Newspace parameters
Level: | \( N \) | \(=\) | \( 38 = 2 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 38.e (of order \(9\), degree \(6\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.24207258022\) |
Analytic rank: | \(0\) |
Dimension: | \(12\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{9})\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
Defining polynomial: |
\( x^{12} - 6 x^{11} - 135 x^{10} + 730 x^{9} + 7953 x^{8} - 36258 x^{7} - 262940 x^{6} + 918855 x^{5} + 5157591 x^{4} - 11890401 x^{3} - 56759508 x^{2} + \cdots + 272110107 \)
|
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{9}]$ |
$q$-expansion
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
\( x^{12} - 6 x^{11} - 135 x^{10} + 730 x^{9} + 7953 x^{8} - 36258 x^{7} - 262940 x^{6} + 918855 x^{5} + 5157591 x^{4} - 11890401 x^{3} - 56759508 x^{2} + \cdots + 272110107 \)
:
\(\beta_{1}\) | \(=\) |
\( \nu \)
|
\(\beta_{2}\) | \(=\) |
\( ( - 177221217078 \nu^{11} + 3257235752468 \nu^{10} + 7448859189417 \nu^{9} - 348800270203041 \nu^{8} + \cdots - 21\!\cdots\!68 ) / 20\!\cdots\!27 \)
|
\(\beta_{3}\) | \(=\) |
\( ( 177221217078 \nu^{11} + 1307802364610 \nu^{10} - 30274049774807 \nu^{9} - 164426429455098 \nu^{8} + \cdots - 19\!\cdots\!62 ) / 20\!\cdots\!27 \)
|
\(\beta_{4}\) | \(=\) |
\( ( - 488144071095 \nu^{11} + 17846740004059 \nu^{10} - 13839005362779 \nu^{9} + \cdots - 20\!\cdots\!46 ) / 20\!\cdots\!27 \)
|
\(\beta_{5}\) | \(=\) |
\( ( 488144071095 \nu^{11} + 12477155222014 \nu^{10} - 137780470767586 \nu^{9} + \cdots - 19\!\cdots\!88 ) / 20\!\cdots\!27 \)
|
\(\beta_{6}\) | \(=\) |
\( ( - 545356750193 \nu^{11} + 639178329540 \nu^{10} + 77274249510048 \nu^{9} - 43408610678418 \nu^{8} + \cdots + 19\!\cdots\!01 ) / 20\!\cdots\!27 \)
|
\(\beta_{7}\) | \(=\) |
\( ( 1581666 \nu^{11} - 8699163 \nu^{10} - 191254825 \nu^{9} + 925890435 \nu^{8} + 9722082222 \nu^{7} - 38409003948 \nu^{6} + \cdots + 11710200330886 ) / 486606417103 \)
|
\(\beta_{8}\) | \(=\) |
\( ( 722577967271 \nu^{11} - 4051943557973 \nu^{10} - 83945461319640 \nu^{9} + 426412366504371 \nu^{8} + \cdots + 74\!\cdots\!83 ) / 20\!\cdots\!27 \)
|
\(\beta_{9}\) | \(=\) |
\( ( 882100559090 \nu^{11} + 8843624294225 \nu^{10} - 174667730262753 \nu^{9} + \cdots - 20\!\cdots\!59 ) / 20\!\cdots\!27 \)
|
\(\beta_{10}\) | \(=\) |
\( ( - 1648551699807 \nu^{11} + 11116259193530 \nu^{10} + 165757370191268 \nu^{9} + \cdots - 21\!\cdots\!01 ) / 20\!\cdots\!27 \)
|
\(\beta_{11}\) | \(=\) |
\( ( - 1648551699807 \nu^{11} + 7017809504347 \nu^{10} + 186249618637183 \nu^{9} - 654710361086352 \nu^{8} + \cdots + 92\!\cdots\!83 ) / 20\!\cdots\!27 \)
|
\(\nu\) | \(=\) |
\( \beta_1 \)
|
\(\nu^{2}\) | \(=\) |
\( -\beta_{11} + \beta_{10} - 3\beta_{8} - 3\beta_{6} - \beta_{3} - 4\beta_{2} + \beta _1 + 26 \)
|
\(\nu^{3}\) | \(=\) |
\( 3 \beta_{10} - 15 \beta_{8} + 10 \beta_{7} + 6 \beta_{6} + 3 \beta_{5} - 3 \beta_{4} + 28 \beta_{3} - 25 \beta_{2} + 25 \beta _1 + 32 \)
|
\(\nu^{4}\) | \(=\) |
\( - 53 \beta_{11} + 59 \beta_{10} + 10 \beta_{9} - 217 \beta_{8} + 15 \beta_{7} - 165 \beta_{6} - 3 \beta_{5} - 15 \beta_{4} + 37 \beta_{3} - 256 \beta_{2} + 44 \beta _1 + 698 \)
|
\(\nu^{5}\) | \(=\) |
\( - 81 \beta_{11} + 194 \beta_{10} + 25 \beta_{9} - 1321 \beta_{8} + 892 \beta_{7} + 381 \beta_{6} + 162 \beta_{5} - 207 \beta_{4} + 1665 \beta_{3} - 1414 \beta_{2} + 615 \beta _1 + 1854 \)
|
\(\nu^{6}\) | \(=\) |
\( - 2280 \beta_{11} + 2604 \beta_{10} + 917 \beta_{9} - 11518 \beta_{8} + 2205 \beta_{7} - 5675 \beta_{6} - 219 \beta_{5} - 1296 \beta_{4} + 5259 \beta_{3} - 11343 \beta_{2} + 1302 \beta _1 + 19896 \)
|
\(\nu^{7}\) | \(=\) |
\( - 6561 \beta_{11} + 9574 \beta_{10} + 3122 \beta_{9} - 76494 \beta_{8} + 52916 \beta_{7} + 19598 \beta_{6} + 5456 \beta_{5} - 10601 \beta_{4} + 78989 \beta_{3} - 60371 \beta_{2} + 14109 \beta _1 + 81501 \)
|
\(\nu^{8}\) | \(=\) |
\( - 93098 \beta_{11} + 103652 \beta_{10} + 56038 \beta_{9} - 531544 \beta_{8} + 177506 \beta_{7} - 126516 \beta_{6} - 14142 \beta_{5} - 73372 \beta_{4} + 339489 \beta_{3} - 419971 \beta_{2} + \cdots + 609442 \)
|
\(\nu^{9}\) | \(=\) |
\( - 366048 \beta_{11} + 423666 \beta_{10} + 233544 \beta_{9} - 3638850 \beta_{8} + 2627538 \beta_{7} + 1015026 \beta_{6} + 112374 \beta_{5} - 475506 \beta_{4} + 3478249 \beta_{3} + \cdots + 3220296 \)
|
\(\nu^{10}\) | \(=\) |
\( - 3739994 \beta_{11} + 3951167 \beta_{10} + 2861082 \beta_{9} - 22427599 \beta_{8} + 10768116 \beta_{7} - 48010 \beta_{6} - 902652 \beta_{5} - 3405306 \beta_{4} + \cdots + 19795826 \)
|
\(\nu^{11}\) | \(=\) |
\( - 17577635 \beta_{11} + 17659182 \beta_{10} + 13629198 \beta_{9} - 153343365 \beta_{8} + 118000897 \beta_{7} + 53163670 \beta_{6} - 854642 \beta_{5} - 19566517 \beta_{4} + \cdots + 119730930 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/38\mathbb{Z}\right)^\times\).
\(n\) | \(21\) |
\(\chi(n)\) | \(-\beta_{6}\) |
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.
Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5.1 |
|
−1.87939 | + | 0.684040i | −1.54081 | + | 8.73839i | 3.06418 | − | 2.57115i | −15.3487 | − | 12.8791i | −3.08163 | − | 17.4768i | 4.71270 | + | 8.16264i | −4.00000 | + | 6.92820i | −48.6136 | − | 17.6939i | 37.6558 | + | 13.7056i | ||||||||||||||||||||||||||||||||||||
5.2 | −1.87939 | + | 0.684040i | 0.0408138 | − | 0.231467i | 3.06418 | − | 2.57115i | 8.45426 | + | 7.09396i | 0.0816277 | + | 0.462933i | 9.26682 | + | 16.0506i | −4.00000 | + | 6.92820i | 25.3198 | + | 9.21565i | −20.7414 | − | 7.54924i | |||||||||||||||||||||||||||||||||||||
9.1 | 0.347296 | + | 1.96962i | −4.43054 | + | 3.71766i | −3.75877 | + | 1.36808i | −5.82452 | − | 2.11995i | −8.86108 | − | 7.43533i | −5.61124 | + | 9.71895i | −4.00000 | − | 6.92820i | 1.12015 | − | 6.35271i | 2.15265 | − | 12.2083i | |||||||||||||||||||||||||||||||||||||
9.2 | 0.347296 | + | 1.96962i | 2.93054 | − | 2.45902i | −3.75877 | + | 1.36808i | 14.2818 | + | 5.19813i | 5.86108 | + | 4.91803i | −0.806634 | + | 1.39713i | −4.00000 | − | 6.92820i | −2.14719 | + | 12.1773i | −5.27832 | + | 29.9349i | |||||||||||||||||||||||||||||||||||||
17.1 | 0.347296 | − | 1.96962i | −4.43054 | − | 3.71766i | −3.75877 | − | 1.36808i | −5.82452 | + | 2.11995i | −8.86108 | + | 7.43533i | −5.61124 | − | 9.71895i | −4.00000 | + | 6.92820i | 1.12015 | + | 6.35271i | 2.15265 | + | 12.2083i | |||||||||||||||||||||||||||||||||||||
17.2 | 0.347296 | − | 1.96962i | 2.93054 | + | 2.45902i | −3.75877 | − | 1.36808i | 14.2818 | − | 5.19813i | 5.86108 | − | 4.91803i | −0.806634 | − | 1.39713i | −4.00000 | + | 6.92820i | −2.14719 | − | 12.1773i | −5.27832 | − | 29.9349i | |||||||||||||||||||||||||||||||||||||
23.1 | −1.87939 | − | 0.684040i | −1.54081 | − | 8.73839i | 3.06418 | + | 2.57115i | −15.3487 | + | 12.8791i | −3.08163 | + | 17.4768i | 4.71270 | − | 8.16264i | −4.00000 | − | 6.92820i | −48.6136 | + | 17.6939i | 37.6558 | − | 13.7056i | |||||||||||||||||||||||||||||||||||||
23.2 | −1.87939 | − | 0.684040i | 0.0408138 | + | 0.231467i | 3.06418 | + | 2.57115i | 8.45426 | − | 7.09396i | 0.0816277 | − | 0.462933i | 9.26682 | − | 16.0506i | −4.00000 | − | 6.92820i | 25.3198 | − | 9.21565i | −20.7414 | + | 7.54924i | |||||||||||||||||||||||||||||||||||||
25.1 | 1.53209 | − | 1.28558i | −6.18284 | − | 2.25037i | 0.694593 | − | 3.93923i | −1.97089 | − | 11.1775i | −12.3657 | + | 4.50074i | 4.35993 | − | 7.55162i | −4.00000 | − | 6.92820i | 12.4801 | + | 10.4721i | −17.3890 | − | 14.5911i | |||||||||||||||||||||||||||||||||||||
25.2 | 1.53209 | − | 1.28558i | 4.68284 | + | 1.70441i | 0.694593 | − | 3.93923i | 0.408053 | + | 2.31419i | 9.36568 | − | 3.40883i | −1.42158 | + | 2.46225i | −4.00000 | − | 6.92820i | −1.65925 | − | 1.39227i | 3.60023 | + | 3.02095i | |||||||||||||||||||||||||||||||||||||
35.1 | 1.53209 | + | 1.28558i | −6.18284 | + | 2.25037i | 0.694593 | + | 3.93923i | −1.97089 | + | 11.1775i | −12.3657 | − | 4.50074i | 4.35993 | + | 7.55162i | −4.00000 | + | 6.92820i | 12.4801 | − | 10.4721i | −17.3890 | + | 14.5911i | |||||||||||||||||||||||||||||||||||||
35.2 | 1.53209 | + | 1.28558i | 4.68284 | − | 1.70441i | 0.694593 | + | 3.93923i | 0.408053 | − | 2.31419i | 9.36568 | + | 3.40883i | −1.42158 | − | 2.46225i | −4.00000 | + | 6.92820i | −1.65925 | + | 1.39227i | 3.60023 | − | 3.02095i | |||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
19.e | even | 9 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 38.4.e.a | ✓ | 12 |
19.e | even | 9 | 1 | inner | 38.4.e.a | ✓ | 12 |
19.e | even | 9 | 1 | 722.4.a.p | 6 | ||
19.f | odd | 18 | 1 | 722.4.a.o | 6 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
38.4.e.a | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
38.4.e.a | ✓ | 12 | 19.e | even | 9 | 1 | inner |
722.4.a.o | 6 | 19.f | odd | 18 | 1 | ||
722.4.a.p | 6 | 19.e | even | 9 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} + 9 T_{3}^{11} + 54 T_{3}^{10} + 18 T_{3}^{9} - 3078 T_{3}^{8} - 12789 T_{3}^{7} + 104059 T_{3}^{6} + 455373 T_{3}^{5} - 1524069 T_{3}^{4} - 7640667 T_{3}^{3} + 41991777 T_{3}^{2} + \cdots + 2289169 \)
acting on \(S_{4}^{\mathrm{new}}(38, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{6} + 8 T^{3} + 64)^{2} \)
$3$
\( T^{12} + 9 T^{11} + 54 T^{10} + \cdots + 2289169 \)
$5$
\( T^{12} - 171 T^{10} + \cdots + 308668025241 \)
$7$
\( T^{12} - 21 T^{11} + \cdots + 6147971281 \)
$11$
\( T^{12} + 9 T^{11} + \cdots + 55\!\cdots\!09 \)
$13$
\( T^{12} - 39 T^{11} + \cdots + 25\!\cdots\!61 \)
$17$
\( T^{12} - 69 T^{11} + \cdots + 41\!\cdots\!89 \)
$19$
\( T^{12} + 462 T^{11} + \cdots + 10\!\cdots\!41 \)
$23$
\( T^{12} + 66 T^{11} + \cdots + 16\!\cdots\!24 \)
$29$
\( T^{12} - 159 T^{11} + \cdots + 42\!\cdots\!49 \)
$31$
\( T^{12} - 72 T^{11} + \cdots + 12\!\cdots\!09 \)
$37$
\( (T^{6} - 558 T^{5} + \cdots - 20292669795592)^{2} \)
$41$
\( T^{12} - 147 T^{11} + \cdots + 38\!\cdots\!69 \)
$43$
\( T^{12} + 117 T^{11} + \cdots + 50\!\cdots\!21 \)
$47$
\( T^{12} - 783 T^{11} + \cdots + 11\!\cdots\!81 \)
$53$
\( T^{12} + 249 T^{11} + \cdots + 10\!\cdots\!69 \)
$59$
\( T^{12} + 4248 T^{11} + \cdots + 36\!\cdots\!09 \)
$61$
\( T^{12} - 3114 T^{11} + \cdots + 28\!\cdots\!49 \)
$67$
\( T^{12} - 3060 T^{11} + \cdots + 46\!\cdots\!24 \)
$71$
\( T^{12} - 1686 T^{11} + \cdots + 82\!\cdots\!04 \)
$73$
\( T^{12} - 1626 T^{11} + \cdots + 20\!\cdots\!64 \)
$79$
\( T^{12} + 327 T^{11} + \cdots + 63\!\cdots\!89 \)
$83$
\( T^{12} - 927 T^{11} + \cdots + 42\!\cdots\!49 \)
$89$
\( T^{12} + 6366 T^{11} + \cdots + 80\!\cdots\!69 \)
$97$
\( T^{12} + 8052 T^{11} + \cdots + 11\!\cdots\!89 \)
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