Properties

Label 2013.2.a.d
Level 2013
Weight 2
Character orbit 2013.a
Self dual yes
Analytic conductor 16.074
Analytic rank 1
Dimension 12
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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - x^{11} - 16 x^{10} + 13 x^{9} + 93 x^{8} - 59 x^{7} - 238 x^{6} + 108 x^{5} + 257 x^{4} - 71 x^{3} - 93 x^{2} + 13 x + 1\)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - x^{11} - 16 x^{10} + 13 x^{9} + 93 x^{8} - 59 x^{7} - 238 x^{6} + 108 x^{5} + 257 x^{4} - 71 x^{3} - 93 x^{2} + 13 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\( -6 \nu^{11} - 4 \nu^{10} + 39 \nu^{9} + 138 \nu^{8} + 125 \nu^{7} - 998 \nu^{6} - 1242 \nu^{5} + 2567 \nu^{4} + 2384 \nu^{3} - 2446 \nu^{2} - 1304 \nu + 517 \)\()/151\)
\(\beta_{4}\)\(=\)\((\)\( 15 \nu^{11} + 10 \nu^{10} - 173 \nu^{9} - 194 \nu^{8} + 518 \nu^{7} + 985 \nu^{6} + 85 \nu^{5} - 1510 \nu^{4} - 1581 \nu^{3} + 226 \nu^{2} + 844 \nu + 293 \)\()/151\)
\(\beta_{5}\)\(=\)\((\)\( 21 \nu^{11} + 14 \nu^{10} - 363 \nu^{9} - 181 \nu^{8} + 2205 \nu^{7} + 775 \nu^{6} - 5921 \nu^{5} - 1359 \nu^{4} + 7209 \nu^{3} + 860 \nu^{2} - 3137 \nu + 78 \)\()/151\)
\(\beta_{6}\)\(=\)\((\)\( 23 \nu^{11} - 35 \nu^{10} - 376 \nu^{9} + 528 \nu^{8} + 2113 \nu^{7} - 2617 \nu^{6} - 4752 \nu^{5} + 4681 \nu^{4} + 3646 \nu^{3} - 2150 \nu^{2} - 387 \nu + 107 \)\()/151\)
\(\beta_{7}\)\(=\)\((\)\( -22 \nu^{11} + 86 \nu^{10} + 294 \nu^{9} - 1155 \nu^{8} - 1404 \nu^{7} + 5300 \nu^{6} + 2694 \nu^{5} - 9513 \nu^{4} - 1577 \nu^{3} + 5326 \nu^{2} + 252 \nu - 168 \)\()/151\)
\(\beta_{8}\)\(=\)\((\)\( 41 \nu^{11} - 23 \nu^{10} - 644 \nu^{9} + 265 \nu^{8} + 3550 \nu^{7} - 982 \nu^{6} - 8123 \nu^{5} + 1057 \nu^{4} + 6913 \nu^{3} + 356 \nu^{2} - 1458 \nu - 387 \)\()/151\)
\(\beta_{9}\)\(=\)\((\)\( 44 \nu^{11} - 21 \nu^{10} - 739 \nu^{9} + 196 \nu^{8} + 4620 \nu^{7} - 483 \nu^{6} - 13089 \nu^{5} + 15989 \nu^{3} + 522 \nu^{2} - 6242 \nu + 185 \)\()/151\)
\(\beta_{10}\)\(=\)\((\)\( -52 \nu^{11} + 66 \nu^{10} + 791 \nu^{9} - 918 \nu^{8} - 4252 \nu^{7} + 4538 \nu^{6} + 9621 \nu^{5} - 9060 \nu^{4} - 8381 \nu^{3} + 5780 \nu^{2} + 1886 \nu - 301 \)\()/151\)
\(\beta_{11}\)\(=\)\((\)\( 52 \nu^{11} - 66 \nu^{10} - 791 \nu^{9} + 918 \nu^{8} + 4252 \nu^{7} - 4538 \nu^{6} - 9621 \nu^{5} + 9060 \nu^{4} + 8532 \nu^{3} - 5931 \nu^{2} - 2490 \nu + 603 \)\()/151\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{11} + \beta_{10} + \beta_{2} + 4 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{10} + \beta_{9} - \beta_{8} + 2 \beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_{3} + 8 \beta_{2} + \beta_{1} + 14\)
\(\nu^{5}\)\(=\)\(8 \beta_{11} + 8 \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + \beta_{4} + 10 \beta_{2} + 20 \beta_{1} + 9\)
\(\nu^{6}\)\(=\)\(3 \beta_{11} + 12 \beta_{10} + 10 \beta_{9} - 11 \beta_{8} + \beta_{7} + 21 \beta_{6} - 10 \beta_{5} + 18 \beta_{4} + 7 \beta_{3} + 57 \beta_{2} + 11 \beta_{1} + 75\)
\(\nu^{7}\)\(=\)\(55 \beta_{11} + 57 \beta_{10} + 14 \beta_{9} - 16 \beta_{8} + 11 \beta_{7} + 28 \beta_{6} - 23 \beta_{5} + 15 \beta_{4} + 87 \beta_{2} + 111 \beta_{1} + 72\)
\(\nu^{8}\)\(=\)\(43 \beta_{11} + 106 \beta_{10} + 79 \beta_{9} - 93 \beta_{8} + 15 \beta_{7} + 170 \beta_{6} - 81 \beta_{5} + 132 \beta_{4} + 41 \beta_{3} + 397 \beta_{2} + 97 \beta_{1} + 437\)
\(\nu^{9}\)\(=\)\(369 \beta_{11} + 402 \beta_{10} + 137 \beta_{9} - 167 \beta_{8} + 91 \beta_{7} + 278 \beta_{6} - 200 \beta_{5} + 157 \beta_{4} + 2 \beta_{3} + 711 \beta_{2} + 660 \beta_{1} + 561\)
\(\nu^{10}\)\(=\)\(433 \beta_{11} + 847 \beta_{10} + 583 \beta_{9} - 717 \beta_{8} + 155 \beta_{7} + 1269 \beta_{6} - 616 \beta_{5} + 922 \beta_{4} + 233 \beta_{3} + 2765 \beta_{2} + 793 \beta_{1} + 2707\)
\(\nu^{11}\)\(=\)\(2487 \beta_{11} + 2847 \beta_{10} + 1168 \beta_{9} - 1471 \beta_{8} + 689 \beta_{7} + 2403 \beta_{6} - 1582 \beta_{5} + 1409 \beta_{4} + 39 \beta_{3} + 5583 \beta_{2} + 4135 \beta_{1} + 4305\)

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
−2.26879
−2.22938
−1.72434
−1.05676
−0.811354
−0.0557908
0.188928
0.852887
1.32082
2.01091
2.07880
2.69409
−2.26879 −1.00000 3.14741 −2.83484 2.26879 −4.72992 −2.60323 1.00000 6.43165
1.2 −2.22938 −1.00000 2.97016 3.25707 2.22938 −0.940570 −2.16285 1.00000 −7.26127
1.3 −1.72434 −1.00000 0.973355 −0.0158764 1.72434 3.08835 1.77029 1.00000 0.0273764
1.4 −1.05676 −1.00000 −0.883253 1.53036 1.05676 −3.09203 3.04691 1.00000 −1.61722
1.5 −0.811354 −1.00000 −1.34170 0.350684 0.811354 3.06527 2.71131 1.00000 −0.284529
1.6 −0.0557908 −1.00000 −1.99689 −3.85264 0.0557908 −0.441454 0.222990 1.00000 0.214942
1.7 0.188928 −1.00000 −1.96431 −1.33997 −0.188928 −0.912886 −0.748968 1.00000 −0.253157
1.8 0.852887 −1.00000 −1.27258 2.61325 −0.852887 −1.65804 −2.79114 1.00000 2.22881
1.9 1.32082 −1.00000 −0.255442 0.125081 −1.32082 1.14683 −2.97903 1.00000 0.165209
1.10 2.01091 −1.00000 2.04375 2.29360 −2.01091 −4.38501 0.0879817 1.00000 4.61222
1.11 2.07880 −1.00000 2.32140 −2.51557 −2.07880 3.31485 0.668123 1.00000 −5.22935
1.12 2.69409 −1.00000 5.25811 −2.61116 −2.69409 −3.45538 8.77762 1.00000 −7.03469
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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.d 12
3.b odd 2 1 6039.2.a.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2013.2.a.d 12 1.a even 1 1 trivial
6039.2.a.e 12 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{12} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2013))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + 8 T^{2} - 9 T^{3} + 37 T^{4} - 45 T^{5} + 130 T^{6} - 166 T^{7} + 377 T^{8} - 491 T^{9} + 931 T^{10} - 1189 T^{11} + 1989 T^{12} - 2378 T^{13} + 3724 T^{14} - 3928 T^{15} + 6032 T^{16} - 5312 T^{17} + 8320 T^{18} - 5760 T^{19} + 9472 T^{20} - 4608 T^{21} + 8192 T^{22} - 2048 T^{23} + 4096 T^{24} \)
$3$ \( ( 1 + T )^{12} \)
$5$ \( 1 + 3 T + 33 T^{2} + 91 T^{3} + 564 T^{4} + 1426 T^{5} + 6531 T^{6} + 15207 T^{7} + 56736 T^{8} + 121503 T^{9} + 388835 T^{10} + 760775 T^{11} + 2154302 T^{12} + 3803875 T^{13} + 9720875 T^{14} + 15187875 T^{15} + 35460000 T^{16} + 47521875 T^{17} + 102046875 T^{18} + 111406250 T^{19} + 220312500 T^{20} + 177734375 T^{21} + 322265625 T^{22} + 146484375 T^{23} + 244140625 T^{24} \)
$7$ \( 1 + 9 T + 75 T^{2} + 426 T^{3} + 2257 T^{4} + 10006 T^{5} + 42027 T^{6} + 157220 T^{7} + 560377 T^{8} + 1823035 T^{9} + 5668638 T^{10} + 16269029 T^{11} + 44727158 T^{12} + 113883203 T^{13} + 277763262 T^{14} + 625301005 T^{15} + 1345465177 T^{16} + 2642396540 T^{17} + 4944434523 T^{18} + 8240371258 T^{19} + 13011155857 T^{20} + 17190636582 T^{21} + 21185643675 T^{22} + 17795940687 T^{23} + 13841287201 T^{24} \)
$11$ \( ( 1 + T )^{12} \)
$13$ \( 1 + T + 97 T^{2} + 92 T^{3} + 4545 T^{4} + 3929 T^{5} + 138095 T^{6} + 103737 T^{7} + 3081009 T^{8} + 1945842 T^{9} + 54034051 T^{10} + 29087982 T^{11} + 773051714 T^{12} + 378143766 T^{13} + 9131754619 T^{14} + 4275014874 T^{15} + 87996698049 T^{16} + 38516821941 T^{17} + 666558188855 T^{18} + 246538923293 T^{19} + 3707496126945 T^{20} + 975613942316 T^{21} + 13372273709353 T^{22} + 1792160394037 T^{23} + 23298085122481 T^{24} \)
$17$ \( 1 - 9 T + 173 T^{2} - 1255 T^{3} + 13720 T^{4} - 83733 T^{5} + 672737 T^{6} - 3543080 T^{7} + 23011888 T^{8} - 106083784 T^{9} + 583441380 T^{10} - 2367329066 T^{11} + 11297633016 T^{12} - 40244594122 T^{13} + 168614558820 T^{14} - 521189630792 T^{15} + 1921975897648 T^{16} - 5030666939560 T^{17} + 16238235756353 T^{18} - 34358888106309 T^{19} + 95707392090520 T^{20} - 148827785003735 T^{21} + 348766944777677 T^{22} - 308447066768697 T^{23} + 582622237229761 T^{24} \)
$19$ \( 1 + 20 T + 289 T^{2} + 3124 T^{3} + 28884 T^{4} + 228638 T^{5} + 1628560 T^{6} + 10438498 T^{7} + 61730866 T^{8} + 336497366 T^{9} + 1718654485 T^{10} + 8195882293 T^{11} + 36888603196 T^{12} + 155721763567 T^{13} + 620434269085 T^{14} + 2308035433394 T^{15} + 8044828187986 T^{16} + 25846754459302 T^{17} + 76617039961360 T^{18} + 204373046661482 T^{19} + 490553234876244 T^{20} + 1008076367861596 T^{21} + 1771878148504489 T^{22} + 2329805177964380 T^{23} + 2213314919066161 T^{24} \)
$23$ \( 1 + 9 T + 174 T^{2} + 1201 T^{3} + 13433 T^{4} + 78562 T^{5} + 668726 T^{6} + 3550447 T^{7} + 25346027 T^{8} + 125061048 T^{9} + 777228859 T^{10} + 3541789790 T^{11} + 19634776696 T^{12} + 81461165170 T^{13} + 411154066411 T^{14} + 1521617771016 T^{15} + 7092857541707 T^{16} + 22851894695321 T^{17} + 98995447907414 T^{18} + 267489896767214 T^{19} + 1051951465279673 T^{20} + 2163184346417063 T^{21} + 7208212951174926 T^{22} + 8575287821225343 T^{23} + 21914624432020321 T^{24} \)
$29$ \( 1 - 18 T + 326 T^{2} - 3622 T^{3} + 39184 T^{4} - 323391 T^{5} + 2622529 T^{6} - 17431968 T^{7} + 116436705 T^{8} - 661332434 T^{9} + 3927443207 T^{10} - 20477068722 T^{11} + 116636944574 T^{12} - 593834992938 T^{13} + 3302979737087 T^{14} - 16129236732826 T^{15} + 82353469149105 T^{16} - 357549693011232 T^{17} + 1559941409198809 T^{18} - 5578454749443819 T^{19} + 19601655445463824 T^{20} - 52544882724597518 T^{21} + 137150558055865526 T^{22} - 219609175782704922 T^{23} + 353814783205469041 T^{24} \)
$31$ \( 1 + 21 T + 333 T^{2} + 3979 T^{3} + 42452 T^{4} + 397509 T^{5} + 3424048 T^{6} + 26877869 T^{7} + 197677279 T^{8} + 1352657357 T^{9} + 8742645356 T^{10} + 52938971212 T^{11} + 303940288824 T^{12} + 1641108107572 T^{13} + 8401682187116 T^{14} + 40297015322387 T^{15} + 182559118379359 T^{16} + 769490570159219 T^{17} + 3038855203920688 T^{18} + 10936511722649499 T^{19} + 36206930321445332 T^{20} + 105203256577309909 T^{21} + 272936219564606733 T^{22} + 533578014824501451 T^{23} + 787662783788549761 T^{24} \)
$37$ \( 1 + 18 T + 464 T^{2} + 6109 T^{3} + 91979 T^{4} + 966128 T^{5} + 10819707 T^{6} + 94823629 T^{7} + 861280056 T^{8} + 6451176732 T^{9} + 49458121400 T^{10} + 320260046983 T^{11} + 2112270198676 T^{12} + 11849621738371 T^{13} + 67708168196600 T^{14} + 326771455005996 T^{15} + 1614177491033016 T^{16} + 6575445651959953 T^{17} + 27760407987542163 T^{18} + 91716344590751024 T^{19} + 323074347692199659 T^{20} + 793936268408125393 T^{21} + 2231183148801881936 T^{22} + 3202517192030287434 T^{23} + 6582952005840035281 T^{24} \)
$41$ \( 1 - 15 T + 269 T^{2} - 2159 T^{3} + 19144 T^{4} - 68668 T^{5} + 273637 T^{6} + 1540565 T^{7} - 5806536 T^{8} + 2472043 T^{9} + 917142791 T^{10} - 9886660827 T^{11} + 77102984112 T^{12} - 405353093907 T^{13} + 1541717031671 T^{14} + 170375675603 T^{15} - 16407882973896 T^{16} + 178484008293565 T^{17} + 1299804274194517 T^{18} - 13373386478860508 T^{19} + 152863408586292424 T^{20} - 706817596356561799 T^{21} + 3610695354430995869 T^{22} - 8254935475743726615 T^{23} + 22563490300366186081 T^{24} \)
$43$ \( 1 + 33 T + 786 T^{2} + 13455 T^{3} + 191905 T^{4} + 2292699 T^{5} + 24054598 T^{6} + 223131068 T^{7} + 1878427460 T^{8} + 14501256530 T^{9} + 105025004902 T^{10} + 723201077622 T^{11} + 4817021641096 T^{12} + 31097646337746 T^{13} + 194191234063798 T^{14} + 1152951402930710 T^{15} + 6421969678675460 T^{16} + 32802150891607124 T^{17} + 152057846955749302 T^{18} + 623198257866407793 T^{19} + 2243024074273019905 T^{20} + 6762383593610222565 T^{21} + 16986625098241419714 T^{22} + 30666693402550349331 T^{23} + 39959630797262576401 T^{24} \)
$47$ \( 1 + 20 T + 519 T^{2} + 7274 T^{3} + 112850 T^{4} + 1252185 T^{5} + 14697139 T^{6} + 136760569 T^{7} + 1325195640 T^{8} + 10691882163 T^{9} + 89448478852 T^{10} + 638258250214 T^{11} + 4720464981838 T^{12} + 29998137760058 T^{13} + 197591689784068 T^{14} + 1110063281809149 T^{15} + 6466531985790840 T^{16} + 31365353654628983 T^{17} + 158423626001243731 T^{18} + 634385872096961655 T^{19} + 2687103699779728850 T^{20} + 8140555061349527158 T^{21} + 27298949630395795431 T^{22} + 49443184301680246060 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \)
$53$ \( 1 + 386 T^{2} - 427 T^{3} + 73251 T^{4} - 150893 T^{5} + 9138826 T^{6} - 25421902 T^{7} + 843858452 T^{8} - 2689153945 T^{9} + 61213957311 T^{10} - 197523968540 T^{11} + 3591646359094 T^{12} - 10468770332620 T^{13} + 171950006086599 T^{14} - 400353171869765 T^{15} + 6658449082195412 T^{16} - 10631324839887686 T^{17} + 202556239759094554 T^{18} - 177255688023424441 T^{19} + 4560584582322604611 T^{20} - 1408999053699510791 T^{21} + 67506563561088036914 T^{22} + \)\(49\!\cdots\!41\)\( T^{24} \)
$59$ \( 1 + 21 T + 593 T^{2} + 8480 T^{3} + 141522 T^{4} + 1578402 T^{5} + 20114270 T^{6} + 189845050 T^{7} + 2055625629 T^{8} + 17259654548 T^{9} + 166172308270 T^{10} + 1260432657194 T^{11} + 10892836503218 T^{12} + 74365526774446 T^{13} + 578445805087870 T^{14} + 3544770591413692 T^{15} + 24908757827445069 T^{16} + 135724839289869950 T^{17} + 848430642399157070 T^{18} + 3928092480941279238 T^{19} + 20779737190638716562 T^{20} + 73462204542193882720 T^{21} + \)\(30\!\cdots\!93\)\( T^{22} + \)\(63\!\cdots\!39\)\( T^{23} + \)\(17\!\cdots\!81\)\( T^{24} \)
$61$ \( ( 1 + T )^{12} \)
$67$ \( 1 + 34 T + 1015 T^{2} + 20841 T^{3} + 385611 T^{4} + 5926892 T^{5} + 83782486 T^{6} + 1045433763 T^{7} + 12148991386 T^{8} + 127955606772 T^{9} + 1264315584635 T^{10} + 11458792405175 T^{11} + 97761523076214 T^{12} + 767739091146725 T^{13} + 5675512659426515 T^{14} + 38484312159567036 T^{15} + 244815795447243706 T^{16} + 1411466371131787641 T^{17} + 7578828137656892134 T^{18} + 35921183127896046116 T^{19} + \)\(15\!\cdots\!51\)\( T^{20} + \)\(56\!\cdots\!27\)\( T^{21} + \)\(18\!\cdots\!35\)\( T^{22} + \)\(41\!\cdots\!22\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \)
$71$ \( 1 + 5 T + 436 T^{2} + 2661 T^{3} + 101370 T^{4} + 701404 T^{5} + 16097289 T^{6} + 120113549 T^{7} + 1936214187 T^{8} + 14781599809 T^{9} + 185383670159 T^{10} + 1368839453739 T^{11} + 14506152869548 T^{12} + 97187601215469 T^{13} + 934519081271519 T^{14} + 5290497169238999 T^{15} + 49202457267718347 T^{16} + 216712390558576699 T^{17} + 2062067291258390169 T^{18} + 6379353659576080964 T^{19} + 65460035462382792570 T^{20} + \)\(12\!\cdots\!91\)\( T^{21} + \)\(14\!\cdots\!36\)\( T^{22} + \)\(11\!\cdots\!55\)\( T^{23} + \)\(16\!\cdots\!41\)\( T^{24} \)
$73$ \( 1 + 2 T + 595 T^{2} + 598 T^{3} + 171665 T^{4} + 44530 T^{5} + 32125678 T^{6} - 7547733 T^{7} + 4371288811 T^{8} - 2179159199 T^{9} + 456343710966 T^{10} - 262228646139 T^{11} + 37446061911870 T^{12} - 19142691168147 T^{13} + 2431855635737814 T^{14} - 847729974117383 T^{15} + 124136913135381451 T^{16} - 15646990873848669 T^{17} + 4861714624139548942 T^{18} + 491940656055389410 T^{19} + \)\(13\!\cdots\!65\)\( T^{20} + 35205208851544211974 T^{21} + \)\(25\!\cdots\!55\)\( T^{22} + \)\(62\!\cdots\!54\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \)
$79$ \( 1 + 31 T + 691 T^{2} + 10467 T^{3} + 141414 T^{4} + 1662705 T^{5} + 19748172 T^{6} + 218165545 T^{7} + 2387782966 T^{8} + 24007153734 T^{9} + 238793154852 T^{10} + 2218538296175 T^{11} + 20439922610232 T^{12} + 175264525397825 T^{13} + 1490308079431332 T^{14} + 11836463069857626 T^{15} + 93004339936120246 T^{16} + 671307686283572455 T^{17} + 4800532882671057612 T^{18} + 31930435490831500095 T^{19} + \)\(21\!\cdots\!54\)\( T^{20} + \)\(12\!\cdots\!73\)\( T^{21} + \)\(65\!\cdots\!91\)\( T^{22} + \)\(23\!\cdots\!49\)\( T^{23} + \)\(59\!\cdots\!41\)\( T^{24} \)
$83$ \( 1 + 32 T + 955 T^{2} + 18166 T^{3} + 328925 T^{4} + 4631812 T^{5} + 63373588 T^{6} + 721281496 T^{7} + 8187904876 T^{8} + 80106634838 T^{9} + 810259467204 T^{10} + 7268466154033 T^{11} + 69811468720106 T^{12} + 603282690784739 T^{13} + 5581877469568356 T^{14} + 45803932414115506 T^{15} + 388584217922673196 T^{16} + 2841157127787841928 T^{17} + 20719384522453177972 T^{18} + \)\(12\!\cdots\!24\)\( T^{19} + \)\(74\!\cdots\!25\)\( T^{20} + \)\(33\!\cdots\!98\)\( T^{21} + \)\(14\!\cdots\!95\)\( T^{22} + \)\(41\!\cdots\!44\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \)
$89$ \( 1 - 27 T + 847 T^{2} - 15572 T^{3} + 288379 T^{4} - 4134838 T^{5} + 58695353 T^{6} - 717071370 T^{7} + 8761760559 T^{8} - 96250677487 T^{9} + 1060385422624 T^{10} - 10571701084571 T^{11} + 104982457822002 T^{12} - 940881396526819 T^{13} + 8399312932604704 T^{14} - 67853743857332903 T^{15} + 549732492577072719 T^{16} - 4004169159255875130 T^{17} + 29170492307351604233 T^{18} - \)\(18\!\cdots\!02\)\( T^{19} + \)\(11\!\cdots\!99\)\( T^{20} - \)\(54\!\cdots\!48\)\( T^{21} + \)\(26\!\cdots\!47\)\( T^{22} - \)\(74\!\cdots\!03\)\( T^{23} + \)\(24\!\cdots\!21\)\( T^{24} \)
$97$ \( 1 + 19 T + 651 T^{2} + 9836 T^{3} + 198464 T^{4} + 2643990 T^{5} + 39962302 T^{6} + 493734300 T^{7} + 6109191987 T^{8} + 71319348041 T^{9} + 759458844371 T^{10} + 8351792047457 T^{11} + 79662888691254 T^{12} + 810123828603329 T^{13} + 7145748266686739 T^{14} + 65091243334623593 T^{15} + 540842374100071347 T^{16} + 4239864430651715100 T^{17} + 33287478818518186558 T^{18} + \)\(21\!\cdots\!70\)\( T^{19} + \)\(15\!\cdots\!04\)\( T^{20} + \)\(74\!\cdots\!12\)\( T^{21} + \)\(48\!\cdots\!99\)\( T^{22} + \)\(13\!\cdots\!07\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \)
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