Properties

Label 2013.2.a.a
Level 2013
Weight 2
Character orbit 2013.a
Self dual yes
Analytic conductor 16.074
Analytic rank 1
Dimension 11
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 2013 = 3 \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2013.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(16.0738859269\)
Analytic rank: \(1\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Defining polynomial: \(x^{11} - 4 x^{10} - 6 x^{9} + 37 x^{8} - 2 x^{7} - 109 x^{6} + 55 x^{5} + 115 x^{4} - 76 x^{3} - 29 x^{2} + 14 x + 3\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{11} - 4 x^{10} - 6 x^{9} + 37 x^{8} - 2 x^{7} - 109 x^{6} + 55 x^{5} + 115 x^{4} - 76 x^{3} - 29 x^{2} + 14 x + 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{10} - 2 \nu^{9} - 10 \nu^{8} + 17 \nu^{7} + 32 \nu^{6} - 45 \nu^{5} - 35 \nu^{4} + 54 \nu^{3} + 14 \nu^{2} - 37 \nu - 6 \)\()/9\)
\(\beta_{3}\)\(=\)\((\)\( 5 \nu^{10} - 16 \nu^{9} - 41 \nu^{8} + 145 \nu^{7} + 97 \nu^{6} - 408 \nu^{5} - 55 \nu^{4} + 378 \nu^{3} - 38 \nu^{2} - 35 \nu + 6 \)\()/9\)
\(\beta_{4}\)\(=\)\((\)\( 5 \nu^{10} - 16 \nu^{9} - 41 \nu^{8} + 145 \nu^{7} + 97 \nu^{6} - 408 \nu^{5} - 64 \nu^{4} + 387 \nu^{3} + 16 \nu^{2} - 62 \nu - 30 \)\()/9\)
\(\beta_{5}\)\(=\)\((\)\( -4 \nu^{10} + 11 \nu^{9} + 40 \nu^{8} - 107 \nu^{7} - 137 \nu^{6} + 339 \nu^{5} + 188 \nu^{4} - 396 \nu^{3} - 83 \nu^{2} + 109 \nu + 6 \)\()/9\)
\(\beta_{6}\)\(=\)\((\)\( 4 \nu^{10} - 11 \nu^{9} - 40 \nu^{8} + 107 \nu^{7} + 137 \nu^{6} - 339 \nu^{5} - 188 \nu^{4} + 396 \nu^{3} + 92 \nu^{2} - 109 \nu - 33 \)\()/9\)
\(\beta_{7}\)\(=\)\((\)\( 7 \nu^{10} - 20 \nu^{9} - 61 \nu^{8} + 179 \nu^{7} + 152 \nu^{6} - 489 \nu^{5} - 53 \nu^{4} + 432 \nu^{3} - 136 \nu^{2} - 19 \nu + 21 \)\()/9\)
\(\beta_{8}\)\(=\)\((\)\( -2 \nu^{10} + 5 \nu^{9} + 20 \nu^{8} - 44 \nu^{7} - 73 \nu^{6} + 119 \nu^{5} + 125 \nu^{4} - 114 \nu^{3} - 94 \nu^{2} + 28 \nu + 15 \)\()/3\)
\(\beta_{9}\)\(=\)\((\)\( 7 \nu^{10} - 14 \nu^{9} - 79 \nu^{8} + 128 \nu^{7} + 314 \nu^{6} - 360 \nu^{5} - 506 \nu^{4} + 342 \nu^{3} + 296 \nu^{2} - 43 \nu - 42 \)\()/9\)
\(\beta_{10}\)\(=\)\((\)\( 16 \nu^{10} - 47 \nu^{9} - 142 \nu^{8} + 431 \nu^{7} + 386 \nu^{6} - 1236 \nu^{5} - 305 \nu^{4} + 1224 \nu^{3} - 46 \nu^{2} - 226 \nu - 6 \)\()/9\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} + \beta_{5} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{10} - \beta_{7} + \beta_{6} + 2 \beta_{5} - \beta_{4} + 4 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(\beta_{10} - \beta_{7} + 7 \beta_{6} + 8 \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{1} + 16\)
\(\nu^{5}\)\(=\)\(8 \beta_{10} + \beta_{9} + \beta_{8} - 9 \beta_{7} + 9 \beta_{6} + 18 \beta_{5} - 8 \beta_{4} + 2 \beta_{3} + 21 \beta_{1} + 19\)
\(\nu^{6}\)\(=\)\(10 \beta_{10} + \beta_{9} + \beta_{8} - 12 \beta_{7} + 45 \beta_{6} + 56 \beta_{5} - 18 \beta_{4} + 11 \beta_{3} + 2 \beta_{2} + 15 \beta_{1} + 96\)
\(\nu^{7}\)\(=\)\(53 \beta_{10} + 10 \beta_{9} + 11 \beta_{8} - 65 \beta_{7} + 72 \beta_{6} + 134 \beta_{5} - 56 \beta_{4} + 25 \beta_{3} + 6 \beta_{2} + 124 \beta_{1} + 154\)
\(\nu^{8}\)\(=\)\(80 \beta_{10} + 14 \beta_{9} + 16 \beta_{8} - 107 \beta_{7} + 292 \beta_{6} + 386 \beta_{5} - 134 \beta_{4} + 96 \beta_{3} + 33 \beta_{2} + 148 \beta_{1} + 613\)
\(\nu^{9}\)\(=\)\(339 \beta_{10} + 80 \beta_{9} + 93 \beta_{8} - 448 \beta_{7} + 551 \beta_{6} + 960 \beta_{5} - 386 \beta_{4} + 236 \beta_{3} + 96 \beta_{2} + 781 \beta_{1} + 1180\)
\(\nu^{10}\)\(=\)\(598 \beta_{10} + 143 \beta_{9} + 172 \beta_{8} - 863 \beta_{7} + 1940 \beta_{6} + 2678 \beta_{5} - 960 \beta_{4} + 780 \beta_{3} + 365 \beta_{2} + 1255 \beta_{1} + 4071\)

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
2.70371
2.39607
1.53948
1.36137
1.14470
0.536504
−0.186189
−0.423080
−1.35090
−1.57504
−2.14662
−2.70371 1.00000 5.31003 −2.83529 −2.70371 2.54457 −8.94935 1.00000 7.66580
1.2 −2.39607 1.00000 3.74115 −0.714370 −2.39607 −0.642303 −4.17191 1.00000 1.71168
1.3 −1.53948 1.00000 0.369988 −0.258725 −1.53948 4.37144 2.50937 1.00000 0.398300
1.4 −1.36137 1.00000 −0.146675 −3.87776 −1.36137 −4.18708 2.92242 1.00000 5.27907
1.5 −1.14470 1.00000 −0.689651 1.33318 −1.14470 −3.52170 3.07886 1.00000 −1.52610
1.6 −0.536504 1.00000 −1.71216 −3.00877 −0.536504 0.211107 1.99159 1.00000 1.61422
1.7 0.186189 1.00000 −1.96533 1.94515 0.186189 0.0328573 −0.738300 1.00000 0.362164
1.8 0.423080 1.00000 −1.82100 −1.36483 0.423080 0.865577 −1.61659 1.00000 −0.577432
1.9 1.35090 1.00000 −0.175068 −0.638467 1.35090 −0.931245 −2.93830 1.00000 −0.862505
1.10 1.57504 1.00000 0.480756 0.664958 1.57504 −4.15209 −2.39287 1.00000 1.04734
1.11 2.14662 1.00000 2.60797 −4.24506 2.14662 0.408872 1.30509 1.00000 −9.11254
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(11\) \(1\)
\(61\) \(-1\)

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 2013.2.a.a 11
3.b odd 2 1 6039.2.a.d 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2013.2.a.a 11 1.a even 1 1 trivial
6039.2.a.d 11 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{11} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2013))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 4 T + 16 T^{2} + 43 T^{3} + 110 T^{4} + 237 T^{5} + 483 T^{6} + 889 T^{7} + 1554 T^{8} + 2513 T^{9} + 3886 T^{10} + 5609 T^{11} + 7772 T^{12} + 10052 T^{13} + 12432 T^{14} + 14224 T^{15} + 15456 T^{16} + 15168 T^{17} + 14080 T^{18} + 11008 T^{19} + 8192 T^{20} + 4096 T^{21} + 2048 T^{22} \)
$3$ \( ( 1 - T )^{11} \)
$5$ \( 1 + 13 T + 110 T^{2} + 692 T^{3} + 3574 T^{4} + 15712 T^{5} + 60545 T^{6} + 207607 T^{7} + 641391 T^{8} + 1797574 T^{9} + 4595382 T^{10} + 10739601 T^{11} + 22976910 T^{12} + 44939350 T^{13} + 80173875 T^{14} + 129754375 T^{15} + 189203125 T^{16} + 245500000 T^{17} + 279218750 T^{18} + 270312500 T^{19} + 214843750 T^{20} + 126953125 T^{21} + 48828125 T^{22} \)
$7$ \( 1 + 5 T + 52 T^{2} + 213 T^{3} + 1237 T^{4} + 4185 T^{5} + 18144 T^{6} + 51991 T^{7} + 189551 T^{8} + 479150 T^{9} + 1566501 T^{10} + 3629137 T^{11} + 10965507 T^{12} + 23478350 T^{13} + 65015993 T^{14} + 124830391 T^{15} + 304946208 T^{16} + 492361065 T^{17} + 1018722691 T^{18} + 1227902613 T^{19} + 2098387564 T^{20} + 1412376245 T^{21} + 1977326743 T^{22} \)
$11$ \( ( 1 + T )^{11} \)
$13$ \( 1 + 3 T + 78 T^{2} + 149 T^{3} + 2849 T^{4} + 2712 T^{5} + 65044 T^{6} - 1789 T^{7} + 1072181 T^{8} - 998985 T^{9} + 14705402 T^{10} - 19849203 T^{11} + 191170226 T^{12} - 168828465 T^{13} + 2355581657 T^{14} - 51095629 T^{15} + 24150381892 T^{16} + 13090306008 T^{17} + 178770524933 T^{18} + 121543877429 T^{19} + 827150951094 T^{20} + 413575475547 T^{21} + 1792160394037 T^{22} \)
$17$ \( 1 + 7 T + 132 T^{2} + 776 T^{3} + 8030 T^{4} + 40947 T^{5} + 305355 T^{6} + 1379147 T^{7} + 8298275 T^{8} + 33705949 T^{9} + 174595051 T^{10} + 641474955 T^{11} + 2968115867 T^{12} + 9741019261 T^{13} + 40769425075 T^{14} + 115187736587 T^{15} + 433560434235 T^{16} + 988361037843 T^{17} + 3295019544190 T^{18} + 5413187774216 T^{19} + 15653599697604 T^{20} + 14111957303143 T^{21} + 34271896307633 T^{22} \)
$19$ \( 1 + 8 T + 134 T^{2} + 712 T^{3} + 6920 T^{4} + 25198 T^{5} + 201182 T^{6} + 516168 T^{7} + 4479068 T^{8} + 9368738 T^{9} + 94516729 T^{10} + 183055555 T^{11} + 1795817851 T^{12} + 3382114418 T^{13} + 30721927412 T^{14} + 67267529928 T^{15} + 498146549018 T^{16} + 1185462109438 T^{17} + 6185592433880 T^{18} + 12092296885192 T^{19} + 43240151502386 T^{20} + 49048530062408 T^{21} + 116490258898219 T^{22} \)
$23$ \( 1 + 15 T + 225 T^{2} + 2156 T^{3} + 19470 T^{4} + 138000 T^{5} + 926178 T^{6} + 5233421 T^{7} + 28727483 T^{8} + 139631979 T^{9} + 695111660 T^{10} + 3230420623 T^{11} + 15987568180 T^{12} + 73865316891 T^{13} + 349527285661 T^{14} + 1464525766061 T^{15} + 5961199287054 T^{16} + 20428952682000 T^{17} + 66291951453090 T^{18} + 168838484265836 T^{19} + 405259348829175 T^{20} + 621397668204735 T^{21} + 952809757913927 T^{22} \)
$29$ \( 1 + 8 T + 181 T^{2} + 1016 T^{3} + 13989 T^{4} + 54615 T^{5} + 647036 T^{6} + 1622755 T^{7} + 21855327 T^{8} + 31449701 T^{9} + 644765272 T^{10} + 649073461 T^{11} + 18698192888 T^{12} + 26449198541 T^{13} + 533029570203 T^{14} + 1147743779155 T^{15} + 13271451804364 T^{16} + 32486275676415 T^{17} + 241308519686601 T^{18} + 508250355568376 T^{19} + 2625793421632289 T^{20} + 3365657866401608 T^{21} + 12200509765705829 T^{22} \)
$31$ \( 1 + 17 T + 310 T^{2} + 3218 T^{3} + 33228 T^{4} + 243289 T^{5} + 1753078 T^{6} + 9348476 T^{7} + 49878537 T^{8} + 190694315 T^{9} + 875576853 T^{10} + 3377656593 T^{11} + 27142882443 T^{12} + 183257236715 T^{13} + 1485931495767 T^{14} + 8633513903996 T^{15} + 50189134776778 T^{16} + 215919883046809 T^{17} + 914189141680308 T^{18} + 2744603358485138 T^{19} + 8196282869808010 T^{20} + 13933680878673617 T^{21} + 25408476896404831 T^{22} \)
$37$ \( 1 + 10 T + 207 T^{2} + 1333 T^{3} + 16830 T^{4} + 69415 T^{5} + 800297 T^{6} + 1988856 T^{7} + 31872061 T^{8} + 57245062 T^{9} + 1311962749 T^{10} + 2206143033 T^{11} + 48542621713 T^{12} + 78368489878 T^{13} + 1614415505833 T^{14} + 3727436349816 T^{15} + 55495760755229 T^{16} + 178099898680735 T^{17} + 1597703492148390 T^{18} + 4682135112076693 T^{19} + 26902080137580939 T^{20} + 48085843724178490 T^{21} + 177917621779460413 T^{22} \)
$41$ \( 1 + 25 T + 444 T^{2} + 5294 T^{3} + 53796 T^{4} + 461206 T^{5} + 3822131 T^{6} + 29662347 T^{7} + 223780129 T^{8} + 1555934808 T^{9} + 10388757942 T^{10} + 66515240905 T^{11} + 425939075622 T^{12} + 2615526412248 T^{13} + 15423150270809 T^{14} + 83818703321067 T^{15} + 442817577384331 T^{16} + 2190776576574646 T^{17} + 10477000917702276 T^{18} + 42272194162966574 T^{19} + 145357578870918684 T^{20} + 335566482753810025 T^{21} + 550329031716248441 T^{22} \)
$43$ \( 1 + 7 T + 257 T^{2} + 1702 T^{3} + 32058 T^{4} + 202021 T^{5} + 2679008 T^{6} + 15738739 T^{7} + 171349212 T^{8} + 915571567 T^{9} + 8888066250 T^{10} + 43079314303 T^{11} + 382186848750 T^{12} + 1692891827383 T^{13} + 13623461798484 T^{14} + 53807616631939 T^{15} + 393836794864544 T^{16} + 1277048084522029 T^{17} + 8713961034868206 T^{18} + 19893316872476902 T^{19} + 129166301267768651 T^{20} + 151280376192989743 T^{21} + 929293739471222707 T^{22} \)
$47$ \( 1 + 30 T + 660 T^{2} + 10476 T^{3} + 142954 T^{4} + 1679519 T^{5} + 17880827 T^{6} + 172567758 T^{7} + 1534962263 T^{8} + 12605212479 T^{9} + 96157731545 T^{10} + 682902362183 T^{11} + 4519413382615 T^{12} + 27844914366111 T^{13} + 159364387031449 T^{14} + 842075609925198 T^{15} + 4100878393480789 T^{16} + 18103896950146751 T^{17} + 72423801562667702 T^{18} + 249447039068608236 T^{19} + 738626112247826220 T^{20} + 1577973967074901470 T^{21} + 2472159215084012303 T^{22} \)
$53$ \( 1 + 18 T + 461 T^{2} + 6111 T^{3} + 90600 T^{4} + 966884 T^{5} + 10734996 T^{6} + 97512162 T^{7} + 901339176 T^{8} + 7233346119 T^{9} + 58832102057 T^{10} + 425427147299 T^{11} + 3118101409021 T^{12} + 20318469248271 T^{13} + 134188672505352 T^{14} + 769417861529922 T^{15} + 4489326944573028 T^{16} + 21430366145852036 T^{17} + 106428829269232200 T^{18} + 380468968103827071 T^{19} + 1521191015820783313 T^{20} + 3147974466579234882 T^{21} + 9269035929372191597 T^{22} \)
$59$ \( 1 + 43 T + 1238 T^{2} + 25369 T^{3} + 424358 T^{4} + 5889111 T^{5} + 71249636 T^{6} + 758289209 T^{7} + 7328225443 T^{8} + 64866807821 T^{9} + 539935788257 T^{10} + 4240654651317 T^{11} + 31856211507163 T^{12} + 225801358024901 T^{13} + 1505063613257897 T^{14} + 9188464087857449 T^{15} + 50938096071305164 T^{16} + 248405844651083151 T^{17} + 1056079166794821202 T^{18} + 3724941371584019449 T^{19} + 10724788823494814482 T^{20} + 21978020391927580243 T^{21} + 30155888444737842659 T^{22} \)
$61$ \( ( 1 - T )^{11} \)
$67$ \( 1 + 30 T + 832 T^{2} + 16439 T^{3} + 285861 T^{4} + 4265651 T^{5} + 57057821 T^{6} + 686646642 T^{7} + 7541342515 T^{8} + 75698105858 T^{9} + 700958561188 T^{10} + 5963213140161 T^{11} + 46964223599596 T^{12} + 339808797196562 T^{13} + 2268156798838945 T^{14} + 13836699567185682 T^{15} + 77035196682811847 T^{16} + 385863888357577019 T^{17} + 1732521080209238103 T^{18} + 6675346551353621399 T^{19} + 22635836617717395904 T^{20} + 54685134136552843470 T^{21} + \)\(12\!\cdots\!83\)\( T^{22} \)
$71$ \( 1 + 7 T + 139 T^{2} + 694 T^{3} + 20409 T^{4} + 107338 T^{5} + 2127578 T^{6} + 10625821 T^{7} + 204961981 T^{8} + 972345748 T^{9} + 16031632136 T^{10} + 69696118649 T^{11} + 1138245881656 T^{12} + 4901594915668 T^{13} + 73358147581691 T^{14} + 270019973615101 T^{15} + 3838638674141878 T^{16} + 13750028275512298 T^{17} + 185622307312601919 T^{18} + 448152950684558134 T^{19} + 6372941599864415309 T^{20} + 22786704857069168407 T^{21} + \)\(23\!\cdots\!71\)\( T^{22} \)
$73$ \( 1 - 6 T + 314 T^{2} - 1146 T^{3} + 45031 T^{4} - 57542 T^{5} + 4317965 T^{6} + 5955239 T^{7} + 332665660 T^{8} + 1327226302 T^{9} + 23359265340 T^{10} + 125872665635 T^{11} + 1705226369820 T^{12} + 7072788963358 T^{13} + 129412597056220 T^{14} + 169118312334599 T^{15} + 8951450581068245 T^{16} - 8708074049121638 T^{17} + 497475402713457007 T^{18} - 924203265310616826 T^{19} + 18485678226396124682 T^{20} - 25785754978221345894 T^{21} + \)\(31\!\cdots\!77\)\( T^{22} \)
$79$ \( 1 - 17 T + 510 T^{2} - 7990 T^{3} + 139448 T^{4} - 1901341 T^{5} + 25726134 T^{6} - 302390608 T^{7} + 3455558864 T^{8} - 35459924532 T^{9} + 352399281068 T^{10} - 3185312378827 T^{11} + 27839543204372 T^{12} - 221305389004212 T^{13} + 1703725286747696 T^{14} - 11778138675239248 T^{15} + 79160765246231466 T^{16} - 462192145767753661 T^{17} + 2677946700301900232 T^{18} - 12121699391153422390 T^{19} + 61124313951135342690 T^{20} - \)\(16\!\cdots\!17\)\( T^{21} + \)\(74\!\cdots\!79\)\( T^{22} \)
$83$ \( 1 + 34 T + 960 T^{2} + 18266 T^{3} + 322939 T^{4} + 4691920 T^{5} + 65456303 T^{6} + 795285212 T^{7} + 9366195719 T^{8} + 98485660756 T^{9} + 1008682396977 T^{10} + 9309164570275 T^{11} + 83720638949091 T^{12} + 678467716948084 T^{13} + 5355468951579853 T^{14} + 37742900877649052 T^{15} + 257835037857522829 T^{16} + 1533978076617478480 T^{17} + 8763289170539153753 T^{18} + 41140369912251722906 T^{19} + \)\(17\!\cdots\!80\)\( T^{20} + \)\(52\!\cdots\!66\)\( T^{21} + \)\(12\!\cdots\!67\)\( T^{22} \)
$89$ \( 1 + 41 T + 1100 T^{2} + 20271 T^{3} + 316687 T^{4} + 4289405 T^{5} + 56485928 T^{6} + 695401739 T^{7} + 8201672365 T^{8} + 87701882052 T^{9} + 891074197783 T^{10} + 8478025448085 T^{11} + 79305603602687 T^{12} + 694686607733892 T^{13} + 5781924765481685 T^{14} + 43631063500157099 T^{15} + 315420779983933672 T^{16} + 2131754034354568205 T^{17} + 14007488754060392423 T^{18} + 79798591680386883951 T^{19} + \)\(38\!\cdots\!00\)\( T^{20} + \)\(12\!\cdots\!41\)\( T^{21} + \)\(27\!\cdots\!89\)\( T^{22} \)
$97$ \( 1 + 41 T + 1360 T^{2} + 30915 T^{3} + 623782 T^{4} + 10465719 T^{5} + 162260178 T^{6} + 2228939233 T^{7} + 28726923179 T^{8} + 336165881822 T^{9} + 3712591176360 T^{10} + 37598782574515 T^{11} + 360121344106920 T^{12} + 3162984782063198 T^{13} + 26218287158547467 T^{14} + 197326387690181473 T^{15} + 1393383358647385746 T^{16} + 8717650938453528951 T^{17} + 50400515488326283366 T^{18} + \)\(24\!\cdots\!15\)\( T^{19} + \)\(10\!\cdots\!20\)\( T^{20} + \)\(30\!\cdots\!09\)\( T^{21} + \)\(71\!\cdots\!53\)\( T^{22} \)
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