Properties

Label 8025.2.a.bf
Level 8025
Weight 2
Character orbit 8025.a
Self dual Yes
Analytic conductor 64.080
Analytic rank 1
Dimension 12
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8025 = 3 \cdot 5^{2} \cdot 107 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8025.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.0799476221\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12}\mathstrut -\mathstrut \) \(3\) \(x^{11}\mathstrut -\mathstrut \) \(15\) \(x^{10}\mathstrut +\mathstrut \) \(49\) \(x^{9}\mathstrut +\mathstrut \) \(71\) \(x^{8}\mathstrut -\mathstrut \) \(278\) \(x^{7}\mathstrut -\mathstrut \) \(92\) \(x^{6}\mathstrut +\mathstrut \) \(649\) \(x^{5}\mathstrut -\mathstrut \) \(127\) \(x^{4}\mathstrut -\mathstrut \) \(529\) \(x^{3}\mathstrut +\mathstrut \) \(267\) \(x^{2}\mathstrut +\mathstrut \) \(15\) \(x\mathstrut -\mathstrut \) \(6\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 5 \nu \)
\(\beta_{4}\)\(=\)\((\)\( \nu^{11} + 24 \nu^{10} - 4 \nu^{9} - 402 \nu^{8} - 101 \nu^{7} + 2336 \nu^{6} + 799 \nu^{5} - 5414 \nu^{4} - 1608 \nu^{3} + 4026 \nu^{2} + 336 \nu - 174 \)\()/49\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{11} + 26 \nu^{10} + 61 \nu^{9} - 411 \nu^{8} - 481 \nu^{7} + 2204 \nu^{6} + 1817 \nu^{5} - 4583 \nu^{4} - 2918 \nu^{3} + 2916 \nu^{2} + 1001 \nu - 66 \)\()/49\)
\(\beta_{6}\)\(=\)\((\)\( 27 \nu^{11} - 38 \nu^{10} - 451 \nu^{9} + 612 \nu^{8} + 2663 \nu^{7} - 3372 \nu^{6} - 6651 \nu^{5} + 7486 \nu^{4} + 6172 \nu^{3} - 5566 \nu^{2} - 581 \nu + 55 \)\()/49\)
\(\beta_{7}\)\(=\)\((\)\( 24 \nu^{11} - 61 \nu^{10} - 390 \nu^{9} + 985 \nu^{8} + 2182 \nu^{7} - 5480 \nu^{6} - 4834 \nu^{5} + 12360 \nu^{4} + 2813 \nu^{3} - 9510 \nu^{2} + 1939 \nu + 283 \)\()/49\)
\(\beta_{8}\)\(=\)\((\)\( 5 \nu^{11} - 6 \nu^{10} - 83 \nu^{9} + 97 \nu^{8} + 482 \nu^{7} - 542 \nu^{6} - 1150 \nu^{5} + 1252 \nu^{4} + 913 \nu^{3} - 1017 \nu^{2} + 91 \nu + 12 \)\()/7\)
\(\beta_{9}\)\(=\)\((\)\( 22 \nu^{11} - 60 \nu^{10} - 382 \nu^{9} + 956 \nu^{8} + 2384 \nu^{7} - 5203 \nu^{6} - 6481 \nu^{5} + 11232 \nu^{4} + 6813 \nu^{3} - 7860 \nu^{2} - 840 \nu + 190 \)\()/49\)
\(\beta_{10}\)\(=\)\((\)\( -5 \nu^{11} + 6 \nu^{10} + 83 \nu^{9} - 97 \nu^{8} - 482 \nu^{7} + 542 \nu^{6} + 1157 \nu^{5} - 1252 \nu^{4} - 976 \nu^{3} + 1024 \nu^{2} + 28 \nu - 33 \)\()/7\)
\(\beta_{11}\)\(=\)\((\)\( -43 \nu^{11} + 95 \nu^{10} + 711 \nu^{9} - 1530 \nu^{8} - 4134 \nu^{7} + 8479 \nu^{6} + 10086 \nu^{5} - 19009 \nu^{4} - 8668 \nu^{3} + 14356 \nu^{2} - 581 \nu - 358 \)\()/49\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(6\) \(\beta_{2}\mathstrut +\mathstrut \) \(16\)
\(\nu^{5}\)\(=\)\(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(9\) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(28\) \(\beta_{1}\)
\(\nu^{6}\)\(=\)\(10\) \(\beta_{11}\mathstrut -\mathstrut \) \(9\) \(\beta_{10}\mathstrut -\mathstrut \) \(9\) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(10\) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(18\) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(37\) \(\beta_{2}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(94\)
\(\nu^{7}\)\(=\)\(-\)\(\beta_{11}\mathstrut +\mathstrut \) \(12\) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(12\) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(68\) \(\beta_{3}\mathstrut -\mathstrut \) \(14\) \(\beta_{2}\mathstrut +\mathstrut \) \(167\) \(\beta_{1}\mathstrut -\mathstrut \) \(5\)
\(\nu^{8}\)\(=\)\(79\) \(\beta_{11}\mathstrut -\mathstrut \) \(67\) \(\beta_{10}\mathstrut -\mathstrut \) \(67\) \(\beta_{9}\mathstrut -\mathstrut \) \(14\) \(\beta_{8}\mathstrut +\mathstrut \) \(79\) \(\beta_{7}\mathstrut +\mathstrut \) \(27\) \(\beta_{6}\mathstrut -\mathstrut \) \(131\) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(28\) \(\beta_{3}\mathstrut +\mathstrut \) \(234\) \(\beta_{2}\mathstrut -\mathstrut \) \(18\) \(\beta_{1}\mathstrut +\mathstrut \) \(577\)
\(\nu^{9}\)\(=\)\(-\)\(18\) \(\beta_{11}\mathstrut +\mathstrut \) \(108\) \(\beta_{10}\mathstrut -\mathstrut \) \(14\) \(\beta_{9}\mathstrut +\mathstrut \) \(104\) \(\beta_{8}\mathstrut -\mathstrut \) \(16\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(10\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(486\) \(\beta_{3}\mathstrut -\mathstrut \) \(140\) \(\beta_{2}\mathstrut +\mathstrut \) \(1035\) \(\beta_{1}\mathstrut -\mathstrut \) \(85\)
\(\nu^{10}\)\(=\)\(577\) \(\beta_{11}\mathstrut -\mathstrut \) \(473\) \(\beta_{10}\mathstrut -\mathstrut \) \(473\) \(\beta_{9}\mathstrut -\mathstrut \) \(136\) \(\beta_{8}\mathstrut +\mathstrut \) \(576\) \(\beta_{7}\mathstrut +\mathstrut \) \(257\) \(\beta_{6}\mathstrut -\mathstrut \) \(897\) \(\beta_{5}\mathstrut -\mathstrut \) \(32\) \(\beta_{4}\mathstrut -\mathstrut \) \(281\) \(\beta_{3}\mathstrut +\mathstrut \) \(1506\) \(\beta_{2}\mathstrut -\mathstrut \) \(214\) \(\beta_{1}\mathstrut +\mathstrut \) \(3634\)
\(\nu^{11}\)\(=\)\(-\)\(209\) \(\beta_{11}\mathstrut +\mathstrut \) \(873\) \(\beta_{10}\mathstrut -\mathstrut \) \(129\) \(\beta_{9}\mathstrut +\mathstrut \) \(801\) \(\beta_{8}\mathstrut -\mathstrut \) \(177\) \(\beta_{7}\mathstrut +\mathstrut \) \(18\) \(\beta_{6}\mathstrut -\mathstrut \) \(55\) \(\beta_{5}\mathstrut +\mathstrut \) \(17\) \(\beta_{4}\mathstrut +\mathstrut \) \(3389\) \(\beta_{3}\mathstrut -\mathstrut \) \(1225\) \(\beta_{2}\mathstrut +\mathstrut \) \(6575\) \(\beta_{1}\mathstrut -\mathstrut \) \(971\)

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
2.51437
2.43355
2.36308
1.79305
0.921208
0.851790
0.142275
−0.155707
−1.37578
−1.73551
−2.11086
−2.64148
−2.51437 −1.00000 4.32208 0 2.51437 0.976551 −5.83856 1.00000 0
1.2 −2.43355 −1.00000 3.92219 0 2.43355 −4.14842 −4.67775 1.00000 0
1.3 −2.36308 −1.00000 3.58415 0 2.36308 1.44739 −3.74348 1.00000 0
1.4 −1.79305 −1.00000 1.21504 0 1.79305 −0.475581 1.40748 1.00000 0
1.5 −0.921208 −1.00000 −1.15138 0 0.921208 −3.92138 2.90307 1.00000 0
1.6 −0.851790 −1.00000 −1.27445 0 0.851790 0.870792 2.78915 1.00000 0
1.7 −0.142275 −1.00000 −1.97976 0 0.142275 1.55684 0.566220 1.00000 0
1.8 0.155707 −1.00000 −1.97576 0 −0.155707 −3.91791 −0.619053 1.00000 0
1.9 1.37578 −1.00000 −0.107224 0 −1.37578 1.29286 −2.89908 1.00000 0
1.10 1.73551 −1.00000 1.01200 0 −1.73551 4.69063 −1.71469 1.00000 0
1.11 2.11086 −1.00000 2.45572 0 −2.11086 −2.18333 0.961964 1.00000 0
1.12 2.64148 −1.00000 4.97740 0 −2.64148 −3.18844 7.86473 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(1\)
\(107\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8025))\):

\(T_{2}^{12} + \cdots\)
\(T_{7}^{12} + \cdots\)
\(T_{11}^{12} - \cdots\)
\(T_{13}^{12} + \cdots\)