Properties

Label 2013.2.a.g
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: \(x^{13} - 4 x^{12} - 11 x^{11} + 57 x^{10} + 28 x^{9} - 290 x^{8} + 51 x^{7} + 644 x^{6} - 259 x^{5} - 640 x^{4} + 274 x^{3} + 256 x^{2} - 74 x - 35\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 5 \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{13} - 4 x^{12} - 11 x^{11} + 57 x^{10} + 28 x^{9} - 290 x^{8} + 51 x^{7} + 644 x^{6} - 259 x^{5} - 640 x^{4} + 274 x^{3} + 256 x^{2} - 74 x - 35\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 5 \nu \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 6 \nu^{2} - \nu + 4 \)
\(\beta_{5}\)\(=\)\((\)\( -8 \nu^{12} + 90 \nu^{11} - 26 \nu^{10} - 1165 \nu^{9} + 1132 \nu^{8} + 4883 \nu^{7} - 5564 \nu^{6} - 6816 \nu^{5} + 8767 \nu^{4} + 884 \nu^{3} - 4652 \nu^{2} + 1882 \nu + 410 \)\()/359\)
\(\beta_{6}\)\(=\)\((\)\( 7 \nu^{12} + 11 \nu^{11} - 67 \nu^{10} - 282 \nu^{9} - 93 \nu^{8} + 2324 \nu^{7} + 2894 \nu^{6} - 8037 \nu^{5} - 10588 \nu^{4} + 11612 \nu^{3} + 12866 \nu^{2} - 5506 \nu - 3500 \)\()/359\)
\(\beta_{7}\)\(=\)\((\)\( 29 \nu^{12} - 57 \nu^{11} - 534 \nu^{10} + 1037 \nu^{9} + 3615 \nu^{8} - 6886 \nu^{7} - 10884 \nu^{6} + 19682 \nu^{5} + 14037 \nu^{4} - 21334 \nu^{3} - 7010 \nu^{2} + 5294 \nu + 1655 \)\()/359\)
\(\beta_{8}\)\(=\)\((\)\( -66 \nu^{12} + 204 \nu^{11} + 683 \nu^{10} - 2521 \nu^{9} - 1431 \nu^{8} + 10039 \nu^{7} - 3900 \nu^{6} - 13511 \nu^{5} + 13003 \nu^{4} + 3344 \nu^{3} - 8582 \nu^{2} + 2064 \nu + 690 \)\()/359\)
\(\beta_{9}\)\(=\)\((\)\( -102 \nu^{12} + 250 \nu^{11} + 1284 \nu^{10} - 3276 \nu^{9} - 4953 \nu^{8} + 14601 \nu^{7} + 4449 \nu^{6} - 25874 \nu^{5} + 6682 \nu^{4} + 17733 \nu^{3} - 8694 \nu^{2} - 3109 \nu + 381 \)\()/359\)
\(\beta_{10}\)\(=\)\((\)\( -71 \nu^{12} + 350 \nu^{11} + 577 \nu^{10} - 4730 \nu^{9} + 533 \nu^{8} + 22021 \nu^{7} - 13660 \nu^{6} - 41824 \nu^{5} + 31900 \nu^{4} + 32078 \nu^{3} - 22798 \nu^{2} - 7440 \nu + 3190 \)\()/359\)
\(\beta_{11}\)\(=\)\((\)\( 96 \nu^{12} - 362 \nu^{11} - 1124 \nu^{10} + 5005 \nu^{9} + 4007 \nu^{8} - 24132 \nu^{7} - 4314 \nu^{6} + 48405 \nu^{5} + 2496 \nu^{4} - 39687 \nu^{3} - 4488 \nu^{2} + 10085 \nu + 1901 \)\()/359\)
\(\beta_{12}\)\(=\)\((\)\( -196 \nu^{12} + 410 \nu^{11} + 2953 \nu^{10} - 5746 \nu^{9} - 16423 \nu^{8} + 28627 \nu^{7} + 42464 \nu^{6} - 61446 \nu^{5} - 54279 \nu^{4} + 56481 \nu^{3} + 31780 \nu^{2} - 17434 \nu - 6469 \)\()/359\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{4} + 6 \beta_{2} + \beta_{1} + 14\)
\(\nu^{5}\)\(=\)\(\beta_{11} + \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 7 \beta_{3} + 27 \beta_{1} + 2\)
\(\nu^{6}\)\(=\)\(2 \beta_{11} + \beta_{9} + \beta_{8} - \beta_{7} + 3 \beta_{6} + 2 \beta_{5} + 10 \beta_{4} + \beta_{3} + 32 \beta_{2} + 12 \beta_{1} + 74\)
\(\nu^{7}\)\(=\)\(\beta_{12} + 15 \beta_{11} + 2 \beta_{10} + 12 \beta_{8} - 11 \beta_{7} + 15 \beta_{6} + 12 \beta_{5} + 14 \beta_{4} + 42 \beta_{3} + 153 \beta_{1} + 25\)
\(\nu^{8}\)\(=\)\(\beta_{12} + 31 \beta_{11} + 2 \beta_{10} + 14 \beta_{9} + 13 \beta_{8} - 14 \beta_{7} + 42 \beta_{6} + 30 \beta_{5} + 80 \beta_{4} + 14 \beta_{3} + 166 \beta_{2} + 109 \beta_{1} + 414\)
\(\nu^{9}\)\(=\)\(15 \beta_{12} + 153 \beta_{11} + 28 \beta_{10} + 4 \beta_{9} + 104 \beta_{8} - 91 \beta_{7} + 153 \beta_{6} + 115 \beta_{5} + 137 \beta_{4} + 246 \beta_{3} - 2 \beta_{2} + 900 \beta_{1} + 233\)
\(\nu^{10}\)\(=\)\(19 \beta_{12} + 328 \beta_{11} + 34 \beta_{10} + 134 \beta_{9} + 123 \beta_{8} - 137 \beta_{7} + 415 \beta_{6} + 312 \beta_{5} + 599 \beta_{4} + 135 \beta_{3} + 852 \beta_{2} + 890 \beta_{1} + 2406\)
\(\nu^{11}\)\(=\)\(153 \beta_{12} + 1334 \beta_{11} + 273 \beta_{10} + 74 \beta_{9} + 797 \beta_{8} - 686 \beta_{7} + 1337 \beta_{6} + 1001 \beta_{5} + 1165 \beta_{4} + 1451 \beta_{3} - 36 \beta_{2} + 5466 \beta_{1} + 1943\)
\(\nu^{12}\)\(=\)\(227 \beta_{12} + 2960 \beta_{11} + 387 \beta_{10} + 1100 \beta_{9} + 1038 \beta_{8} - 1168 \beta_{7} + 3572 \beta_{6} + 2782 \beta_{5} + 4363 \beta_{4} + 1129 \beta_{3} + 4344 \beta_{2} + 6882 \beta_{1} + 14427\)

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
−2.33467
−2.17077
−1.38103
−1.17594
−0.638829
−0.312603
0.669508
0.931518
1.31287
1.88247
2.25716
2.30577
2.65453
−2.33467 1.00000 3.45069 0.0293558 −2.33467 −2.70286 −3.38688 1.00000 −0.0685361
1.2 −2.17077 1.00000 2.71223 2.18891 −2.17077 3.77754 −1.54609 1.00000 −4.75162
1.3 −1.38103 1.00000 −0.0927606 −0.547866 −1.38103 1.24123 2.89016 1.00000 0.756618
1.4 −1.17594 1.00000 −0.617176 3.63888 −1.17594 −0.818481 3.07763 1.00000 −4.27908
1.5 −0.638829 1.00000 −1.59190 −2.55169 −0.638829 −2.98273 2.29461 1.00000 1.63009
1.6 −0.312603 1.00000 −1.90228 −0.566394 −0.312603 3.64999 1.21987 1.00000 0.177056
1.7 0.669508 1.00000 −1.55176 2.40049 0.669508 3.10183 −2.37793 1.00000 1.60715
1.8 0.931518 1.00000 −1.13227 1.13816 0.931518 −2.87358 −2.91777 1.00000 1.06022
1.9 1.31287 1.00000 −0.276371 −3.61438 1.31287 −0.837447 −2.98858 1.00000 −4.74521
1.10 1.88247 1.00000 1.54370 3.12160 1.88247 2.89419 −0.858967 1.00000 5.87632
1.11 2.25716 1.00000 3.09478 1.69786 2.25716 0.963803 2.47110 1.00000 3.83236
1.12 2.30577 1.00000 3.31659 −2.09861 2.30577 3.60849 3.03576 1.00000 −4.83892
1.13 2.65453 1.00000 5.04652 2.16368 2.65453 −2.02198 8.08709 1.00000 5.74356
\(n\): 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.g 13
3.b odd 2 1 6039.2.a.h 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2013.2.a.g 13 1.a even 1 1 trivial
6039.2.a.h 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2013))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 4 T + 15 T^{2} - 39 T^{3} + 98 T^{4} - 206 T^{5} + 423 T^{6} - 776 T^{7} + 1407 T^{8} - 2352 T^{9} + 3888 T^{10} - 6000 T^{11} + 9138 T^{12} - 13027 T^{13} + 18276 T^{14} - 24000 T^{15} + 31104 T^{16} - 37632 T^{17} + 45024 T^{18} - 49664 T^{19} + 54144 T^{20} - 52736 T^{21} + 50176 T^{22} - 39936 T^{23} + 30720 T^{24} - 16384 T^{25} + 8192 T^{26} \)
$3$ \( ( 1 - T )^{13} \)
$5$ \( 1 - 7 T + 56 T^{2} - 258 T^{3} + 1255 T^{4} - 4503 T^{5} + 16859 T^{6} - 50885 T^{7} + 160423 T^{8} - 425008 T^{9} + 1180717 T^{10} - 2815444 T^{11} + 7070565 T^{12} - 15390246 T^{13} + 35352825 T^{14} - 70386100 T^{15} + 147589625 T^{16} - 265630000 T^{17} + 501321875 T^{18} - 795078125 T^{19} + 1317109375 T^{20} - 1758984375 T^{21} + 2451171875 T^{22} - 2519531250 T^{23} + 2734375000 T^{24} - 1708984375 T^{25} + 1220703125 T^{26} \)
$7$ \( 1 - 7 T + 70 T^{2} - 369 T^{3} + 2288 T^{4} - 9950 T^{5} + 47642 T^{6} - 178052 T^{7} + 711086 T^{8} - 2332387 T^{9} + 8053671 T^{10} - 23432188 T^{11} + 71259100 T^{12} - 184614114 T^{13} + 498813700 T^{14} - 1148177212 T^{15} + 2762409153 T^{16} - 5600061187 T^{17} + 11951222402 T^{18} - 20947639748 T^{19} + 39235235606 T^{20} - 57359769950 T^{21} + 92329052816 T^{22} - 104233366881 T^{23} + 138412872010 T^{24} - 96889010407 T^{25} + 96889010407 T^{26} \)
$11$ \( ( 1 - T )^{13} \)
$13$ \( 1 - 9 T + 142 T^{2} - 963 T^{3} + 8890 T^{4} - 49469 T^{5} + 343406 T^{6} - 1641094 T^{7} + 9394370 T^{8} - 39521641 T^{9} + 194950678 T^{10} - 730889714 T^{11} + 3178238861 T^{12} - 10658263028 T^{13} + 41317105193 T^{14} - 123520361666 T^{15} + 428306639566 T^{16} - 1128777588601 T^{17} + 3488063820410 T^{18} - 7921247289046 T^{19} + 21548217228902 T^{20} - 40353383037149 T^{21} + 94273999425970 T^{22} - 132757727650587 T^{23} + 254486775953254 T^{24} - 209682766102329 T^{25} + 302875106592253 T^{26} \)
$17$ \( 1 - 19 T + 286 T^{2} - 3030 T^{3} + 27921 T^{4} - 216056 T^{5} + 1519141 T^{6} - 9542471 T^{7} + 55860051 T^{8} - 300909000 T^{9} + 1531481120 T^{10} - 7271209038 T^{11} + 32798268168 T^{12} - 138656460496 T^{13} + 557570558856 T^{14} - 2101379411982 T^{15} + 7524166742560 T^{16} - 25132220589000 T^{17} + 79313284432707 T^{18} - 230332052192999 T^{19} + 623362302039893 T^{20} - 1507154249672696 T^{21} + 3311092099672737 T^{22} - 6108461518360470 T^{23} + 9801762343983038 T^{24} - 11069822507365459 T^{25} + 9904578032905937 T^{26} \)
$19$ \( 1 - 14 T + 248 T^{2} - 2424 T^{3} + 25701 T^{4} - 198294 T^{5} + 1592044 T^{6} - 10285272 T^{7} + 68058462 T^{8} - 379790666 T^{9} + 2155393863 T^{10} - 10559591841 T^{11} + 52449577423 T^{12} - 227152092088 T^{13} + 996541971037 T^{14} - 3812012654601 T^{15} + 14783846506317 T^{16} - 49494699383786 T^{17} + 168519489699738 T^{18} - 483879682564632 T^{19} + 1423083138844516 T^{20} - 3367738649652054 T^{21} + 8293396520618079 T^{22} - 14861704608909624 T^{23} + 28889584206758312 T^{24} - 30986408866926254 T^{25} + 42052983462257059 T^{26} \)
$23$ \( 1 - 5 T + 215 T^{2} - 1100 T^{3} + 22269 T^{4} - 114329 T^{5} + 1478099 T^{6} - 7463909 T^{7} + 70504439 T^{8} - 342737121 T^{9} + 2566070728 T^{10} - 11715614720 T^{11} + 73674321281 T^{12} - 306620883056 T^{13} + 1694509389463 T^{14} - 6197560186880 T^{15} + 31221382547576 T^{16} - 95911898677761 T^{17} + 453790752426577 T^{18} - 1104926404230101 T^{19} + 5032669088385253 T^{20} - 8953216636191449 T^{21} + 40109868618119547 T^{22} - 45569162335013900 T^{23} + 204854097951494305 T^{24} - 109573122160101605 T^{25} + 504036361936467383 T^{26} \)
$29$ \( 1 - 10 T + 271 T^{2} - 2320 T^{3} + 34320 T^{4} - 250863 T^{5} + 2687247 T^{6} - 16896957 T^{7} + 146990240 T^{8} - 807948642 T^{9} + 6107409596 T^{10} - 30083037390 T^{11} + 206972882469 T^{12} - 938460561856 T^{13} + 6002213591601 T^{14} - 25299834444990 T^{15} + 148953612636844 T^{16} - 571446723462402 T^{17} + 3014938714185760 T^{18} - 10050704077534197 T^{19} + 46354678361731323 T^{20} - 125493315894635343 T^{21} + 497885249891824080 T^{22} - 976040781256466320 T^{23} + 3306338146506279659 T^{24} - 3538147832054690410 T^{25} + 10260628712958602189 T^{26} \)
$31$ \( 1 + T + 288 T^{2} + 128 T^{3} + 40625 T^{4} + 2854 T^{5} + 3714170 T^{6} - 635768 T^{7} + 245031489 T^{8} - 75359216 T^{9} + 12295404095 T^{10} - 4425200321 T^{11} + 482112034156 T^{12} - 167128670114 T^{13} + 14945473058836 T^{14} - 4252617508481 T^{15} + 366292383394145 T^{16} - 69595818519536 T^{17} + 7015043498335839 T^{18} - 564246440262008 T^{19} + 102186525952652870 T^{20} + 2434151020856614 T^{21} + 1074109650277259375 T^{22} + 104912420733542528 T^{23} + 7317641346164591328 T^{24} + 787662783788549761 T^{25} + 24417546297445042591 T^{26} \)
$37$ \( 1 + 8 T + 275 T^{2} + 1701 T^{3} + 34615 T^{4} + 165433 T^{5} + 2730912 T^{6} + 9899925 T^{7} + 157492443 T^{8} + 426504573 T^{9} + 7393508744 T^{10} + 15442255673 T^{11} + 303399072842 T^{12} + 552624316190 T^{13} + 11225765695154 T^{14} + 21140448016337 T^{15} + 374503398409832 T^{16} + 799338237038253 T^{17} + 10921149195216951 T^{18} + 25400499019619325 T^{19} + 259250602445035296 T^{20} + 581080013500512793 T^{21} + 4498625623006590355 T^{22} + 8179402017482761149 T^{23} + 48927345989351613575 T^{24} + 52663616046720282248 T^{25} + \)\(24\!\cdots\!97\)\( T^{26} \)
$41$ \( 1 - 21 T + 540 T^{2} - 7636 T^{3} + 116825 T^{4} - 1280643 T^{5} + 14810923 T^{6} - 135199551 T^{7} + 1296033361 T^{8} - 10264106178 T^{9} + 85277910821 T^{10} - 598271745082 T^{11} + 4396906526603 T^{12} - 27518809751956 T^{13} + 180273167590723 T^{14} - 1005694803482842 T^{15} + 5877438891694141 T^{16} - 29003910937651458 T^{17} + 150153501574721561 T^{18} - 642211960586395791 T^{19} + 2884490554372402163 T^{20} - 10225838600197204803 T^{21} + 38246394485574493825 T^{22} - \)\(10\!\cdots\!36\)\( T^{23} + \)\(29\!\cdots\!40\)\( T^{24} - \)\(47\!\cdots\!01\)\( T^{25} + \)\(92\!\cdots\!21\)\( T^{26} \)
$43$ \( 1 - 11 T + 343 T^{2} - 2734 T^{3} + 54061 T^{4} - 355546 T^{5} + 5728635 T^{6} - 33015751 T^{7} + 460695872 T^{8} - 2369020056 T^{9} + 29408375578 T^{10} - 136593716802 T^{11} + 1532816968042 T^{12} - 6469732775516 T^{13} + 65911129625806 T^{14} - 252561782366898 T^{15} + 2338171717080046 T^{16} - 8099208136472856 T^{17} + 67726182839247296 T^{18} - 208704548406384799 T^{19} + 1557149609238948945 T^{20} - 4155692855899925146 T^{21} + 27170659193917669423 T^{22} - 59085792644519136766 T^{23} + \)\(31\!\cdots\!01\)\( T^{24} - \)\(43\!\cdots\!11\)\( T^{25} + \)\(17\!\cdots\!43\)\( T^{26} \)
$47$ \( 1 - 22 T + 642 T^{2} - 10096 T^{3} + 174563 T^{4} - 2175831 T^{5} + 28224583 T^{6} - 293657148 T^{7} + 3111592527 T^{8} - 27806395138 T^{9} + 250636904182 T^{10} - 1953859409661 T^{11} + 15297994055552 T^{12} - 104712788542460 T^{13} + 719005720610944 T^{14} - 4316075435941149 T^{15} + 26021875302887786 T^{16} - 135686338033390978 T^{17} + 713628209885962689 T^{18} - 3165393631192021692 T^{19} + 14299226313226941929 T^{20} - 51809335668546098391 T^{21} + \)\(19\!\cdots\!21\)\( T^{22} - \)\(53\!\cdots\!04\)\( T^{23} + \)\(15\!\cdots\!26\)\( T^{24} - \)\(25\!\cdots\!02\)\( T^{25} + \)\(54\!\cdots\!27\)\( T^{26} \)
$53$ \( 1 - 16 T + 703 T^{2} - 9307 T^{3} + 224125 T^{4} - 2520330 T^{5} + 43323009 T^{6} - 420329087 T^{7} + 5696644844 T^{8} - 48096154711 T^{9} + 539874071107 T^{10} - 3978515326043 T^{11} + 38008878473513 T^{12} - 243974894241370 T^{13} + 2014470559096189 T^{14} - 11175649550854787 T^{15} + 80374832084196839 T^{16} - 379501794920205991 T^{17} + 2382311198982488092 T^{18} - 9316325677290859223 T^{19} + 50892021283558609533 T^{20} - \)\(15\!\cdots\!30\)\( T^{21} + \)\(73\!\cdots\!25\)\( T^{22} - \)\(16\!\cdots\!43\)\( T^{23} + \)\(65\!\cdots\!91\)\( T^{24} - \)\(78\!\cdots\!56\)\( T^{25} + \)\(26\!\cdots\!73\)\( T^{26} \)
$59$ \( 1 - 19 T + 588 T^{2} - 8387 T^{3} + 150323 T^{4} - 1725136 T^{5} + 23246896 T^{6} - 226730256 T^{7} + 2557136489 T^{8} - 22130373950 T^{9} + 220805866149 T^{10} - 1736747698856 T^{11} + 15678767927188 T^{12} - 112576920082032 T^{13} + 925047307704092 T^{14} - 6045618739717736 T^{15} + 45348887983815471 T^{16} - 268161730217145950 T^{17} + 1828159011845646211 T^{18} - 9563603190640542096 T^{19} + 57853422247832871824 T^{20} - \)\(25\!\cdots\!56\)\( T^{21} + \)\(13\!\cdots\!97\)\( T^{22} - \)\(42\!\cdots\!87\)\( T^{23} + \)\(17\!\cdots\!92\)\( T^{24} - \)\(33\!\cdots\!39\)\( T^{25} + \)\(10\!\cdots\!79\)\( T^{26} \)
$61$ \( ( 1 - T )^{13} \)
$67$ \( 1 - 12 T + 476 T^{2} - 5497 T^{3} + 116752 T^{4} - 1262381 T^{5} + 19411279 T^{6} - 192068545 T^{7} + 2419350272 T^{8} - 21779688147 T^{9} + 238485194313 T^{10} - 1959319395389 T^{11} + 19212910079827 T^{12} - 144341874973268 T^{13} + 1287264975348409 T^{14} - 8795384765901221 T^{15} + 71727522497160819 T^{16} - 438885131192462787 T^{17} + 3266425544854479104 T^{18} - 17374209846253774105 T^{19} + \)\(11\!\cdots\!17\)\( T^{20} - \)\(51\!\cdots\!21\)\( T^{21} + \)\(31\!\cdots\!44\)\( T^{22} - \)\(10\!\cdots\!53\)\( T^{23} + \)\(58\!\cdots\!08\)\( T^{24} - \)\(98\!\cdots\!32\)\( T^{25} + \)\(54\!\cdots\!87\)\( T^{26} \)
$71$ \( 1 - 5 T + 667 T^{2} - 2790 T^{3} + 210426 T^{4} - 721315 T^{5} + 41963243 T^{6} - 115239987 T^{7} + 5974373412 T^{8} - 12982883096 T^{9} + 650776182024 T^{10} - 1141711263752 T^{11} + 56611739724711 T^{12} - 85847839551638 T^{13} + 4019433520454481 T^{14} - 5755366480573832 T^{15} + 232919954084391864 T^{16} - 329916883695844376 T^{17} + 10779139863764415612 T^{18} - 14762275053752349027 T^{19} + \)\(38\!\cdots\!13\)\( T^{20} - \)\(46\!\cdots\!15\)\( T^{21} + \)\(96\!\cdots\!06\)\( T^{22} - \)\(90\!\cdots\!90\)\( T^{23} + \)\(15\!\cdots\!57\)\( T^{24} - \)\(82\!\cdots\!05\)\( T^{25} + \)\(11\!\cdots\!11\)\( T^{26} \)
$73$ \( 1 - 18 T + 722 T^{2} - 11164 T^{3} + 249956 T^{4} - 3378036 T^{5} + 55268985 T^{6} - 658789573 T^{7} + 8728233621 T^{8} - 92269947656 T^{9} + 1040846671509 T^{10} - 9788272584286 T^{11} + 96495170021146 T^{12} - 807214488838654 T^{13} + 7044147411543658 T^{14} - 52161704601660094 T^{15} + 404907049610416653 T^{16} - 2620304210592473096 T^{17} + 18094253176762628253 T^{18} - 99697410317215684597 T^{19} + \)\(61\!\cdots\!45\)\( T^{20} - \)\(27\!\cdots\!16\)\( T^{21} + \)\(14\!\cdots\!28\)\( T^{22} - \)\(47\!\cdots\!36\)\( T^{23} + \)\(22\!\cdots\!94\)\( T^{24} - \)\(41\!\cdots\!78\)\( T^{25} + \)\(16\!\cdots\!33\)\( T^{26} \)
$79$ \( 1 + T + 592 T^{2} + 1682 T^{3} + 175369 T^{4} + 744808 T^{5} + 34871614 T^{6} + 181072780 T^{7} + 5201860224 T^{8} + 29584351201 T^{9} + 613761565080 T^{10} + 3530100791245 T^{11} + 58889113102736 T^{12} + 318873705758666 T^{13} + 4652239935116144 T^{14} + 22031359038160045 T^{15} + 302608388285478120 T^{16} + 1152312875611397281 T^{17} + 16006417288962773376 T^{18} + 44016521354313818380 T^{19} + \)\(66\!\cdots\!26\)\( T^{20} + \)\(11\!\cdots\!88\)\( T^{21} + \)\(21\!\cdots\!11\)\( T^{22} + \)\(15\!\cdots\!82\)\( T^{23} + \)\(44\!\cdots\!68\)\( T^{24} + \)\(59\!\cdots\!41\)\( T^{25} + \)\(46\!\cdots\!39\)\( T^{26} \)
$83$ \( 1 - 48 T + 1946 T^{2} - 54074 T^{3} + 1322286 T^{4} - 26651308 T^{5} + 485319219 T^{6} - 7734534804 T^{7} + 113100793526 T^{8} - 1486614970798 T^{9} + 18090592348036 T^{10} - 200478795669543 T^{11} + 2066582253204772 T^{12} - 19506872219703810 T^{13} + 171526327015996076 T^{14} - 1381098423367481727 T^{15} + 10343965526906460332 T^{16} - 70552250487537110158 T^{17} + \)\(44\!\cdots\!18\)\( T^{18} - \)\(25\!\cdots\!76\)\( T^{19} + \)\(13\!\cdots\!13\)\( T^{20} - \)\(60\!\cdots\!28\)\( T^{21} + \)\(24\!\cdots\!58\)\( T^{22} - \)\(83\!\cdots\!26\)\( T^{23} + \)\(25\!\cdots\!82\)\( T^{24} - \)\(51\!\cdots\!28\)\( T^{25} + \)\(88\!\cdots\!63\)\( T^{26} \)
$89$ \( 1 - 15 T + 786 T^{2} - 11565 T^{3} + 310984 T^{4} - 4296636 T^{5} + 81310600 T^{6} - 1023800184 T^{7} + 15480612414 T^{8} - 175057827139 T^{9} + 2244424047975 T^{10} - 22673108801304 T^{11} + 253614549212220 T^{12} - 2281130993837004 T^{13} + 22571694879887580 T^{14} - 179593694815128984 T^{15} + 1582249376676887775 T^{16} - 10983520379291478499 T^{17} + 86444660026703399886 T^{18} - \)\(50\!\cdots\!24\)\( T^{19} + \)\(35\!\cdots\!00\)\( T^{20} - \)\(16\!\cdots\!16\)\( T^{21} + \)\(10\!\cdots\!56\)\( T^{22} - \)\(36\!\cdots\!65\)\( T^{23} + \)\(21\!\cdots\!54\)\( T^{24} - \)\(37\!\cdots\!15\)\( T^{25} + \)\(21\!\cdots\!69\)\( T^{26} \)
$97$ \( 1 + 17 T + 938 T^{2} + 14141 T^{3} + 424719 T^{4} + 5744714 T^{5} + 123323034 T^{6} + 1505382982 T^{7} + 25668744301 T^{8} + 283066250203 T^{9} + 4047350409404 T^{10} + 40194221467436 T^{11} + 497446404474603 T^{12} + 4416178399171106 T^{13} + 48252301234036491 T^{14} + 378187429787105324 T^{15} + 3693907440201976892 T^{16} + 25059651605837694043 T^{17} + \)\(22\!\cdots\!57\)\( T^{18} + \)\(12\!\cdots\!78\)\( T^{19} + \)\(99\!\cdots\!42\)\( T^{20} + \)\(45\!\cdots\!54\)\( T^{21} + \)\(32\!\cdots\!23\)\( T^{22} + \)\(10\!\cdots\!09\)\( T^{23} + \)\(67\!\cdots\!14\)\( T^{24} + \)\(11\!\cdots\!97\)\( T^{25} + \)\(67\!\cdots\!77\)\( T^{26} \)
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