Properties

Label 8004.2.a.e
Level 8004
Weight 2
Character orbit 8004.a
Self dual Yes
Analytic conductor 63.912
Analytic rank 1
Dimension 9
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8004.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(63.9122617778\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(- q^{3}\) \( + \beta_{1} q^{5} \) \( + ( 1 - \beta_{3} ) q^{7} \) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(- q^{3}\) \( + \beta_{1} q^{5} \) \( + ( 1 - \beta_{3} ) q^{7} \) \(+ q^{9}\) \( + ( \beta_{3} + \beta_{5} + \beta_{6} + \beta_{8} ) q^{11} \) \( + ( -1 - \beta_{1} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{13} \) \( -\beta_{1} q^{15} \) \( + ( \beta_{3} + \beta_{5} + \beta_{7} ) q^{17} \) \( + ( \beta_{1} - \beta_{3} - \beta_{4} - \beta_{7} ) q^{19} \) \( + ( -1 + \beta_{3} ) q^{21} \) \(- q^{23}\) \( + ( -\beta_{2} - \beta_{4} + \beta_{7} - \beta_{8} ) q^{25} \) \(- q^{27}\) \(+ q^{29}\) \( + ( -2 \beta_{1} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{8} ) q^{31} \) \( + ( -\beta_{3} - \beta_{5} - \beta_{6} - \beta_{8} ) q^{33} \) \( + ( -1 + \beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} ) q^{35} \) \( + ( -1 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{6} ) q^{37} \) \( + ( 1 + \beta_{1} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{39} \) \( + ( -2 + \beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{6} - \beta_{8} ) q^{41} \) \( + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} - 3 \beta_{5} + \beta_{7} ) q^{43} \) \( + \beta_{1} q^{45} \) \( + ( \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} - \beta_{8} ) q^{47} \) \( + ( -2 - 2 \beta_{3} - \beta_{6} ) q^{49} \) \( + ( -\beta_{3} - \beta_{5} - \beta_{7} ) q^{51} \) \( + ( -3 - \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{8} ) q^{53} \) \( + ( 1 - 2 \beta_{1} - \beta_{2} + 2 \beta_{4} - \beta_{5} + \beta_{7} - \beta_{8} ) q^{55} \) \( + ( -\beta_{1} + \beta_{3} + \beta_{4} + \beta_{7} ) q^{57} \) \( + ( -1 + \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{59} \) \( + ( -1 - \beta_{1} + \beta_{3} - \beta_{4} - \beta_{7} + \beta_{8} ) q^{61} \) \( + ( 1 - \beta_{3} ) q^{63} \) \( + ( -3 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{65} \) \( + ( 2 + \beta_{1} - 2 \beta_{2} + \beta_{3} + 3 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} ) q^{67} \) \(+ q^{69}\) \( + ( -1 - \beta_{1} + \beta_{5} - \beta_{7} - \beta_{8} ) q^{71} \) \( + ( -\beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{73} \) \( + ( \beta_{2} + \beta_{4} - \beta_{7} + \beta_{8} ) q^{75} \) \( + ( \beta_{1} - \beta_{2} + 2 \beta_{3} + 4 \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{77} \) \( + ( -2 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{5} - \beta_{6} - 2 \beta_{8} ) q^{79} \) \(+ q^{81}\) \( + ( -1 - \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} ) q^{83} \) \( + ( 1 - 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} ) q^{85} \) \(- q^{87}\) \( + ( -3 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} + \beta_{5} + 3 \beta_{6} - \beta_{7} + 2 \beta_{8} ) q^{89} \) \( + ( -3 - \beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{8} ) q^{91} \) \( + ( 2 \beta_{1} - 2 \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{8} ) q^{93} \) \( + ( 2 + \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} ) q^{95} \) \( + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} - 3 \beta_{7} - \beta_{8} ) q^{97} \) \( + ( \beta_{3} + \beta_{5} + \beta_{6} + \beta_{8} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(9q \) \(\mathstrut -\mathstrut 9q^{3} \) \(\mathstrut -\mathstrut 3q^{5} \) \(\mathstrut +\mathstrut 7q^{7} \) \(\mathstrut +\mathstrut 9q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(9q \) \(\mathstrut -\mathstrut 9q^{3} \) \(\mathstrut -\mathstrut 3q^{5} \) \(\mathstrut +\mathstrut 7q^{7} \) \(\mathstrut +\mathstrut 9q^{9} \) \(\mathstrut -\mathstrut 2q^{11} \) \(\mathstrut -\mathstrut 7q^{13} \) \(\mathstrut +\mathstrut 3q^{15} \) \(\mathstrut -\mathstrut q^{19} \) \(\mathstrut -\mathstrut 7q^{21} \) \(\mathstrut -\mathstrut 9q^{23} \) \(\mathstrut +\mathstrut 2q^{25} \) \(\mathstrut -\mathstrut 9q^{27} \) \(\mathstrut +\mathstrut 9q^{29} \) \(\mathstrut +\mathstrut 8q^{31} \) \(\mathstrut +\mathstrut 2q^{33} \) \(\mathstrut -\mathstrut 5q^{35} \) \(\mathstrut -\mathstrut 8q^{37} \) \(\mathstrut +\mathstrut 7q^{39} \) \(\mathstrut -\mathstrut 19q^{41} \) \(\mathstrut -\mathstrut 3q^{43} \) \(\mathstrut -\mathstrut 3q^{45} \) \(\mathstrut -\mathstrut 3q^{47} \) \(\mathstrut -\mathstrut 18q^{49} \) \(\mathstrut -\mathstrut 17q^{53} \) \(\mathstrut +\mathstrut 9q^{55} \) \(\mathstrut +\mathstrut q^{57} \) \(\mathstrut -\mathstrut 10q^{59} \) \(\mathstrut +\mathstrut q^{61} \) \(\mathstrut +\mathstrut 7q^{63} \) \(\mathstrut -\mathstrut 16q^{65} \) \(\mathstrut +\mathstrut 12q^{67} \) \(\mathstrut +\mathstrut 9q^{69} \) \(\mathstrut -\mathstrut 7q^{71} \) \(\mathstrut +\mathstrut 13q^{73} \) \(\mathstrut -\mathstrut 2q^{75} \) \(\mathstrut -\mathstrut 15q^{77} \) \(\mathstrut -\mathstrut 10q^{79} \) \(\mathstrut +\mathstrut 9q^{81} \) \(\mathstrut +\mathstrut 9q^{83} \) \(\mathstrut -\mathstrut 6q^{85} \) \(\mathstrut -\mathstrut 9q^{87} \) \(\mathstrut -\mathstrut 5q^{89} \) \(\mathstrut -\mathstrut 18q^{91} \) \(\mathstrut -\mathstrut 8q^{93} \) \(\mathstrut +\mathstrut 31q^{95} \) \(\mathstrut -\mathstrut 7q^{97} \) \(\mathstrut -\mathstrut 2q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9}\mathstrut -\mathstrut \) \(3\) \(x^{8}\mathstrut -\mathstrut \) \(13\) \(x^{7}\mathstrut +\mathstrut \) \(32\) \(x^{6}\mathstrut +\mathstrut \) \(40\) \(x^{5}\mathstrut -\mathstrut \) \(79\) \(x^{4}\mathstrut -\mathstrut \) \(39\) \(x^{3}\mathstrut +\mathstrut \) \(58\) \(x^{2}\mathstrut +\mathstrut \) \(9\) \(x\mathstrut -\mathstrut \) \(11\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 93 \nu^{8} - 7 \nu^{7} - 1649 \nu^{6} - 1742 \nu^{5} + 8484 \nu^{4} + 13848 \nu^{3} - 21492 \nu^{2} - 13414 \nu + 16720 \)\()/4877\)
\(\beta_{2}\)\(=\)\((\)\( -676 \nu^{8} - 54 \nu^{7} + 14451 \nu^{6} + 5373 \nu^{5} - 80862 \nu^{4} - 26245 \nu^{3} + 118464 \nu^{2} + 29593 \nu - 29868 \)\()/4877\)
\(\beta_{3}\)\(=\)\((\)\( -1248 \nu^{8} + 4027 \nu^{7} + 15049 \nu^{6} - 42227 \nu^{5} - 37863 \nu^{4} + 93731 \nu^{3} + 26624 \nu^{2} - 41031 \nu - 4120 \)\()/4877\)
\(\beta_{4}\)\(=\)\((\)\( 1613 \nu^{8} - 4474 \nu^{7} - 21416 \nu^{6} + 45249 \nu^{5} + 67542 \nu^{4} - 97748 \nu^{3} - 66924 \nu^{2} + 53254 \nu + 16986 \)\()/4877\)
\(\beta_{5}\)\(=\)\((\)\( 1706 \nu^{8} - 4481 \nu^{7} - 23065 \nu^{6} + 43507 \nu^{5} + 76026 \nu^{4} - 83900 \nu^{3} - 83539 \nu^{2} + 34963 \nu + 19075 \)\()/4877\)
\(\beta_{6}\)\(=\)\((\)\( -2082 \nu^{8} + 5663 \nu^{7} + 27005 \nu^{6} - 53822 \nu^{5} - 79649 \nu^{4} + 92100 \nu^{3} + 68801 \nu^{2} - 18907 \nu - 12313 \)\()/4877\)
\(\beta_{7}\)\(=\)\((\)\( 2159 \nu^{8} - 5931 \nu^{7} - 29524 \nu^{6} + 60980 \nu^{5} + 102091 \nu^{4} - 136012 \nu^{3} - 122465 \nu^{2} + 64804 \nu + 35858 \)\()/4877\)
\(\beta_{8}\)\(=\)\((\)\( 2535 \nu^{8} - 7113 \nu^{7} - 33464 \nu^{6} + 71295 \nu^{5} + 105714 \nu^{4} - 144212 \nu^{3} - 107727 \nu^{2} + 58502 \nu + 24219 \)\()/4877\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(3\) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(7\)\()/2\)
\(\nu^{3}\)\(=\)\(4\) \(\beta_{8}\mathstrut -\mathstrut \) \(5\) \(\beta_{7}\mathstrut +\mathstrut \) \(5\) \(\beta_{6}\mathstrut +\mathstrut \) \(6\) \(\beta_{5}\mathstrut -\mathstrut \) \(3\) \(\beta_{4}\mathstrut -\mathstrut \) \(4\) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(7\)
\(\nu^{4}\)\(=\)\((\)\(13\) \(\beta_{8}\mathstrut -\mathstrut \) \(19\) \(\beta_{7}\mathstrut +\mathstrut \) \(23\) \(\beta_{6}\mathstrut +\mathstrut \) \(53\) \(\beta_{5}\mathstrut -\mathstrut \) \(32\) \(\beta_{4}\mathstrut -\mathstrut \) \(14\) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut -\mathstrut \) \(18\) \(\beta_{1}\mathstrut +\mathstrut \) \(75\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(83\) \(\beta_{8}\mathstrut -\mathstrut \) \(129\) \(\beta_{7}\mathstrut +\mathstrut \) \(135\) \(\beta_{6}\mathstrut +\mathstrut \) \(193\) \(\beta_{5}\mathstrut -\mathstrut \) \(100\) \(\beta_{4}\mathstrut -\mathstrut \) \(130\) \(\beta_{3}\mathstrut -\mathstrut \) \(30\) \(\beta_{2}\mathstrut -\mathstrut \) \(14\) \(\beta_{1}\mathstrut +\mathstrut \) \(225\)\()/2\)
\(\nu^{6}\)\(=\)\(96\) \(\beta_{8}\mathstrut -\mathstrut \) \(174\) \(\beta_{7}\mathstrut +\mathstrut \) \(201\) \(\beta_{6}\mathstrut +\mathstrut \) \(413\) \(\beta_{5}\mathstrut -\mathstrut \) \(231\) \(\beta_{4}\mathstrut -\mathstrut \) \(167\) \(\beta_{3}\mathstrut -\mathstrut \) \(29\) \(\beta_{2}\mathstrut -\mathstrut \) \(99\) \(\beta_{1}\mathstrut +\mathstrut \) \(520\)
\(\nu^{7}\)\(=\)\((\)\(1033\) \(\beta_{8}\mathstrut -\mathstrut \) \(1823\) \(\beta_{7}\mathstrut +\mathstrut \) \(1943\) \(\beta_{6}\mathstrut +\mathstrut \) \(3085\) \(\beta_{5}\mathstrut -\mathstrut \) \(1588\) \(\beta_{4}\mathstrut -\mathstrut \) \(1928\) \(\beta_{3}\mathstrut -\mathstrut \) \(424\) \(\beta_{2}\mathstrut -\mathstrut \) \(342\) \(\beta_{1}\mathstrut +\mathstrut \) \(3591\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(3035\) \(\beta_{8}\mathstrut -\mathstrut \) \(5877\) \(\beta_{7}\mathstrut +\mathstrut \) \(6591\) \(\beta_{6}\mathstrut +\mathstrut \) \(12709\) \(\beta_{5}\mathstrut -\mathstrut \) \(6834\) \(\beta_{4}\mathstrut -\mathstrut \) \(6034\) \(\beta_{3}\mathstrut -\mathstrut \) \(1142\) \(\beta_{2}\mathstrut -\mathstrut \) \(2514\) \(\beta_{1}\mathstrut +\mathstrut \) \(15401\)\()/2\)

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
−2.86352
−1.47135
3.93556
2.25739
0.761439
−0.975163
1.39711
0.511914
−0.553378
0 −1.00000 0 −3.70748 0 −1.59469 0 1.00000 0
1.2 0 −1.00000 0 −3.48372 0 2.18462 0 1.00000 0
1.3 0 −1.00000 0 −2.36120 0 2.82157 0 1.00000 0
1.4 0 −1.00000 0 −0.650765 0 −1.33403 0 1.00000 0
1.5 0 −1.00000 0 0.461945 0 0.732054 0 1.00000 0
1.6 0 −1.00000 0 0.900888 0 3.90501 0 1.00000 0
1.7 0 −1.00000 0 1.20115 0 −1.61263 0 1.00000 0
1.8 0 −1.00000 0 1.34732 0 2.91839 0 1.00000 0
1.9 0 −1.00000 0 3.29187 0 −1.02028 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
\(2\) \(-1\)
\(3\) \(1\)
\(23\) \(1\)
\(29\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8004))\):

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