Properties

Label 8046.2.a.o
Level 8046
Weight 2
Character orbit 8046.a
Self dual Yes
Analytic conductor 64.248
Analytic rank 0
Dimension 12
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8046.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.2476334663\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + \beta_{1} q^{5} + ( 1 + \beta_{6} ) q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} + \beta_{1} q^{5} + ( 1 + \beta_{6} ) q^{7} + q^{8} + \beta_{1} q^{10} + ( 1 - \beta_{5} ) q^{11} + ( 1 - \beta_{5} + \beta_{9} ) q^{13} + ( 1 + \beta_{6} ) q^{14} + q^{16} + ( 1 + \beta_{7} ) q^{17} -\beta_{11} q^{19} + \beta_{1} q^{20} + ( 1 - \beta_{5} ) q^{22} + ( -\beta_{2} - \beta_{7} + \beta_{8} + \beta_{10} ) q^{23} + ( -1 + \beta_{1} - \beta_{2} + \beta_{4} - \beta_{7} + \beta_{8} - \beta_{11} ) q^{25} + ( 1 - \beta_{5} + \beta_{9} ) q^{26} + ( 1 + \beta_{6} ) q^{28} + ( 2 - \beta_{8} ) q^{29} + ( 1 + \beta_{1} - \beta_{4} - \beta_{6} - \beta_{8} + \beta_{11} ) q^{31} + q^{32} + ( 1 + \beta_{7} ) q^{34} + ( 1 + \beta_{1} + \beta_{4} + \beta_{6} - \beta_{7} - \beta_{10} - \beta_{11} ) q^{35} + ( 1 + \beta_{7} - \beta_{9} + \beta_{11} ) q^{37} -\beta_{11} q^{38} + \beta_{1} q^{40} + ( 2 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{9} - \beta_{10} ) q^{41} + ( \beta_{1} + \beta_{2} + \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{43} + ( 1 - \beta_{5} ) q^{44} + ( -\beta_{2} - \beta_{7} + \beta_{8} + \beta_{10} ) q^{46} + ( 3 - 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{6} + \beta_{7} + \beta_{9} - \beta_{10} ) q^{47} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{10} + \beta_{11} ) q^{49} + ( -1 + \beta_{1} - \beta_{2} + \beta_{4} - \beta_{7} + \beta_{8} - \beta_{11} ) q^{50} + ( 1 - \beta_{5} + \beta_{9} ) q^{52} + ( 2 - \beta_{1} - \beta_{3} + \beta_{6} - \beta_{7} + \beta_{10} - \beta_{11} ) q^{53} + ( \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{11} ) q^{55} + ( 1 + \beta_{6} ) q^{56} + ( 2 - \beta_{8} ) q^{58} + ( 2 - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} ) q^{59} + ( \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} ) q^{61} + ( 1 + \beta_{1} - \beta_{4} - \beta_{6} - \beta_{8} + \beta_{11} ) q^{62} + q^{64} + ( 1 - \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} ) q^{65} + ( 1 - \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} ) q^{67} + ( 1 + \beta_{7} ) q^{68} + ( 1 + \beta_{1} + \beta_{4} + \beta_{6} - \beta_{7} - \beta_{10} - \beta_{11} ) q^{70} + ( 4 + \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} - 2 \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{71} + ( -\beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} ) q^{73} + ( 1 + \beta_{7} - \beta_{9} + \beta_{11} ) q^{74} -\beta_{11} q^{76} + ( 1 + 2 \beta_{3} - \beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} + 2 \beta_{11} ) q^{77} + ( 1 + \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{9} - \beta_{10} ) q^{79} + \beta_{1} q^{80} + ( 2 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{9} - \beta_{10} ) q^{82} + ( 2 - 2 \beta_{3} + \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{9} ) q^{83} + ( 2 \beta_{1} - \beta_{3} - \beta_{6} - \beta_{8} - \beta_{9} + \beta_{10} ) q^{85} + ( \beta_{1} + \beta_{2} + \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{86} + ( 1 - \beta_{5} ) q^{88} + ( 3 - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{7} - 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{89} + ( 1 - \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{9} - \beta_{10} - \beta_{11} ) q^{91} + ( -\beta_{2} - \beta_{7} + \beta_{8} + \beta_{10} ) q^{92} + ( 3 - 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{6} + \beta_{7} + \beta_{9} - \beta_{10} ) q^{94} + ( 2 \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} ) q^{95} + ( \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{10} - \beta_{11} ) q^{97} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{10} + \beta_{11} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 12q^{2} + 12q^{4} + 3q^{5} + 6q^{7} + 12q^{8} + O(q^{10}) \) \( 12q + 12q^{2} + 12q^{4} + 3q^{5} + 6q^{7} + 12q^{8} + 3q^{10} + 10q^{11} + 5q^{13} + 6q^{14} + 12q^{16} + 8q^{17} + 2q^{19} + 3q^{20} + 10q^{22} + 9q^{23} + 7q^{25} + 5q^{26} + 6q^{28} + 19q^{29} + 10q^{31} + 12q^{32} + 8q^{34} + 20q^{35} + 11q^{37} + 2q^{38} + 3q^{40} + 8q^{41} + 13q^{43} + 10q^{44} + 9q^{46} + 11q^{47} + 2q^{49} + 7q^{50} + 5q^{52} + 24q^{53} + 3q^{55} + 6q^{56} + 19q^{58} + 10q^{59} + 10q^{62} + 12q^{64} + 28q^{65} + 21q^{67} + 8q^{68} + 20q^{70} + 37q^{71} - 2q^{73} + 11q^{74} + 2q^{76} + 2q^{77} + 7q^{79} + 3q^{80} + 8q^{82} + 22q^{83} + 15q^{85} + 13q^{86} + 10q^{88} + 40q^{89} + q^{91} + 9q^{92} + 11q^{94} + 11q^{95} + 7q^{97} + 2q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 3 x^{11} - 29 x^{10} + 76 x^{9} + 320 x^{8} - 724 x^{7} - 1643 x^{6} + 3265 x^{5} + 3921 x^{4} - 6927 x^{3} - 3639 x^{2} + 5508 x + 423\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-1293103 \nu^{11} + 6001945 \nu^{10} + 23580251 \nu^{9} - 131132508 \nu^{8} - 90139960 \nu^{7} + 1024446702 \nu^{6} - 573136895 \nu^{5} - 3491861196 \nu^{4} + 4215813919 \nu^{3} + 4581336765 \nu^{2} - 6050385741 \nu - 735740835\)\()/ 174942609 \)
\(\beta_{3}\)\(=\)\((\)\(-1915090 \nu^{11} - 1046362 \nu^{10} + 74238453 \nu^{9} + 35401589 \nu^{8} - 1005031345 \nu^{7} - 425190469 \nu^{6} + 5944816403 \nu^{5} + 1979959133 \nu^{4} - 15609041924 \nu^{3} - 3485888106 \nu^{2} + 14934980991 \nu + 1318582776\)\()/ 174942609 \)
\(\beta_{4}\)\(=\)\((\)\(-4181218 \nu^{11} + 5656554 \nu^{10} + 132467602 \nu^{9} - 109619327 \nu^{8} - 1543743823 \nu^{7} + 649333723 \nu^{6} + 8064151794 \nu^{5} - 1288637477 \nu^{4} - 19088618612 \nu^{3} - 59111379 \nu^{2} + 16748632692 \nu + 1376131821\)\()/ 174942609 \)
\(\beta_{5}\)\(=\)\((\)\(-4389972 \nu^{11} + 4775951 \nu^{10} + 138192352 \nu^{9} - 72357264 \nu^{8} - 1596499871 \nu^{7} + 185639731 \nu^{6} + 8187073056 \nu^{5} + 784010463 \nu^{4} - 18702206507 \nu^{3} - 2790478839 \nu^{2} + 15392930346 \nu + 1169371344\)\()/ 174942609 \)
\(\beta_{6}\)\(=\)\((\)\(6533895 \nu^{11} - 11760004 \nu^{10} - 202559541 \nu^{9} + 251802455 \nu^{8} + 2345517923 \nu^{7} - 1816569674 \nu^{6} - 12292593347 \nu^{5} + 5268511082 \nu^{4} + 28827085403 \nu^{3} - 4670185662 \nu^{2} - 24116897136 \nu - 1422840759\)\()/ 174942609 \)
\(\beta_{7}\)\(=\)\((\)\(-6869077 \nu^{11} + 5839142 \nu^{10} + 230548446 \nu^{9} - 91270315 \nu^{8} - 2856540142 \nu^{7} + 274308218 \nu^{6} + 15985954088 \nu^{5} + 676618097 \nu^{4} - 40668047126 \nu^{3} - 3142259250 \nu^{2} + 37843096842 \nu + 2092764192\)\()/ 174942609 \)
\(\beta_{8}\)\(=\)\((\)\(9158236 \nu^{11} - 12464738 \nu^{10} - 290204756 \nu^{9} + 242859302 \nu^{8} + 3388516467 \nu^{7} - 1488120542 \nu^{6} - 17686820067 \nu^{5} + 3338509649 \nu^{4} + 41281617595 \nu^{3} - 819687051 \nu^{2} - 34744019517 \nu - 3556778949\)\()/ 174942609 \)
\(\beta_{9}\)\(=\)\((\)\(-10768077 \nu^{11} + 13709156 \nu^{10} + 343745338 \nu^{9} - 248488722 \nu^{8} - 4077208325 \nu^{7} + 1283221540 \nu^{6} + 21816188427 \nu^{5} - 1622126526 \nu^{4} - 52604128721 \nu^{3} - 2624930826 \nu^{2} + 45986566413 \nu + 4392624474\)\()/ 174942609 \)
\(\beta_{10}\)\(=\)\((\)\(-11759125 \nu^{11} + 19783265 \nu^{10} + 355999031 \nu^{9} - 376860878 \nu^{8} - 4047108810 \nu^{7} + 2253401432 \nu^{6} + 20885745918 \nu^{5} - 4769542871 \nu^{4} - 48828823276 \nu^{3} + 1245833994 \nu^{2} + 41813328450 \nu + 3637207113\)\()/ 174942609 \)
\(\beta_{11}\)\(=\)\((\)\(13139198 \nu^{11} - 18649271 \nu^{10} - 411865851 \nu^{9} + 355642798 \nu^{8} + 4791452746 \nu^{7} - 2137541739 \nu^{6} - 25035485466 \nu^{5} + 4865115271 \nu^{4} + 58645232190 \nu^{3} - 2492818554 \nu^{2} - 49613155317 \nu - 2837900049\)\()/ 174942609 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{11} + \beta_{8} - \beta_{7} + \beta_{4} - \beta_{2} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(-3 \beta_{11} - \beta_{10} + 2 \beta_{9} + 2 \beta_{8} - 3 \beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - 3 \beta_{2} + 10 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-18 \beta_{11} - 2 \beta_{10} + 6 \beta_{9} + 16 \beta_{8} - 19 \beta_{7} + 4 \beta_{6} - 5 \beta_{5} + 18 \beta_{4} - 3 \beta_{3} - 17 \beta_{2} + 20 \beta_{1} + 27\)
\(\nu^{5}\)\(=\)\(-62 \beta_{11} - 20 \beta_{10} + 42 \beta_{9} + 47 \beta_{8} - 61 \beta_{7} + 23 \beta_{6} - 26 \beta_{5} + 53 \beta_{4} - 35 \beta_{3} - 56 \beta_{2} + 122 \beta_{1} + 13\)
\(\nu^{6}\)\(=\)\(-296 \beta_{11} - 61 \beta_{10} + 153 \beta_{9} + 256 \beta_{8} - 304 \beta_{7} + 94 \beta_{6} - 119 \beta_{5} + 295 \beta_{4} - 81 \beta_{3} - 254 \beta_{2} + 336 \beta_{1} + 248\)
\(\nu^{7}\)\(=\)\(-1096 \beta_{11} - 361 \beta_{10} + 776 \beta_{9} + 897 \beta_{8} - 1076 \beta_{7} + 428 \beta_{6} - 516 \beta_{5} + 1042 \beta_{4} - 538 \beta_{3} - 905 \beta_{2} + 1656 \beta_{1} + 333\)
\(\nu^{8}\)\(=\)\(-4821 \beta_{11} - 1304 \beta_{10} + 3055 \beta_{9} + 4200 \beta_{8} - 4846 \beta_{7} + 1764 \beta_{6} - 2258 \beta_{5} + 4857 \beta_{4} - 1606 \beta_{3} - 3827 \beta_{2} + 5403 \beta_{1} + 2803\)
\(\nu^{9}\)\(=\)\(-18627 \beta_{11} - 6354 \beta_{10} + 13806 \beta_{9} + 15966 \beta_{8} - 18280 \beta_{7} + 7495 \beta_{6} - 9404 \beta_{5} + 18623 \beta_{4} - 8326 \beta_{3} - 14361 \beta_{2} + 24022 \beta_{1} + 6252\)
\(\nu^{10}\)\(=\)\(-78834 \beta_{11} - 24603 \beta_{10} + 55891 \beta_{9} + 69732 \beta_{8} - 78265 \beta_{7} + 30911 \beta_{6} - 40046 \beta_{5} + 80628 \beta_{4} - 28744 \beta_{3} - 59098 \beta_{2} + 86165 \beta_{1} + 36254\)
\(\nu^{11}\)\(=\)\(-312025 \beta_{11} - 109982 \beta_{10} + 240193 \beta_{9} + 275200 \beta_{8} - 305996 \beta_{7} + 128162 \beta_{6} - 164919 \beta_{5} + 320329 \beta_{4} - 131955 \beta_{3} - 228964 \beta_{2} + 364213 \beta_{1} + 106224\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.89416
−2.72416
−2.41916
−1.76721
−1.46524
−0.0737297
1.41673
1.51858
1.84210
2.15566
3.34249
4.06809
1.00000 0 1.00000 −2.89416 0 0.647278 1.00000 0 −2.89416
1.2 1.00000 0 1.00000 −2.72416 0 −2.55489 1.00000 0 −2.72416
1.3 1.00000 0 1.00000 −2.41916 0 0.521062 1.00000 0 −2.41916
1.4 1.00000 0 1.00000 −1.76721 0 2.53166 1.00000 0 −1.76721
1.5 1.00000 0 1.00000 −1.46524 0 −3.53999 1.00000 0 −1.46524
1.6 1.00000 0 1.00000 −0.0737297 0 2.82077 1.00000 0 −0.0737297
1.7 1.00000 0 1.00000 1.41673 0 −1.27536 1.00000 0 1.41673
1.8 1.00000 0 1.00000 1.51858 0 4.35237 1.00000 0 1.51858
1.9 1.00000 0 1.00000 1.84210 0 2.73011 1.00000 0 1.84210
1.10 1.00000 0 1.00000 2.15566 0 −3.96511 1.00000 0 2.15566
1.11 1.00000 0 1.00000 3.34249 0 1.09959 1.00000 0 3.34249
1.12 1.00000 0 1.00000 4.06809 0 2.63250 1.00000 0 4.06809
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(149\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8046))\):

\(T_{5}^{12} - \cdots\)
\(T_{11}^{12} - \cdots\)