Properties

Label 8022.2.a.p
Level 8022
Weight 2
Character orbit 8022.a
Self dual Yes
Analytic conductor 64.056
Analytic rank 1
Dimension 9
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8022 = 2 \cdot 3 \cdot 7 \cdot 191 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8022.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.0559925015\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9}\mathstrut -\mathstrut \) \(17\) \(x^{7}\mathstrut -\mathstrut \) \(12\) \(x^{6}\mathstrut +\mathstrut \) \(72\) \(x^{5}\mathstrut +\mathstrut \) \(81\) \(x^{4}\mathstrut -\mathstrut \) \(67\) \(x^{3}\mathstrut -\mathstrut \) \(105\) \(x^{2}\mathstrut -\mathstrut \) \(17\) \(x\mathstrut +\mathstrut \) \(6\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -96 \nu^{8} + 151 \nu^{7} + 1380 \nu^{6} - 1032 \nu^{5} - 5048 \nu^{4} + 494 \nu^{3} + 4877 \nu^{2} + 1233 \nu - 177 \)\()/107\)
\(\beta_{2}\)\(=\)\((\)\( -143 \nu^{8} + 177 \nu^{7} + 2176 \nu^{6} - 922 \nu^{5} - 8634 \nu^{4} - 1286 \nu^{3} + 9252 \nu^{2} + 3873 \nu - 481 \)\()/107\)
\(\beta_{3}\)\(=\)\((\)\( -202 \nu^{8} + 262 \nu^{7} + 3091 \nu^{6} - 1583 \nu^{5} - 12423 \nu^{4} - 249 \nu^{3} + 13501 \nu^{2} + 3524 \nu - 874 \)\()/107\)
\(\beta_{4}\)\(=\)\((\)\( 202 \nu^{8} - 262 \nu^{7} - 3091 \nu^{6} + 1583 \nu^{5} + 12423 \nu^{4} + 249 \nu^{3} - 13394 \nu^{2} - 3524 \nu + 446 \)\()/107\)
\(\beta_{5}\)\(=\)\((\)\( 405 \nu^{8} - 520 \nu^{7} - 6183 \nu^{6} + 3043 \nu^{5} + 24747 \nu^{4} + 1146 \nu^{3} - 26831 \nu^{2} - 7753 \nu + 1586 \)\()/107\)
\(\beta_{6}\)\(=\)\((\)\( -439 \nu^{8} + 598 \nu^{7} + 6645 \nu^{6} - 3783 \nu^{5} - 26410 \nu^{4} + 469 \nu^{3} + 28721 \nu^{2} + 6578 \nu - 1963 \)\()/107\)
\(\beta_{7}\)\(=\)\((\)\( -616 \nu^{8} + 853 \nu^{7} + 9283 \nu^{6} - 5445 \nu^{5} - 36707 \nu^{4} + 798 \nu^{3} + 39756 \nu^{2} + 9704 \nu - 2500 \)\()/107\)
\(\beta_{8}\)\(=\)\((\)\( -623 \nu^{8} + 825 \nu^{7} + 9504 \nu^{6} - 5119 \nu^{5} - 38154 \nu^{4} + 114 \nu^{3} + 41876 \nu^{2} + 10038 \nu - 2864 \)\()/107\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7}\mathstrut -\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(1\)\()/4\)
\(\nu^{2}\)\(=\)\(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(8\) \(\beta_{8}\mathstrut +\mathstrut \) \(7\) \(\beta_{7}\mathstrut -\mathstrut \) \(6\) \(\beta_{6}\mathstrut -\mathstrut \) \(11\) \(\beta_{5}\mathstrut +\mathstrut \) \(11\) \(\beta_{4}\mathstrut +\mathstrut \) \(15\) \(\beta_{3}\mathstrut -\mathstrut \) \(11\) \(\beta_{2}\mathstrut -\mathstrut \) \(4\) \(\beta_{1}\mathstrut +\mathstrut \) \(23\)\()/4\)
\(\nu^{4}\)\(=\)\(-\)\(4\) \(\beta_{8}\mathstrut +\mathstrut \) \(2\) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(11\) \(\beta_{4}\mathstrut +\mathstrut \) \(18\) \(\beta_{3}\mathstrut -\mathstrut \) \(3\) \(\beta_{2}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(37\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(124\) \(\beta_{8}\mathstrut +\mathstrut \) \(81\) \(\beta_{7}\mathstrut -\mathstrut \) \(34\) \(\beta_{6}\mathstrut -\mathstrut \) \(113\) \(\beta_{5}\mathstrut +\mathstrut \) \(145\) \(\beta_{4}\mathstrut +\mathstrut \) \(241\) \(\beta_{3}\mathstrut -\mathstrut \) \(117\) \(\beta_{2}\mathstrut -\mathstrut \) \(64\) \(\beta_{1}\mathstrut +\mathstrut \) \(357\)\()/4\)
\(\nu^{6}\)\(=\)\(-\)\(81\) \(\beta_{8}\mathstrut +\mathstrut \) \(44\) \(\beta_{7}\mathstrut -\mathstrut \) \(15\) \(\beta_{6}\mathstrut -\mathstrut \) \(43\) \(\beta_{5}\mathstrut +\mathstrut \) \(141\) \(\beta_{4}\mathstrut +\mathstrut \) \(260\) \(\beta_{3}\mathstrut -\mathstrut \) \(59\) \(\beta_{2}\mathstrut -\mathstrut \) \(32\) \(\beta_{1}\mathstrut +\mathstrut \) \(440\)
\(\nu^{7}\)\(=\)\((\)\(-\)\(1696\) \(\beta_{8}\mathstrut +\mathstrut \) \(1043\) \(\beta_{7}\mathstrut -\mathstrut \) \(326\) \(\beta_{6}\mathstrut -\mathstrut \) \(1275\) \(\beta_{5}\mathstrut +\mathstrut \) \(1999\) \(\beta_{4}\mathstrut +\mathstrut \) \(3607\) \(\beta_{3}\mathstrut -\mathstrut \) \(1403\) \(\beta_{2}\mathstrut -\mathstrut \) \(868\) \(\beta_{1}\mathstrut +\mathstrut \) \(5279\)\()/4\)
\(\nu^{8}\)\(=\)\(-\)\(1298\) \(\beta_{8}\mathstrut +\mathstrut \) \(732\) \(\beta_{7}\mathstrut -\mathstrut \) \(214\) \(\beta_{6}\mathstrut -\mathstrut \) \(728\) \(\beta_{5}\mathstrut +\mathstrut \) \(1913\) \(\beta_{4}\mathstrut +\mathstrut \) \(3635\) \(\beta_{3}\mathstrut -\mathstrut \) \(945\) \(\beta_{2}\mathstrut -\mathstrut \) \(583\) \(\beta_{1}\mathstrut +\mathstrut \) \(5730\)

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
−1.29242
−1.49428
−2.74795
−0.470396
3.74341
−1.37137
0.166868
2.17782
1.28832
1.00000 −1.00000 1.00000 −3.75450 −1.00000 1.00000 1.00000 1.00000 −3.75450
1.2 1.00000 −1.00000 1.00000 −3.72931 −1.00000 1.00000 1.00000 1.00000 −3.72931
1.3 1.00000 −1.00000 1.00000 −2.04654 −1.00000 1.00000 1.00000 1.00000 −2.04654
1.4 1.00000 −1.00000 1.00000 −1.03764 −1.00000 1.00000 1.00000 1.00000 −1.03764
1.5 1.00000 −1.00000 1.00000 −0.149044 −1.00000 1.00000 1.00000 1.00000 −0.149044
1.6 1.00000 −1.00000 1.00000 −0.137007 −1.00000 1.00000 1.00000 1.00000 −0.137007
1.7 1.00000 −1.00000 1.00000 0.787345 −1.00000 1.00000 1.00000 1.00000 0.787345
1.8 1.00000 −1.00000 1.00000 2.12019 −1.00000 1.00000 1.00000 1.00000 2.12019
1.9 1.00000 −1.00000 1.00000 3.94651 −1.00000 1.00000 1.00000 1.00000 3.94651
\(n\): 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
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(-1\)
\(191\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8022))\):

\(T_{5}^{9} + \cdots\)
\(T_{11}^{9} + \cdots\)
\(T_{13}^{9} + \cdots\)