Properties

Label 8034.2.a.q
Level 8034
Weight 2
Character orbit 8034.a
Self dual Yes
Analytic conductor 64.152
Analytic rank 1
Dimension 8
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8034.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.1518129839\)
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.

\(f(q)\) \(=\) \( q\) \(+ q^{2}\) \(+ q^{3}\) \(+ q^{4}\) \( + ( -1 + \beta_{1} ) q^{5} \) \(+ q^{6}\) \( + ( -\beta_{1} - \beta_{6} ) q^{7} \) \(+ q^{8}\) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(+ q^{2}\) \(+ q^{3}\) \(+ q^{4}\) \( + ( -1 + \beta_{1} ) q^{5} \) \(+ q^{6}\) \( + ( -\beta_{1} - \beta_{6} ) q^{7} \) \(+ q^{8}\) \(+ q^{9}\) \( + ( -1 + \beta_{1} ) q^{10} \) \( + ( -2 + \beta_{6} ) q^{11} \) \(+ q^{12}\) \(- q^{13}\) \( + ( -\beta_{1} - \beta_{6} ) q^{14} \) \( + ( -1 + \beta_{1} ) q^{15} \) \(+ q^{16}\) \( + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{17} \) \(+ q^{18}\) \( + ( -1 - \beta_{1} - 2 \beta_{2} - \beta_{6} ) q^{19} \) \( + ( -1 + \beta_{1} ) q^{20} \) \( + ( -\beta_{1} - \beta_{6} ) q^{21} \) \( + ( -2 + \beta_{6} ) q^{22} \) \( + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} + 3 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{23} \) \(+ q^{24}\) \( + ( -1 - 2 \beta_{1} + \beta_{3} + \beta_{4} ) q^{25} \) \(- q^{26}\) \(+ q^{27}\) \( + ( -\beta_{1} - \beta_{6} ) q^{28} \) \( + ( -2 + \beta_{1} - \beta_{2} - 2 \beta_{7} ) q^{29} \) \( + ( -1 + \beta_{1} ) q^{30} \) \( + ( -1 - \beta_{1} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{31} \) \(+ q^{32}\) \( + ( -2 + \beta_{6} ) q^{33} \) \( + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{34} \) \( + ( -2 + \beta_{2} - \beta_{4} + \beta_{6} - \beta_{7} ) q^{35} \) \(+ q^{36}\) \( + ( -4 + \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{37} \) \( + ( -1 - \beta_{1} - 2 \beta_{2} - \beta_{6} ) q^{38} \) \(- q^{39}\) \( + ( -1 + \beta_{1} ) q^{40} \) \( + ( -1 - 2 \beta_{1} + 2 \beta_{2} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{41} \) \( + ( -\beta_{1} - \beta_{6} ) q^{42} \) \( + ( -1 - \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{43} \) \( + ( -2 + \beta_{6} ) q^{44} \) \( + ( -1 + \beta_{1} ) q^{45} \) \( + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} + 3 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{46} \) \( + ( -2 \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} + 2 \beta_{7} ) q^{47} \) \(+ q^{48}\) \( + ( 1 + \beta_{1} - 4 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{49} \) \( + ( -1 - 2 \beta_{1} + \beta_{3} + \beta_{4} ) q^{50} \) \( + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{51} \) \(- q^{52}\) \( + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{53} \) \(+ q^{54}\) \( + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{6} + \beta_{7} ) q^{55} \) \( + ( -\beta_{1} - \beta_{6} ) q^{56} \) \( + ( -1 - \beta_{1} - 2 \beta_{2} - \beta_{6} ) q^{57} \) \( + ( -2 + \beta_{1} - \beta_{2} - 2 \beta_{7} ) q^{58} \) \( + ( -4 + 2 \beta_{1} - \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 3 \beta_{7} ) q^{59} \) \( + ( -1 + \beta_{1} ) q^{60} \) \( + ( -3 - \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - 3 \beta_{7} ) q^{61} \) \( + ( -1 - \beta_{1} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{62} \) \( + ( -\beta_{1} - \beta_{6} ) q^{63} \) \(+ q^{64}\) \( + ( 1 - \beta_{1} ) q^{65} \) \( + ( -2 + \beta_{6} ) q^{66} \) \( + ( -2 + 2 \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{4} - 2 \beta_{6} + \beta_{7} ) q^{67} \) \( + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{68} \) \( + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} + 3 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{69} \) \( + ( -2 + \beta_{2} - \beta_{4} + \beta_{6} - \beta_{7} ) q^{70} \) \( + ( -3 + \beta_{1} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{71} \) \(+ q^{72}\) \( + ( -3 + \beta_{5} - \beta_{6} ) q^{73} \) \( + ( -4 + \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{74} \) \( + ( -1 - 2 \beta_{1} + \beta_{3} + \beta_{4} ) q^{75} \) \( + ( -1 - \beta_{1} - 2 \beta_{2} - \beta_{6} ) q^{76} \) \( + ( -6 + 2 \beta_{1} + 3 \beta_{2} + \beta_{3} - \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{77} \) \(- q^{78}\) \( + ( -1 - \beta_{3} - 3 \beta_{4} + \beta_{6} ) q^{79} \) \( + ( -1 + \beta_{1} ) q^{80} \) \(+ q^{81}\) \( + ( -1 - 2 \beta_{1} + 2 \beta_{2} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{82} \) \( + ( -3 - 2 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{5} - 2 \beta_{7} ) q^{83} \) \( + ( -\beta_{1} - \beta_{6} ) q^{84} \) \( + ( 4 - 5 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{85} \) \( + ( -1 - \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{86} \) \( + ( -2 + \beta_{1} - \beta_{2} - 2 \beta_{7} ) q^{87} \) \( + ( -2 + \beta_{6} ) q^{88} \) \( + ( 1 - \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} + \beta_{6} + 3 \beta_{7} ) q^{89} \) \( + ( -1 + \beta_{1} ) q^{90} \) \( + ( \beta_{1} + \beta_{6} ) q^{91} \) \( + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} + 3 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{92} \) \( + ( -1 - \beta_{1} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{93} \) \( + ( -2 \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} + 2 \beta_{7} ) q^{94} \) \( + ( -3 + \beta_{1} + \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{95} \) \(+ q^{96}\) \( + ( -4 - 3 \beta_{1} + \beta_{3} + \beta_{4} - 2 \beta_{7} ) q^{97} \) \( + ( 1 + \beta_{1} - 4 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{98} \) \( + ( -2 + \beta_{6} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut +\mathstrut 8q^{2} \) \(\mathstrut +\mathstrut 8q^{3} \) \(\mathstrut +\mathstrut 8q^{4} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut +\mathstrut 8q^{6} \) \(\mathstrut -\mathstrut 3q^{7} \) \(\mathstrut +\mathstrut 8q^{8} \) \(\mathstrut +\mathstrut 8q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut +\mathstrut 8q^{2} \) \(\mathstrut +\mathstrut 8q^{3} \) \(\mathstrut +\mathstrut 8q^{4} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut +\mathstrut 8q^{6} \) \(\mathstrut -\mathstrut 3q^{7} \) \(\mathstrut +\mathstrut 8q^{8} \) \(\mathstrut +\mathstrut 8q^{9} \) \(\mathstrut -\mathstrut 6q^{10} \) \(\mathstrut -\mathstrut 15q^{11} \) \(\mathstrut +\mathstrut 8q^{12} \) \(\mathstrut -\mathstrut 8q^{13} \) \(\mathstrut -\mathstrut 3q^{14} \) \(\mathstrut -\mathstrut 6q^{15} \) \(\mathstrut +\mathstrut 8q^{16} \) \(\mathstrut -\mathstrut 11q^{17} \) \(\mathstrut +\mathstrut 8q^{18} \) \(\mathstrut -\mathstrut 15q^{19} \) \(\mathstrut -\mathstrut 6q^{20} \) \(\mathstrut -\mathstrut 3q^{21} \) \(\mathstrut -\mathstrut 15q^{22} \) \(\mathstrut +\mathstrut q^{23} \) \(\mathstrut +\mathstrut 8q^{24} \) \(\mathstrut -\mathstrut 10q^{25} \) \(\mathstrut -\mathstrut 8q^{26} \) \(\mathstrut +\mathstrut 8q^{27} \) \(\mathstrut -\mathstrut 3q^{28} \) \(\mathstrut -\mathstrut 10q^{29} \) \(\mathstrut -\mathstrut 6q^{30} \) \(\mathstrut -\mathstrut 3q^{31} \) \(\mathstrut +\mathstrut 8q^{32} \) \(\mathstrut -\mathstrut 15q^{33} \) \(\mathstrut -\mathstrut 11q^{34} \) \(\mathstrut -\mathstrut 12q^{35} \) \(\mathstrut +\mathstrut 8q^{36} \) \(\mathstrut -\mathstrut 26q^{37} \) \(\mathstrut -\mathstrut 15q^{38} \) \(\mathstrut -\mathstrut 8q^{39} \) \(\mathstrut -\mathstrut 6q^{40} \) \(\mathstrut -\mathstrut 12q^{41} \) \(\mathstrut -\mathstrut 3q^{42} \) \(\mathstrut -\mathstrut 4q^{43} \) \(\mathstrut -\mathstrut 15q^{44} \) \(\mathstrut -\mathstrut 6q^{45} \) \(\mathstrut +\mathstrut q^{46} \) \(\mathstrut -\mathstrut 6q^{47} \) \(\mathstrut +\mathstrut 8q^{48} \) \(\mathstrut -\mathstrut 5q^{49} \) \(\mathstrut -\mathstrut 10q^{50} \) \(\mathstrut -\mathstrut 11q^{51} \) \(\mathstrut -\mathstrut 8q^{52} \) \(\mathstrut -\mathstrut 4q^{53} \) \(\mathstrut +\mathstrut 8q^{54} \) \(\mathstrut -\mathstrut 3q^{56} \) \(\mathstrut -\mathstrut 15q^{57} \) \(\mathstrut -\mathstrut 10q^{58} \) \(\mathstrut -\mathstrut 19q^{59} \) \(\mathstrut -\mathstrut 6q^{60} \) \(\mathstrut -\mathstrut 14q^{61} \) \(\mathstrut -\mathstrut 3q^{62} \) \(\mathstrut -\mathstrut 3q^{63} \) \(\mathstrut +\mathstrut 8q^{64} \) \(\mathstrut +\mathstrut 6q^{65} \) \(\mathstrut -\mathstrut 15q^{66} \) \(\mathstrut -\mathstrut 13q^{67} \) \(\mathstrut -\mathstrut 11q^{68} \) \(\mathstrut +\mathstrut q^{69} \) \(\mathstrut -\mathstrut 12q^{70} \) \(\mathstrut -\mathstrut 31q^{71} \) \(\mathstrut +\mathstrut 8q^{72} \) \(\mathstrut -\mathstrut 27q^{73} \) \(\mathstrut -\mathstrut 26q^{74} \) \(\mathstrut -\mathstrut 10q^{75} \) \(\mathstrut -\mathstrut 15q^{76} \) \(\mathstrut -\mathstrut 30q^{77} \) \(\mathstrut -\mathstrut 8q^{78} \) \(\mathstrut -\mathstrut 13q^{79} \) \(\mathstrut -\mathstrut 6q^{80} \) \(\mathstrut +\mathstrut 8q^{81} \) \(\mathstrut -\mathstrut 12q^{82} \) \(\mathstrut -\mathstrut 28q^{83} \) \(\mathstrut -\mathstrut 3q^{84} \) \(\mathstrut +\mathstrut 15q^{85} \) \(\mathstrut -\mathstrut 4q^{86} \) \(\mathstrut -\mathstrut 10q^{87} \) \(\mathstrut -\mathstrut 15q^{88} \) \(\mathstrut -\mathstrut 2q^{89} \) \(\mathstrut -\mathstrut 6q^{90} \) \(\mathstrut +\mathstrut 3q^{91} \) \(\mathstrut +\mathstrut q^{92} \) \(\mathstrut -\mathstrut 3q^{93} \) \(\mathstrut -\mathstrut 6q^{94} \) \(\mathstrut -\mathstrut 18q^{95} \) \(\mathstrut +\mathstrut 8q^{96} \) \(\mathstrut -\mathstrut 30q^{97} \) \(\mathstrut -\mathstrut 5q^{98} \) \(\mathstrut -\mathstrut 15q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut -\mathstrut \) \(2\) \(x^{7}\mathstrut -\mathstrut \) \(11\) \(x^{6}\mathstrut +\mathstrut \) \(21\) \(x^{5}\mathstrut +\mathstrut \) \(23\) \(x^{4}\mathstrut -\mathstrut \) \(29\) \(x^{3}\mathstrut -\mathstrut \) \(27\) \(x^{2}\mathstrut +\mathstrut \) \(x\mathstrut +\mathstrut \) \(3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{7} - 3 \nu^{6} - 9 \nu^{5} + 31 \nu^{4} + 3 \nu^{3} - 41 \nu^{2} - 6 \nu + 8 \)
\(\beta_{3}\)\(=\)\( 2 \nu^{7} - 6 \nu^{6} - 17 \nu^{5} + 60 \nu^{4} - 4 \nu^{3} - 63 \nu^{2} - 3 \nu + 6 \)
\(\beta_{4}\)\(=\)\( -2 \nu^{7} + 6 \nu^{6} + 17 \nu^{5} - 60 \nu^{4} + 4 \nu^{3} + 64 \nu^{2} + 3 \nu - 9 \)
\(\beta_{5}\)\(=\)\( -2 \nu^{7} + 5 \nu^{6} + 19 \nu^{5} - 51 \nu^{4} - 15 \nu^{3} + 61 \nu^{2} + 14 \nu - 9 \)
\(\beta_{6}\)\(=\)\( -6 \nu^{7} + 17 \nu^{6} + 53 \nu^{5} - 171 \nu^{4} - 8 \nu^{3} + 188 \nu^{2} + 26 \nu - 24 \)
\(\beta_{7}\)\(=\)\( 8 \nu^{7} - 22 \nu^{6} - 71 \nu^{5} + 221 \nu^{4} + 13 \nu^{3} - 240 \nu^{2} - 28 \nu + 33 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(6\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(10\) \(\beta_{4}\mathstrut +\mathstrut \) \(9\) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut -\mathstrut \) \(3\) \(\beta_{1}\mathstrut +\mathstrut \) \(20\)
\(\nu^{5}\)\(=\)\(2\) \(\beta_{7}\mathstrut -\mathstrut \) \(8\) \(\beta_{6}\mathstrut +\mathstrut \) \(12\) \(\beta_{5}\mathstrut +\mathstrut \) \(11\) \(\beta_{4}\mathstrut -\mathstrut \) \(10\) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(45\) \(\beta_{1}\mathstrut -\mathstrut \) \(7\)
\(\nu^{6}\)\(=\)\(13\) \(\beta_{7}\mathstrut +\mathstrut \) \(12\) \(\beta_{6}\mathstrut +\mathstrut \) \(13\) \(\beta_{5}\mathstrut +\mathstrut \) \(91\) \(\beta_{4}\mathstrut +\mathstrut \) \(77\) \(\beta_{3}\mathstrut +\mathstrut \) \(22\) \(\beta_{2}\mathstrut -\mathstrut \) \(40\) \(\beta_{1}\mathstrut +\mathstrut \) \(157\)
\(\nu^{7}\)\(=\)\(26\) \(\beta_{7}\mathstrut -\mathstrut \) \(64\) \(\beta_{6}\mathstrut +\mathstrut \) \(113\) \(\beta_{5}\mathstrut +\mathstrut \) \(100\) \(\beta_{4}\mathstrut -\mathstrut \) \(94\) \(\beta_{3}\mathstrut +\mathstrut \) \(23\) \(\beta_{2}\mathstrut +\mathstrut \) \(366\) \(\beta_{1}\mathstrut -\mathstrut \) \(97\)

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
−2.96271
−0.811382
−0.749852
−0.501271
0.315468
1.79755
2.03483
2.87737
1.00000 1.00000 1.00000 −3.96271 1.00000 0.133176 1.00000 1.00000 −3.96271
1.2 1.00000 1.00000 1.00000 −1.81138 1.00000 4.37785 1.00000 1.00000 −1.81138
1.3 1.00000 1.00000 1.00000 −1.74985 1.00000 −2.02962 1.00000 1.00000 −1.74985
1.4 1.00000 1.00000 1.00000 −1.50127 1.00000 1.44399 1.00000 1.00000 −1.50127
1.5 1.00000 1.00000 1.00000 −0.684532 1.00000 −1.46314 1.00000 1.00000 −0.684532
1.6 1.00000 1.00000 1.00000 0.797548 1.00000 −4.52438 1.00000 1.00000 0.797548
1.7 1.00000 1.00000 1.00000 1.03483 1.00000 0.662784 1.00000 1.00000 1.03483
1.8 1.00000 1.00000 1.00000 1.87737 1.00000 −1.60066 1.00000 1.00000 1.87737
\(n\): 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
\(2\) \(-1\)
\(3\) \(-1\)
\(13\) \(1\)
\(103\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

\(T_{5}^{8} + \cdots\)
\(T_{7}^{8} + \cdots\)