Properties

Label 8016.2.a.z
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: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q\) \(+ q^{3}\) \( + ( \beta_{1} + \beta_{2} - \beta_{5} ) q^{5} \) \( + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{7} \) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(+ q^{3}\) \( + ( \beta_{1} + \beta_{2} - \beta_{5} ) q^{5} \) \( + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{7} \) \(+ q^{9}\) \( + ( 1 - 2 \beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{11} \) \( + ( -\beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{13} \) \( + ( \beta_{1} + \beta_{2} - \beta_{5} ) q^{15} \) \( + ( -2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{6} - \beta_{7} ) q^{17} \) \( + ( -5 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} ) q^{19} \) \( + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{21} \) \( + ( -\beta_{2} - \beta_{4} + 3 \beta_{5} + \beta_{6} ) q^{23} \) \( + ( 3 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{25} \) \(+ q^{27}\) \( + ( -1 + \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} ) q^{29} \) \( + ( -1 - 4 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} + 4 \beta_{5} + 2 \beta_{7} ) q^{31} \) \( + ( 1 - 2 \beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{33} \) \( + ( -4 + 2 \beta_{1} + \beta_{2} - \beta_{4} - 2 \beta_{5} + \beta_{7} ) q^{35} \) \( + ( 3 - 3 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{7} ) q^{37} \) \( + ( -\beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{39} \) \( + ( -\beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{41} \) \( + ( -4 + 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{43} \) \( + ( \beta_{1} + \beta_{2} - \beta_{5} ) q^{45} \) \( + ( 2 - 2 \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - 4 \beta_{7} ) q^{47} \) \( + ( 2 - 2 \beta_{1} + 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{49} \) \( + ( -2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{6} - \beta_{7} ) q^{51} \) \( + ( -5 + 2 \beta_{1} + 3 \beta_{3} - 2 \beta_{5} - \beta_{6} ) q^{53} \) \( + ( -3 - 3 \beta_{1} - \beta_{3} + 3 \beta_{4} + 2 \beta_{6} ) q^{55} \) \( + ( -5 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} ) q^{57} \) \( + ( -4 - \beta_{1} + \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{59} \) \( + ( \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{7} ) q^{61} \) \( + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{63} \) \( + ( 2 - 6 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + 8 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} ) q^{65} \) \( + ( -3 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + 3 \beta_{5} + 3 \beta_{7} ) q^{67} \) \( + ( -\beta_{2} - \beta_{4} + 3 \beta_{5} + \beta_{6} ) q^{69} \) \( + ( -1 - \beta_{1} + 5 \beta_{2} - \beta_{3} - 2 \beta_{4} - 4 \beta_{5} - \beta_{7} ) q^{71} \) \( + ( -2 + 5 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - 3 \beta_{6} - 2 \beta_{7} ) q^{73} \) \( + ( 3 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{75} \) \( + ( -2 - \beta_{2} - \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{77} \) \( + ( 4 \beta_{1} - \beta_{2} - \beta_{3} - 3 \beta_{4} + \beta_{5} - 4 \beta_{6} + \beta_{7} ) q^{79} \) \(+ q^{81}\) \( + ( -1 + \beta_{2} + 3 \beta_{4} - 3 \beta_{5} - \beta_{6} - \beta_{7} ) q^{83} \) \( + ( 1 - 8 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + \beta_{4} + 6 \beta_{5} + 4 \beta_{6} + \beta_{7} ) q^{85} \) \( + ( -1 + \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} ) q^{87} \) \( + ( 1 + 5 \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{89} \) \( + ( -4 + 2 \beta_{3} - \beta_{4} - 5 \beta_{5} - \beta_{6} - 4 \beta_{7} ) q^{91} \) \( + ( -1 - 4 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} + 4 \beta_{5} + 2 \beta_{7} ) q^{93} \) \( + ( 2 - 5 \beta_{1} - 5 \beta_{2} - \beta_{3} - \beta_{4} + 8 \beta_{5} + \beta_{7} ) q^{95} \) \( + ( -6 + 4 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} - 4 \beta_{4} - 7 \beta_{5} - \beta_{6} - 3 \beta_{7} ) q^{97} \) \( + ( 1 - 2 \beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut +\mathstrut 8q^{3} \) \(\mathstrut +\mathstrut q^{5} \) \(\mathstrut +\mathstrut 8q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut +\mathstrut 8q^{3} \) \(\mathstrut +\mathstrut q^{5} \) \(\mathstrut +\mathstrut 8q^{9} \) \(\mathstrut -\mathstrut 5q^{11} \) \(\mathstrut +\mathstrut q^{15} \) \(\mathstrut -\mathstrut 7q^{17} \) \(\mathstrut -\mathstrut 24q^{19} \) \(\mathstrut -\mathstrut q^{23} \) \(\mathstrut +\mathstrut 3q^{25} \) \(\mathstrut +\mathstrut 8q^{27} \) \(\mathstrut -\mathstrut 11q^{29} \) \(\mathstrut -\mathstrut 30q^{31} \) \(\mathstrut -\mathstrut 5q^{33} \) \(\mathstrut -\mathstrut 26q^{35} \) \(\mathstrut +\mathstrut 11q^{37} \) \(\mathstrut +\mathstrut 10q^{41} \) \(\mathstrut -\mathstrut 24q^{43} \) \(\mathstrut +\mathstrut q^{45} \) \(\mathstrut +\mathstrut 3q^{47} \) \(\mathstrut +\mathstrut 6q^{49} \) \(\mathstrut -\mathstrut 7q^{51} \) \(\mathstrut -\mathstrut 25q^{53} \) \(\mathstrut -\mathstrut 25q^{55} \) \(\mathstrut -\mathstrut 24q^{57} \) \(\mathstrut -\mathstrut 45q^{59} \) \(\mathstrut +\mathstrut 16q^{61} \) \(\mathstrut -\mathstrut 10q^{65} \) \(\mathstrut -\mathstrut 18q^{67} \) \(\mathstrut -\mathstrut q^{69} \) \(\mathstrut -\mathstrut 21q^{71} \) \(\mathstrut -\mathstrut 8q^{73} \) \(\mathstrut +\mathstrut 3q^{75} \) \(\mathstrut -\mathstrut 18q^{77} \) \(\mathstrut -\mathstrut 10q^{79} \) \(\mathstrut +\mathstrut 8q^{81} \) \(\mathstrut -\mathstrut 7q^{83} \) \(\mathstrut -\mathstrut 11q^{85} \) \(\mathstrut -\mathstrut 11q^{87} \) \(\mathstrut +\mathstrut 26q^{89} \) \(\mathstrut -\mathstrut 15q^{91} \) \(\mathstrut -\mathstrut 30q^{93} \) \(\mathstrut -\mathstrut q^{95} \) \(\mathstrut -\mathstrut 3q^{97} \) \(\mathstrut -\mathstrut 5q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut -\mathstrut \) \(2\) \(x^{7}\mathstrut -\mathstrut \) \(8\) \(x^{6}\mathstrut +\mathstrut \) \(15\) \(x^{5}\mathstrut +\mathstrut \) \(19\) \(x^{4}\mathstrut -\mathstrut \) \(31\) \(x^{3}\mathstrut -\mathstrut \) \(13\) \(x^{2}\mathstrut +\mathstrut \) \(14\) \(x\mathstrut +\mathstrut \) \(1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 4 \nu \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 5 \nu^{2} + 3 \)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} - \nu^{6} - 7 \nu^{5} + 6 \nu^{4} + 13 \nu^{3} - 8 \nu^{2} - 5 \nu - 1 \)\()/2\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} + \nu^{6} - 9 \nu^{5} - 8 \nu^{4} + 23 \nu^{3} + 16 \nu^{2} - 13 \nu - 3 \)\()/2\)
\(\beta_{7}\)\(=\)\( -\nu^{7} + \nu^{6} + 8 \nu^{5} - 7 \nu^{4} - 18 \nu^{3} + 13 \nu^{2} + 9 \nu - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{4}\mathstrut +\mathstrut \) \(5\) \(\beta_{2}\mathstrut +\mathstrut \) \(12\)
\(\nu^{5}\)\(=\)\(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(5\) \(\beta_{3}\mathstrut +\mathstrut \) \(16\) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{6}\)\(=\)\(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(8\) \(\beta_{4}\mathstrut +\mathstrut \) \(23\) \(\beta_{2}\mathstrut +\mathstrut \) \(51\)
\(\nu^{7}\)\(=\)\(8\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(17\) \(\beta_{5}\mathstrut +\mathstrut \) \(9\) \(\beta_{4}\mathstrut +\mathstrut \) \(22\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(65\) \(\beta_{1}\mathstrut +\mathstrut \) \(18\)

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
−1.63639
−0.0678707
−0.853788
0.688556
−2.05189
1.60046
2.22210
2.09883
0 1.00000 0 −3.62511 0 1.70865 0 1.00000 0
1.2 0 1.00000 0 −2.71255 0 5.14356 0 1.00000 0
1.3 0 1.00000 0 −0.621044 0 0.794861 0 1.00000 0
1.4 0 1.00000 0 0.0425261 0 −1.43576 0 1.00000 0
1.5 0 1.00000 0 0.925650 0 −2.76498 0 1.00000 0
1.6 0 1.00000 0 1.28584 0 0.120874 0 1.00000 0
1.7 0 1.00000 0 1.56175 0 −4.60397 0 1.00000 0
1.8 0 1.00000 0 4.14293 0 1.03675 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8016))\):

\(T_{5}^{8} - \cdots\)
\(T_{7}^{8} \) \(\mathstrut -\mathstrut 31 T_{7}^{6} \) \(\mathstrut -\mathstrut 7 T_{7}^{5} \) \(\mathstrut +\mathstrut 175 T_{7}^{4} \) \(\mathstrut -\mathstrut 59 T_{7}^{3} \) \(\mathstrut -\mathstrut 224 T_{7}^{2} \) \(\mathstrut +\mathstrut 160 T_{7} \) \(\mathstrut -\mathstrut 16 \)
\(T_{11}^{8} + \cdots\)
\(T_{13}^{8} \) \(\mathstrut -\mathstrut 61 T_{13}^{6} \) \(\mathstrut -\mathstrut 10 T_{13}^{5} \) \(\mathstrut +\mathstrut 1088 T_{13}^{4} \) \(\mathstrut -\mathstrut 237 T_{13}^{3} \) \(\mathstrut -\mathstrut 6922 T_{13}^{2} \) \(\mathstrut +\mathstrut 3828 T_{13} \) \(\mathstrut +\mathstrut 9224 \)