Properties

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

\(f(q)\) \(=\) \( q + q^{2} + q^{4} -\beta_{1} q^{5} + ( -1 - \beta_{5} ) q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} -\beta_{1} q^{5} + ( -1 - \beta_{5} ) q^{7} + q^{8} -\beta_{1} q^{10} + ( -1 + \beta_{6} ) q^{11} + ( -1 + \beta_{1} - \beta_{4} + \beta_{5} ) q^{13} + ( -1 - \beta_{5} ) q^{14} + q^{16} + ( 1 + \beta_{1} - \beta_{3} ) q^{17} + ( -1 + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{19} -\beta_{1} q^{20} + ( -1 + \beta_{6} ) q^{22} + ( -3 - \beta_{2} + \beta_{3} + \beta_{7} ) q^{23} + ( \beta_{4} + \beta_{5} ) q^{25} + ( -1 + \beta_{1} - \beta_{4} + \beta_{5} ) q^{26} + ( -1 - \beta_{5} ) q^{28} + ( \beta_{1} + \beta_{2} - 2 \beta_{6} ) q^{29} + ( -1 - \beta_{3} - \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{31} + q^{32} + ( 1 + \beta_{1} - \beta_{3} ) q^{34} + ( 1 + 3 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{35} + ( -\beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{37} + ( -1 + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{38} -\beta_{1} q^{40} + ( 2 \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{41} + ( -2 - \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{43} + ( -1 + \beta_{6} ) q^{44} + ( -3 - \beta_{2} + \beta_{3} + \beta_{7} ) q^{46} + ( -3 - 2 \beta_{2} + \beta_{7} ) q^{47} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} ) q^{49} + ( \beta_{4} + \beta_{5} ) q^{50} + ( -1 + \beta_{1} - \beta_{4} + \beta_{5} ) q^{52} + ( 2 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{53} + ( -1 - \beta_{2} ) q^{55} + ( -1 - \beta_{5} ) q^{56} + ( \beta_{1} + \beta_{2} - 2 \beta_{6} ) q^{58} + ( 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{59} + ( -3 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{6} ) q^{61} + ( -1 - \beta_{3} - \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{62} + q^{64} + ( -5 + \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{65} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{67} + ( 1 + \beta_{1} - \beta_{3} ) q^{68} + ( 1 + 3 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{70} + ( -1 - \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{7} ) q^{71} + ( -\beta_{1} + 3 \beta_{4} - \beta_{7} ) q^{73} + ( -\beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{74} + ( -1 + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{76} + ( -2 - \beta_{1} - \beta_{2} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{77} + ( -4 - 3 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{79} -\beta_{1} q^{80} + ( 2 \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{82} + ( -3 - \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{83} + ( -5 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{85} + ( -2 - \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{86} + ( -1 + \beta_{6} ) q^{88} + ( 4 + 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{89} + ( -3 - 3 \beta_{1} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{91} + ( -3 - \beta_{2} + \beta_{3} + \beta_{7} ) q^{92} + ( -3 - 2 \beta_{2} + \beta_{7} ) q^{94} + ( 1 - 2 \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} ) q^{95} + ( -1 - 3 \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{97} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{2} + 8q^{4} - 5q^{7} + 8q^{8} + O(q^{10}) \) \( 8q + 8q^{2} + 8q^{4} - 5q^{7} + 8q^{8} - 6q^{11} - 8q^{13} - 5q^{14} + 8q^{16} + 5q^{17} - 14q^{19} - 6q^{22} - 21q^{23} - 6q^{25} - 8q^{26} - 5q^{28} - 3q^{29} - 4q^{31} + 8q^{32} + 5q^{34} + 2q^{35} - 3q^{37} - 14q^{38} - 7q^{41} - 12q^{43} - 6q^{44} - 21q^{46} - 25q^{47} - 7q^{49} - 6q^{50} - 8q^{52} - 3q^{53} - 9q^{55} - 5q^{56} - 3q^{58} - 2q^{59} - 17q^{61} - 4q^{62} + 8q^{64} - 32q^{65} - 14q^{67} + 5q^{68} + 2q^{70} - 7q^{71} - 10q^{73} - 3q^{74} - 14q^{76} - 12q^{77} - 33q^{79} - 7q^{82} - 13q^{83} - 33q^{85} - 12q^{86} - 6q^{88} + 22q^{89} - 22q^{91} - 21q^{92} - 25q^{94} + 14q^{95} - 11q^{97} - 7q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 17 x^{6} - 2 x^{5} + 71 x^{4} - 18 x^{3} - 81 x^{2} + 36 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{7} + 17 \nu^{5} + 5 \nu^{4} - 71 \nu^{3} - 18 \nu^{2} + 63 \nu + 12 \)\()/6\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} + 2 \nu^{6} - 14 \nu^{5} - 33 \nu^{4} + 25 \nu^{3} + 64 \nu^{2} - 21 \nu - 9 \)\()/6\)
\(\beta_{4}\)\(=\)\((\)\( 2 \nu^{7} + 2 \nu^{6} - 31 \nu^{5} - 35 \nu^{4} + 93 \nu^{3} + 55 \nu^{2} - 75 \nu - 18 \)\()/6\)
\(\beta_{5}\)\(=\)\((\)\( -2 \nu^{7} - 2 \nu^{6} + 31 \nu^{5} + 35 \nu^{4} - 93 \nu^{3} - 49 \nu^{2} + 75 \nu - 12 \)\()/6\)
\(\beta_{6}\)\(=\)\((\)\( -2 \nu^{7} - \nu^{6} + 34 \nu^{5} + 21 \nu^{4} - 137 \nu^{3} - 35 \nu^{2} + 144 \nu - 3 \)\()/6\)
\(\beta_{7}\)\(=\)\((\)\( 5 \nu^{7} + 6 \nu^{6} - 79 \nu^{5} - 103 \nu^{4} + 247 \nu^{3} + 192 \nu^{2} - 213 \nu - 48 \)\()/6\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} + \beta_{4} + 5\)
\(\nu^{3}\)\(=\)\(-\beta_{7} - 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + \beta_{2} + 9 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(-\beta_{7} - 2 \beta_{6} + 11 \beta_{5} + 13 \beta_{4} + 3 \beta_{2} + 6 \beta_{1} + 46\)
\(\nu^{5}\)\(=\)\(-14 \beta_{7} - 24 \beta_{6} + 30 \beta_{5} + 34 \beta_{4} + 26 \beta_{3} + 12 \beta_{2} + 94 \beta_{1} + 53\)
\(\nu^{6}\)\(=\)\(-16 \beta_{7} - 38 \beta_{6} + 132 \beta_{5} + 154 \beta_{4} + 10 \beta_{3} + 50 \beta_{2} + 129 \beta_{1} + 494\)
\(\nu^{7}\)\(=\)\(-172 \beta_{7} - 276 \beta_{6} + 405 \beta_{5} + 483 \beta_{4} + 300 \beta_{3} + 142 \beta_{2} + 1052 \beta_{1} + 911\)

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
3.58447
1.40004
1.17278
0.854224
−0.181211
−1.29304
−2.60917
−2.92810
1.00000 0 1.00000 −3.58447 0 −4.00108 1.00000 0 −3.58447
1.2 1.00000 0 1.00000 −1.40004 0 −2.13611 1.00000 0 −1.40004
1.3 1.00000 0 1.00000 −1.17278 0 1.96150 1.00000 0 −1.17278
1.4 1.00000 0 1.00000 −0.854224 0 0.727081 1.00000 0 −0.854224
1.5 1.00000 0 1.00000 0.181211 0 3.43581 1.00000 0 0.181211
1.6 1.00000 0 1.00000 1.29304 0 −0.779884 1.00000 0 1.29304
1.7 1.00000 0 1.00000 2.60917 0 −0.922875 1.00000 0 2.60917
1.8 1.00000 0 1.00000 2.92810 0 −3.28444 1.00000 0 2.92810
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8046))\):

\( T_{5}^{8} - 17 T_{5}^{6} + 2 T_{5}^{5} + 71 T_{5}^{4} + 18 T_{5}^{3} - 81 T_{5}^{2} - 36 T_{5} + 9 \)
\(T_{11}^{8} + \cdots\)