Properties

Label 4002.2.a.bj
Level 4002
Weight 2
Character orbit 4002.a
Self dual Yes
Analytic conductor 31.956
Analytic rank 0
Dimension 8
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4002.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(31.9561308889\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut -\mathstrut \) \(x^{7}\mathstrut -\mathstrut \) \(26\) \(x^{6}\mathstrut +\mathstrut \) \(4\) \(x^{5}\mathstrut +\mathstrut \) \(209\) \(x^{4}\mathstrut +\mathstrut \) \(113\) \(x^{3}\mathstrut -\mathstrut \) \(436\) \(x^{2}\mathstrut -\mathstrut \) \(360\) \(x\mathstrut -\mathstrut \) \(16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 45 \nu^{7} - 144 \nu^{6} - 770 \nu^{5} + 1874 \nu^{4} + 3119 \nu^{3} - 3940 \nu^{2} + 1944 \nu + 1488 \)\()/1664\)
\(\beta_{3}\)\(=\)\((\)\( 89 \nu^{7} - 368 \nu^{6} - 1338 \nu^{5} + 5130 \nu^{4} + 4819 \nu^{3} - 13108 \nu^{2} + 184 \nu - 3824 \)\()/1664\)
\(\beta_{4}\)\(=\)\((\)\( -53 \nu^{7} + 128 \nu^{6} + 1138 \nu^{5} - 1634 \nu^{4} - 7815 \nu^{3} + 2676 \nu^{2} + 15016 \nu + 4016 \)\()/832\)
\(\beta_{5}\)\(=\)\((\)\( 9 \nu^{7} - 34 \nu^{6} - 154 \nu^{5} + 510 \nu^{4} + 759 \nu^{3} - 1698 \nu^{2} - 984 \nu + 464 \)\()/104\)
\(\beta_{6}\)\(=\)\((\)\( -251 \nu^{7} + 720 \nu^{6} + 4942 \nu^{5} - 10046 \nu^{4} - 29193 \nu^{3} + 25628 \nu^{2} + 42072 \nu + 5456 \)\()/1664\)
\(\beta_{7}\)\(=\)\((\)\( 311 \nu^{7} - 912 \nu^{6} - 6246 \nu^{5} + 12822 \nu^{4} + 38621 \nu^{3} - 32268 \nu^{2} - 61112 \nu - 8464 \)\()/1664\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(7\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(3\) \(\beta_{2}\mathstrut +\mathstrut \) \(10\) \(\beta_{1}\mathstrut +\mathstrut \) \(8\)
\(\nu^{4}\)\(=\)\(-\)\(2\) \(\beta_{7}\mathstrut -\mathstrut \) \(17\) \(\beta_{6}\mathstrut -\mathstrut \) \(12\) \(\beta_{5}\mathstrut +\mathstrut \) \(15\) \(\beta_{4}\mathstrut +\mathstrut \) \(12\) \(\beta_{3}\mathstrut -\mathstrut \) \(31\) \(\beta_{2}\mathstrut +\mathstrut \) \(7\) \(\beta_{1}\mathstrut +\mathstrut \) \(82\)
\(\nu^{5}\)\(=\)\(-\)\(27\) \(\beta_{7}\mathstrut -\mathstrut \) \(37\) \(\beta_{6}\mathstrut -\mathstrut \) \(7\) \(\beta_{5}\mathstrut -\mathstrut \) \(9\) \(\beta_{4}\mathstrut +\mathstrut \) \(26\) \(\beta_{3}\mathstrut -\mathstrut \) \(70\) \(\beta_{2}\mathstrut +\mathstrut \) \(119\) \(\beta_{1}\mathstrut +\mathstrut \) \(181\)
\(\nu^{6}\)\(=\)\(-\)\(78\) \(\beta_{7}\mathstrut -\mathstrut \) \(293\) \(\beta_{6}\mathstrut -\mathstrut \) \(157\) \(\beta_{5}\mathstrut +\mathstrut \) \(189\) \(\beta_{4}\mathstrut +\mathstrut \) \(163\) \(\beta_{3}\mathstrut -\mathstrut \) \(470\) \(\beta_{2}\mathstrut +\mathstrut \) \(178\) \(\beta_{1}\mathstrut +\mathstrut \) \(1147\)
\(\nu^{7}\)\(=\)\(-\)\(559\) \(\beta_{7}\mathstrut -\mathstrut \) \(881\) \(\beta_{6}\mathstrut -\mathstrut \) \(210\) \(\beta_{5}\mathstrut -\mathstrut \) \(17\) \(\beta_{4}\mathstrut +\mathstrut \) \(485\) \(\beta_{3}\mathstrut -\mathstrut \) \(1341\) \(\beta_{2}\mathstrut +\mathstrut \) \(1578\) \(\beta_{1}\mathstrut +\mathstrut \) \(3378\)

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
−3.17046
−2.26152
−2.09236
−0.869754
−0.0471692
1.78927
3.49512
4.15688
1.00000 −1.00000 1.00000 −3.17046 −1.00000 −4.05963 1.00000 1.00000 −3.17046
1.2 1.00000 −1.00000 1.00000 −2.26152 −1.00000 3.32565 1.00000 1.00000 −2.26152
1.3 1.00000 −1.00000 1.00000 −2.09236 −1.00000 1.21381 1.00000 1.00000 −2.09236
1.4 1.00000 −1.00000 1.00000 −0.869754 −1.00000 3.97135 1.00000 1.00000 −0.869754
1.5 1.00000 −1.00000 1.00000 −0.0471692 −1.00000 −3.98374 1.00000 1.00000 −0.0471692
1.6 1.00000 −1.00000 1.00000 1.78927 −1.00000 0.126532 1.00000 1.00000 1.78927
1.7 1.00000 −1.00000 1.00000 3.49512 −1.00000 −1.05589 1.00000 1.00000 3.49512
1.8 1.00000 −1.00000 1.00000 4.15688 −1.00000 0.461913 1.00000 1.00000 4.15688
\(n\): 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
\(2\) \(-1\)
\(3\) \(1\)
\(23\) \(-1\)
\(29\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4002))\):

\(T_{5}^{8} - \cdots\)
\(T_{7}^{8} \) \(\mathstrut -\mathstrut 31 T_{7}^{6} \) \(\mathstrut +\mathstrut 10 T_{7}^{5} \) \(\mathstrut +\mathstrut 256 T_{7}^{4} \) \(\mathstrut -\mathstrut 168 T_{7}^{3} \) \(\mathstrut -\mathstrut 248 T_{7}^{2} \) \(\mathstrut +\mathstrut 160 T_{7} \) \(\mathstrut -\mathstrut 16 \)
\(T_{11}^{8} - \cdots\)