Properties

Label 8008.2.a.s
Level 8008
Weight 2
Character orbit 8008.a
Self dual Yes
Analytic conductor 63.944
Analytic rank 1
Dimension 10
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8008.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(63.9442019386\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{3} \) \( + \beta_{9} q^{5} \) \(+ q^{7}\) \( + ( 1 + \beta_{5} - \beta_{6} ) q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{3} \) \( + \beta_{9} q^{5} \) \(+ q^{7}\) \( + ( 1 + \beta_{5} - \beta_{6} ) q^{9} \) \(+ q^{11}\) \(- q^{13}\) \( + ( -1 + \beta_{1} + \beta_{6} - \beta_{9} ) q^{15} \) \( + ( -1 - \beta_{2} - \beta_{4} - \beta_{9} ) q^{17} \) \( + ( 1 + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{19} \) \( -\beta_{1} q^{21} \) \( + ( -2 + 2 \beta_{1} - \beta_{5} + \beta_{7} + \beta_{8} ) q^{23} \) \( + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{25} \) \( + ( -2 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} ) q^{27} \) \( + ( -2 + 2 \beta_{1} + \beta_{2} - \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{8} ) q^{29} \) \( + ( -2 + \beta_{2} + \beta_{3} - \beta_{6} - \beta_{8} + \beta_{9} ) q^{31} \) \( -\beta_{1} q^{33} \) \( + \beta_{9} q^{35} \) \( + ( 1 + \beta_{3} + 2 \beta_{4} - \beta_{6} - \beta_{7} ) q^{37} \) \( + \beta_{1} q^{39} \) \( + ( -1 - \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{8} - \beta_{9} ) q^{41} \) \( + ( 1 + \beta_{1} + \beta_{2} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{43} \) \( + ( -2 + 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} ) q^{45} \) \( + ( -4 - \beta_{5} - \beta_{7} + \beta_{8} ) q^{47} \) \(+ q^{49}\) \( + ( -1 + 4 \beta_{1} + 2 \beta_{2} - \beta_{4} - \beta_{5} + \beta_{7} + \beta_{9} ) q^{51} \) \( + ( -3 - 2 \beta_{2} + \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} - 3 \beta_{9} ) q^{53} \) \( + \beta_{9} q^{55} \) \( + ( -3 - \beta_{1} + \beta_{2} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{8} ) q^{57} \) \( + ( -2 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{9} ) q^{59} \) \( + ( -2 + \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{61} \) \( + ( 1 + \beta_{5} - \beta_{6} ) q^{63} \) \( -\beta_{9} q^{65} \) \( + ( -2 + \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{67} \) \( + ( -4 + 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} + 2 \beta_{6} + \beta_{8} - 2 \beta_{9} ) q^{69} \) \( + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{7} ) q^{71} \) \( + ( -1 - 2 \beta_{1} + \beta_{2} + 2 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{73} \) \( + ( -1 - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - \beta_{8} ) q^{75} \) \(+ q^{77}\) \( + ( -1 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} + 2 \beta_{8} ) q^{79} \) \( + ( 2 + 4 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} - 3 \beta_{6} + 4 \beta_{7} + 2 \beta_{9} ) q^{81} \) \( + ( -2 + 2 \beta_{1} + \beta_{2} - \beta_{3} - 3 \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{83} \) \( + ( -3 + \beta_{1} - \beta_{3} - \beta_{4} + \beta_{6} + 2 \beta_{7} - 2 \beta_{9} ) q^{85} \) \( + ( 1 + 3 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{87} \) \( + ( -4 \beta_{1} - 2 \beta_{2} + \beta_{4} + 2 \beta_{5} - 3 \beta_{7} + \beta_{8} - 2 \beta_{9} ) q^{89} \) \(- q^{91}\) \( + ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{4} + \beta_{6} - 3 \beta_{7} - \beta_{8} - 2 \beta_{9} ) q^{93} \) \( + ( -3 + 2 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{95} \) \( + ( -2 - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{9} ) q^{97} \) \( + ( 1 + \beta_{5} - \beta_{6} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(10q \) \(\mathstrut -\mathstrut 3q^{3} \) \(\mathstrut -\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut 10q^{7} \) \(\mathstrut +\mathstrut 9q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(10q \) \(\mathstrut -\mathstrut 3q^{3} \) \(\mathstrut -\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut 10q^{7} \) \(\mathstrut +\mathstrut 9q^{9} \) \(\mathstrut +\mathstrut 10q^{11} \) \(\mathstrut -\mathstrut 10q^{13} \) \(\mathstrut -\mathstrut 5q^{15} \) \(\mathstrut -\mathstrut 11q^{17} \) \(\mathstrut +\mathstrut 2q^{19} \) \(\mathstrut -\mathstrut 3q^{21} \) \(\mathstrut -\mathstrut 8q^{23} \) \(\mathstrut +\mathstrut 2q^{25} \) \(\mathstrut -\mathstrut 15q^{27} \) \(\mathstrut -\mathstrut 8q^{29} \) \(\mathstrut -\mathstrut 23q^{31} \) \(\mathstrut -\mathstrut 3q^{33} \) \(\mathstrut -\mathstrut 4q^{35} \) \(\mathstrut +\mathstrut 10q^{37} \) \(\mathstrut +\mathstrut 3q^{39} \) \(\mathstrut -\mathstrut 18q^{41} \) \(\mathstrut +\mathstrut 12q^{43} \) \(\mathstrut -\mathstrut 10q^{45} \) \(\mathstrut -\mathstrut 36q^{47} \) \(\mathstrut +\mathstrut 10q^{49} \) \(\mathstrut +\mathstrut 9q^{51} \) \(\mathstrut -\mathstrut 21q^{53} \) \(\mathstrut -\mathstrut 4q^{55} \) \(\mathstrut -\mathstrut 30q^{57} \) \(\mathstrut -\mathstrut 13q^{59} \) \(\mathstrut -\mathstrut 2q^{61} \) \(\mathstrut +\mathstrut 9q^{63} \) \(\mathstrut +\mathstrut 4q^{65} \) \(\mathstrut -\mathstrut 4q^{67} \) \(\mathstrut -\mathstrut 26q^{69} \) \(\mathstrut -\mathstrut 24q^{71} \) \(\mathstrut -\mathstrut 23q^{73} \) \(\mathstrut -\mathstrut 28q^{75} \) \(\mathstrut +\mathstrut 10q^{77} \) \(\mathstrut +\mathstrut 14q^{79} \) \(\mathstrut +\mathstrut 30q^{81} \) \(\mathstrut -\mathstrut 9q^{83} \) \(\mathstrut -\mathstrut 17q^{85} \) \(\mathstrut +\mathstrut 7q^{87} \) \(\mathstrut -\mathstrut 18q^{89} \) \(\mathstrut -\mathstrut 10q^{91} \) \(\mathstrut +\mathstrut q^{93} \) \(\mathstrut -\mathstrut 4q^{95} \) \(\mathstrut -\mathstrut 9q^{97} \) \(\mathstrut +\mathstrut 9q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10}\mathstrut -\mathstrut \) \(3\) \(x^{9}\mathstrut -\mathstrut \) \(15\) \(x^{8}\mathstrut +\mathstrut \) \(43\) \(x^{7}\mathstrut +\mathstrut \) \(66\) \(x^{6}\mathstrut -\mathstrut \) \(173\) \(x^{5}\mathstrut -\mathstrut \) \(127\) \(x^{4}\mathstrut +\mathstrut \) \(246\) \(x^{3}\mathstrut +\mathstrut \) \(99\) \(x^{2}\mathstrut -\mathstrut \) \(82\) \(x\mathstrut +\mathstrut \) \(8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( -2 \nu^{9} + 41 \nu^{7} + 7 \nu^{6} - 255 \nu^{5} - 64 \nu^{4} + 501 \nu^{3} + 157 \nu^{2} - 173 \nu + 16 \)
\(\beta_{3}\)\(=\)\( -5 \nu^{9} + 15 \nu^{8} + 62 \nu^{7} - 180 \nu^{6} - 159 \nu^{5} + 452 \nu^{4} + 108 \nu^{3} - 245 \nu^{2} + 42 \nu + 3 \)
\(\beta_{4}\)\(=\)\((\)\( -11 \nu^{9} + 29 \nu^{8} + 147 \nu^{7} - 343 \nu^{6} - 474 \nu^{5} + 827 \nu^{4} + 525 \nu^{3} - 356 \nu^{2} - \nu + 10 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( -19 \nu^{9} + 41 \nu^{8} + 277 \nu^{7} - 469 \nu^{6} - 1090 \nu^{5} + 1015 \nu^{4} + 1547 \nu^{3} - 140 \nu^{2} - 241 \nu + 18 \)\()/2\)
\(\beta_{6}\)\(=\)\((\)\( -19 \nu^{9} + 41 \nu^{8} + 277 \nu^{7} - 469 \nu^{6} - 1090 \nu^{5} + 1015 \nu^{4} + 1547 \nu^{3} - 142 \nu^{2} - 241 \nu + 26 \)\()/2\)
\(\beta_{7}\)\(=\)\((\)\( -19 \nu^{9} + 33 \nu^{8} + 299 \nu^{7} - 365 \nu^{6} - 1350 \nu^{5} + 705 \nu^{4} + 2169 \nu^{3} + 156 \nu^{2} - 505 \nu + 62 \)\()/2\)
\(\beta_{8}\)\(=\)\((\)\( 13 \nu^{9} - 9 \nu^{8} - 241 \nu^{7} + 71 \nu^{6} + 1352 \nu^{5} + 71 \nu^{4} - 2493 \nu^{3} - 684 \nu^{2} + 735 \nu - 76 \)\()/2\)
\(\beta_{9}\)\(=\)\((\)\( 15 \nu^{9} - 11 \nu^{8} - 277 \nu^{7} + 91 \nu^{6} + 1550 \nu^{5} + 45 \nu^{4} - 2875 \nu^{3} - 732 \nu^{2} + 895 \nu - 96 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(8\) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{4}\)\(=\)\(2\) \(\beta_{9}\mathstrut +\mathstrut \) \(4\) \(\beta_{7}\mathstrut -\mathstrut \) \(12\) \(\beta_{6}\mathstrut +\mathstrut \) \(10\) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\mathstrut +\mathstrut \) \(29\)
\(\nu^{5}\)\(=\)\(18\) \(\beta_{9}\mathstrut -\mathstrut \) \(14\) \(\beta_{8}\mathstrut +\mathstrut \) \(6\) \(\beta_{7}\mathstrut -\mathstrut \) \(17\) \(\beta_{6}\mathstrut +\mathstrut \) \(14\) \(\beta_{5}\mathstrut -\mathstrut \) \(27\) \(\beta_{4}\mathstrut +\mathstrut \) \(26\) \(\beta_{3}\mathstrut +\mathstrut \) \(17\) \(\beta_{2}\mathstrut +\mathstrut \) \(79\) \(\beta_{1}\mathstrut +\mathstrut \) \(26\)
\(\nu^{6}\)\(=\)\(42\) \(\beta_{9}\mathstrut -\mathstrut \) \(6\) \(\beta_{8}\mathstrut +\mathstrut \) \(63\) \(\beta_{7}\mathstrut -\mathstrut \) \(132\) \(\beta_{6}\mathstrut +\mathstrut \) \(102\) \(\beta_{5}\mathstrut -\mathstrut \) \(39\) \(\beta_{4}\mathstrut +\mathstrut \) \(23\) \(\beta_{3}\mathstrut +\mathstrut \) \(31\) \(\beta_{2}\mathstrut +\mathstrut \) \(80\) \(\beta_{1}\mathstrut +\mathstrut \) \(263\)
\(\nu^{7}\)\(=\)\(245\) \(\beta_{9}\mathstrut -\mathstrut \) \(159\) \(\beta_{8}\mathstrut +\mathstrut \) \(123\) \(\beta_{7}\mathstrut -\mathstrut \) \(233\) \(\beta_{6}\mathstrut +\mathstrut \) \(174\) \(\beta_{5}\mathstrut -\mathstrut \) \(314\) \(\beta_{4}\mathstrut +\mathstrut \) \(291\) \(\beta_{3}\mathstrut +\mathstrut \) \(234\) \(\beta_{2}\mathstrut +\mathstrut \) \(846\) \(\beta_{1}\mathstrut +\mathstrut \) \(321\)
\(\nu^{8}\)\(=\)\(635\) \(\beta_{9}\mathstrut -\mathstrut \) \(138\) \(\beta_{8}\mathstrut +\mathstrut \) \(807\) \(\beta_{7}\mathstrut -\mathstrut \) \(1454\) \(\beta_{6}\mathstrut +\mathstrut \) \(1077\) \(\beta_{5}\mathstrut -\mathstrut \) \(571\) \(\beta_{4}\mathstrut +\mathstrut \) \(371\) \(\beta_{3}\mathstrut +\mathstrut \) \(533\) \(\beta_{2}\mathstrut +\mathstrut \) \(1233\) \(\beta_{1}\mathstrut +\mathstrut \) \(2642\)
\(\nu^{9}\)\(=\)\(3061\) \(\beta_{9}\mathstrut -\mathstrut \) \(1746\) \(\beta_{8}\mathstrut +\mathstrut \) \(1849\) \(\beta_{7}\mathstrut -\mathstrut \) \(3016\) \(\beta_{6}\mathstrut +\mathstrut \) \(2148\) \(\beta_{5}\mathstrut -\mathstrut \) \(3568\) \(\beta_{4}\mathstrut +\mathstrut \) \(3200\) \(\beta_{3}\mathstrut +\mathstrut \) \(2956\) \(\beta_{2}\mathstrut +\mathstrut \) \(9340\) \(\beta_{1}\mathstrut +\mathstrut \) \(4081\)

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
3.44977
2.46329
1.79551
1.79314
0.364324
0.119595
−1.01146
−1.48379
−1.49066
−2.99972
0 −3.44977 0 0.464152 0 1.00000 0 8.90091 0
1.2 0 −2.46329 0 −1.35093 0 1.00000 0 3.06778 0
1.3 0 −1.79551 0 3.68086 0 1.00000 0 0.223871 0
1.4 0 −1.79314 0 −4.09989 0 1.00000 0 0.215350 0
1.5 0 −0.364324 0 2.29944 0 1.00000 0 −2.86727 0
1.6 0 −0.119595 0 −2.15143 0 1.00000 0 −2.98570 0
1.7 0 1.01146 0 −0.0780254 0 1.00000 0 −1.97695 0
1.8 0 1.48379 0 1.24336 0 1.00000 0 −0.798370 0
1.9 0 1.49066 0 −2.23108 0 1.00000 0 −0.777924 0
1.10 0 2.99972 0 −1.77646 0 1.00000 0 5.99829 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(-1\)
\(11\) \(-1\)
\(13\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\):

\(T_{3}^{10} + \cdots\)
\(T_{5}^{10} + \cdots\)
\(T_{17}^{10} + \cdots\)