Properties

Label 2013.2.a.e
Level $2013$
Weight $2$
Character orbit 2013.a
Self dual yes
Analytic conductor $16.074$
Analytic rank $0$
Dimension $13$
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: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Defining polynomial: \(x^{13} - 2 x^{12} - 19 x^{11} + 35 x^{10} + 136 x^{9} - 220 x^{8} - 469 x^{7} + 610 x^{6} + 841 x^{5} - 760 x^{4} - 742 x^{3} + 366 x^{2} + 236 x - 47\)
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{13} - 2 x^{12} - 19 x^{11} + 35 x^{10} + 136 x^{9} - 220 x^{8} - 469 x^{7} + 610 x^{6} + 841 x^{5} - 760 x^{4} - 742 x^{3} + 366 x^{2} + 236 x - 47\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\( -38 \nu^{12} + 47 \nu^{11} + 646 \nu^{10} - 577 \nu^{9} - 3970 \nu^{8} + 1615 \nu^{7} + 11430 \nu^{6} + 2278 \nu^{5} - 18122 \nu^{4} - 10328 \nu^{3} + 14993 \nu^{2} + 9272 \nu - 4830 \)\()/1261\)
\(\beta_{4}\)\(=\)\((\)\( 80 \nu^{12} - 192 \nu^{11} - 1217 \nu^{10} + 2967 \nu^{9} + 5684 \nu^{8} - 15191 \nu^{7} - 6221 \nu^{6} + 29789 \nu^{5} - 11934 \nu^{4} - 24751 \nu^{3} + 21328 \nu^{2} + 12785 \nu - 1964 \)\()/1261\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{12} + 275 \nu^{11} - 352 \nu^{10} - 5029 \nu^{9} + 5943 \nu^{8} + 33414 \nu^{7} - 31352 \nu^{6} - 99199 \nu^{5} + 60892 \nu^{4} + 131903 \nu^{3} - 38492 \nu^{2} - 60160 \nu + 5338 \)\()/1261\)
\(\beta_{6}\)\(=\)\((\)\( 74 \nu^{12} - 448 \nu^{11} - 738 \nu^{10} + 7586 \nu^{9} - 695 \nu^{8} - 44875 \nu^{7} + 24220 \nu^{6} + 111281 \nu^{5} - 66105 \nu^{4} - 114506 \nu^{3} + 52816 \nu^{2} + 35335 \nu - 9293 \)\()/1261\)
\(\beta_{7}\)\(=\)\((\)\( 112 \nu^{12} - 495 \nu^{11} - 1384 \nu^{10} + 8163 \nu^{9} + 3275 \nu^{8} - 46490 \nu^{7} + 12790 \nu^{6} + 109003 \nu^{5} - 47983 \nu^{4} - 105439 \nu^{3} + 39084 \nu^{2} + 32368 \nu - 8246 \)\()/1261\)
\(\beta_{8}\)\(=\)\((\)\( 128 \nu^{12} - 42 \nu^{11} - 2670 \nu^{10} + 699 \nu^{9} + 20498 \nu^{8} - 4257 \nu^{7} - 69946 \nu^{6} + 11200 \nu^{5} + 100321 \nu^{4} - 10180 \nu^{3} - 46002 \nu^{2} + 1840 \nu + 1171 \)\()/1261\)
\(\beta_{9}\)\(=\)\((\)\( 162 \nu^{12} - 524 \nu^{11} - 2507 \nu^{10} + 8823 \nu^{9} + 12632 \nu^{8} - 52242 \nu^{7} - 22121 \nu^{6} + 132371 \nu^{5} + 6149 \nu^{4} - 147391 \nu^{3} + 16859 \nu^{2} + 57426 \nu - 11183 \)\()/1261\)
\(\beta_{10}\)\(=\)\((\)\( -191 \nu^{12} + 448 \nu^{11} + 3260 \nu^{10} - 7625 \nu^{9} - 19377 \nu^{8} + 45850 \nu^{7} + 47579 \nu^{6} - 118535 \nu^{5} - 45357 \nu^{4} + 134188 \nu^{3} + 6321 \nu^{2} - 53288 \nu + 8136 \)\()/1261\)
\(\beta_{11}\)\(=\)\((\)\( 187 \nu^{12} - 883 \nu^{11} - 2451 \nu^{10} + 15172 \nu^{9} + 7541 \nu^{8} - 92220 \nu^{7} + 11676 \nu^{6} + 240723 \nu^{5} - 65676 \nu^{4} - 270378 \nu^{3} + 62706 \nu^{2} + 97970 \nu - 17754 \)\()/1261\)
\(\beta_{12}\)\(=\)\((\)\( -344 \nu^{12} + 953 \nu^{11} + 5640 \nu^{10} - 16038 \nu^{9} - 31573 \nu^{8} + 94362 \nu^{7} + 70611 \nu^{6} - 233199 \nu^{5} - 59488 \nu^{4} + 240692 \nu^{3} + 6554 \nu^{2} - 79409 \nu + 8700 \)\()/1261\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(-\beta_{7} + \beta_{6} - \beta_{3} + \beta_{2} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-\beta_{12} + \beta_{10} - \beta_{9} - \beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} + 9 \beta_{2} + 14\)
\(\nu^{5}\)\(=\)\(-\beta_{12} - \beta_{11} - \beta_{8} - 10 \beta_{7} + 9 \beta_{6} - \beta_{5} + \beta_{4} - 9 \beta_{3} + 11 \beta_{2} + 29 \beta_{1} - 1\)
\(\nu^{6}\)\(=\)\(-10 \beta_{12} - 2 \beta_{11} + 11 \beta_{10} - 8 \beta_{9} + \beta_{8} - 11 \beta_{7} + 3 \beta_{6} - 11 \beta_{5} + 10 \beta_{4} - 10 \beta_{3} + 69 \beta_{2} - \beta_{1} + 79\)
\(\nu^{7}\)\(=\)\(-13 \beta_{12} - 16 \beta_{11} - \beta_{10} - 12 \beta_{8} - 79 \beta_{7} + 72 \beta_{6} - 13 \beta_{5} + 10 \beta_{4} - 67 \beta_{3} + 94 \beta_{2} + 181 \beta_{1} - 8\)
\(\nu^{8}\)\(=\)\(-81 \beta_{12} - 33 \beta_{11} + 92 \beta_{10} - 51 \beta_{9} + 13 \beta_{8} - 92 \beta_{7} + 44 \beta_{6} - 97 \beta_{5} + 76 \beta_{4} - 83 \beta_{3} + 506 \beta_{2} - 5 \beta_{1} + 488\)
\(\nu^{9}\)\(=\)\(-120 \beta_{12} - 176 \beta_{11} - 22 \beta_{10} + \beta_{9} - 105 \beta_{8} - 580 \beta_{7} + 560 \beta_{6} - 128 \beta_{5} + 71 \beta_{4} - 478 \beta_{3} + 739 \beta_{2} + 1179 \beta_{1} - 29\)
\(\nu^{10}\)\(=\)\(-606 \beta_{12} - 375 \beta_{11} + 688 \beta_{10} - 303 \beta_{9} + 127 \beta_{8} - 697 \beta_{7} + 471 \beta_{6} - 793 \beta_{5} + 523 \beta_{4} - 655 \beta_{3} + 3644 \beta_{2} + 34 \beta_{1} + 3165\)
\(\nu^{11}\)\(=\)\(-967 \beta_{12} - 1670 \beta_{11} - 303 \beta_{10} + 23 \beta_{9} - 815 \beta_{8} - 4127 \beta_{7} + 4302 \beta_{6} - 1137 \beta_{5} + 427 \beta_{4} - 3386 \beta_{3} + 5608 \beta_{2} + 7898 \beta_{1} + 124\)
\(\nu^{12}\)\(=\)\(-4357 \beta_{12} - 3662 \beta_{11} + 4833 \beta_{10} - 1739 \beta_{9} + 1118 \beta_{8} - 5052 \beta_{7} + 4458 \beta_{6} - 6254 \beta_{5} + 3417 \beta_{4} - 5073 \beta_{3} + 26039 \beta_{2} + 982 \beta_{1} + 21158\)

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
−2.59890
−2.37960
−1.46794
−1.33092
−0.948254
−0.805107
0.171582
0.822526
1.50067
1.61181
2.14727
2.53773
2.73913
−2.59890 −1.00000 4.75428 −3.50938 2.59890 0.252288 −7.15810 1.00000 9.12053
1.2 −2.37960 −1.00000 3.66252 2.55185 2.37960 1.67712 −3.95613 1.00000 −6.07239
1.3 −1.46794 −1.00000 0.154851 3.84216 1.46794 2.46223 2.70857 1.00000 −5.64006
1.4 −1.33092 −1.00000 −0.228660 −2.36819 1.33092 −2.92169 2.96616 1.00000 3.15187
1.5 −0.948254 −1.00000 −1.10081 −2.25122 0.948254 5.24025 2.94036 1.00000 2.13473
1.6 −0.805107 −1.00000 −1.35180 1.06503 0.805107 −0.203035 2.69856 1.00000 −0.857467
1.7 0.171582 −1.00000 −1.97056 −0.133072 −0.171582 0.615329 −0.681275 1.00000 −0.0228327
1.8 0.822526 −1.00000 −1.32345 2.11599 −0.822526 0.404149 −2.73363 1.00000 1.74046
1.9 1.50067 −1.00000 0.252024 −0.569898 −1.50067 −3.97554 −2.62314 1.00000 −0.855232
1.10 1.61181 −1.00000 0.597924 4.27930 −1.61181 1.98799 −2.25988 1.00000 6.89741
1.11 2.14727 −1.00000 2.61077 −3.62125 −2.14727 2.48742 1.31149 1.00000 −7.77580
1.12 2.53773 −1.00000 4.44007 0.994065 −2.53773 4.88646 6.19224 1.00000 2.52267
1.13 2.73913 −1.00000 5.50285 0.604616 −2.73913 −1.91298 9.59477 1.00000 1.65612
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
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.e 13
3.b odd 2 1 6039.2.a.i 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2013.2.a.e 13 1.a even 1 1 trivial
6039.2.a.i 13 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{13} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2013))\).