) |
\( - 294471 \beta_{18} + 363040 \beta_{17} - 995885 \beta_{16} - 82489 \beta_{15} - 458763 \beta_{14} + \cdots + 7712550 \)
|
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 |
|
−2.51452 | −1.00000 | 4.32283 | 3.89235 | 2.51452 | 0 | −5.84082 | 1.00000 | −9.78740 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.48306 | −1.00000 | 4.16558 | −3.12763 | 2.48306 | 0 | −5.37727 | 1.00000 | 7.76609 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.38608 | −1.00000 | 3.69339 | 3.36735 | 2.38608 | 0 | −4.04057 | 1.00000 | −8.03478 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.70252 | −1.00000 | 0.898581 | 0.715975 | 1.70252 | 0 | 1.87519 | 1.00000 | −1.21896 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.56798 | −1.00000 | 0.458569 | 1.03695 | 1.56798 | 0 | 2.41694 | 1.00000 | −1.62592 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.929382 | −1.00000 | −1.13625 | 0.382022 | 0.929382 | 0 | 2.91477 | 1.00000 | −0.355045 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.912298 | −1.00000 | −1.16771 | −2.90082 | 0.912298 | 0 | 2.88990 | 1.00000 | 2.64641 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −0.710718 | −1.00000 | −1.49488 | −1.85767 | 0.710718 | 0 | 2.48387 | 1.00000 | 1.32028 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −0.190364 | −1.00000 | −1.96376 | 3.09829 | 0.190364 | 0 | 0.754556 | 1.00000 | −0.589803 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0.642777 | −1.00000 | −1.58684 | −4.23924 | −0.642777 | 0 | −2.30554 | 1.00000 | −2.72489 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0.746083 | −1.00000 | −1.44336 | −1.32326 | −0.746083 | 0 | −2.56903 | 1.00000 | −0.987265 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0.973554 | −1.00000 | −1.05219 | −0.151186 | −0.973554 | 0 | −2.97147 | 1.00000 | −0.147188 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 1.22692 | −1.00000 | −0.494677 | 3.84235 | −1.22692 | 0 | −3.06076 | 1.00000 | 4.71425 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 1.66531 | −1.00000 | 0.773271 | −2.48363 | −1.66531 | 0 | −2.04289 | 1.00000 | −4.13602 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 1.90477 | −1.00000 | 1.62814 | 2.77895 | −1.90477 | 0 | −0.708301 | 1.00000 | 5.29325 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 2.18101 | −1.00000 | 2.75681 | −2.14822 | −2.18101 | 0 | 1.65062 | 1.00000 | −4.68530 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 2.62196 | −1.00000 | 4.87466 | 4.35407 | −2.62196 | 0 | 7.53724 | 1.00000 | 11.4162 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.18 | 2.69923 | −1.00000 | 5.28587 | 0.294682 | −2.69923 | 0 | 8.86932 | 1.00000 | 0.795416 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.19 | 2.73532 | −1.00000 | 5.48196 | −3.53133 | −2.73532 | 0 | 9.52425 | 1.00000 | −9.65931 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(7\) | \( -1 \) |
| \(61\) | \( +1 \) |
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 | 8967.2.a.bi | ✓ | 19 |
| 7.b | odd | 2 | 1 | 8967.2.a.bj | yes | 19 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8967.2.a.bi | ✓ | 19 | 1.a | even | 1 | 1 | trivial |
| 8967.2.a.bj | yes | 19 | 7.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8967))\):
|
\( T_{2}^{19} - 4 T_{2}^{18} - 23 T_{2}^{17} + 104 T_{2}^{16} + 198 T_{2}^{15} - 1098 T_{2}^{14} + \cdots + 350 \)
|
|
\( T_{5}^{19} - 2 T_{5}^{18} - 70 T_{5}^{17} + 116 T_{5}^{16} + 2050 T_{5}^{15} - 2636 T_{5}^{14} + \cdots + 42496 \)
|
|
\( T_{11}^{19} - 4 T_{11}^{18} - 129 T_{11}^{17} + 484 T_{11}^{16} + 6947 T_{11}^{15} - 23754 T_{11}^{14} + \cdots + 54995584 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{19} - 4 T^{18} + \cdots + 350 \)
|
| $3$ |
\( (T + 1)^{19} \)
|
| $5$ |
\( T^{19} - 2 T^{18} + \cdots + 42496 \)
|
| $7$ |
\( T^{19} \)
|
| $11$ |
\( T^{19} - 4 T^{18} + \cdots + 54995584 \)
|
| $13$ |
\( T^{19} + \cdots - 1172040192 \)
|
| $17$ |
\( T^{19} + 5 T^{18} + \cdots + 26322016 \)
|
| $19$ |
\( T^{19} + 18 T^{18} + \cdots - 5748832 \)
|
| $23$ |
\( T^{19} + \cdots + 7070726464 \)
|
| $29$ |
\( T^{19} + \cdots + 3763435480 \)
|
| $31$ |
\( T^{19} + 29 T^{18} + \cdots - 8552448 \)
|
| $37$ |
\( T^{19} + \cdots - 138414080 \)
|
| $41$ |
\( T^{19} + \cdots + 1101803724800 \)
|
| $43$ |
\( T^{19} + \cdots + 1419275638784 \)
|
| $47$ |
\( T^{19} + \cdots - 3691678269440 \)
|
| $53$ |
\( T^{19} + \cdots - 9760985488 \)
|
| $59$ |
\( T^{19} + \cdots - 9015770244868 \)
|
| $61$ |
\( (T + 1)^{19} \)
|
| $67$ |
\( T^{19} + \cdots - 973469113286656 \)
|
| $71$ |
\( T^{19} + \cdots - 32922172686400 \)
|
| $73$ |
\( T^{19} + \cdots + 231639423123456 \)
|
| $79$ |
\( T^{19} + \cdots - 4061839970304 \)
|
| $83$ |
\( T^{19} + \cdots + 428202786816 \)
|
| $89$ |
\( T^{19} + \cdots + 171635482664960 \)
|
| $97$ |
\( T^{19} + \cdots + 893809767974400 \)
|
| show more | |
| show less |