Properties

Label 8004.2.a.f
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: \(\Z[a_1, \ldots, a_{13}]\)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(+ q^{3}\) \( -\beta_{1} q^{5} \) \( + ( -1 - \beta_{6} ) q^{7} \) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(+ q^{3}\) \( -\beta_{1} q^{5} \) \( + ( -1 - \beta_{6} ) q^{7} \) \(+ q^{9}\) \( + ( -1 - \beta_{3} ) q^{11} \) \( + ( \beta_{1} + \beta_{3} + \beta_{6} - \beta_{7} ) q^{13} \) \( -\beta_{1} q^{15} \) \( + ( \beta_{1} + \beta_{7} ) q^{17} \) \( + ( -1 - \beta_{1} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{19} \) \( + ( -1 - \beta_{6} ) q^{21} \) \(- q^{23}\) \( + ( -1 + 2 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{25} \) \(+ q^{27}\) \(- q^{29}\) \( + ( \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{31} \) \( + ( -1 - \beta_{3} ) q^{33} \) \( + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{6} - \beta_{8} ) q^{35} \) \( + ( 1 - 2 \beta_{2} - \beta_{4} + \beta_{6} ) q^{37} \) \( + ( \beta_{1} + \beta_{3} + \beta_{6} - \beta_{7} ) q^{39} \) \( + ( -3 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} ) q^{41} \) \( + ( -1 - \beta_{1} + \beta_{5} + \beta_{6} - \beta_{8} ) q^{43} \) \( -\beta_{1} q^{45} \) \( + ( -2 + 2 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{7} + 2 \beta_{8} ) q^{47} \) \( + ( -1 - \beta_{3} + 2 \beta_{5} + \beta_{8} ) q^{49} \) \( + ( \beta_{1} + \beta_{7} ) q^{51} \) \( + ( 1 + \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - \beta_{8} ) q^{53} \) \( + ( -3 + \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{6} + \beta_{8} ) q^{55} \) \( + ( -1 - \beta_{1} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{57} \) \( + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{7} + \beta_{8} ) q^{59} \) \( + ( -\beta_{2} + 2 \beta_{4} + 2 \beta_{6} + \beta_{7} - 2 \beta_{8} ) q^{61} \) \( + ( -1 - \beta_{6} ) q^{63} \) \( + ( -2 - 4 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{65} \) \( + ( -3 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} ) q^{67} \) \(- q^{69}\) \( + ( 3 + 2 \beta_{1} + 3 \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} - 3 \beta_{8} ) q^{71} \) \( + ( -5 - \beta_{1} + 3 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} - \beta_{5} - 4 \beta_{6} ) q^{73} \) \( + ( -1 + 2 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{75} \) \( + ( -1 - \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{8} ) q^{77} \) \( + ( 1 - \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{8} ) q^{79} \) \(+ q^{81}\) \( + ( 3 - 2 \beta_{2} + 4 \beta_{3} + \beta_{4} - \beta_{5} + 4 \beta_{6} - 2 \beta_{8} ) q^{83} \) \( + ( -5 - \beta_{2} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{85} \) \(- q^{87}\) \( + ( -3 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} ) q^{89} \) \( + ( -2 - 2 \beta_{1} + \beta_{4} - \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - \beta_{8} ) q^{91} \) \( + ( \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{93} \) \( + ( 2 + 5 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} ) q^{95} \) \( + ( -4 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{8} ) q^{97} \) \( + ( -1 - \beta_{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(9q \) \(\mathstrut +\mathstrut 9q^{3} \) \(\mathstrut -\mathstrut q^{5} \) \(\mathstrut -\mathstrut 5q^{7} \) \(\mathstrut +\mathstrut 9q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(9q \) \(\mathstrut +\mathstrut 9q^{3} \) \(\mathstrut -\mathstrut q^{5} \) \(\mathstrut -\mathstrut 5q^{7} \) \(\mathstrut +\mathstrut 9q^{9} \) \(\mathstrut -\mathstrut 8q^{11} \) \(\mathstrut -\mathstrut q^{13} \) \(\mathstrut -\mathstrut q^{15} \) \(\mathstrut -\mathstrut 2q^{17} \) \(\mathstrut -\mathstrut 11q^{19} \) \(\mathstrut -\mathstrut 5q^{21} \) \(\mathstrut -\mathstrut 9q^{23} \) \(\mathstrut -\mathstrut 10q^{25} \) \(\mathstrut +\mathstrut 9q^{27} \) \(\mathstrut -\mathstrut 9q^{29} \) \(\mathstrut -\mathstrut 8q^{33} \) \(\mathstrut +\mathstrut q^{35} \) \(\mathstrut -\mathstrut 2q^{37} \) \(\mathstrut -\mathstrut q^{39} \) \(\mathstrut -\mathstrut 3q^{41} \) \(\mathstrut -\mathstrut 19q^{43} \) \(\mathstrut -\mathstrut q^{45} \) \(\mathstrut -\mathstrut 3q^{47} \) \(\mathstrut -\mathstrut 6q^{49} \) \(\mathstrut -\mathstrut 2q^{51} \) \(\mathstrut -\mathstrut 9q^{53} \) \(\mathstrut -\mathstrut 7q^{55} \) \(\mathstrut -\mathstrut 11q^{57} \) \(\mathstrut -\mathstrut 2q^{59} \) \(\mathstrut -\mathstrut 25q^{61} \) \(\mathstrut -\mathstrut 5q^{63} \) \(\mathstrut -\mathstrut 12q^{65} \) \(\mathstrut -\mathstrut 20q^{67} \) \(\mathstrut -\mathstrut 9q^{69} \) \(\mathstrut +\mathstrut 9q^{71} \) \(\mathstrut -\mathstrut 11q^{73} \) \(\mathstrut -\mathstrut 10q^{75} \) \(\mathstrut -\mathstrut 19q^{77} \) \(\mathstrut +\mathstrut 4q^{79} \) \(\mathstrut +\mathstrut 9q^{81} \) \(\mathstrut -\mathstrut 9q^{83} \) \(\mathstrut -\mathstrut 50q^{85} \) \(\mathstrut -\mathstrut 9q^{87} \) \(\mathstrut -\mathstrut 29q^{89} \) \(\mathstrut -\mathstrut 38q^{91} \) \(\mathstrut +\mathstrut 23q^{95} \) \(\mathstrut -\mathstrut 43q^{97} \) \(\mathstrut -\mathstrut 8q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9}\mathstrut -\mathstrut \) \(x^{8}\mathstrut -\mathstrut \) \(17\) \(x^{7}\mathstrut +\mathstrut \) \(4\) \(x^{6}\mathstrut +\mathstrut \) \(75\) \(x^{5}\mathstrut +\mathstrut \) \(x^{4}\mathstrut -\mathstrut \) \(118\) \(x^{3}\mathstrut -\mathstrut \) \(26\) \(x^{2}\mathstrut +\mathstrut \) \(60\) \(x\mathstrut +\mathstrut \) \(24\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -11 \nu^{8} - 2 \nu^{7} + 180 \nu^{6} + 161 \nu^{5} - 559 \nu^{4} - 480 \nu^{3} + 553 \nu^{2} + 258 \nu - 182 \)\()/34\)
\(\beta_{3}\)\(=\)\((\)\( -5 \nu^{8} - 4 \nu^{7} + 88 \nu^{6} + 118 \nu^{5} - 302 \nu^{4} - 382 \nu^{3} + 307 \nu^{2} + 278 \nu - 41 \)\()/17\)
\(\beta_{4}\)\(=\)\((\)\( -21 \nu^{8} + 24 \nu^{7} + 322 \nu^{6} - 113 \nu^{5} - 1061 \nu^{4} + 252 \nu^{3} + 895 \nu^{2} - 104 \nu - 162 \)\()/34\)
\(\beta_{5}\)\(=\)\((\)\( 22 \nu^{8} - 13 \nu^{7} - 343 \nu^{6} - 67 \nu^{5} + 1050 \nu^{4} + 229 \nu^{3} - 749 \nu^{2} - 74 \nu + 24 \)\()/34\)
\(\beta_{6}\)\(=\)\((\)\( 4 \nu^{8} + 10 \nu^{7} - 67 \nu^{6} - 210 \nu^{5} + 109 \nu^{4} + 683 \nu^{3} + 193 \nu^{2} - 559 \nu - 297 \)\()/17\)
\(\beta_{7}\)\(=\)\((\)\( -23 \nu^{8} + 36 \nu^{7} + 364 \nu^{6} - 297 \nu^{5} - 1447 \nu^{4} + 854 \nu^{3} + 1861 \nu^{2} - 598 \nu - 702 \)\()/34\)
\(\beta_{8}\)\(=\)\((\)\( -16 \nu^{8} + 28 \nu^{7} + 251 \nu^{6} - 248 \nu^{5} - 1014 \nu^{4} + 685 \nu^{3} + 1319 \nu^{2} - 484 \nu - 495 \)\()/17\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(\beta_{8}\mathstrut -\mathstrut \) \(3\) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(12\) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{4}\)\(=\)\(\beta_{8}\mathstrut -\mathstrut \) \(16\) \(\beta_{7}\mathstrut +\mathstrut \) \(15\) \(\beta_{6}\mathstrut -\mathstrut \) \(4\) \(\beta_{5}\mathstrut +\mathstrut \) \(12\) \(\beta_{4}\mathstrut +\mathstrut \) \(16\) \(\beta_{3}\mathstrut -\mathstrut \) \(4\) \(\beta_{2}\mathstrut +\mathstrut \) \(37\) \(\beta_{1}\mathstrut +\mathstrut \) \(38\)
\(\nu^{5}\)\(=\)\(17\) \(\beta_{8}\mathstrut -\mathstrut \) \(58\) \(\beta_{7}\mathstrut +\mathstrut \) \(39\) \(\beta_{6}\mathstrut -\mathstrut \) \(28\) \(\beta_{5}\mathstrut +\mathstrut \) \(12\) \(\beta_{4}\mathstrut +\mathstrut \) \(41\) \(\beta_{3}\mathstrut -\mathstrut \) \(16\) \(\beta_{2}\mathstrut +\mathstrut \) \(173\) \(\beta_{1}\mathstrut +\mathstrut \) \(69\)
\(\nu^{6}\)\(=\)\(30\) \(\beta_{8}\mathstrut -\mathstrut \) \(246\) \(\beta_{7}\mathstrut +\mathstrut \) \(215\) \(\beta_{6}\mathstrut -\mathstrut \) \(82\) \(\beta_{5}\mathstrut +\mathstrut \) \(147\) \(\beta_{4}\mathstrut +\mathstrut \) \(233\) \(\beta_{3}\mathstrut -\mathstrut \) \(73\) \(\beta_{2}\mathstrut +\mathstrut \) \(612\) \(\beta_{1}\mathstrut +\mathstrut \) \(480\)
\(\nu^{7}\)\(=\)\(238\) \(\beta_{8}\mathstrut -\mathstrut \) \(944\) \(\beta_{7}\mathstrut +\mathstrut \) \(675\) \(\beta_{6}\mathstrut -\mathstrut \) \(402\) \(\beta_{5}\mathstrut +\mathstrut \) \(300\) \(\beta_{4}\mathstrut +\mathstrut \) \(719\) \(\beta_{3}\mathstrut -\mathstrut \) \(258\) \(\beta_{2}\mathstrut +\mathstrut \) \(2611\) \(\beta_{1}\mathstrut +\mathstrut \) \(1298\)
\(\nu^{8}\)\(=\)\(602\) \(\beta_{8}\mathstrut -\mathstrut \) \(3809\) \(\beta_{7}\mathstrut +\mathstrut \) \(3167\) \(\beta_{6}\mathstrut -\mathstrut \) \(1388\) \(\beta_{5}\mathstrut +\mathstrut \) \(1967\) \(\beta_{4}\mathstrut +\mathstrut \) \(3432\) \(\beta_{3}\mathstrut -\mathstrut \) \(1138\) \(\beta_{2}\mathstrut +\mathstrut \) \(9792\) \(\beta_{1}\mathstrut +\mathstrut \) \(6751\)

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
3.95129
1.58421
1.51482
1.05365
−0.509166
−0.808455
−1.09926
−1.91513
−2.77196
0 1.00000 0 −3.95129 0 −1.61455 0 1.00000 0
1.2 0 1.00000 0 −1.58421 0 1.44549 0 1.00000 0
1.3 0 1.00000 0 −1.51482 0 −4.32415 0 1.00000 0
1.4 0 1.00000 0 −1.05365 0 3.84396 0 1.00000 0
1.5 0 1.00000 0 0.509166 0 1.30726 0 1.00000 0
1.6 0 1.00000 0 0.808455 0 −2.11918 0 1.00000 0
1.7 0 1.00000 0 1.09926 0 −1.62400 0 1.00000 0
1.8 0 1.00000 0 1.91513 0 −2.97719 0 1.00000 0
1.9 0 1.00000 0 2.77196 0 1.06236 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8004))\):

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