Properties

Label 2013.2.a.b
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: \(x^{11} - 2 x^{10} - 14 x^{9} + 27 x^{8} + 66 x^{7} - 125 x^{6} - 115 x^{5} + 227 x^{4} + 40 x^{3} - 129 x^{2} + 26 x - 1\)
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 \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{11} - 2 x^{10} - 14 x^{9} + 27 x^{8} + 66 x^{7} - 125 x^{6} - 115 x^{5} + 227 x^{4} + 40 x^{3} - 129 x^{2} + 26 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\( 6 \nu^{10} - 3 \nu^{9} - 80 \nu^{8} + 42 \nu^{7} + 340 \nu^{6} - 257 \nu^{5} - 506 \nu^{4} + 688 \nu^{3} + 167 \nu^{2} - 532 \nu + 72 \)\()/17\)
\(\beta_{4}\)\(=\)\((\)\( 7 \nu^{10} + 5 \nu^{9} - 99 \nu^{8} - 70 \nu^{7} + 459 \nu^{6} + 264 \nu^{5} - 834 \nu^{4} - 257 \nu^{3} + 583 \nu^{2} + 48 \nu - 86 \)\()/17\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{10} - 9 \nu^{9} - 19 \nu^{8} + 126 \nu^{7} + 136 \nu^{6} - 584 \nu^{5} - 379 \nu^{4} + 1061 \nu^{3} + 280 \nu^{2} - 678 \nu + 46 \)\()/17\)
\(\beta_{6}\)\(=\)\((\)\( -13 \nu^{10} - 2 \nu^{9} + 179 \nu^{8} + 45 \nu^{7} - 799 \nu^{6} - 211 \nu^{5} + 1323 \nu^{4} + 266 \nu^{3} - 699 \nu^{2} - 111 \nu + 31 \)\()/17\)
\(\beta_{7}\)\(=\)\((\)\( -20 \nu^{10} - 7 \nu^{9} + 278 \nu^{8} + 115 \nu^{7} - 1258 \nu^{6} - 475 \nu^{5} + 2157 \nu^{4} + 506 \nu^{3} - 1265 \nu^{2} - 74 \nu + 66 \)\()/17\)
\(\beta_{8}\)\(=\)\((\)\( 8 \nu^{10} - 21 \nu^{9} - 118 \nu^{8} + 277 \nu^{7} + 612 \nu^{6} - 1204 \nu^{5} - 1247 \nu^{4} + 1960 \nu^{3} + 676 \nu^{2} - 1004 \nu + 113 \)\()/17\)
\(\beta_{9}\)\(=\)\((\)\( 23 \nu^{10} - 3 \nu^{9} - 318 \nu^{8} + 25 \nu^{7} + 1428 \nu^{6} - 189 \nu^{5} - 2359 \nu^{4} + 705 \nu^{3} + 1119 \nu^{2} - 583 \nu + 89 \)\()/17\)
\(\beta_{10}\)\(=\)\((\)\( 20 \nu^{10} - 10 \nu^{9} - 278 \nu^{8} + 123 \nu^{7} + 1275 \nu^{6} - 613 \nu^{5} - 2225 \nu^{4} + 1347 \nu^{3} + 1231 \nu^{2} - 878 \nu + 36 \)\()/17\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(-\beta_{7} + \beta_{6} - \beta_{4} + \beta_{2} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-\beta_{10} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} + 6 \beta_{2} + \beta_{1} + 14\)
\(\nu^{5}\)\(=\)\(\beta_{8} - 9 \beta_{7} + 10 \beta_{6} - 2 \beta_{5} - 8 \beta_{4} + 9 \beta_{2} + 28 \beta_{1} + 2\)
\(\nu^{6}\)\(=\)\(-9 \beta_{10} - 2 \beta_{9} + 11 \beta_{8} - 13 \beta_{7} + 13 \beta_{6} - 12 \beta_{5} - \beta_{4} + 11 \beta_{3} + 37 \beta_{2} + 13 \beta_{1} + 75\)
\(\nu^{7}\)\(=\)\(-\beta_{10} + 13 \beta_{8} - 68 \beta_{7} + 81 \beta_{6} - 25 \beta_{5} - 54 \beta_{4} + 2 \beta_{3} + 70 \beta_{2} + 167 \beta_{1} + 28\)
\(\nu^{8}\)\(=\)\(-65 \beta_{10} - 25 \beta_{9} + 92 \beta_{8} - 123 \beta_{7} + 125 \beta_{6} - 108 \beta_{5} - 20 \beta_{4} + 92 \beta_{3} + 242 \beta_{2} + 128 \beta_{1} + 434\)
\(\nu^{9}\)\(=\)\(-20 \beta_{10} - 2 \beta_{9} + 125 \beta_{8} - 495 \beta_{7} + 612 \beta_{6} - 230 \beta_{5} - 354 \beta_{4} + 35 \beta_{3} + 524 \beta_{2} + 1044 \beta_{1} + 283\)
\(\nu^{10}\)\(=\)\(-444 \beta_{10} - 221 \beta_{9} + 702 \beta_{8} - 1030 \beta_{7} + 1067 \beta_{6} - 870 \beta_{5} - 237 \beta_{4} + 694 \beta_{3} + 1651 \beta_{2} + 1122 \beta_{1} + 2653\)

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
2.70023
2.05394
2.04468
1.21038
0.942842
0.175402
0.0512060
−1.03852
−1.65258
−2.09663
−2.39095
−2.70023 −1.00000 5.29122 2.21957 2.70023 −2.11895 −8.88702 1.00000 −5.99333
1.2 −2.05394 −1.00000 2.21866 −0.0248942 2.05394 1.52855 −0.449108 1.00000 0.0511312
1.3 −2.04468 −1.00000 2.18073 −1.44717 2.04468 −4.07149 −0.369530 1.00000 2.95900
1.4 −1.21038 −1.00000 −0.534974 −4.03730 1.21038 0.314382 3.06829 1.00000 4.88668
1.5 −0.942842 −1.00000 −1.11105 3.49194 0.942842 1.81837 2.93323 1.00000 −3.29234
1.6 −0.175402 −1.00000 −1.96923 −1.94648 0.175402 −2.33819 0.696212 1.00000 0.341417
1.7 −0.0512060 −1.00000 −1.99738 3.74211 0.0512060 −4.65643 0.204690 1.00000 −0.191618
1.8 1.03852 −1.00000 −0.921477 −1.07279 −1.03852 1.21131 −3.03401 1.00000 −1.11412
1.9 1.65258 −1.00000 0.731020 2.07534 −1.65258 −0.262572 −2.09709 1.00000 3.42966
1.10 2.09663 −1.00000 2.39586 −1.65820 −2.09663 −1.74412 0.829977 1.00000 −3.47663
1.11 2.39095 −1.00000 3.71663 −2.34211 −2.39095 −0.680854 4.10437 1.00000 −5.59986
\(n\): 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.b 11
3.b odd 2 1 6039.2.a.c 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2013.2.a.b 11 1.a even 1 1 trivial
6039.2.a.c 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

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 8 T^{2} + 13 T^{3} + 34 T^{4} + 53 T^{5} + 113 T^{6} + 169 T^{7} + 306 T^{8} + 437 T^{9} + 706 T^{10} + 957 T^{11} + 1412 T^{12} + 1748 T^{13} + 2448 T^{14} + 2704 T^{15} + 3616 T^{16} + 3392 T^{17} + 4352 T^{18} + 3328 T^{19} + 4096 T^{20} + 2048 T^{21} + 2048 T^{22} \)
$3$ \( ( 1 + T )^{11} \)
$5$ \( 1 + T + 22 T^{2} + 10 T^{3} + 250 T^{4} + 92 T^{5} + 2061 T^{6} + 941 T^{7} + 13377 T^{8} + 8028 T^{9} + 75394 T^{10} + 50051 T^{11} + 376970 T^{12} + 200700 T^{13} + 1672125 T^{14} + 588125 T^{15} + 6440625 T^{16} + 1437500 T^{17} + 19531250 T^{18} + 3906250 T^{19} + 42968750 T^{20} + 9765625 T^{21} + 48828125 T^{22} \)
$7$ \( 1 + 11 T + 108 T^{2} + 729 T^{3} + 4381 T^{4} + 21873 T^{5} + 98878 T^{6} + 393345 T^{7} + 1434035 T^{8} + 4707756 T^{9} + 14259573 T^{10} + 39234779 T^{11} + 99817011 T^{12} + 230680044 T^{13} + 491874005 T^{14} + 944421345 T^{15} + 1661842546 T^{16} + 2573336577 T^{17} + 3607941883 T^{18} + 4202539929 T^{19} + 4358189556 T^{20} + 3107227739 T^{21} + 1977326743 T^{22} \)
$11$ \( ( 1 - T )^{11} \)
$13$ \( 1 + 13 T + 150 T^{2} + 1197 T^{3} + 8477 T^{4} + 50644 T^{5} + 275926 T^{6} + 1352043 T^{7} + 6176229 T^{8} + 26038009 T^{9} + 103599080 T^{10} + 383806651 T^{11} + 1346788040 T^{12} + 4400423521 T^{13} + 13569175113 T^{14} + 38615700123 T^{15} + 102449392318 T^{16} + 244448914996 T^{17} + 531919178609 T^{18} + 976429673037 T^{19} + 1590674905950 T^{20} + 1792160394037 T^{21} + 1792160394037 T^{22} \)
$17$ \( 1 + 13 T + 156 T^{2} + 1360 T^{3} + 10974 T^{4} + 74673 T^{5} + 474033 T^{6} + 2699445 T^{7} + 14398643 T^{8} + 70304093 T^{9} + 323291325 T^{10} + 1374162515 T^{11} + 5495952525 T^{12} + 20317882877 T^{13} + 70740533059 T^{14} + 225460345845 T^{15} + 673059073281 T^{16} + 1802424689937 T^{17} + 4503056597502 T^{18} + 9487030119760 T^{19} + 18499708733532 T^{20} + 26207920705837 T^{21} + 34271896307633 T^{22} \)
$19$ \( 1 + 12 T + 192 T^{2} + 1600 T^{3} + 14502 T^{4} + 94088 T^{5} + 619600 T^{6} + 3372332 T^{7} + 17991486 T^{8} + 87117584 T^{9} + 405441617 T^{10} + 1808732257 T^{11} + 7703390723 T^{12} + 31449447824 T^{13} + 123403602474 T^{14} + 439485678572 T^{15} + 1534190940400 T^{16} + 4426452851528 T^{17} + 12962927958978 T^{18} + 27173700865600 T^{19} + 61956037973568 T^{20} + 73572795093612 T^{21} + 116490258898219 T^{22} \)
$23$ \( 1 + 3 T + 145 T^{2} + 198 T^{3} + 9728 T^{4} + 454 T^{5} + 429546 T^{6} - 396773 T^{7} + 14564661 T^{8} - 21122959 T^{9} + 403088352 T^{10} - 613049799 T^{11} + 9271032096 T^{12} - 11174045311 T^{13} + 177208230387 T^{14} - 111033353093 T^{15} + 2764705390278 T^{16} + 67208293606 T^{17} + 33122141948416 T^{18} + 15505575085638 T^{19} + 261167135912135 T^{20} + 124279533640947 T^{21} + 952809757913927 T^{22} \)
$29$ \( 1 - 2 T + 89 T^{2} - 122 T^{3} + 4647 T^{4} - 7017 T^{5} + 218100 T^{6} - 326437 T^{7} + 8599199 T^{8} - 11638759 T^{9} + 284538994 T^{10} - 368929043 T^{11} + 8251630826 T^{12} - 9788196319 T^{13} + 209725864411 T^{14} - 230882687797 T^{15} + 4473481596900 T^{16} - 4173875243457 T^{17} + 80160175207923 T^{18} - 61030062381242 T^{19} + 1291135991852341 T^{20} - 841414466600402 T^{21} + 12200509765705829 T^{22} \)
$31$ \( 1 - T + 222 T^{2} - 230 T^{3} + 24244 T^{4} - 24897 T^{5} + 1733170 T^{6} - 1711674 T^{7} + 90227039 T^{8} - 83314829 T^{9} + 3588768283 T^{10} - 2991531987 T^{11} + 111251816773 T^{12} - 80065550669 T^{13} + 2687953718849 T^{14} - 1580766884154 T^{15} + 49619185638670 T^{16} - 22096179145857 T^{17} + 667015816507084 T^{18} - 196164938611430 T^{19} + 5869596119668962 T^{20} - 819628286980801 T^{21} + 25408476896404831 T^{22} \)
$37$ \( 1 + 14 T + 291 T^{2} + 3245 T^{3} + 40744 T^{4} + 377581 T^{5} + 3641975 T^{6} + 28907272 T^{7} + 232333899 T^{8} + 1607659582 T^{9} + 11160234151 T^{10} + 67840750825 T^{11} + 412928663587 T^{12} + 2200885967758 T^{13} + 11768408986047 T^{14} + 54176881798792 T^{15} + 252548957795075 T^{16} + 968769543236629 T^{17} + 3867904401906952 T^{18} + 11397995827973645 T^{19} + 37818866280367407 T^{20} + 67320181213849886 T^{21} + 177917621779460413 T^{22} \)
$41$ \( 1 - 3 T + 200 T^{2} - 948 T^{3} + 21068 T^{4} - 116950 T^{5} + 1665137 T^{6} - 8954847 T^{7} + 104162277 T^{8} - 528618594 T^{9} + 5171339938 T^{10} - 24698308195 T^{11} + 212024937458 T^{12} - 888607856514 T^{13} + 7178968293117 T^{14} - 25304257413567 T^{15} + 192916446964537 T^{16} - 555524690984950 T^{17} + 4103083042124908 T^{18} - 7569709117206708 T^{19} + 65476386878792200 T^{20} - 40267977930457203 T^{21} + 550329031716248441 T^{22} \)
$43$ \( 1 + 21 T + 411 T^{2} + 5608 T^{3} + 71396 T^{4} + 760537 T^{5} + 7616040 T^{6} + 68041789 T^{7} + 573679600 T^{8} + 4423116819 T^{9} + 32274274640 T^{10} + 217535136353 T^{11} + 1387793809520 T^{12} + 8178342998331 T^{13} + 45611543957200 T^{14} + 232621336274989 T^{15} + 1119622182225720 T^{16} + 4807630489197313 T^{17} + 19406761558595372 T^{18} + 65547427156786408 T^{19} + 206565563506042473 T^{20} + 453841128578969229 T^{21} + 929293739471222707 T^{22} \)
$47$ \( 1 + 16 T + 408 T^{2} + 4882 T^{3} + 75334 T^{4} + 741371 T^{5} + 8694921 T^{6} + 73191812 T^{7} + 709299677 T^{8} + 5211390271 T^{9} + 43378149237 T^{10} + 280028273333 T^{11} + 2038773014139 T^{12} + 11511961108639 T^{13} + 73641620365171 T^{14} + 357152694371972 T^{15} + 1994136717609447 T^{16} + 7991397647676059 T^{17} + 38165946156959642 T^{18} + 116246701482717202 T^{19} + 456605233025928936 T^{20} + 841586115773280784 T^{21} + 2472159215084012303 T^{22} \)
$53$ \( 1 + 171 T^{2} - 447 T^{3} + 19622 T^{4} - 55268 T^{5} + 1874340 T^{6} - 4981360 T^{7} + 138786662 T^{8} - 376151679 T^{9} + 8696488649 T^{10} - 21797865799 T^{11} + 460913898397 T^{12} - 1056610066311 T^{13} + 20662141878574 T^{14} - 39305326434160 T^{15} + 783840540349620 T^{16} - 1224979910877572 T^{17} + 23050181985881614 T^{18} - 27830081613878367 T^{19} + 564259574198164743 T^{20} + 9269035929372191597 T^{22} \)
$59$ \( 1 - 3 T + 504 T^{2} - 1533 T^{3} + 119624 T^{4} - 367229 T^{5} + 17792142 T^{6} - 53981851 T^{7} + 1858021409 T^{8} - 5363837721 T^{9} + 144074909069 T^{10} - 375090942553 T^{11} + 8500419635071 T^{12} - 18671519106801 T^{13} + 381598578959011 T^{14} - 654117576015211 T^{15} + 12720034647058458 T^{16} - 15489915188450789 T^{17} + 297702445219988056 T^{18} - 225091060847424093 T^{19} + 4366149892602089256 T^{20} - 1533350259901924203 T^{21} + 30155888444737842659 T^{22} \)
$61$ \( ( 1 - T )^{11} \)
$67$ \( 1 + 24 T + 746 T^{2} + 11525 T^{3} + 205735 T^{4} + 2338279 T^{5} + 30851917 T^{6} + 278464946 T^{7} + 3068071825 T^{8} + 23594529666 T^{9} + 236961820908 T^{10} + 1668482725813 T^{11} + 15876442000836 T^{12} + 105915843670674 T^{13} + 922762486302475 T^{14} + 5611380821104466 T^{15} + 41653947740780119 T^{16} + 211516935399747151 T^{17} + 1246900502121127405 T^{18} + 4679929983840287525 T^{19} + 20296074659636030462 T^{20} + 43748107309242274776 T^{21} + \)\(12\!\cdots\!83\)\( T^{22} \)
$71$ \( 1 - 7 T + 423 T^{2} - 2938 T^{3} + 94891 T^{4} - 634016 T^{5} + 14512884 T^{6} - 91820873 T^{7} + 1664655219 T^{8} - 9737118634 T^{9} + 148962046506 T^{10} - 787502321349 T^{11} + 10576305301926 T^{12} - 49084815033994 T^{13} + 595798414087509 T^{14} - 2333322733817513 T^{15} + 26184571280458284 T^{16} - 81217629610456736 T^{17} + 863045046949880381 T^{18} - 1897223874800045818 T^{19} + 19393915803903940113 T^{20} - 22786704857069168407 T^{21} + \)\(23\!\cdots\!71\)\( T^{22} \)
$73$ \( 1 + 42 T + 1222 T^{2} + 25622 T^{3} + 447305 T^{4} + 6597184 T^{5} + 86476323 T^{6} + 1015901577 T^{7} + 10992487226 T^{8} + 109895882058 T^{9} + 1031664342334 T^{10} + 9069041561577 T^{11} + 75311496990382 T^{12} + 585635155487082 T^{13} + 4276264403196842 T^{14} + 28849817815926057 T^{15} + 179271608678392539 T^{16} + 998379736326170176 T^{17} + 4941556594584683585 T^{18} + 20663120474510143382 T^{19} + 71941078957503389686 T^{20} + \)\(18\!\cdots\!58\)\( T^{21} + \)\(31\!\cdots\!77\)\( T^{22} \)
$79$ \( 1 + 11 T + 690 T^{2} + 6490 T^{3} + 218178 T^{4} + 1770975 T^{5} + 42469286 T^{6} + 300331876 T^{7} + 5775405374 T^{8} + 35916525340 T^{9} + 588476371774 T^{10} + 3234511194975 T^{11} + 46489633370146 T^{12} + 224155034646940 T^{13} + 2847500090191586 T^{14} + 11697950897081956 T^{15} + 130680388247261114 T^{16} + 430501806541302975 T^{17} + 4189870454782198302 T^{18} + 9846036176293580890 T^{19} + 82697601228006640110 T^{20} + \)\(10\!\cdots\!11\)\( T^{21} + \)\(74\!\cdots\!79\)\( T^{22} \)
$83$ \( 1 + 34 T + 1164 T^{2} + 24674 T^{3} + 499533 T^{4} + 7832010 T^{5} + 117485561 T^{6} + 1476053272 T^{7} + 17943932359 T^{8} + 189853527918 T^{9} + 1966784410629 T^{10} + 18060985019151 T^{11} + 163243106082207 T^{12} + 1307900953827102 T^{13} + 10260107251755533 T^{14} + 70051009995676312 T^{15} + 462780399744655723 T^{16} + 2560600273629741690 T^{17} + 13555352959001344191 T^{18} + 55573058535798697634 T^{19} + \)\(21\!\cdots\!92\)\( T^{20} + \)\(52\!\cdots\!66\)\( T^{21} + \)\(12\!\cdots\!67\)\( T^{22} \)
$89$ \( 1 - 29 T + 1006 T^{2} - 20891 T^{3} + 434731 T^{4} - 7113247 T^{5} + 111586710 T^{6} - 1505638719 T^{7} + 19231282025 T^{8} - 219069317588 T^{9} + 2354328494051 T^{10} - 22873997682689 T^{11} + 209535235970539 T^{12} - 1735248064614548 T^{13} + 13557457657882225 T^{14} - 94467147366429279 T^{15} + 623106822358322790 T^{16} - 3535150676984460367 T^{17} + 19228732450468217699 T^{18} - 82239276739922174171 T^{19} + \)\(35\!\cdots\!54\)\( T^{20} - \)\(90\!\cdots\!29\)\( T^{21} + \)\(27\!\cdots\!89\)\( T^{22} \)
$97$ \( 1 + 45 T + 1376 T^{2} + 29931 T^{3} + 545438 T^{4} + 8375059 T^{5} + 116449942 T^{6} + 1465766429 T^{7} + 17473897571 T^{8} + 194927586130 T^{9} + 2087374168964 T^{10} + 20998963042063 T^{11} + 202475294389508 T^{12} + 1834073657897170 T^{13} + 15947954517817283 T^{14} + 129763248073307549 T^{15} + 999995274861915094 T^{16} + 6976189686628665811 T^{17} + 44070454689172998494 T^{18} + \)\(23\!\cdots\!91\)\( T^{19} + \)\(10\!\cdots\!92\)\( T^{20} + \)\(33\!\cdots\!05\)\( T^{21} + \)\(71\!\cdots\!53\)\( T^{22} \)
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