Properties

Label 38.4.e.a
Level $38$
Weight $4$
Character orbit 38.e
Analytic conductor $2.242$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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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: \( x^{12} - 6 x^{11} - 135 x^{10} + 730 x^{9} + 7953 x^{8} - 36258 x^{7} - 262940 x^{6} + 918855 x^{5} + 5157591 x^{4} - 11890401 x^{3} - 56759508 x^{2} + \cdots + 272110107 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q - 2 \beta_{3} q^{2} + ( - 2 \beta_{8} - \beta_{7} + 2 \beta_{6} - \beta_{4} + 2 \beta_{2} - 1) q^{3} - 4 \beta_{6} q^{4} + (\beta_{10} + \beta_{9} + 3 \beta_{8} + \beta_{6} + \beta_{5} - 3 \beta_{3} - 2 \beta_1 + 1) q^{5} + (2 \beta_{8} + 4 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{2}) q^{6} + (\beta_{9} - 7 \beta_{8} - 4 \beta_{7} + 2 \beta_{6} + 2 \beta_{3} + 6 \beta_{2} - \beta_1) q^{7} + ( - 8 \beta_{7} - 8) q^{8} + (3 \beta_{11} + 3 \beta_{10} + 3 \beta_{7} + 3 \beta_{4} - 3 \beta_{3} - 7 \beta_{2} - 3 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_{3} q^{2} + ( - 2 \beta_{8} - \beta_{7} + 2 \beta_{6} - \beta_{4} + 2 \beta_{2} - 1) q^{3} - 4 \beta_{6} q^{4} + (\beta_{10} + \beta_{9} + 3 \beta_{8} + \beta_{6} + \beta_{5} - 3 \beta_{3} - 2 \beta_1 + 1) q^{5} + (2 \beta_{8} + 4 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{2}) q^{6} + (\beta_{9} - 7 \beta_{8} - 4 \beta_{7} + 2 \beta_{6} + 2 \beta_{3} + 6 \beta_{2} - \beta_1) q^{7} + ( - 8 \beta_{7} - 8) q^{8} + (3 \beta_{11} + 3 \beta_{10} + 3 \beta_{7} + 3 \beta_{4} - 3 \beta_{3} - 7 \beta_{2} - 3 \beta_1) q^{9} + ( - 4 \beta_{11} - 2 \beta_{9} + 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{4} + 12 \beta_{2} + \cdots + 2) q^{10}+ \cdots + ( - 55 \beta_{11} - 138 \beta_{10} + 55 \beta_{9} + 183 \beta_{8} + \cdots + 443) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 9 q^{3} - 18 q^{6} + 21 q^{7} - 48 q^{8} - 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 9 q^{3} - 18 q^{6} + 21 q^{7} - 48 q^{8} - 27 q^{9} - 9 q^{11} + 36 q^{12} + 39 q^{13} - 138 q^{14} + 423 q^{15} + 69 q^{17} + 132 q^{18} - 462 q^{19} - 216 q^{20} - 279 q^{21} + 204 q^{22} - 66 q^{23} - 72 q^{24} + 342 q^{25} + 48 q^{26} + 189 q^{27} + 192 q^{28} + 159 q^{29} + 72 q^{31} - 1560 q^{33} + 408 q^{34} - 135 q^{35} - 108 q^{36} + 1116 q^{37} - 294 q^{38} - 1248 q^{39} + 147 q^{41} + 414 q^{42} - 117 q^{43} + 408 q^{44} + 1296 q^{45} + 528 q^{46} + 783 q^{47} + 288 q^{48} + 1413 q^{49} - 354 q^{50} - 2301 q^{51} - 348 q^{52} - 249 q^{53} - 540 q^{54} + 2187 q^{55} - 336 q^{56} - 2670 q^{57} - 1932 q^{58} - 4248 q^{59} + 324 q^{60} + 3114 q^{61} - 438 q^{62} + 363 q^{63} - 384 q^{64} + 495 q^{65} + 822 q^{66} + 3060 q^{67} + 408 q^{68} - 237 q^{69} - 270 q^{70} + 1686 q^{71} + 432 q^{72} + 1626 q^{73} + 90 q^{74} - 1854 q^{75} - 1416 q^{77} - 108 q^{78} - 327 q^{79} + 3483 q^{81} + 294 q^{82} + 927 q^{83} + 204 q^{84} - 3294 q^{85} + 1188 q^{86} + 2892 q^{87} - 72 q^{88} - 6366 q^{89} - 5076 q^{90} + 840 q^{91} - 156 q^{92} + 870 q^{93} + 3432 q^{94} + 513 q^{95} - 576 q^{96} - 8052 q^{97} + 378 q^{98} + 4494 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} - 135 x^{10} + 730 x^{9} + 7953 x^{8} - 36258 x^{7} - 262940 x^{6} + 918855 x^{5} + 5157591 x^{4} - 11890401 x^{3} - 56759508 x^{2} + \cdots + 272110107 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 177221217078 \nu^{11} + 3257235752468 \nu^{10} + 7448859189417 \nu^{9} - 348800270203041 \nu^{8} + \cdots - 21\!\cdots\!68 ) / 20\!\cdots\!27 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 177221217078 \nu^{11} + 1307802364610 \nu^{10} - 30274049774807 \nu^{9} - 164426429455098 \nu^{8} + \cdots - 19\!\cdots\!62 ) / 20\!\cdots\!27 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 488144071095 \nu^{11} + 17846740004059 \nu^{10} - 13839005362779 \nu^{9} + \cdots - 20\!\cdots\!46 ) / 20\!\cdots\!27 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 488144071095 \nu^{11} + 12477155222014 \nu^{10} - 137780470767586 \nu^{9} + \cdots - 19\!\cdots\!88 ) / 20\!\cdots\!27 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 545356750193 \nu^{11} + 639178329540 \nu^{10} + 77274249510048 \nu^{9} - 43408610678418 \nu^{8} + \cdots + 19\!\cdots\!01 ) / 20\!\cdots\!27 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1581666 \nu^{11} - 8699163 \nu^{10} - 191254825 \nu^{9} + 925890435 \nu^{8} + 9722082222 \nu^{7} - 38409003948 \nu^{6} + \cdots + 11710200330886 ) / 486606417103 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 722577967271 \nu^{11} - 4051943557973 \nu^{10} - 83945461319640 \nu^{9} + 426412366504371 \nu^{8} + \cdots + 74\!\cdots\!83 ) / 20\!\cdots\!27 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 882100559090 \nu^{11} + 8843624294225 \nu^{10} - 174667730262753 \nu^{9} + \cdots - 20\!\cdots\!59 ) / 20\!\cdots\!27 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1648551699807 \nu^{11} + 11116259193530 \nu^{10} + 165757370191268 \nu^{9} + \cdots - 21\!\cdots\!01 ) / 20\!\cdots\!27 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1648551699807 \nu^{11} + 7017809504347 \nu^{10} + 186249618637183 \nu^{9} - 654710361086352 \nu^{8} + \cdots + 92\!\cdots\!83 ) / 20\!\cdots\!27 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} + \beta_{10} - 3\beta_{8} - 3\beta_{6} - \beta_{3} - 4\beta_{2} + \beta _1 + 26 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3 \beta_{10} - 15 \beta_{8} + 10 \beta_{7} + 6 \beta_{6} + 3 \beta_{5} - 3 \beta_{4} + 28 \beta_{3} - 25 \beta_{2} + 25 \beta _1 + 32 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 53 \beta_{11} + 59 \beta_{10} + 10 \beta_{9} - 217 \beta_{8} + 15 \beta_{7} - 165 \beta_{6} - 3 \beta_{5} - 15 \beta_{4} + 37 \beta_{3} - 256 \beta_{2} + 44 \beta _1 + 698 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 81 \beta_{11} + 194 \beta_{10} + 25 \beta_{9} - 1321 \beta_{8} + 892 \beta_{7} + 381 \beta_{6} + 162 \beta_{5} - 207 \beta_{4} + 1665 \beta_{3} - 1414 \beta_{2} + 615 \beta _1 + 1854 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2280 \beta_{11} + 2604 \beta_{10} + 917 \beta_{9} - 11518 \beta_{8} + 2205 \beta_{7} - 5675 \beta_{6} - 219 \beta_{5} - 1296 \beta_{4} + 5259 \beta_{3} - 11343 \beta_{2} + 1302 \beta _1 + 19896 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 6561 \beta_{11} + 9574 \beta_{10} + 3122 \beta_{9} - 76494 \beta_{8} + 52916 \beta_{7} + 19598 \beta_{6} + 5456 \beta_{5} - 10601 \beta_{4} + 78989 \beta_{3} - 60371 \beta_{2} + 14109 \beta _1 + 81501 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 93098 \beta_{11} + 103652 \beta_{10} + 56038 \beta_{9} - 531544 \beta_{8} + 177506 \beta_{7} - 126516 \beta_{6} - 14142 \beta_{5} - 73372 \beta_{4} + 339489 \beta_{3} - 419971 \beta_{2} + \cdots + 609442 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 366048 \beta_{11} + 423666 \beta_{10} + 233544 \beta_{9} - 3638850 \beta_{8} + 2627538 \beta_{7} + 1015026 \beta_{6} + 112374 \beta_{5} - 475506 \beta_{4} + 3478249 \beta_{3} + \cdots + 3220296 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 3739994 \beta_{11} + 3951167 \beta_{10} + 2861082 \beta_{9} - 22427599 \beta_{8} + 10768116 \beta_{7} - 48010 \beta_{6} - 902652 \beta_{5} - 3405306 \beta_{4} + \cdots + 19795826 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 17577635 \beta_{11} + 17659182 \beta_{10} + 13629198 \beta_{9} - 153343365 \beta_{8} + 118000897 \beta_{7} + 53163670 \beta_{6} - 854642 \beta_{5} - 19566517 \beta_{4} + \cdots + 119730930 \) Copy content Toggle raw display

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
5.05412 0.342020i
−4.05412 0.342020i
5.30460 0.984808i
−4.30460 0.984808i
5.30460 + 0.984808i
−4.30460 + 0.984808i
5.05412 + 0.342020i
−4.05412 + 0.342020i
−5.28151 + 0.642788i
6.28151 + 0.642788i
−5.28151 0.642788i
6.28151 0.642788i
−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
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 35.2
Significant digits:
Format:

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 \(S_{4}^{\mathrm{new}}(38, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + 8 T^{3} + 64)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} + 9 T^{11} + 54 T^{10} + \cdots + 2289169 \) Copy content Toggle raw display
$5$ \( T^{12} - 171 T^{10} + \cdots + 308668025241 \) Copy content Toggle raw display
$7$ \( T^{12} - 21 T^{11} + \cdots + 6147971281 \) Copy content Toggle raw display
$11$ \( T^{12} + 9 T^{11} + \cdots + 55\!\cdots\!09 \) Copy content Toggle raw display
$13$ \( T^{12} - 39 T^{11} + \cdots + 25\!\cdots\!61 \) Copy content Toggle raw display
$17$ \( T^{12} - 69 T^{11} + \cdots + 41\!\cdots\!89 \) Copy content Toggle raw display
$19$ \( T^{12} + 462 T^{11} + \cdots + 10\!\cdots\!41 \) Copy content Toggle raw display
$23$ \( T^{12} + 66 T^{11} + \cdots + 16\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{12} - 159 T^{11} + \cdots + 42\!\cdots\!49 \) Copy content Toggle raw display
$31$ \( T^{12} - 72 T^{11} + \cdots + 12\!\cdots\!09 \) Copy content Toggle raw display
$37$ \( (T^{6} - 558 T^{5} + \cdots - 20292669795592)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} - 147 T^{11} + \cdots + 38\!\cdots\!69 \) Copy content Toggle raw display
$43$ \( T^{12} + 117 T^{11} + \cdots + 50\!\cdots\!21 \) Copy content Toggle raw display
$47$ \( T^{12} - 783 T^{11} + \cdots + 11\!\cdots\!81 \) Copy content Toggle raw display
$53$ \( T^{12} + 249 T^{11} + \cdots + 10\!\cdots\!69 \) Copy content Toggle raw display
$59$ \( T^{12} + 4248 T^{11} + \cdots + 36\!\cdots\!09 \) Copy content Toggle raw display
$61$ \( T^{12} - 3114 T^{11} + \cdots + 28\!\cdots\!49 \) Copy content Toggle raw display
$67$ \( T^{12} - 3060 T^{11} + \cdots + 46\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{12} - 1686 T^{11} + \cdots + 82\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( T^{12} - 1626 T^{11} + \cdots + 20\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( T^{12} + 327 T^{11} + \cdots + 63\!\cdots\!89 \) Copy content Toggle raw display
$83$ \( T^{12} - 927 T^{11} + \cdots + 42\!\cdots\!49 \) Copy content Toggle raw display
$89$ \( T^{12} + 6366 T^{11} + \cdots + 80\!\cdots\!69 \) Copy content Toggle raw display
$97$ \( T^{12} + 8052 T^{11} + \cdots + 11\!\cdots\!89 \) Copy content Toggle raw display
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