Properties

Label 8030.2.a.bc
Level 8030
Weight 2
Character orbit 8030.a
Self dual Yes
Analytic conductor 64.120
Analytic rank 1
Dimension 11
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8030 = 2 \cdot 5 \cdot 11 \cdot 73 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8030.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.1198728231\)
Analytic rank: \(1\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{11}\mathstrut -\mathstrut \) \(5\) \(x^{10}\mathstrut -\mathstrut \) \(10\) \(x^{9}\mathstrut +\mathstrut \) \(71\) \(x^{8}\mathstrut +\mathstrut \) \(28\) \(x^{7}\mathstrut -\mathstrut \) \(360\) \(x^{6}\mathstrut -\mathstrut \) \(60\) \(x^{5}\mathstrut +\mathstrut \) \(788\) \(x^{4}\mathstrut +\mathstrut \) \(309\) \(x^{3}\mathstrut -\mathstrut \) \(632\) \(x^{2}\mathstrut -\mathstrut \) \(503\) \(x\mathstrut -\mathstrut \) \(95\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(\beta_{3}\)\(=\)\((\)\( -4 \nu^{10} + 17 \nu^{9} + 33 \nu^{8} - 200 \nu^{7} - 25 \nu^{6} + 730 \nu^{5} - 121 \nu^{4} - 932 \nu^{3} - 434 \nu^{2} + 583 \nu + 573 \)\()/79\)
\(\beta_{4}\)\(=\)\((\)\( -9 \nu^{10} + 58 \nu^{9} + 15 \nu^{8} - 687 \nu^{7} + 635 \nu^{6} + 2630 \nu^{5} - 2820 \nu^{4} - 4072 \nu^{3} + 2144 \nu^{2} + 3346 \nu + 993 \)\()/79\)
\(\beta_{5}\)\(=\)\((\)\( 35 \nu^{10} - 129 \nu^{9} - 427 \nu^{8} + 1671 \nu^{7} + 1779 \nu^{6} - 6980 \nu^{5} - 3385 \nu^{4} + 10051 \nu^{3} + 3521 \nu^{2} - 3225 \nu - 649 \)\()/158\)
\(\beta_{6}\)\(=\)\((\)\( 19 \nu^{10} - 140 \nu^{9} + 21 \nu^{8} + 1740 \nu^{7} - 2113 \nu^{6} - 7299 \nu^{5} + 9640 \nu^{4} + 12959 \nu^{3} - 10302 \nu^{2} - 10531 \nu - 1438 \)\()/158\)
\(\beta_{7}\)\(=\)\((\)\( 39 \nu^{10} - 225 \nu^{9} - 223 \nu^{8} + 2977 \nu^{7} - 1119 \nu^{6} - 13714 \nu^{5} + 7717 \nu^{4} + 27257 \nu^{3} - 6789 \nu^{2} - 22531 \nu - 6199 \)\()/158\)
\(\beta_{8}\)\(=\)\((\)\( 41 \nu^{10} - 273 \nu^{9} - 121 \nu^{8} + 3551 \nu^{7} - 2489 \nu^{6} - 15896 \nu^{5} + 12557 \nu^{4} + 30409 \nu^{3} - 11075 \nu^{2} - 24995 \nu - 6209 \)\()/158\)
\(\beta_{9}\)\(=\)\((\)\( 44 \nu^{10} - 266 \nu^{9} - 205 \nu^{8} + 3464 \nu^{7} - 1779 \nu^{6} - 15535 \nu^{5} + 10337 \nu^{4} + 29607 \nu^{3} - 9051 \nu^{2} - 23556 \nu - 6066 \)\()/79\)
\(\beta_{10}\)\(=\)\((\)\( 44 \nu^{10} - 266 \nu^{9} - 205 \nu^{8} + 3464 \nu^{7} - 1779 \nu^{6} - 15535 \nu^{5} + 10337 \nu^{4} + 29686 \nu^{3} - 9051 \nu^{2} - 24030 \nu - 6303 \)\()/79\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(6\) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{4}\)\(=\)\(2\) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(9\) \(\beta_{1}\mathstrut +\mathstrut \) \(25\)
\(\nu^{5}\)\(=\)\(12\) \(\beta_{10}\mathstrut -\mathstrut \) \(10\) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(3\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(3\) \(\beta_{2}\mathstrut +\mathstrut \) \(43\) \(\beta_{1}\mathstrut +\mathstrut \) \(32\)
\(\nu^{6}\)\(=\)\(30\) \(\beta_{10}\mathstrut -\mathstrut \) \(13\) \(\beta_{9}\mathstrut -\mathstrut \) \(15\) \(\beta_{8}\mathstrut -\mathstrut \) \(18\) \(\beta_{7}\mathstrut -\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(11\) \(\beta_{3}\mathstrut +\mathstrut \) \(48\) \(\beta_{2}\mathstrut +\mathstrut \) \(80\) \(\beta_{1}\mathstrut +\mathstrut \) \(177\)
\(\nu^{7}\)\(=\)\(123\) \(\beta_{10}\mathstrut -\mathstrut \) \(85\) \(\beta_{9}\mathstrut -\mathstrut \) \(23\) \(\beta_{8}\mathstrut -\mathstrut \) \(51\) \(\beta_{7}\mathstrut -\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(16\) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(41\) \(\beta_{2}\mathstrut +\mathstrut \) \(332\) \(\beta_{1}\mathstrut +\mathstrut \) \(304\)
\(\nu^{8}\)\(=\)\(337\) \(\beta_{10}\mathstrut -\mathstrut \) \(138\) \(\beta_{9}\mathstrut -\mathstrut \) \(167\) \(\beta_{8}\mathstrut -\mathstrut \) \(216\) \(\beta_{7}\mathstrut -\mathstrut \) \(26\) \(\beta_{6}\mathstrut -\mathstrut \) \(17\) \(\beta_{5}\mathstrut +\mathstrut \) \(24\) \(\beta_{4}\mathstrut +\mathstrut \) \(90\) \(\beta_{3}\mathstrut +\mathstrut \) \(339\) \(\beta_{2}\mathstrut +\mathstrut \) \(717\) \(\beta_{1}\mathstrut +\mathstrut \) \(1349\)
\(\nu^{9}\)\(=\)\(1195\) \(\beta_{10}\mathstrut -\mathstrut \) \(708\) \(\beta_{9}\mathstrut -\mathstrut \) \(331\) \(\beta_{8}\mathstrut -\mathstrut \) \(609\) \(\beta_{7}\mathstrut -\mathstrut \) \(32\) \(\beta_{6}\mathstrut -\mathstrut \) \(40\) \(\beta_{5}\mathstrut +\mathstrut \) \(183\) \(\beta_{4}\mathstrut +\mathstrut \) \(29\) \(\beta_{3}\mathstrut +\mathstrut \) \(424\) \(\beta_{2}\mathstrut +\mathstrut \) \(2685\) \(\beta_{1}\mathstrut +\mathstrut \) \(2808\)
\(\nu^{10}\)\(=\)\(3418\) \(\beta_{10}\mathstrut -\mathstrut \) \(1378\) \(\beta_{9}\mathstrut -\mathstrut \) \(1693\) \(\beta_{8}\mathstrut -\mathstrut \) \(2225\) \(\beta_{7}\mathstrut -\mathstrut \) \(238\) \(\beta_{6}\mathstrut -\mathstrut \) \(204\) \(\beta_{5}\mathstrut +\mathstrut \) \(352\) \(\beta_{4}\mathstrut +\mathstrut \) \(647\) \(\beta_{3}\mathstrut +\mathstrut \) \(2476\) \(\beta_{2}\mathstrut +\mathstrut \) \(6441\) \(\beta_{1}\mathstrut +\mathstrut \) \(10851\)

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
3.03286
2.97446
2.54167
2.29138
1.62999
−0.328152
−0.722992
−0.858907
−1.10547
−2.03519
−2.41966
−1.00000 −3.03286 1.00000 1.00000 3.03286 −1.46211 −1.00000 6.19822 −1.00000
1.2 −1.00000 −2.97446 1.00000 1.00000 2.97446 −1.89665 −1.00000 5.84742 −1.00000
1.3 −1.00000 −2.54167 1.00000 1.00000 2.54167 1.74685 −1.00000 3.46009 −1.00000
1.4 −1.00000 −2.29138 1.00000 1.00000 2.29138 3.99723 −1.00000 2.25044 −1.00000
1.5 −1.00000 −1.62999 1.00000 1.00000 1.62999 −2.80536 −1.00000 −0.343117 −1.00000
1.6 −1.00000 0.328152 1.00000 1.00000 −0.328152 −2.28236 −1.00000 −2.89232 −1.00000
1.7 −1.00000 0.722992 1.00000 1.00000 −0.722992 3.72469 −1.00000 −2.47728 −1.00000
1.8 −1.00000 0.858907 1.00000 1.00000 −0.858907 0.0500818 −1.00000 −2.26228 −1.00000
1.9 −1.00000 1.10547 1.00000 1.00000 −1.10547 −0.823192 −1.00000 −1.77794 −1.00000
1.10 −1.00000 2.03519 1.00000 1.00000 −2.03519 −2.33086 −1.00000 1.14199 −1.00000
1.11 −1.00000 2.41966 1.00000 1.00000 −2.41966 1.08168 −1.00000 2.85477 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(11\) \(-1\)
\(73\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8030))\):

\(T_{3}^{11} + \cdots\)
\(T_{7}^{11} + \cdots\)