Properties

Label 8016.2.a.v
Level 8016
Weight 2
Character orbit 8016.a
Self dual Yes
Analytic conductor 64.008
Analytic rank 0
Dimension 7
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: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + ( 1 - \beta_{1} ) q^{5} + ( -1 + \beta_{3} ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + ( 1 - \beta_{1} ) q^{5} + ( -1 + \beta_{3} ) q^{7} + q^{9} + ( -1 - \beta_{2} - \beta_{5} + \beta_{6} ) q^{11} + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} ) q^{13} + ( -1 + \beta_{1} ) q^{15} + ( \beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} ) q^{17} + ( 1 + \beta_{2} - \beta_{5} - \beta_{6} ) q^{19} + ( 1 - \beta_{3} ) q^{21} + ( -\beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{23} + ( 1 + \beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{25} - q^{27} + ( -1 - \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{29} + ( -1 - \beta_{1} - \beta_{2} - \beta_{4} ) q^{31} + ( 1 + \beta_{2} + \beta_{5} - \beta_{6} ) q^{33} + ( 1 + 2 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{6} ) q^{35} + ( -1 + \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{37} + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} ) q^{39} + ( -2 - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{41} + ( -\beta_{1} + \beta_{3} + \beta_{4} + \beta_{6} ) q^{43} + ( 1 - \beta_{1} ) q^{45} + ( 1 + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{47} + ( 2 - \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{49} + ( -\beta_{1} - \beta_{3} + \beta_{5} - \beta_{6} ) q^{51} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{53} + ( -2 - \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{55} + ( -1 - \beta_{2} + \beta_{5} + \beta_{6} ) q^{57} + ( 3 - \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{59} + ( 3 + \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{61} + ( -1 + \beta_{3} ) q^{63} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{65} + ( 3 + 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} ) q^{67} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{69} + ( 4 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{71} + ( 4 - \beta_{1} + 3 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{73} + ( -1 - \beta_{1} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{75} + ( -4 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{77} + ( -\beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{79} + q^{81} + ( -1 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - \beta_{5} + \beta_{6} ) q^{83} + ( -3 - 4 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + \beta_{5} + \beta_{6} ) q^{85} + ( 1 + \beta_{2} + 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{87} + ( -1 - 2 \beta_{1} - 3 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} ) q^{89} + ( 2 - 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + \beta_{4} + 4 \beta_{5} ) q^{91} + ( 1 + \beta_{1} + \beta_{2} + \beta_{4} ) q^{93} + ( 6 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 4 \beta_{6} ) q^{95} + ( -2 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{97} + ( -1 - \beta_{2} - \beta_{5} + \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q - 7q^{3} + 5q^{5} - 7q^{7} + 7q^{9} + O(q^{10}) \) \( 7q - 7q^{3} + 5q^{5} - 7q^{7} + 7q^{9} + 6q^{13} - 5q^{15} + 6q^{17} + 2q^{19} + 7q^{21} + 12q^{25} - 7q^{27} - 4q^{29} - 7q^{31} + 13q^{35} - 3q^{37} - 6q^{39} - 12q^{41} + 2q^{43} + 5q^{45} + 11q^{47} + 10q^{49} - 6q^{51} + q^{53} + 2q^{55} - 2q^{57} + 19q^{59} + 12q^{61} - 7q^{63} - 10q^{65} + 17q^{67} + 20q^{71} + 10q^{73} - 12q^{75} - 24q^{77} - 2q^{79} + 7q^{81} + 7q^{83} - 18q^{85} + 4q^{87} - 3q^{89} - 4q^{91} + 7q^{93} + 24q^{95} - 3q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - 2 x^{6} - 20 x^{5} + 2 x^{4} + 87 x^{3} + 46 x^{2} - 48 x - 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 5 \nu^{6} - 21 \nu^{5} - 33 \nu^{4} + 41 \nu^{3} + 12 \nu^{2} + 162 \nu - 14 \)\()/104\)
\(\beta_{3}\)\(=\)\((\)\( -3 \nu^{6} + 23 \nu^{5} - \nu^{4} - 243 \nu^{3} + 128 \nu^{2} + 610 \nu - 158 \)\()/52\)
\(\beta_{4}\)\(=\)\((\)\( -25 \nu^{6} + 105 \nu^{5} + 269 \nu^{4} - 621 \nu^{3} - 892 \nu^{2} + 542 \nu + 486 \)\()/104\)
\(\beta_{5}\)\(=\)\((\)\( -17 \nu^{6} + 61 \nu^{5} + 237 \nu^{4} - 389 \nu^{3} - 800 \nu^{2} + 406 \nu + 162 \)\()/52\)
\(\beta_{6}\)\(=\)\((\)\( -59 \nu^{6} + 227 \nu^{5} + 743 \nu^{4} - 1399 \nu^{3} - 2388 \nu^{2} + 1042 \nu + 290 \)\()/104\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} - \beta_{5} - \beta_{4} + 3 \beta_{1} + 5\)
\(\nu^{3}\)\(=\)\(4 \beta_{6} - 5 \beta_{5} - 2 \beta_{4} - \beta_{3} + 2 \beta_{2} + 18 \beta_{1} + 11\)
\(\nu^{4}\)\(=\)\(24 \beta_{6} - 28 \beta_{5} - 15 \beta_{4} - 4 \beta_{3} + 13 \beta_{2} + 83 \beta_{1} + 80\)
\(\nu^{5}\)\(=\)\(119 \beta_{6} - 148 \beta_{5} - 59 \beta_{4} - 24 \beta_{3} + 74 \beta_{2} + 437 \beta_{1} + 342\)
\(\nu^{6}\)\(=\)\(623 \beta_{6} - 763 \beta_{5} - 328 \beta_{4} - 119 \beta_{3} + 401 \beta_{2} + 2196 \beta_{1} + 1865\)

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
5.12652
2.30288
0.561628
−0.0402333
−1.55552
−2.09849
−2.29679
0 −1.00000 0 −4.12652 0 −3.19234 0 1.00000 0
1.2 0 −1.00000 0 −1.30288 0 −1.53952 0 1.00000 0
1.3 0 −1.00000 0 0.438372 0 2.51945 0 1.00000 0
1.4 0 −1.00000 0 1.04023 0 −4.50614 0 1.00000 0
1.5 0 −1.00000 0 2.55552 0 −3.69934 0 1.00000 0
1.6 0 −1.00000 0 3.09849 0 2.06938 0 1.00000 0
1.7 0 −1.00000 0 3.29679 0 1.34851 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8016))\):

\( T_{5}^{7} - 5 T_{5}^{6} - 11 T_{5}^{5} + 93 T_{5}^{4} - 110 T_{5}^{3} - 110 T_{5}^{2} + 208 T_{5} - 64 \)
\( T_{7}^{7} + 7 T_{7}^{6} - 5 T_{7}^{5} - 99 T_{7}^{4} - 28 T_{7}^{3} + 448 T_{7}^{2} + 96 T_{7} - 576 \)
\( T_{11}^{7} - 48 T_{11}^{5} - 12 T_{11}^{4} + 624 T_{11}^{3} + 368 T_{11}^{2} - 1344 T_{11} - 1152 \)
\( T_{13}^{7} - 6 T_{13}^{6} - 30 T_{13}^{5} + 244 T_{13}^{4} - 288 T_{13}^{3} - 552 T_{13}^{2} + 1088 T_{13} - 384 \)