Properties

Label 1502.2.a.e
Level 1502
Weight 2
Character orbit 1502.a
Self dual yes
Analytic conductor 11.994
Analytic rank 1
Dimension 11
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1502 = 2 \cdot 751 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1502.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.9935303836\)
Analytic rank: \(1\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{11} - 4 x^{10} - 9 x^{9} + 58 x^{8} - 40 x^{7} - 146 x^{6} + 237 x^{5} - 47 x^{4} - 89 x^{3} + 39 x^{2} - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{10} - 3 \nu^{9} - 12 \nu^{8} + 46 \nu^{7} + 6 \nu^{6} - 140 \nu^{5} + 97 \nu^{4} + 50 \nu^{3} - 39 \nu^{2} - 1 \)
\(\beta_{3}\)\(=\)\((\)\( -159 \nu^{10} + 469 \nu^{9} + 1907 \nu^{8} - 7174 \nu^{7} - 957 \nu^{6} + 21503 \nu^{5} - 15476 \nu^{4} - 6481 \nu^{3} + 6560 \nu^{2} - 578 \nu - 175 \)\()/29\)
\(\beta_{4}\)\(=\)\((\)\( -297 \nu^{10} + 829 \nu^{9} + 3665 \nu^{8} - 12780 \nu^{7} - 3422 \nu^{6} + 38954 \nu^{5} - 23775 \nu^{4} - 13811 \nu^{3} + 10037 \nu^{2} - 202 \nu - 305 \)\()/29\)
\(\beta_{5}\)\(=\)\((\)\( -466 \nu^{10} + 1285 \nu^{9} + 5799 \nu^{8} - 19850 \nu^{7} - 6119 \nu^{6} + 60844 \nu^{5} - 34869 \nu^{4} - 22622 \nu^{3} + 14360 \nu^{2} - 8 \nu - 344 \)\()/29\)
\(\beta_{6}\)\(=\)\((\)\( -524 \nu^{10} + 1430 \nu^{9} + 6553 \nu^{8} - 22112 \nu^{7} - 7395 \nu^{6} + 67862 \nu^{5} - 37537 \nu^{4} - 25377 \nu^{3} + 15317 \nu^{2} - 211 \nu - 431 \)\()/29\)
\(\beta_{7}\)\(=\)\((\)\( 666 \nu^{10} - 1837 \nu^{9} - 8280 \nu^{8} + 28355 \nu^{7} + 8642 \nu^{6} - 86651 \nu^{5} + 50005 \nu^{4} + 31319 \nu^{3} - 20225 \nu^{2} + 489 \nu + 495 \)\()/29\)
\(\beta_{8}\)\(=\)\((\)\( 738 \nu^{10} - 2074 \nu^{9} - 9085 \nu^{8} + 31933 \nu^{7} + 8149 \nu^{6} - 97076 \nu^{5} + 60000 \nu^{4} + 33595 \nu^{3} - 24726 \nu^{2} + 864 \nu + 670 \)\()/29\)
\(\beta_{9}\)\(=\)\((\)\( 800 \nu^{10} - 2237 \nu^{9} - 9866 \nu^{8} + 34455 \nu^{7} + 9135 \nu^{6} - 104736 \nu^{5} + 63995 \nu^{4} + 36180 \nu^{3} - 26070 \nu^{2} + 938 \nu + 691 \)\()/29\)
\(\beta_{10}\)\(=\)\((\)\( -858 \nu^{10} + 2353 \nu^{9} + 10707 \nu^{8} - 36369 \nu^{7} - 11745 \nu^{6} + 111580 \nu^{5} - 62603 \nu^{4} - 41748 \nu^{3} + 25548 \nu^{2} - 10 \nu - 691 \)\()/29\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} - \beta_{2} + 2\)
\(\nu^{3}\)\(=\)\(-\beta_{10} + \beta_{9} - 2 \beta_{7} + \beta_{4} - \beta_{2} + 8 \beta_{1} - 4\)
\(\nu^{4}\)\(=\)\(-\beta_{10} - 2 \beta_{9} - 8 \beta_{8} + 11 \beta_{7} + 10 \beta_{6} - 6 \beta_{5} - 2 \beta_{4} - 9 \beta_{3} - 9 \beta_{2} - 5 \beta_{1} + 14\)
\(\nu^{5}\)\(=\)\(-10 \beta_{10} + 14 \beta_{9} - 28 \beta_{7} - 7 \beta_{6} + 2 \beta_{5} + 12 \beta_{4} + \beta_{3} - 5 \beta_{2} + 69 \beta_{1} - 47\)
\(\nu^{6}\)\(=\)\(-9 \beta_{10} - 30 \beta_{9} - 69 \beta_{8} + 118 \beta_{7} + 96 \beta_{6} - 41 \beta_{5} - 31 \beta_{4} - 80 \beta_{3} - 73 \beta_{2} - 90 \beta_{1} + 139\)
\(\nu^{7}\)\(=\)\(-89 \beta_{10} + 158 \beta_{9} + 22 \beta_{8} - 334 \beta_{7} - 128 \beta_{6} + 41 \beta_{5} + 128 \beta_{4} + 42 \beta_{3} + 6 \beta_{2} + 644 \beta_{1} - 503\)
\(\nu^{8}\)\(=\)\(-47 \beta_{10} - 373 \beta_{9} - 624 \beta_{8} + 1264 \beta_{7} + 936 \beta_{6} - 332 \beta_{5} - 384 \beta_{4} - 732 \beta_{3} - 618 \beta_{2} - 1220 \beta_{1} + 1508\)
\(\nu^{9}\)\(=\)\(-779 \beta_{10} + 1695 \beta_{9} + 497 \beta_{8} - 3793 \beta_{7} - 1752 \beta_{6} + 580 \beta_{5} + 1361 \beta_{4} + 773 \beta_{3} + 494 \beta_{2} + 6357 \beta_{1} - 5375\)
\(\nu^{10}\)\(=\)\(-6 \beta_{10} - 4375 \beta_{9} - 5858 \beta_{8} + 13597 \beta_{7} + 9377 \beta_{6} - 3061 \beta_{5} - 4403 \beta_{4} - 6943 \beta_{3} - 5587 \beta_{2} - 14908 \beta_{1} + 16616\)

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
−0.141525
2.65496
1.22814
−3.30464
−0.628353
0.440910
−1.88979
1.90813
2.26367
0.235096
1.23340
−1.00000 −3.13913 1.00000 −0.691662 3.13913 −0.0689252 −1.00000 6.85413 0.691662
1.2 −1.00000 −2.56973 1.00000 3.54277 2.56973 2.20818 −1.00000 3.60353 −3.54277
1.3 −1.00000 −2.24044 1.00000 −4.22616 2.24044 −3.19230 −1.00000 2.01957 4.22616
1.4 −1.00000 −1.42377 1.00000 0.121552 1.42377 1.78140 −1.00000 −0.972877 −0.121552
1.5 −1.00000 −1.01328 1.00000 2.69149 1.01328 −4.75435 −1.00000 −1.97326 −2.69149
1.6 −1.00000 −0.0490252 1.00000 −2.08781 0.0490252 −0.273638 −1.00000 −2.99760 2.08781
1.7 −1.00000 0.195838 1.00000 −1.22685 −0.195838 4.43512 −1.00000 −2.96165 1.22685
1.8 −1.00000 0.785790 1.00000 1.57015 −0.785790 −1.48223 −1.00000 −2.38253 −1.57015
1.9 −1.00000 1.04934 1.00000 2.15494 −1.04934 −0.540539 −1.00000 −1.89889 −2.15494
1.10 −1.00000 2.13096 1.00000 0.446946 −2.13096 −1.57697 −1.00000 1.54099 −0.446946
1.11 −1.00000 2.27345 1.00000 −1.29537 −2.27345 −2.53574 −1.00000 2.16859 1.29537
\(n\): 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.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1502.2.a.e 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1502.2.a.e 11 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(751\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{11} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1502))\).