Properties

Label 8016.2.a.x
Level 8016
Weight 2
Character orbit 8016.a
Self dual Yes
Analytic conductor 64.008
Analytic rank 1
Dimension 8
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8016.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.0080822603\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q\) \(- q^{3}\) \( + ( 1 + \beta_{2} ) q^{5} \) \( + ( 1 - \beta_{1} + \beta_{3} + \beta_{5} ) q^{7} \) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(- q^{3}\) \( + ( 1 + \beta_{2} ) q^{5} \) \( + ( 1 - \beta_{1} + \beta_{3} + \beta_{5} ) q^{7} \) \(+ q^{9}\) \( + ( -2 + \beta_{1} + \beta_{4} ) q^{11} \) \( + ( -\beta_{5} - \beta_{7} ) q^{13} \) \( + ( -1 - \beta_{2} ) q^{15} \) \( + ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} ) q^{17} \) \( + ( -2 + \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{19} \) \( + ( -1 + \beta_{1} - \beta_{3} - \beta_{5} ) q^{21} \) \( + ( -1 + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{23} \) \( + ( \beta_{2} - \beta_{3} + \beta_{6} - \beta_{7} ) q^{25} \) \(- q^{27}\) \( + ( 1 - \beta_{1} - 2 \beta_{3} - \beta_{7} ) q^{29} \) \( + ( -1 + 2 \beta_{1} - \beta_{2} - 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{31} \) \( + ( 2 - \beta_{1} - \beta_{4} ) q^{33} \) \( + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{35} \) \( + ( -1 + \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{37} \) \( + ( \beta_{5} + \beta_{7} ) q^{39} \) \( + ( 1 + \beta_{1} - 2 \beta_{3} + 3 \beta_{6} - \beta_{7} ) q^{41} \) \( + ( \beta_{2} - \beta_{3} - \beta_{4} ) q^{43} \) \( + ( 1 + \beta_{2} ) q^{45} \) \( + ( -2 + \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{47} \) \( + ( -3 \beta_{1} + 3 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} ) q^{49} \) \( + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{6} - \beta_{7} ) q^{51} \) \( + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{53} \) \( + ( -1 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{7} ) q^{55} \) \( + ( 2 - \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{57} \) \( + ( -4 + \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{5} + \beta_{6} ) q^{59} \) \( + ( -1 + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{61} \) \( + ( 1 - \beta_{1} + \beta_{3} + \beta_{5} ) q^{63} \) \( + ( 1 - \beta_{4} - 2 \beta_{5} - 2 \beta_{7} ) q^{65} \) \( + ( -1 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{67} \) \( + ( 1 - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{69} \) \( + ( -2 - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{71} \) \( + ( -3 - 4 \beta_{2} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{73} \) \( + ( -\beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} ) q^{75} \) \( + ( -5 + 5 \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{4} - 3 \beta_{5} + \beta_{6} ) q^{77} \) \( + ( -2 - 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{79} \) \(+ q^{81}\) \( + ( -5 + \beta_{1} + 2 \beta_{2} - 2 \beta_{4} - \beta_{5} + 2 \beta_{7} ) q^{83} \) \( + ( -4 - 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{85} \) \( + ( -1 + \beta_{1} + 2 \beta_{3} + \beta_{7} ) q^{87} \) \( + ( \beta_{1} - \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{89} \) \( + ( -1 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{91} \) \( + ( 1 - 2 \beta_{1} + \beta_{2} + 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{93} \) \( + ( -1 + \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - \beta_{5} + \beta_{6} ) q^{95} \) \( + ( -6 + 3 \beta_{1} - 3 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{97} \) \( + ( -2 + \beta_{1} + \beta_{4} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut -\mathstrut 8q^{3} \) \(\mathstrut +\mathstrut 7q^{5} \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 8q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut -\mathstrut 8q^{3} \) \(\mathstrut +\mathstrut 7q^{5} \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 8q^{9} \) \(\mathstrut -\mathstrut 13q^{11} \) \(\mathstrut -\mathstrut 7q^{15} \) \(\mathstrut +\mathstrut 11q^{17} \) \(\mathstrut -\mathstrut 12q^{19} \) \(\mathstrut -\mathstrut 4q^{21} \) \(\mathstrut -\mathstrut 7q^{23} \) \(\mathstrut -\mathstrut 5q^{25} \) \(\mathstrut -\mathstrut 8q^{27} \) \(\mathstrut +\mathstrut q^{29} \) \(\mathstrut +\mathstrut 2q^{31} \) \(\mathstrut +\mathstrut 13q^{33} \) \(\mathstrut +\mathstrut 4q^{35} \) \(\mathstrut -\mathstrut 9q^{37} \) \(\mathstrut +\mathstrut 4q^{41} \) \(\mathstrut -\mathstrut 2q^{43} \) \(\mathstrut +\mathstrut 7q^{45} \) \(\mathstrut -\mathstrut 17q^{47} \) \(\mathstrut -\mathstrut 2q^{49} \) \(\mathstrut -\mathstrut 11q^{51} \) \(\mathstrut +\mathstrut 9q^{53} \) \(\mathstrut -\mathstrut 7q^{55} \) \(\mathstrut +\mathstrut 12q^{57} \) \(\mathstrut -\mathstrut 29q^{59} \) \(\mathstrut -\mathstrut 12q^{61} \) \(\mathstrut +\mathstrut 4q^{63} \) \(\mathstrut +\mathstrut 8q^{65} \) \(\mathstrut +\mathstrut 7q^{69} \) \(\mathstrut -\mathstrut 13q^{71} \) \(\mathstrut -\mathstrut 20q^{73} \) \(\mathstrut +\mathstrut 5q^{75} \) \(\mathstrut -\mathstrut 22q^{77} \) \(\mathstrut -\mathstrut 8q^{79} \) \(\mathstrut +\mathstrut 8q^{81} \) \(\mathstrut -\mathstrut 33q^{83} \) \(\mathstrut -\mathstrut 31q^{85} \) \(\mathstrut -\mathstrut q^{87} \) \(\mathstrut +\mathstrut 4q^{89} \) \(\mathstrut -\mathstrut q^{91} \) \(\mathstrut -\mathstrut 2q^{93} \) \(\mathstrut -\mathstrut 3q^{95} \) \(\mathstrut -\mathstrut 31q^{97} \) \(\mathstrut -\mathstrut 13q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut -\mathstrut \) \(3\) \(x^{7}\mathstrut -\mathstrut \) \(8\) \(x^{6}\mathstrut +\mathstrut \) \(28\) \(x^{5}\mathstrut +\mathstrut \) \(9\) \(x^{4}\mathstrut -\mathstrut \) \(64\) \(x^{3}\mathstrut +\mathstrut \) \(17\) \(x^{2}\mathstrut +\mathstrut \) \(23\) \(x\mathstrut +\mathstrut \) \(1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -3 \nu^{7} + 38 \nu^{5} + 2 \nu^{4} - 147 \nu^{3} - 11 \nu^{2} + 168 \nu + 15 \)\()/14\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} + 15 \nu^{5} + 3 \nu^{4} - 70 \nu^{3} - 20 \nu^{2} + 98 \nu + 19 \)\()/7\)
\(\beta_{4}\)\(=\)\((\)\( 2 \nu^{7} - 7 \nu^{6} - 16 \nu^{5} + 64 \nu^{4} + 21 \nu^{3} - 135 \nu^{2} + 21 \nu + 25 \)\()/7\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{7} + 7 \nu^{6} + 24 \nu^{5} - 61 \nu^{4} - 28 \nu^{3} + 122 \nu^{2} - 42 \nu - 27 \)\()/7\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{7} + 2 \nu^{6} + 10 \nu^{5} - 18 \nu^{4} - 25 \nu^{3} + 37 \nu^{2} + 8 \nu - 5 \)\()/2\)
\(\beta_{7}\)\(=\)\((\)\( 4 \nu^{7} - 7 \nu^{6} - 39 \nu^{5} + 58 \nu^{4} + 98 \nu^{3} - 95 \nu^{2} - 56 \nu - 13 \)\()/7\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(6\) \(\beta_{7}\mathstrut +\mathstrut \) \(7\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(8\) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(15\)
\(\nu^{5}\)\(=\)\(\beta_{7}\mathstrut +\mathstrut \) \(9\) \(\beta_{6}\mathstrut +\mathstrut \) \(8\) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut -\mathstrut \) \(9\) \(\beta_{2}\mathstrut +\mathstrut \) \(28\) \(\beta_{1}\)
\(\nu^{6}\)\(=\)\(37\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(45\) \(\beta_{5}\mathstrut +\mathstrut \) \(8\) \(\beta_{4}\mathstrut +\mathstrut \) \(57\) \(\beta_{3}\mathstrut -\mathstrut \) \(21\) \(\beta_{2}\mathstrut -\mathstrut \) \(8\) \(\beta_{1}\mathstrut +\mathstrut \) \(84\)
\(\nu^{7}\)\(=\)\(13\) \(\beta_{7}\mathstrut +\mathstrut \) \(65\) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(53\) \(\beta_{4}\mathstrut +\mathstrut \) \(27\) \(\beta_{3}\mathstrut -\mathstrut \) \(71\) \(\beta_{2}\mathstrut +\mathstrut \) \(165\) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)

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
−0.459587
2.63734
1.60389
−2.45154
−0.0452510
−1.71120
2.18779
1.23856
0 −1.00000 0 −2.63865 0 0.604857 0 1.00000 0
1.2 0 −1.00000 0 −1.29727 0 2.24503 0 1.00000 0
1.3 0 −1.00000 0 −0.122001 0 −3.47013 0 1.00000 0
1.4 0 −1.00000 0 1.48532 0 5.10210 0 1.00000 0
1.5 0 −1.00000 0 1.52778 0 −0.428515 0 1.00000 0
1.6 0 −1.00000 0 2.45529 0 −2.43610 0 1.00000 0
1.7 0 −1.00000 0 2.52133 0 0.307143 0 1.00000 0
1.8 0 −1.00000 0 3.06821 0 2.07562 0 1.00000 0
\(n\): 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
\(2\) \(-1\)
\(3\) \(1\)
\(167\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8016))\):

\(T_{5}^{8} - \cdots\)
\(T_{7}^{8} - \cdots\)
\(T_{11}^{8} + \cdots\)
\(T_{13}^{8} \) \(\mathstrut -\mathstrut 37 T_{13}^{6} \) \(\mathstrut -\mathstrut 52 T_{13}^{5} \) \(\mathstrut +\mathstrut 320 T_{13}^{4} \) \(\mathstrut +\mathstrut 973 T_{13}^{3} \) \(\mathstrut +\mathstrut 994 T_{13}^{2} \) \(\mathstrut +\mathstrut 412 T_{13} \) \(\mathstrut +\mathstrut 56 \)