Properties

Label 8022.2.a.q
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: \( 1 \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9}\mathstrut -\mathstrut \) \(4\) \(x^{8}\mathstrut -\mathstrut \) \(11\) \(x^{7}\mathstrut +\mathstrut \) \(36\) \(x^{6}\mathstrut +\mathstrut \) \(50\) \(x^{5}\mathstrut -\mathstrut \) \(70\) \(x^{4}\mathstrut -\mathstrut \) \(73\) \(x^{3}\mathstrut +\mathstrut \) \(14\) \(x^{2}\mathstrut +\mathstrut \) \(22\) \(x\mathstrut +\mathstrut \) \(4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -38 \nu^{8} + 156 \nu^{7} + 422 \nu^{6} - 1489 \nu^{5} - 1973 \nu^{4} + 3562 \nu^{3} + 3221 \nu^{2} - 2178 \nu - 1056 \)\()/194\)
\(\beta_{3}\)\(=\)\((\)\( 112 \nu^{8} - 470 \nu^{7} - 1157 \nu^{6} + 4358 \nu^{5} + 4789 \nu^{4} - 9503 \nu^{3} - 6221 \nu^{2} + 3734 \nu + 1734 \)\()/194\)
\(\beta_{4}\)\(=\)\((\)\( -121 \nu^{8} + 558 \nu^{7} + 1017 \nu^{6} - 5091 \nu^{5} - 3181 \nu^{4} + 11286 \nu^{3} + 2892 \nu^{2} - 4694 \nu - 912 \)\()/194\)
\(\beta_{5}\)\(=\)\((\)\( -218 \nu^{8} + 946 \nu^{7} + 2084 \nu^{6} - 8583 \nu^{5} - 8031 \nu^{4} + 18076 \nu^{3} + 9973 \nu^{2} - 6052 \nu - 2852 \)\()/194\)
\(\beta_{6}\)\(=\)\((\)\( -231 \nu^{8} + 1030 \nu^{7} + 2065 \nu^{6} - 9243 \nu^{5} - 7325 \nu^{4} + 19412 \nu^{3} + 8290 \nu^{2} - 6986 \nu - 2570 \)\()/194\)
\(\beta_{7}\)\(=\)\((\)\( 257 \nu^{8} - 1101 \nu^{7} - 2512 \nu^{6} + 9981 \nu^{5} + 9890 \nu^{4} - 20920 \nu^{3} - 12102 \nu^{2} + 7690 \nu + 3170 \)\()/194\)
\(\beta_{8}\)\(=\)\((\)\( 376 \nu^{8} - 1661 \nu^{7} - 3420 \nu^{6} + 14866 \nu^{5} + 12426 \nu^{4} - 30829 \nu^{3} - 13977 \nu^{2} + 10360 \nu + 3424 \)\()/194\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(3\) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(2\) \(\beta_{8}\mathstrut -\mathstrut \) \(4\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(4\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(15\) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{4}\)\(=\)\(13\) \(\beta_{8}\mathstrut -\mathstrut \) \(21\) \(\beta_{7}\mathstrut +\mathstrut \) \(6\) \(\beta_{6}\mathstrut +\mathstrut \) \(5\) \(\beta_{4}\mathstrut +\mathstrut \) \(23\) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(55\) \(\beta_{1}\mathstrut +\mathstrut \) \(22\)
\(\nu^{5}\)\(=\)\(42\) \(\beta_{8}\mathstrut -\mathstrut \) \(86\) \(\beta_{7}\mathstrut +\mathstrut \) \(7\) \(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(25\) \(\beta_{4}\mathstrut +\mathstrut \) \(100\) \(\beta_{3}\mathstrut +\mathstrut \) \(18\) \(\beta_{2}\mathstrut +\mathstrut \) \(238\) \(\beta_{1}\mathstrut +\mathstrut \) \(49\)
\(\nu^{6}\)\(=\)\(196\) \(\beta_{8}\mathstrut -\mathstrut \) \(382\) \(\beta_{7}\mathstrut +\mathstrut \) \(20\) \(\beta_{6}\mathstrut +\mathstrut \) \(25\) \(\beta_{5}\mathstrut +\mathstrut \) \(118\) \(\beta_{4}\mathstrut +\mathstrut \) \(458\) \(\beta_{3}\mathstrut +\mathstrut \) \(65\) \(\beta_{2}\mathstrut +\mathstrut \) \(945\) \(\beta_{1}\mathstrut +\mathstrut \) \(230\)
\(\nu^{7}\)\(=\)\(749\) \(\beta_{8}\mathstrut -\mathstrut \) \(1589\) \(\beta_{7}\mathstrut -\mathstrut \) \(36\) \(\beta_{6}\mathstrut +\mathstrut \) \(121\) \(\beta_{5}\mathstrut +\mathstrut \) \(523\) \(\beta_{4}\mathstrut +\mathstrut \) \(1973\) \(\beta_{3}\mathstrut +\mathstrut \) \(339\) \(\beta_{2}\mathstrut +\mathstrut \) \(3952\) \(\beta_{1}\mathstrut +\mathstrut \) \(716\)
\(\nu^{8}\)\(=\)\(3203\) \(\beta_{8}\mathstrut -\mathstrut \) \(6765\) \(\beta_{7}\mathstrut -\mathstrut \) \(333\) \(\beta_{6}\mathstrut +\mathstrut \) \(759\) \(\beta_{5}\mathstrut +\mathstrut \) \(2312\) \(\beta_{4}\mathstrut +\mathstrut \) \(8533\) \(\beta_{3}\mathstrut +\mathstrut \) \(1393\) \(\beta_{2}\mathstrut +\mathstrut \) \(16140\) \(\beta_{1}\mathstrut +\mathstrut \) \(2939\)

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
3.07793
0.643056
−0.290914
−2.20868
4.15530
−0.411372
1.48321
−1.73248
−0.716055
1.00000 −1.00000 1.00000 −3.16835 −1.00000 −1.00000 1.00000 1.00000 −3.16835
1.2 1.00000 −1.00000 1.00000 −2.31300 −1.00000 −1.00000 1.00000 1.00000 −2.31300
1.3 1.00000 −1.00000 1.00000 −0.279858 −1.00000 −1.00000 1.00000 1.00000 −0.279858
1.4 1.00000 −1.00000 1.00000 0.330054 −1.00000 −1.00000 1.00000 1.00000 0.330054
1.5 1.00000 −1.00000 1.00000 0.491351 −1.00000 −1.00000 1.00000 1.00000 0.491351
1.6 1.00000 −1.00000 1.00000 1.51454 −1.00000 −1.00000 1.00000 1.00000 1.51454
1.7 1.00000 −1.00000 1.00000 2.29662 −1.00000 −1.00000 1.00000 1.00000 2.29662
1.8 1.00000 −1.00000 1.00000 2.78783 −1.00000 −1.00000 1.00000 1.00000 2.78783
1.9 1.00000 −1.00000 1.00000 4.34081 −1.00000 −1.00000 1.00000 1.00000 4.34081
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8022))\):

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