Properties

Label 8022.2.a.r
Level 8022
Weight 2
Character orbit 8022.a
Self dual Yes
Analytic conductor 64.056
Analytic rank 0
Dimension 10
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: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10}\mathstrut -\mathstrut \) \(2\) \(x^{9}\mathstrut -\mathstrut \) \(19\) \(x^{8}\mathstrut +\mathstrut \) \(28\) \(x^{7}\mathstrut +\mathstrut \) \(114\) \(x^{6}\mathstrut -\mathstrut \) \(110\) \(x^{5}\mathstrut -\mathstrut \) \(282\) \(x^{4}\mathstrut +\mathstrut \) \(149\) \(x^{3}\mathstrut +\mathstrut \) \(285\) \(x^{2}\mathstrut -\mathstrut \) \(49\) \(x\mathstrut -\mathstrut \) \(79\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 229 \nu^{9} - 2973 \nu^{8} + 1250 \nu^{7} + 49952 \nu^{6} - 46396 \nu^{5} - 247296 \nu^{4} + 163808 \nu^{3} + 461973 \nu^{2} - 136188 \nu - 238207 \)\()/18602\)
\(\beta_{3}\)\(=\)\((\)\( -489 \nu^{9} - 2587 \nu^{8} + 13902 \nu^{7} + 49542 \nu^{6} - 94908 \nu^{5} - 251022 \nu^{4} + 128906 \nu^{3} + 341569 \nu^{2} + 57434 \nu - 28117 \)\()/18602\)
\(\beta_{4}\)\(=\)\((\)\( -1035 \nu^{9} + 4339 \nu^{8} + 15958 \nu^{7} - 70776 \nu^{6} - 71692 \nu^{5} + 337626 \nu^{4} + 135320 \nu^{3} - 558531 \nu^{2} - 94928 \nu + 234811 \)\()/18602\)
\(\beta_{5}\)\(=\)\((\)\( 1237 \nu^{9} - 5743 \nu^{8} - 15424 \nu^{7} + 86773 \nu^{6} + 35234 \nu^{5} - 353133 \nu^{4} - 1142 \nu^{3} + 450947 \nu^{2} - 9444 \nu - 119763 \)\()/9301\)
\(\beta_{6}\)\(=\)\((\)\( 1517 \nu^{9} - 5479 \nu^{8} - 23156 \nu^{7} + 84083 \nu^{6} + 95113 \nu^{5} - 362380 \nu^{4} - 138938 \nu^{3} + 507824 \nu^{2} + 75368 \nu - 159203 \)\()/9301\)
\(\beta_{7}\)\(=\)\((\)\( 1818 \nu^{9} - 3335 \nu^{8} - 32398 \nu^{7} + 41662 \nu^{6} + 164831 \nu^{5} - 130262 \nu^{4} - 280558 \nu^{3} + 129262 \nu^{2} + 139568 \nu - 34183 \)\()/9301\)
\(\beta_{8}\)\(=\)\((\)\( 5003 \nu^{9} - 9633 \nu^{8} - 87796 \nu^{7} + 120350 \nu^{6} + 435548 \nu^{5} - 364498 \nu^{4} - 699314 \nu^{3} + 298399 \nu^{2} + 225360 \nu - 17763 \)\()/18602\)
\(\beta_{9}\)\(=\)\((\)\( 5605 \nu^{9} - 5345 \nu^{8} - 106280 \nu^{7} + 35508 \nu^{6} + 574984 \nu^{5} + 99738 \nu^{4} - 982554 \nu^{3} - 440123 \nu^{2} + 353760 \nu + 157869 \)\()/18602\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(3\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(9\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(12\) \(\beta_{9}\mathstrut -\mathstrut \) \(10\) \(\beta_{8}\mathstrut -\mathstrut \) \(14\) \(\beta_{7}\mathstrut +\mathstrut \) \(14\) \(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(3\) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\mathstrut +\mathstrut \) \(29\)
\(\nu^{5}\)\(=\)\(21\) \(\beta_{9}\mathstrut +\mathstrut \) \(9\) \(\beta_{8}\mathstrut -\mathstrut \) \(47\) \(\beta_{7}\mathstrut +\mathstrut \) \(22\) \(\beta_{6}\mathstrut -\mathstrut \) \(19\) \(\beta_{5}\mathstrut +\mathstrut \) \(5\) \(\beta_{4}\mathstrut +\mathstrut \) \(22\) \(\beta_{3}\mathstrut +\mathstrut \) \(19\) \(\beta_{2}\mathstrut +\mathstrut \) \(96\) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{6}\)\(=\)\(148\) \(\beta_{9}\mathstrut -\mathstrut \) \(100\) \(\beta_{8}\mathstrut -\mathstrut \) \(190\) \(\beta_{7}\mathstrut +\mathstrut \) \(184\) \(\beta_{6}\mathstrut -\mathstrut \) \(49\) \(\beta_{5}\mathstrut +\mathstrut \) \(57\) \(\beta_{4}\mathstrut +\mathstrut \) \(50\) \(\beta_{3}\mathstrut +\mathstrut \) \(35\) \(\beta_{2}\mathstrut +\mathstrut \) \(100\) \(\beta_{1}\mathstrut +\mathstrut \) \(273\)
\(\nu^{7}\)\(=\)\(346\) \(\beta_{9}\mathstrut +\mathstrut \) \(57\) \(\beta_{8}\mathstrut -\mathstrut \) \(673\) \(\beta_{7}\mathstrut +\mathstrut \) \(389\) \(\beta_{6}\mathstrut -\mathstrut \) \(291\) \(\beta_{5}\mathstrut +\mathstrut \) \(133\) \(\beta_{4}\mathstrut +\mathstrut \) \(341\) \(\beta_{3}\mathstrut +\mathstrut \) \(291\) \(\beta_{2}\mathstrut +\mathstrut \) \(1137\) \(\beta_{1}\mathstrut +\mathstrut \) \(119\)
\(\nu^{8}\)\(=\)\(1929\) \(\beta_{9}\mathstrut -\mathstrut \) \(1061\) \(\beta_{8}\mathstrut -\mathstrut \) \(2638\) \(\beta_{7}\mathstrut +\mathstrut \) \(2442\) \(\beta_{6}\mathstrut -\mathstrut \) \(863\) \(\beta_{5}\mathstrut +\mathstrut \) \(873\) \(\beta_{4}\mathstrut +\mathstrut \) \(906\) \(\beta_{3}\mathstrut +\mathstrut \) \(701\) \(\beta_{2}\mathstrut +\mathstrut \) \(1880\) \(\beta_{1}\mathstrut +\mathstrut \) \(2934\)
\(\nu^{9}\)\(=\)\(5352\) \(\beta_{9}\mathstrut +\mathstrut \) \(54\) \(\beta_{8}\mathstrut -\mathstrut \) \(9607\) \(\beta_{7}\mathstrut +\mathstrut \) \(6287\) \(\beta_{6}\mathstrut -\mathstrut \) \(4221\) \(\beta_{5}\mathstrut +\mathstrut \) \(2427\) \(\beta_{4}\mathstrut +\mathstrut \) \(4896\) \(\beta_{3}\mathstrut +\mathstrut \) \(4173\) \(\beta_{2}\mathstrut +\mathstrut \) \(14314\) \(\beta_{1}\mathstrut +\mathstrut \) \(2787\)

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
3.78925
2.49471
1.57301
1.48373
0.667588
−0.632147
−1.09237
−1.38859
−1.81749
−3.07769
1.00000 −1.00000 1.00000 −2.78925 −1.00000 1.00000 1.00000 1.00000 −2.78925
1.2 1.00000 −1.00000 1.00000 −1.49471 −1.00000 1.00000 1.00000 1.00000 −1.49471
1.3 1.00000 −1.00000 1.00000 −0.573010 −1.00000 1.00000 1.00000 1.00000 −0.573010
1.4 1.00000 −1.00000 1.00000 −0.483732 −1.00000 1.00000 1.00000 1.00000 −0.483732
1.5 1.00000 −1.00000 1.00000 0.332412 −1.00000 1.00000 1.00000 1.00000 0.332412
1.6 1.00000 −1.00000 1.00000 1.63215 −1.00000 1.00000 1.00000 1.00000 1.63215
1.7 1.00000 −1.00000 1.00000 2.09237 −1.00000 1.00000 1.00000 1.00000 2.09237
1.8 1.00000 −1.00000 1.00000 2.38859 −1.00000 1.00000 1.00000 1.00000 2.38859
1.9 1.00000 −1.00000 1.00000 2.81749 −1.00000 1.00000 1.00000 1.00000 2.81749
1.10 1.00000 −1.00000 1.00000 4.07769 −1.00000 1.00000 1.00000 1.00000 4.07769
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8022))\):

\(T_{5}^{10} - \cdots\)
\(T_{11}^{10} - \cdots\)
\(T_{13}^{10} - \cdots\)