Properties

Label 8046.2.a.h
Level 8046
Weight 2
Character orbit 8046.a
Self dual Yes
Analytic conductor 64.248
Analytic rank 1
Dimension 9
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: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + ( -\beta_{1} + \beta_{5} - \beta_{7} ) q^{5} + ( -\beta_{1} + \beta_{3} ) q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} + ( -\beta_{1} + \beta_{5} - \beta_{7} ) q^{5} + ( -\beta_{1} + \beta_{3} ) q^{7} + q^{8} + ( -\beta_{1} + \beta_{5} - \beta_{7} ) q^{10} + ( -1 + \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{11} + ( -1 + \beta_{5} - \beta_{6} ) q^{13} + ( -\beta_{1} + \beta_{3} ) q^{14} + q^{16} + ( -1 + 2 \beta_{1} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{17} + ( -2 + 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{19} + ( -\beta_{1} + \beta_{5} - \beta_{7} ) q^{20} + ( -1 + \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{22} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} ) q^{23} + ( \beta_{1} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{25} + ( -1 + \beta_{5} - \beta_{6} ) q^{26} + ( -\beta_{1} + \beta_{3} ) q^{28} + ( -1 + 2 \beta_{1} - \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{29} + ( -2 - \beta_{1} - 2 \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} ) q^{31} + q^{32} + ( -1 + 2 \beta_{1} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{34} + ( -\beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{8} ) q^{35} + ( -2 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{37} + ( -2 + 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{38} + ( -\beta_{1} + \beta_{5} - \beta_{7} ) q^{40} + ( 2 \beta_{1} + \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{7} - \beta_{8} ) q^{41} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{43} + ( -1 + \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{44} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} ) q^{46} + ( \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{47} + ( -2 + 2 \beta_{1} - 3 \beta_{3} - 2 \beta_{4} - \beta_{5} + 3 \beta_{6} - \beta_{7} + 2 \beta_{8} ) q^{49} + ( \beta_{1} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{50} + ( -1 + \beta_{5} - \beta_{6} ) q^{52} + ( 1 - 5 \beta_{1} + 2 \beta_{4} - \beta_{8} ) q^{53} + ( -4 + \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} ) q^{55} + ( -\beta_{1} + \beta_{3} ) q^{56} + ( -1 + 2 \beta_{1} - \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{58} + ( -2 + \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{7} ) q^{59} + ( -1 - \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} ) q^{61} + ( -2 - \beta_{1} - 2 \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} ) q^{62} + q^{64} + ( 1 + 4 \beta_{1} - 2 \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{65} + ( -1 - \beta_{1} + \beta_{2} + \beta_{4} + 2 \beta_{5} - 3 \beta_{8} ) q^{67} + ( -1 + 2 \beta_{1} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{68} + ( -\beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{8} ) q^{70} + ( -3 - 3 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} - \beta_{4} + 3 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{71} + ( 2 \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} + 2 \beta_{8} ) q^{73} + ( -2 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{74} + ( -2 + 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{76} + ( -2 + \beta_{1} + 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{8} ) q^{77} + ( -5 + \beta_{1} - \beta_{2} - \beta_{3} - 3 \beta_{4} - 3 \beta_{6} - \beta_{7} + 2 \beta_{8} ) q^{79} + ( -\beta_{1} + \beta_{5} - \beta_{7} ) q^{80} + ( 2 \beta_{1} + \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{7} - \beta_{8} ) q^{82} + ( -3 + 5 \beta_{1} + \beta_{2} - \beta_{5} + 4 \beta_{7} + \beta_{8} ) q^{83} + ( -4 + \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{85} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{86} + ( -1 + \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{88} + ( -1 - 3 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{7} + \beta_{8} ) q^{89} + ( -3 + \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{8} ) q^{91} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} ) q^{92} + ( \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{94} + ( -6 + 3 \beta_{1} - 3 \beta_{2} + \beta_{3} - \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + 2 \beta_{8} ) q^{95} + ( -3 - \beta_{2} + \beta_{4} + 2 \beta_{7} + 2 \beta_{8} ) q^{97} + ( -2 + 2 \beta_{1} - 3 \beta_{3} - 2 \beta_{4} - \beta_{5} + 3 \beta_{6} - \beta_{7} + 2 \beta_{8} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q + 9q^{2} + 9q^{4} - 4q^{5} - 4q^{7} + 9q^{8} + O(q^{10}) \) \( 9q + 9q^{2} + 9q^{4} - 4q^{5} - 4q^{7} + 9q^{8} - 4q^{10} - 4q^{11} - 8q^{13} - 4q^{14} + 9q^{16} - q^{17} - 10q^{19} - 4q^{20} - 4q^{22} - 8q^{23} - 3q^{25} - 8q^{26} - 4q^{28} - 4q^{29} - 17q^{31} + 9q^{32} - q^{34} - 10q^{35} - 11q^{37} - 10q^{38} - 4q^{40} - 16q^{43} - 4q^{44} - 8q^{46} - 7q^{47} - 5q^{49} - 3q^{50} - 8q^{52} - 12q^{53} - 23q^{55} - 4q^{56} - 4q^{58} - 6q^{59} - 13q^{61} - 17q^{62} + 9q^{64} + 24q^{65} - 14q^{67} - q^{68} - 10q^{70} - 30q^{71} - 12q^{73} - 11q^{74} - 10q^{76} - 12q^{77} - 35q^{79} - 4q^{80} - 5q^{83} - 27q^{85} - 16q^{86} - 4q^{88} - 23q^{89} - 28q^{91} - 8q^{92} - 7q^{94} - 32q^{95} - 21q^{97} - 5q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - 3 x^{8} - 9 x^{7} + 25 x^{6} + 29 x^{5} - 58 x^{4} - 43 x^{3} + 34 x^{2} + 25 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\((\)\( 15 \nu^{8} - 69 \nu^{7} - 40 \nu^{6} + 516 \nu^{5} - 329 \nu^{4} - 975 \nu^{3} + 992 \nu^{2} + 355 \nu - 347 \)\()/77\)
\(\beta_{4}\)\(=\)\((\)\( -19 \nu^{8} + 72 \nu^{7} + 102 \nu^{6} - 515 \nu^{5} - 35 \nu^{4} + 850 \nu^{3} - 235 \nu^{2} - 116 \nu + 34 \)\()/77\)
\(\beta_{5}\)\(=\)\((\)\( -24 \nu^{8} + 95 \nu^{7} + 141 \nu^{6} - 764 \nu^{5} - 105 \nu^{4} + 1714 \nu^{3} - 232 \nu^{2} - 1030 \nu - 30 \)\()/77\)
\(\beta_{6}\)\(=\)\((\)\( 24 \nu^{8} - 95 \nu^{7} - 141 \nu^{6} + 764 \nu^{5} + 105 \nu^{4} - 1637 \nu^{3} + 155 \nu^{2} + 722 \nu + 107 \)\()/77\)
\(\beta_{7}\)\(=\)\((\)\( -32 \nu^{8} + 101 \nu^{7} + 265 \nu^{6} - 839 \nu^{5} - 679 \nu^{4} + 1926 \nu^{3} + 589 \nu^{2} - 1168 \nu - 194 \)\()/77\)
\(\beta_{8}\)\(=\)\((\)\( 47 \nu^{8} - 170 \nu^{7} - 305 \nu^{6} + 1355 \nu^{5} + 427 \nu^{4} - 2978 \nu^{3} - 59 \nu^{2} + 1600 \nu + 232 \)\()/77\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{6} + \beta_{5} + \beta_{2} + 5 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(\beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} + 7 \beta_{2} + 10 \beta_{1} + 15\)
\(\nu^{5}\)\(=\)\(2 \beta_{8} + \beta_{7} + 8 \beta_{6} + 9 \beta_{5} + 2 \beta_{4} + 11 \beta_{2} + 33 \beta_{1} + 21\)
\(\nu^{6}\)\(=\)\(13 \beta_{8} + 10 \beta_{7} + 10 \beta_{6} + 16 \beta_{5} + 3 \beta_{4} - 6 \beta_{3} + 48 \beta_{2} + 82 \beta_{1} + 94\)
\(\nu^{7}\)\(=\)\(32 \beta_{8} + 13 \beta_{7} + 51 \beta_{6} + 80 \beta_{5} + 23 \beta_{4} + 3 \beta_{3} + 95 \beta_{2} + 240 \beta_{1} + 185\)
\(\nu^{8}\)\(=\)\(135 \beta_{8} + 74 \beta_{7} + 73 \beta_{6} + 188 \beta_{5} + 45 \beta_{4} - 19 \beta_{3} + 339 \beta_{2} + 642 \beta_{1} + 663\)

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
1.05829
2.58022
−2.13687
−0.836154
−0.584549
1.67390
2.89499
−0.0929510
−1.55687
1.00000 0 1.00000 −3.80777 0 0.515673 1.00000 0 −3.80777
1.2 1.00000 0 1.00000 −2.72526 0 −2.79628 1.00000 0 −2.72526
1.3 1.00000 0 1.00000 −1.65274 0 0.0358535 1.00000 0 −1.65274
1.4 1.00000 0 1.00000 −1.63508 0 4.18187 1.00000 0 −1.63508
1.5 1.00000 0 1.00000 −0.683695 0 −0.639069 1.00000 0 −0.683695
1.6 1.00000 0 1.00000 0.102080 0 0.350570 1.00000 0 0.102080
1.7 1.00000 0 1.00000 1.72525 0 −2.60583 1.00000 0 1.72525
1.8 1.00000 0 1.00000 1.96687 0 −4.72097 1.00000 0 1.96687
1.9 1.00000 0 1.00000 2.71034 0 1.67819 1.00000 0 2.71034
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8046))\):

\(T_{5}^{9} + \cdots\)
\(T_{11}^{9} + \cdots\)