Properties

Label 4002.2.a.bk
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut -\mathstrut \) \(2\) \(x^{7}\mathstrut -\mathstrut \) \(30\) \(x^{6}\mathstrut +\mathstrut \) \(52\) \(x^{5}\mathstrut +\mathstrut \) \(267\) \(x^{4}\mathstrut -\mathstrut \) \(352\) \(x^{3}\mathstrut -\mathstrut \) \(632\) \(x^{2}\mathstrut +\mathstrut \) \(240\) \(x\mathstrut +\mathstrut \) \(288\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 73 \nu^{7} - 1010 \nu^{6} - 2490 \nu^{5} + 23896 \nu^{4} + 27735 \nu^{3} - 142756 \nu^{2} - 87080 \nu + 110232 \)\()/21048\)
\(\beta_{3}\)\(=\)\((\)\( 45 \nu^{7} - 94 \nu^{6} - 718 \nu^{5} + 2092 \nu^{4} + 1383 \nu^{3} - 14092 \nu^{2} + 1968 \nu + 26528 \)\()/7016\)
\(\beta_{4}\)\(=\)\((\)\( 101 \nu^{7} - 172 \nu^{6} - 2508 \nu^{5} + 1850 \nu^{4} + 17253 \nu^{3} + 9220 \nu^{2} - 28792 \nu - 30576 \)\()/10524\)
\(\beta_{5}\)\(=\)\((\)\( -68 \nu^{7} + 220 \nu^{6} + 1923 \nu^{5} - 6221 \nu^{4} - 13257 \nu^{3} + 47741 \nu^{2} - 986 \nu - 51936 \)\()/5262\)
\(\beta_{6}\)\(=\)\((\)\( 407 \nu^{7} - 1162 \nu^{6} - 9846 \nu^{5} + 31160 \nu^{4} + 57177 \nu^{3} - 212192 \nu^{2} + 9848 \nu + 97896 \)\()/21048\)
\(\beta_{7}\)\(=\)\((\)\( -269 \nu^{7} + 406 \nu^{6} + 7878 \nu^{5} - 11648 \nu^{4} - 64863 \nu^{3} + 89228 \nu^{2} + 110548 \nu - 71736 \)\()/10524\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(9\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(12\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(-\)\(\beta_{7}\mathstrut +\mathstrut \) \(16\) \(\beta_{6}\mathstrut +\mathstrut \) \(16\) \(\beta_{5}\mathstrut -\mathstrut \) \(4\) \(\beta_{4}\mathstrut -\mathstrut \) \(14\) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(118\)
\(\nu^{5}\)\(=\)\(-\)\(14\) \(\beta_{7}\mathstrut +\mathstrut \) \(5\) \(\beta_{6}\mathstrut +\mathstrut \) \(37\) \(\beta_{5}\mathstrut -\mathstrut \) \(19\) \(\beta_{4}\mathstrut +\mathstrut \) \(24\) \(\beta_{3}\mathstrut +\mathstrut \) \(15\) \(\beta_{2}\mathstrut +\mathstrut \) \(155\) \(\beta_{1}\mathstrut +\mathstrut \) \(22\)
\(\nu^{6}\)\(=\)\(-\)\(32\) \(\beta_{7}\mathstrut +\mathstrut \) \(235\) \(\beta_{6}\mathstrut +\mathstrut \) \(245\) \(\beta_{5}\mathstrut -\mathstrut \) \(96\) \(\beta_{4}\mathstrut -\mathstrut \) \(189\) \(\beta_{3}\mathstrut -\mathstrut \) \(18\) \(\beta_{2}\mathstrut -\mathstrut \) \(12\) \(\beta_{1}\mathstrut +\mathstrut \) \(1637\)
\(\nu^{7}\)\(=\)\(-\)\(213\) \(\beta_{7}\mathstrut +\mathstrut \) \(140\) \(\beta_{6}\mathstrut +\mathstrut \) \(610\) \(\beta_{5}\mathstrut -\mathstrut \) \(287\) \(\beta_{4}\mathstrut +\mathstrut \) \(451\) \(\beta_{3}\mathstrut +\mathstrut \) \(171\) \(\beta_{2}\mathstrut +\mathstrut \) \(2082\) \(\beta_{1}\mathstrut +\mathstrut \) \(483\)

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
−3.78725
−3.42323
−1.12580
−0.652958
0.815784
2.56063
3.65713
3.95569
1.00000 −1.00000 1.00000 −3.78725 −1.00000 3.57193 1.00000 1.00000 −3.78725
1.2 1.00000 −1.00000 1.00000 −3.42323 −1.00000 −1.25167 1.00000 1.00000 −3.42323
1.3 1.00000 −1.00000 1.00000 −1.12580 −1.00000 −0.350474 1.00000 1.00000 −1.12580
1.4 1.00000 −1.00000 1.00000 −0.652958 −1.00000 −3.89658 1.00000 1.00000 −0.652958
1.5 1.00000 −1.00000 1.00000 0.815784 −1.00000 2.48945 1.00000 1.00000 0.815784
1.6 1.00000 −1.00000 1.00000 2.56063 −1.00000 3.94138 1.00000 1.00000 2.56063
1.7 1.00000 −1.00000 1.00000 3.65713 −1.00000 −4.08293 1.00000 1.00000 3.65713
1.8 1.00000 −1.00000 1.00000 3.95569 −1.00000 4.57890 1.00000 1.00000 3.95569
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4002))\):

\(T_{5}^{8} - \cdots\)
\(T_{7}^{8} - \cdots\)
\(T_{11}^{8} - \cdots\)