Properties

Label 8016.2.a.ba
Level 8016
Weight 2
Character orbit 8016.a
Self dual Yes
Analytic conductor 64.008
Analytic rank 0
Dimension 9
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: \(9\)
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.

\(f(q)\) \(=\) \( q - q^{3} + ( -1 + \beta_{1} ) q^{5} + ( 2 - \beta_{5} - \beta_{7} ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + ( -1 + \beta_{1} ) q^{5} + ( 2 - \beta_{5} - \beta_{7} ) q^{7} + q^{9} + ( -2 + \beta_{1} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{8} ) q^{11} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{7} + \beta_{8} ) q^{13} + ( 1 - \beta_{1} ) q^{15} + ( -1 - \beta_{2} - \beta_{6} - \beta_{7} ) q^{17} + ( 1 - \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{8} ) q^{19} + ( -2 + \beta_{5} + \beta_{7} ) q^{21} + ( \beta_{1} - \beta_{6} + \beta_{7} - \beta_{8} ) q^{23} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{25} - q^{27} + ( -5 + \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{8} ) q^{29} + ( 3 + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} ) q^{31} + ( 2 - \beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{8} ) q^{33} + ( 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} ) q^{35} + ( 1 - \beta_{2} - \beta_{5} + \beta_{6} - 3 \beta_{7} + \beta_{8} ) q^{37} + ( \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{7} - \beta_{8} ) q^{39} + ( -3 + 2 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{6} + \beta_{7} + \beta_{8} ) q^{41} + ( 4 + 2 \beta_{3} - \beta_{7} ) q^{43} + ( -1 + \beta_{1} ) q^{45} + ( 1 + \beta_{1} - \beta_{3} - \beta_{6} - \beta_{7} ) q^{47} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{7} - \beta_{8} ) q^{49} + ( 1 + \beta_{2} + \beta_{6} + \beta_{7} ) q^{51} + ( -3 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} ) q^{53} + ( 3 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} ) q^{55} + ( -1 + \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} ) q^{57} + ( -2 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{8} ) q^{59} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{61} + ( 2 - \beta_{5} - \beta_{7} ) q^{63} + ( 2 + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - \beta_{8} ) q^{65} + ( -1 + 2 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{8} ) q^{67} + ( -\beta_{1} + \beta_{6} - \beta_{7} + \beta_{8} ) q^{69} + ( \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{8} ) q^{71} + ( 2 + \beta_{1} - \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{73} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{75} + ( -3 + \beta_{1} - \beta_{2} - \beta_{7} + 2 \beta_{8} ) q^{77} + ( 4 - 2 \beta_{2} - 2 \beta_{4} - \beta_{7} + 2 \beta_{8} ) q^{79} + q^{81} + ( 2 + 3 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - 3 \beta_{8} ) q^{83} + ( -2 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{8} ) q^{85} + ( 5 - \beta_{2} + \beta_{3} - 3 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} - 2 \beta_{8} ) q^{87} + ( -6 + 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} + \beta_{5} - 4 \beta_{6} + \beta_{7} + \beta_{8} ) q^{89} + ( 3 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{91} + ( -3 - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{93} + ( 5 + 3 \beta_{1} + 4 \beta_{2} + \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + \beta_{6} - 3 \beta_{7} - 3 \beta_{8} ) q^{95} + ( 4 - \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} + 4 \beta_{6} - 3 \beta_{7} - \beta_{8} ) q^{97} + ( -2 + \beta_{1} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{8} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q - 9q^{3} - 6q^{5} + 11q^{7} + 9q^{9} + O(q^{10}) \) \( 9q - 9q^{3} - 6q^{5} + 11q^{7} + 9q^{9} - q^{11} - 4q^{13} + 6q^{15} - 9q^{17} + 8q^{19} - 11q^{21} + 7q^{23} - q^{25} - 9q^{27} - 9q^{29} + 25q^{31} + q^{33} - 5q^{35} - 6q^{37} + 4q^{39} - 4q^{41} + 24q^{43} - 6q^{45} + 16q^{47} + 4q^{49} + 9q^{51} - 26q^{53} + 29q^{55} - 8q^{57} + 4q^{59} - 20q^{61} + 11q^{63} - 8q^{65} + 25q^{67} - 7q^{69} + 15q^{71} - 10q^{73} + q^{75} - 20q^{77} + 34q^{79} + 9q^{81} + 4q^{83} - 13q^{85} + 9q^{87} + 13q^{89} + 21q^{91} - 25q^{93} + 7q^{95} - 4q^{97} - q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - 3 x^{8} - 16 x^{7} + 45 x^{6} + 67 x^{5} - 166 x^{4} - 83 x^{3} + 152 x^{2} + 51 x - 10\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -253 \nu^{8} + 10903 \nu^{7} - 17171 \nu^{6} - 175482 \nu^{5} + 247955 \nu^{4} + 716568 \nu^{3} - 563660 \nu^{2} - 635601 \nu + 68945 \)\()/102665\)
\(\beta_{3}\)\(=\)\((\)\( -428 \nu^{8} + 9923 \nu^{7} - 2266 \nu^{6} - 178372 \nu^{5} + 67645 \nu^{4} + 878658 \nu^{3} - 38080 \nu^{2} - 988001 \nu - 301330 \)\()/102665\)
\(\beta_{4}\)\(=\)\((\)\( -3031 \nu^{8} + 7666 \nu^{7} + 48718 \nu^{6} - 99334 \nu^{5} - 199070 \nu^{4} + 207921 \nu^{3} + 208150 \nu^{2} + 200063 \nu - 130065 \)\()/102665\)
\(\beta_{5}\)\(=\)\((\)\( 3811 \nu^{8} - 23831 \nu^{7} - 35953 \nu^{6} + 387944 \nu^{5} - 79645 \nu^{4} - 1646101 \nu^{3} + 520415 \nu^{2} + 1605297 \nu + 255125 \)\()/102665\)
\(\beta_{6}\)\(=\)\((\)\( 5684 \nu^{8} - 5129 \nu^{7} - 118627 \nu^{6} + 65121 \nu^{5} + 751965 \nu^{4} - 184819 \nu^{3} - 1469865 \nu^{2} + 296533 \nu + 445920 \)\()/102665\)
\(\beta_{7}\)\(=\)\((\)\( -6589 \nu^{8} + 20594 \nu^{7} + 101842 \nu^{6} - 311796 \nu^{5} - 367380 \nu^{4} + 1137454 \nu^{3} + 148730 \nu^{2} - 769633 \nu + 59190 \)\()/102665\)
\(\beta_{8}\)\(=\)\((\)\( 16128 \nu^{8} - 24668 \nu^{7} - 293609 \nu^{6} + 290982 \nu^{5} + 1500975 \nu^{4} - 446023 \nu^{3} - 2000020 \nu^{2} - 396149 \nu + 286160 \)\()/102665\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{7} - \beta_{5} + \beta_{4} - \beta_{2} + 5\)
\(\nu^{3}\)\(=\)\(-2 \beta_{8} - 3 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 3 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 8 \beta_{1} + 7\)
\(\nu^{4}\)\(=\)\(-3 \beta_{8} - 12 \beta_{7} + 6 \beta_{6} - 12 \beta_{5} + 7 \beta_{4} + 5 \beta_{3} - 17 \beta_{2} - 2 \beta_{1} + 54\)
\(\nu^{5}\)\(=\)\(-30 \beta_{8} - 45 \beta_{7} + 33 \beta_{6} - 27 \beta_{5} - 36 \beta_{4} + 34 \beta_{3} - 32 \beta_{2} + 73 \beta_{1} + 109\)
\(\nu^{6}\)\(=\)\(-56 \beta_{8} - 145 \beta_{7} + 106 \beta_{6} - 132 \beta_{5} + 54 \beta_{4} + 105 \beta_{3} - 225 \beta_{2} - 33 \beta_{1} + 635\)
\(\nu^{7}\)\(=\)\(-391 \beta_{8} - 576 \beta_{7} + 466 \beta_{6} - 324 \beta_{5} - 395 \beta_{4} + 506 \beta_{3} - 459 \beta_{2} + 691 \beta_{1} + 1496\)
\(\nu^{8}\)\(=\)\(-846 \beta_{8} - 1799 \beta_{7} + 1544 \beta_{6} - 1474 \beta_{5} + 418 \beta_{4} + 1662 \beta_{3} - 2818 \beta_{2} - 429 \beta_{1} + 7652\)

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
−3.17025
−1.99382
−0.985991
−0.482381
0.141931
1.41388
1.58856
2.94484
3.54323
0 −1.00000 0 −4.17025 0 1.78655 0 1.00000 0
1.2 0 −1.00000 0 −2.99382 0 4.65862 0 1.00000 0
1.3 0 −1.00000 0 −1.98599 0 0.0909950 0 1.00000 0
1.4 0 −1.00000 0 −1.48238 0 1.03908 0 1.00000 0
1.5 0 −1.00000 0 −0.858069 0 −2.33225 0 1.00000 0
1.6 0 −1.00000 0 0.413878 0 −2.72537 0 1.00000 0
1.7 0 −1.00000 0 0.588564 0 1.23518 0 1.00000 0
1.8 0 −1.00000 0 1.94484 0 3.19705 0 1.00000 0
1.9 0 −1.00000 0 2.54323 0 4.05015 0 1.00000 0
\(n\): 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
\(2\) \(1\)
\(3\) \(1\)
\(167\) \(-1\)

Hecke kernels

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

\(T_{5}^{9} + \cdots\)
\(T_{7}^{9} - \cdots\)
\(T_{11}^{9} + \cdots\)
\(T_{13}^{9} + \cdots\)