Properties

Label 8008.2.a.o
Level 8008
Weight 2
Character orbit 8008.a
Self dual Yes
Analytic conductor 63.944
Analytic rank 1
Dimension 9
CM No
Inner twists 1

Related objects

Downloads

Learn more about

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: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{3} \) \( + \beta_{4} q^{5} \) \(+ q^{7}\) \( + \beta_{2} q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{3} \) \( + \beta_{4} q^{5} \) \(+ q^{7}\) \( + \beta_{2} q^{9} \) \(- q^{11}\) \(+ q^{13}\) \( -\beta_{3} q^{15} \) \( + ( -1 + \beta_{1} - 2 \beta_{2} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} ) q^{17} \) \( + ( \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{19} \) \( -\beta_{1} q^{21} \) \( + ( -2 + \beta_{3} - \beta_{8} ) q^{23} \) \( + ( -1 - \beta_{4} - \beta_{8} ) q^{25} \) \( + ( -1 + \beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} ) q^{27} \) \( + ( -1 + \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + \beta_{8} ) q^{29} \) \( + ( 2 + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{7} ) q^{31} \) \( + \beta_{1} q^{33} \) \( + \beta_{4} q^{35} \) \( + ( -2 - 2 \beta_{1} - \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} ) q^{37} \) \( -\beta_{1} q^{39} \) \( + ( -3 + \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{8} ) q^{41} \) \( + ( \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} - 3 \beta_{5} + 2 \beta_{7} + \beta_{8} ) q^{43} \) \( + ( 2 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{45} \) \( + ( -2 + 2 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + 3 \beta_{6} + \beta_{8} ) q^{47} \) \(+ q^{49}\) \( + ( -1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{51} \) \( + ( -2 + \beta_{2} - \beta_{4} + \beta_{6} - \beta_{7} ) q^{53} \) \( -\beta_{4} q^{55} \) \( + ( -4 - \beta_{1} - 3 \beta_{2} - \beta_{4} - 2 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} - \beta_{8} ) q^{57} \) \( + ( -1 + 3 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{5} + \beta_{7} + \beta_{8} ) q^{59} \) \( + ( 1 - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{8} ) q^{61} \) \( + \beta_{2} q^{63} \) \( + \beta_{4} q^{65} \) \( + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{5} - \beta_{6} - \beta_{8} ) q^{67} \) \( + ( -1 - \beta_{3} - 2 \beta_{4} - 2 \beta_{6} + \beta_{7} ) q^{69} \) \( + ( 2 + \beta_{2} - \beta_{3} - 2 \beta_{7} + 2 \beta_{8} ) q^{71} \) \( + ( -2 + \beta_{2} - 2 \beta_{4} - \beta_{6} - \beta_{8} ) q^{73} \) \( + ( -1 + \beta_{1} + \beta_{3} + \beta_{5} ) q^{75} \) \(- q^{77}\) \( + ( -2 - \beta_{2} + \beta_{3} + 2 \beta_{5} - \beta_{7} ) q^{79} \) \( + ( -3 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{81} \) \( + ( -2 + \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{83} \) \( + ( -\beta_{4} - \beta_{5} - 2 \beta_{7} - \beta_{8} ) q^{85} \) \( + ( 4 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - 3 \beta_{5} + \beta_{7} + \beta_{8} ) q^{87} \) \( + ( -3 - \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{89} \) \(+ q^{91}\) \( + ( -1 - \beta_{1} + 2 \beta_{4} + \beta_{6} - \beta_{7} ) q^{93} \) \( + ( -2 + \beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{5} - \beta_{7} ) q^{95} \) \( + ( -1 + 2 \beta_{1} - 3 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} + 3 \beta_{8} ) q^{97} \) \( -\beta_{2} q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(9q \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut 9q^{7} \) \(\mathstrut +\mathstrut q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(9q \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut 9q^{7} \) \(\mathstrut +\mathstrut q^{9} \) \(\mathstrut -\mathstrut 9q^{11} \) \(\mathstrut +\mathstrut 9q^{13} \) \(\mathstrut -\mathstrut 4q^{15} \) \(\mathstrut -\mathstrut 9q^{17} \) \(\mathstrut +\mathstrut 9q^{19} \) \(\mathstrut -\mathstrut 2q^{21} \) \(\mathstrut -\mathstrut 10q^{23} \) \(\mathstrut -\mathstrut q^{25} \) \(\mathstrut -\mathstrut 8q^{27} \) \(\mathstrut -\mathstrut 13q^{29} \) \(\mathstrut +\mathstrut 7q^{31} \) \(\mathstrut +\mathstrut 2q^{33} \) \(\mathstrut -\mathstrut 4q^{35} \) \(\mathstrut -\mathstrut 15q^{37} \) \(\mathstrut -\mathstrut 2q^{39} \) \(\mathstrut -\mathstrut 18q^{41} \) \(\mathstrut +\mathstrut 7q^{43} \) \(\mathstrut +\mathstrut 9q^{45} \) \(\mathstrut -\mathstrut 11q^{47} \) \(\mathstrut +\mathstrut 9q^{49} \) \(\mathstrut +\mathstrut q^{51} \) \(\mathstrut -\mathstrut 16q^{53} \) \(\mathstrut +\mathstrut 4q^{55} \) \(\mathstrut -\mathstrut 26q^{57} \) \(\mathstrut -\mathstrut 2q^{59} \) \(\mathstrut +\mathstrut 2q^{61} \) \(\mathstrut +\mathstrut q^{63} \) \(\mathstrut -\mathstrut 4q^{65} \) \(\mathstrut +\mathstrut 13q^{67} \) \(\mathstrut -\mathstrut 3q^{69} \) \(\mathstrut -\mathstrut q^{71} \) \(\mathstrut -\mathstrut 6q^{73} \) \(\mathstrut -\mathstrut 4q^{75} \) \(\mathstrut -\mathstrut 9q^{77} \) \(\mathstrut -\mathstrut 21q^{79} \) \(\mathstrut -\mathstrut 23q^{81} \) \(\mathstrut -\mathstrut 30q^{83} \) \(\mathstrut +\mathstrut q^{85} \) \(\mathstrut +\mathstrut q^{87} \) \(\mathstrut -\mathstrut 36q^{89} \) \(\mathstrut +\mathstrut 9q^{91} \) \(\mathstrut -\mathstrut 22q^{93} \) \(\mathstrut -\mathstrut 23q^{95} \) \(\mathstrut -\mathstrut 5q^{97} \) \(\mathstrut -\mathstrut q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9}\mathstrut -\mathstrut \) \(2\) \(x^{8}\mathstrut -\mathstrut \) \(12\) \(x^{7}\mathstrut +\mathstrut \) \(20\) \(x^{6}\mathstrut +\mathstrut \) \(47\) \(x^{5}\mathstrut -\mathstrut \) \(55\) \(x^{4}\mathstrut -\mathstrut \) \(68\) \(x^{3}\mathstrut +\mathstrut \) \(37\) \(x^{2}\mathstrut +\mathstrut \) \(21\) \(x\mathstrut -\mathstrut \) \(9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{8} + 4 \nu^{7} + 11 \nu^{6} - 42 \nu^{5} - 40 \nu^{4} + 114 \nu^{3} + 57 \nu^{2} - 46 \nu - 6 \)\()/7\)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{8} + \nu^{7} + 36 \nu^{6} - 7 \nu^{5} - 220 \nu^{4} - 10 \nu^{3} + 478 \nu^{2} + 97 \nu - 180 \)\()/21\)
\(\beta_{5}\)\(=\)\((\)\( -2 \nu^{8} + \nu^{7} + 29 \nu^{6} - 7 \nu^{5} - 136 \nu^{4} + 4 \nu^{3} + 212 \nu^{2} + 27 \nu - 47 \)\()/7\)
\(\beta_{6}\)\(=\)\((\)\( 2 \nu^{8} - \nu^{7} - 29 \nu^{6} + 7 \nu^{5} + 136 \nu^{4} + 3 \nu^{3} - 219 \nu^{2} - 62 \nu + 61 \)\()/7\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{8} + 2 \nu^{7} + 12 \nu^{6} - 20 \nu^{5} - 47 \nu^{4} + 55 \nu^{3} + 68 \nu^{2} - 34 \nu - 18 \)\()/3\)
\(\beta_{8}\)\(=\)\((\)\( -6 \nu^{8} + 10 \nu^{7} + 73 \nu^{6} - 91 \nu^{5} - 289 \nu^{4} + 194 \nu^{3} + 412 \nu^{2} - 10 \nu - 99 \)\()/7\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(-\)\(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(8\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(15\)
\(\nu^{5}\)\(=\)\(\beta_{8}\mathstrut -\mathstrut \) \(3\) \(\beta_{7}\mathstrut +\mathstrut \) \(11\) \(\beta_{6}\mathstrut +\mathstrut \) \(11\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(12\) \(\beta_{2}\mathstrut +\mathstrut \) \(29\) \(\beta_{1}\mathstrut +\mathstrut \) \(11\)
\(\nu^{6}\)\(=\)\(-\)\(12\) \(\beta_{7}\mathstrut +\mathstrut \) \(14\) \(\beta_{6}\mathstrut +\mathstrut \) \(25\) \(\beta_{5}\mathstrut -\mathstrut \) \(9\) \(\beta_{4}\mathstrut +\mathstrut \) \(12\) \(\beta_{3}\mathstrut +\mathstrut \) \(60\) \(\beta_{2}\mathstrut +\mathstrut \) \(12\) \(\beta_{1}\mathstrut +\mathstrut \) \(87\)
\(\nu^{7}\)\(=\)\(11\) \(\beta_{8}\mathstrut -\mathstrut \) \(37\) \(\beta_{7}\mathstrut +\mathstrut \) \(95\) \(\beta_{6}\mathstrut +\mathstrut \) \(97\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(17\) \(\beta_{3}\mathstrut +\mathstrut \) \(110\) \(\beta_{2}\mathstrut +\mathstrut \) \(180\) \(\beta_{1}\mathstrut +\mathstrut \) \(93\)
\(\nu^{8}\)\(=\)\(2\) \(\beta_{8}\mathstrut -\mathstrut \) \(114\) \(\beta_{7}\mathstrut +\mathstrut \) \(146\) \(\beta_{6}\mathstrut +\mathstrut \) \(235\) \(\beta_{5}\mathstrut -\mathstrut \) \(63\) \(\beta_{4}\mathstrut +\mathstrut \) \(111\) \(\beta_{3}\mathstrut +\mathstrut \) \(447\) \(\beta_{2}\mathstrut +\mathstrut \) \(118\) \(\beta_{1}\mathstrut +\mathstrut \) \(546\)

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
2.76592
2.33191
1.93909
0.559729
0.390946
−0.706153
−1.07573
−1.79806
−2.40766
0 −2.76592 0 1.47621 0 1.00000 0 4.65032 0
1.2 0 −2.33191 0 −2.59697 0 1.00000 0 2.43780 0
1.3 0 −1.93909 0 2.25153 0 1.00000 0 0.760073 0
1.4 0 −0.559729 0 0.0677230 0 1.00000 0 −2.68670 0
1.5 0 −0.390946 0 −3.55681 0 1.00000 0 −2.84716 0
1.6 0 0.706153 0 −2.65916 0 1.00000 0 −2.50135 0
1.7 0 1.07573 0 2.25022 0 1.00000 0 −1.84282 0
1.8 0 1.79806 0 0.876486 0 1.00000 0 0.233011 0
1.9 0 2.40766 0 −2.10923 0 1.00000 0 2.79683 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
\(2\) \(1\)
\(7\) \(-1\)
\(11\) \(1\)
\(13\) \(-1\)

Hecke kernels

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

\(T_{3}^{9} + \cdots\)
\(T_{5}^{9} + \cdots\)
\(T_{17}^{9} + \cdots\)