Properties

Label 115.2.e.a
Level $115$
Weight $2$
Character orbit 115.e
Analytic conductor $0.918$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 115.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.918279623245\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 6 x^{18} + 3 x^{16} + 80 x^{14} - 600 x^{12} + 3500 x^{10} - 15000 x^{8} + 50000 x^{6} + 46875 x^{4} - 2343750 x^{2} + 9765625\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{3} q^{2} + \beta_{2} q^{3} + \beta_{4} q^{4} + \beta_{18} q^{5} + ( -1 + \beta_{3} - \beta_{5} - \beta_{10} ) q^{6} -\beta_{1} q^{7} + ( \beta_{2} - \beta_{4} + \beta_{8} - \beta_{10} - \beta_{14} ) q^{8} + ( -\beta_{2} + \beta_{4} - \beta_{11} + \beta_{14} ) q^{9} +O(q^{10})\) \( q -\beta_{3} q^{2} + \beta_{2} q^{3} + \beta_{4} q^{4} + \beta_{18} q^{5} + ( -1 + \beta_{3} - \beta_{5} - \beta_{10} ) q^{6} -\beta_{1} q^{7} + ( \beta_{2} - \beta_{4} + \beta_{8} - \beta_{10} - \beta_{14} ) q^{8} + ( -\beta_{2} + \beta_{4} - \beta_{11} + \beta_{14} ) q^{9} -\beta_{12} q^{10} -\beta_{9} q^{11} + ( -1 + \beta_{3} - \beta_{4} - \beta_{6} - \beta_{8} + \beta_{11} ) q^{12} + ( -\beta_{2} + \beta_{10} + \beta_{14} ) q^{13} + ( \beta_{1} - \beta_{7} + \beta_{12} + \beta_{13} - \beta_{18} ) q^{14} + ( \beta_{1} - \beta_{7} + \beta_{16} + \beta_{17} - \beta_{18} - \beta_{19} ) q^{15} + ( 1 - \beta_{3} + \beta_{5} + \beta_{10} ) q^{16} + ( -\beta_{1} + \beta_{7} + \beta_{9} - \beta_{16} - \beta_{17} + \beta_{19} ) q^{17} + ( -\beta_{4} + \beta_{8} ) q^{18} + ( -\beta_{15} + \beta_{16} + \beta_{17} ) q^{19} + ( \beta_{1} + \beta_{15} ) q^{20} + ( \beta_{7} + \beta_{9} + \beta_{12} - \beta_{15} - \beta_{16} - \beta_{17} - \beta_{18} ) q^{21} + ( \beta_{7} - \beta_{16} - \beta_{17} ) q^{22} + ( -\beta_{2} + \beta_{4} - \beta_{8} - \beta_{12} - \beta_{13} + \beta_{15} + \beta_{18} ) q^{23} + ( -\beta_{2} + \beta_{4} + \beta_{6} - \beta_{11} - \beta_{14} ) q^{24} + ( \beta_{3} + 2 \beta_{5} + 2 \beta_{10} + \beta_{14} ) q^{25} + ( -1 - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{10} + \beta_{11} ) q^{26} + ( 1 - \beta_{4} + \beta_{6} - \beta_{8} - \beta_{11} ) q^{27} + ( -\beta_{1} + \beta_{7} + \beta_{12} - \beta_{13} - \beta_{17} + 2 \beta_{18} + \beta_{19} ) q^{28} + ( \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{11} - 2 \beta_{14} ) q^{29} + ( -\beta_{1} - \beta_{13} + \beta_{15} + \beta_{18} + \beta_{19} ) q^{30} + ( -1 - \beta_{2} + 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{8} + \beta_{11} ) q^{31} + ( 1 - \beta_{3} - \beta_{4} + \beta_{6} - \beta_{8} + 2 \beta_{10} + 3 \beta_{11} - 2 \beta_{14} ) q^{32} + ( \beta_{1} - \beta_{12} - \beta_{13} + \beta_{16} + 2 \beta_{17} - 2 \beta_{18} - \beta_{19} ) q^{33} + ( \beta_{1} - \beta_{7} + \beta_{12} + \beta_{13} - \beta_{15} + \beta_{16} + \beta_{17} - \beta_{18} - \beta_{19} ) q^{34} + ( -1 + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} + 3 \beta_{6} + 3 \beta_{8} - 2 \beta_{10} - 2 \beta_{11} - \beta_{14} ) q^{35} + ( \beta_{2} - 2 \beta_{3} + 2 \beta_{5} - \beta_{11} ) q^{36} + ( \beta_{1} - \beta_{7} - \beta_{9} + \beta_{12} + \beta_{15} + \beta_{16} + \beta_{17} - \beta_{18} - \beta_{19} ) q^{37} + ( \beta_{1} - \beta_{7} - \beta_{9} - \beta_{12} + \beta_{13} + \beta_{17} - 2 \beta_{18} ) q^{38} + ( -\beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{14} ) q^{39} + ( -2 \beta_{1} + \beta_{7} + \beta_{9} - \beta_{13} - \beta_{16} - \beta_{17} + \beta_{18} ) q^{40} + ( -1 - 2 \beta_{8} ) q^{41} + ( -\beta_{9} - \beta_{12} - \beta_{15} + \beta_{18} - \beta_{19} ) q^{42} + ( -\beta_{7} - \beta_{16} - \beta_{17} ) q^{43} + ( \beta_{7} - \beta_{12} + \beta_{18} + \beta_{19} ) q^{44} + ( \beta_{1} + \beta_{19} ) q^{45} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} + 2 \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} + \beta_{13} ) q^{46} + ( -2 + 2 \beta_{3} + \beta_{4} - 2 \beta_{6} + \beta_{8} - 2 \beta_{10} + \beta_{11} + 2 \beta_{14} ) q^{47} + ( \beta_{4} - 3 \beta_{5} - \beta_{8} ) q^{48} + ( -2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{11} + \beta_{14} ) q^{49} + ( 1 - \beta_{2} - \beta_{4} - 3 \beta_{6} + 2 \beta_{8} + 2 \beta_{11} ) q^{50} + ( -3 \beta_{1} + \beta_{7} + \beta_{12} + \beta_{13} - 2 \beta_{16} + 3 \beta_{18} + \beta_{19} ) q^{51} + ( 2 - \beta_{3} + \beta_{4} + 2 \beta_{6} + \beta_{8} - \beta_{10} - \beta_{11} + \beta_{14} ) q^{52} + ( -\beta_{1} + \beta_{9} + \beta_{12} + 2 \beta_{13} - \beta_{16} - 2 \beta_{17} + 2 \beta_{18} ) q^{53} + ( -\beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{11} + 3 \beta_{14} ) q^{54} + ( 1 + 4 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} - 3 \beta_{6} + 2 \beta_{8} - 3 \beta_{10} - 3 \beta_{11} + \beta_{14} ) q^{55} + ( \beta_{1} - \beta_{7} - \beta_{12} + \beta_{13} - \beta_{15} + \beta_{16} - \beta_{17} - 3 \beta_{18} - \beta_{19} ) q^{56} + ( 2 \beta_{1} - \beta_{7} + \beta_{16} + \beta_{17} ) q^{57} + ( 3 + \beta_{4} + 3 \beta_{5} - 3 \beta_{6} - \beta_{8} + \beta_{10} + \beta_{14} ) q^{58} + ( 3 \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + 3 \beta_{11} - 3 \beta_{14} ) q^{59} + ( -\beta_{1} - \beta_{7} - \beta_{9} ) q^{60} + ( 2 \beta_{1} - \beta_{7} - \beta_{9} - \beta_{12} + 2 \beta_{16} - 3 \beta_{18} - \beta_{19} ) q^{61} + ( -3 + 3 \beta_{3} - 3 \beta_{6} - \beta_{10} - 2 \beta_{11} + \beta_{14} ) q^{62} + ( -\beta_{1} + \beta_{7} + \beta_{15} - \beta_{17} + 3 \beta_{18} + \beta_{19} ) q^{63} + ( \beta_{2} + 3 \beta_{4} - \beta_{6} + \beta_{11} + \beta_{14} ) q^{64} + ( \beta_{1} - \beta_{9} - \beta_{15} + \beta_{16} + \beta_{17} - \beta_{18} ) q^{65} + ( -\beta_{1} + \beta_{13} + \beta_{15} + \beta_{16} + \beta_{17} ) q^{66} + ( -\beta_{1} + \beta_{7} - \beta_{12} - \beta_{15} - \beta_{16} - \beta_{17} + \beta_{18} ) q^{67} + ( \beta_{9} + \beta_{12} - \beta_{15} - \beta_{18} - \beta_{19} ) q^{68} + ( -\beta_{1} + 2 \beta_{2} + \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{11} - \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} - \beta_{17} + \beta_{19} ) q^{69} + ( -2 + 2 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - 4 \beta_{5} + \beta_{6} - 4 \beta_{8} + \beta_{10} + \beta_{11} - 2 \beta_{14} ) q^{70} + ( 1 - \beta_{3} + \beta_{5} + 2 \beta_{8} - 4 \beta_{10} ) q^{71} + ( 3 + 2 \beta_{3} + 3 \beta_{6} - \beta_{10} + \beta_{14} ) q^{72} + ( -3 - \beta_{2} + \beta_{4} + 3 \beta_{6} - \beta_{8} - \beta_{10} - \beta_{14} ) q^{73} + ( -3 \beta_{1} + \beta_{7} + \beta_{9} + \beta_{12} - \beta_{13} - 4 \beta_{17} + \beta_{18} + \beta_{19} ) q^{74} + ( 2 - 2 \beta_{2} - 4 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} - \beta_{6} - \beta_{8} + 2 \beta_{10} - \beta_{11} + \beta_{14} ) q^{75} + ( \beta_{1} - 3 \beta_{13} + 2 \beta_{15} + 2 \beta_{17} + 4 \beta_{18} + \beta_{19} ) q^{76} + ( 1 - 6 \beta_{3} + 2 \beta_{4} + \beta_{6} + 2 \beta_{8} + 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{14} ) q^{77} + ( -3 - 3 \beta_{5} + 3 \beta_{6} - \beta_{10} - \beta_{14} ) q^{78} + ( 2 \beta_{1} - \beta_{7} - \beta_{9} - \beta_{12} + 4 \beta_{17} - \beta_{18} - \beta_{19} ) q^{79} + ( \beta_{1} + \beta_{13} - \beta_{15} - \beta_{18} - \beta_{19} ) q^{80} + ( 5 - 2 \beta_{2} - \beta_{3} + \beta_{5} - 2 \beta_{8} + 4 \beta_{10} + 2 \beta_{11} ) q^{81} + ( 5 \beta_{3} + 2 \beta_{4} + 2 \beta_{8} - 2 \beta_{10} - 2 \beta_{11} + 2 \beta_{14} ) q^{82} + ( -\beta_{12} + \beta_{13} + \beta_{15} + \beta_{18} ) q^{83} + ( 2 \beta_{1} - \beta_{7} - \beta_{9} - \beta_{12} + 4 \beta_{17} - \beta_{18} - 2 \beta_{19} ) q^{84} + ( -3 - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + 4 \beta_{6} - \beta_{8} - \beta_{10} + 4 \beta_{11} - 3 \beta_{14} ) q^{85} + ( \beta_{7} + \beta_{12} - 2 \beta_{13} - 2 \beta_{16} + 3 \beta_{18} + \beta_{19} ) q^{86} + ( 2 - 2 \beta_{3} + \beta_{4} + 2 \beta_{6} + \beta_{8} + 2 \beta_{10} - \beta_{11} - 2 \beta_{14} ) q^{87} + ( \beta_{1} + \beta_{12} + \beta_{13} - 2 \beta_{15} + \beta_{16} + 2 \beta_{17} - 4 \beta_{18} - \beta_{19} ) q^{88} + ( -2 \beta_{1} - 2 \beta_{13} + \beta_{15} - \beta_{16} - \beta_{17} ) q^{89} + ( -\beta_{1} + \beta_{7} - \beta_{13} - \beta_{15} ) q^{90} + ( -\beta_{1} + \beta_{9} + \beta_{13} ) q^{91} + ( -2 - 3 \beta_{3} - \beta_{4} - 2 \beta_{6} - \beta_{8} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{14} + \beta_{15} - \beta_{18} ) q^{92} + ( 1 + 3 \beta_{2} - 3 \beta_{4} + 6 \beta_{5} - \beta_{6} + 3 \beta_{8} + \beta_{10} + \beta_{14} ) q^{93} + ( -\beta_{2} - \beta_{3} - 4 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{11} - 3 \beta_{14} ) q^{94} + ( 3 + 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} + \beta_{5} - 4 \beta_{6} + \beta_{8} + \beta_{10} + \beta_{11} - 2 \beta_{14} ) q^{95} + ( -4 + 3 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} + \beta_{8} - 3 \beta_{11} ) q^{96} + ( -\beta_{1} + \beta_{7} + \beta_{9} - \beta_{13} + \beta_{15} - 2 \beta_{16} + \beta_{17} + \beta_{18} + 2 \beta_{19} ) q^{97} + ( -1 - 2 \beta_{2} + 2 \beta_{4} + 7 \beta_{5} + \beta_{6} - 2 \beta_{8} + 3 \beta_{10} + 3 \beta_{14} ) q^{98} + ( -\beta_{1} + \beta_{7} + \beta_{9} + \beta_{12} + \beta_{13} + \beta_{15} - \beta_{16} - 5 \beta_{17} + \beta_{18} + \beta_{19} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q - 4q^{2} - 8q^{3} - 8q^{6} + 4q^{8} + O(q^{10}) \) \( 20q - 4q^{2} - 8q^{3} - 8q^{6} + 4q^{8} - 16q^{12} + 4q^{13} + 8q^{16} + 8q^{18} - 12q^{25} - 16q^{26} + 4q^{27} - 4q^{31} + 24q^{32} - 8q^{35} - 32q^{36} - 36q^{41} + 32q^{46} - 8q^{47} + 4q^{48} + 60q^{50} + 40q^{52} - 12q^{55} + 36q^{58} - 60q^{62} - 76q^{70} + 44q^{71} + 72q^{72} - 56q^{73} + 28q^{75} - 12q^{77} - 44q^{78} + 92q^{81} + 28q^{82} - 4q^{85} + 24q^{87} - 72q^{92} - 8q^{93} + 64q^{95} - 104q^{96} - 60q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 6 x^{18} + 3 x^{16} + 80 x^{14} - 600 x^{12} + 3500 x^{10} - 15000 x^{8} + 50000 x^{6} + 46875 x^{4} - 2343750 x^{2} + 9765625\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -301 \nu^{19} - 1894 \nu^{17} + 116922 \nu^{15} - 115180 \nu^{13} + 1787100 \nu^{11} - 9514750 \nu^{9} - 77653750 \nu^{7} + 153262500 \nu^{5} + 439953125 \nu^{3} + 450000000 \nu \)\()/ 10070312500 \)
\(\beta_{2}\)\(=\)\((\)\( 63 \nu^{18} - 7728 \nu^{16} + 15664 \nu^{14} - 116510 \nu^{12} + 556450 \nu^{10} + 1058000 \nu^{8} - 11445000 \nu^{6} + 176056250 \nu^{4} - 598453125 \nu^{2} - 12500000 \)\()/ 2014062500 \)
\(\beta_{3}\)\(=\)\((\)\( 531 \nu^{18} - 1886 \nu^{16} - 19082 \nu^{14} + 62380 \nu^{12} - 326350 \nu^{10} + 2664750 \nu^{8} - 22771250 \nu^{6} - 4012500 \nu^{4} + 507234375 \nu^{2} - 1725000000 \)\()/ 2014062500 \)
\(\beta_{4}\)\(=\)\((\)\( -423 \nu^{18} - 1712 \nu^{16} + 6231 \nu^{14} + 25785 \nu^{12} + 326675 \nu^{10} - 2886750 \nu^{8} - 5726875 \nu^{6} + 19646875 \nu^{4} - 110687500 \nu^{2} + 869531250 \)\()/ 1007031250 \)
\(\beta_{5}\)\(=\)\((\)\( -777 \nu^{18} - 938 \nu^{16} - 606 \nu^{14} + 118540 \nu^{12} - 555550 \nu^{10} - 778250 \nu^{8} + 11498750 \nu^{6} - 41412500 \nu^{4} + 386859375 \nu^{2} - 854687500 \)\()/ 2014062500 \)
\(\beta_{6}\)\(=\)\((\)\( 31 \nu^{18} + 31 \nu^{16} - 1124 \nu^{14} + 6146 \nu^{12} - 13260 \nu^{10} + 30550 \nu^{8} + 76500 \nu^{6} + 523750 \nu^{4} + 17015625 \nu^{2} - 95578125 \)\()/80562500\)
\(\beta_{7}\)\(=\)\((\)\( -244 \nu^{19} - 2661 \nu^{17} + 9018 \nu^{15} - 91270 \nu^{13} + 5150 \nu^{11} + 1901000 \nu^{9} - 24683750 \nu^{7} - 17106250 \nu^{5} + 81843750 \nu^{3} + 664453125 \nu \)\()/ 2014062500 \)
\(\beta_{8}\)\(=\)\((\)\( 634 \nu^{18} - 2529 \nu^{16} - 16248 \nu^{14} + 70045 \nu^{12} - 24900 \nu^{10} + 1659000 \nu^{8} - 3110000 \nu^{6} - 51409375 \nu^{4} + 71281250 \nu^{2} - 817578125 \)\()/ 1007031250 \)
\(\beta_{9}\)\(=\)\((\)\( 968 \nu^{19} - 1458 \nu^{17} + 58054 \nu^{15} + 324865 \nu^{13} - 242175 \nu^{11} + 5365500 \nu^{9} - 10170000 \nu^{7} - 65678125 \nu^{5} - 157828125 \nu^{3} + 1040625000 \nu \)\()/ 5035156250 \)
\(\beta_{10}\)\(=\)\((\)\( 1616 \nu^{18} - 7971 \nu^{16} - 752 \nu^{14} + 89080 \nu^{12} - 716100 \nu^{10} + 5372250 \nu^{8} - 27902500 \nu^{6} + 73800000 \nu^{4} - 413000000 \nu^{2} - 3238671875 \)\()/ 2014062500 \)
\(\beta_{11}\)\(=\)\((\)\( 1711 \nu^{18} - 5966 \nu^{16} - 12542 \nu^{14} - 3970 \nu^{12} - 387600 \nu^{10} - 881500 \nu^{8} - 2171250 \nu^{6} + 55893750 \nu^{4} + 138953125 \nu^{2} - 927343750 \)\()/ 2014062500 \)
\(\beta_{12}\)\(=\)\((\)\( 2188 \nu^{19} - 32553 \nu^{17} - 16886 \nu^{15} + 159890 \nu^{13} + 1650700 \nu^{11} - 6230750 \nu^{9} - 52276250 \nu^{7} + 396868750 \nu^{5} - 932750000 \nu^{3} + 4543359375 \nu \)\()/ 10070312500 \)
\(\beta_{13}\)\(=\)\((\)\( 3232 \nu^{19} - 18867 \nu^{17} + 144046 \nu^{15} - 411740 \nu^{13} - 3934700 \nu^{11} + 8753250 \nu^{9} - 45861250 \nu^{7} + 423412500 \nu^{5} - 364437500 \nu^{3} - 1577734375 \nu \)\()/ 10070312500 \)
\(\beta_{14}\)\(=\)\((\)\( 2947 \nu^{18} - 11232 \nu^{16} + 37266 \nu^{14} - 65140 \nu^{12} - 223950 \nu^{10} + 6193250 \nu^{8} - 21761250 \nu^{6} + 197037500 \nu^{4} - 213265625 \nu^{2} - 2172656250 \)\()/ 2014062500 \)
\(\beta_{15}\)\(=\)\((\)\( -777 \nu^{19} - 938 \nu^{17} - 606 \nu^{15} + 118540 \nu^{13} - 555550 \nu^{11} - 778250 \nu^{9} + 11498750 \nu^{7} - 41412500 \nu^{5} + 386859375 \nu^{3} + 1159375000 \nu \)\()/ 2014062500 \)
\(\beta_{16}\)\(=\)\((\)\( 4416 \nu^{19} - 13221 \nu^{17} - 33902 \nu^{15} - 123770 \nu^{13} - 1090100 \nu^{11} + 7297250 \nu^{9} + 378750 \nu^{7} - 348481250 \nu^{5} + 106687500 \nu^{3} + 2330859375 \nu \)\()/ 10070312500 \)
\(\beta_{17}\)\(=\)\((\)\( \nu^{19} - 6 \nu^{17} + 3 \nu^{15} + 80 \nu^{13} - 600 \nu^{11} + 3500 \nu^{9} - 15000 \nu^{7} + 50000 \nu^{5} + 46875 \nu^{3} - 2343750 \nu \)\()/1953125\)
\(\beta_{18}\)\(=\)\((\)\( 6117 \nu^{19} - 17327 \nu^{17} + 37726 \nu^{15} - 213140 \nu^{13} + 171050 \nu^{11} + 13122000 \nu^{9} - 72661250 \nu^{7} + 353662500 \nu^{5} + 614078125 \nu^{3} - 3701953125 \nu \)\()/ 10070312500 \)
\(\beta_{19}\)\(=\)\((\)\( 3374 \nu^{19} + 6556 \nu^{17} - 63178 \nu^{15} + 359695 \nu^{13} - 2058525 \nu^{11} - 4089750 \nu^{9} - 8091250 \nu^{7} - 70565625 \nu^{5} + 636828125 \nu^{3} - 5507031250 \nu \)\()/ 5035156250 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{19} + 2 \beta_{18} - 4 \beta_{17} - \beta_{16} + \beta_{15} + \beta_{13} + 2 \beta_{12} + 2 \beta_{9} + 2 \beta_{7} - 3 \beta_{1}\)\()/5\)
\(\nu^{2}\)\(=\)\(\beta_{14} - 2 \beta_{10} + \beta_{5} + 2 \beta_{3}\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{19} + 18 \beta_{18} - \beta_{17} + \beta_{16} + 14 \beta_{15} - 6 \beta_{13} - 7 \beta_{12} - 7 \beta_{9} - 2 \beta_{7} + 8 \beta_{1}\)\()/5\)
\(\nu^{4}\)\(=\)\(-\beta_{11} - 7 \beta_{8} + 12 \beta_{6} - \beta_{4} + 7 \beta_{2} + 9\)
\(\nu^{5}\)\(=\)\((\)\(11 \beta_{19} + 62 \beta_{18} - 34 \beta_{17} - 86 \beta_{16} + 11 \beta_{15} + 21 \beta_{13} + 27 \beta_{12} + 2 \beta_{9} + 2 \beta_{7} - 53 \beta_{1}\)\()/5\)
\(\nu^{6}\)\(=\)\(7 \beta_{14} + 4 \beta_{11} - 24 \beta_{10} + 18 \beta_{8} + 22 \beta_{6} + 32 \beta_{5} - 26 \beta_{4} - 51 \beta_{3} + 22 \beta_{2} + 4\)
\(\nu^{7}\)\(=\)\((\)\(-186 \beta_{19} + 58 \beta_{18} + 69 \beta_{17} + 186 \beta_{16} + 94 \beta_{15} + 14 \beta_{13} - 102 \beta_{12} - 102 \beta_{9} - 392 \beta_{7} + 48 \beta_{1}\)\()/5\)
\(\nu^{8}\)\(=\)\(62 \beta_{14} - 248 \beta_{11} - 34 \beta_{10} + 104 \beta_{8} + 46 \beta_{6} + 62 \beta_{5} - 178 \beta_{4} + 34 \beta_{3} + 136 \beta_{2} + 247\)
\(\nu^{9}\)\(=\)\((\)\(-469 \beta_{19} + 1422 \beta_{18} - 54 \beta_{17} - 931 \beta_{16} + 11 \beta_{15} - 399 \beta_{13} - 558 \beta_{12} + 892 \beta_{9} + 752 \beta_{7} - 1993 \beta_{1}\)\()/5\)
\(\nu^{10}\)\(=\)\(701 \beta_{14} - 1170 \beta_{11} - 1332 \beta_{10} + 1130 \beta_{8} + 750 \beta_{6} - 749 \beta_{5} + 590 \beta_{4} - 68 \beta_{3} + 440 \beta_{2} - 1000\)
\(\nu^{11}\)\(=\)\((\)\(-4111 \beta_{19} + 4698 \beta_{18} + 1839 \beta_{17} + 1711 \beta_{16} - 1196 \beta_{15} - 7466 \beta_{13} - 777 \beta_{12} - 77 \beta_{9} - 3922 \beta_{7} + 4938 \beta_{1}\)\()/5\)
\(\nu^{12}\)\(=\)\(2460 \beta_{14} - 6571 \beta_{11} + 2280 \beta_{10} + 1453 \beta_{8} + 8552 \beta_{6} + 1760 \beta_{5} + 4929 \beta_{4} - 4680 \beta_{3} - 2203 \beta_{2} + 7089\)
\(\nu^{13}\)\(=\)\((\)\(10471 \beta_{19} - 1518 \beta_{18} + 3776 \beta_{17} - 25046 \beta_{16} + 7871 \beta_{15} - 14869 \beta_{13} + 16497 \beta_{12} + 35722 \beta_{9} + 22 \beta_{7} - 10533 \beta_{1}\)\()/5\)
\(\nu^{14}\)\(=\)\(33147 \beta_{14} - 22676 \beta_{11} - 5254 \beta_{10} - 5192 \beta_{8} - 14668 \beta_{6} + 13922 \beta_{5} + 17444 \beta_{4} - 9321 \beta_{3} - 30468 \beta_{2} - 22076\)
\(\nu^{15}\)\(=\)\((\)\(-86796 \beta_{19} - 55412 \beta_{18} + 54359 \beta_{17} + 99796 \beta_{16} + 9084 \beta_{15} + 72504 \beta_{13} - 105372 \beta_{12} + 73128 \beta_{9} - 139212 \beta_{7} + 253228 \beta_{1}\)\()/5\)
\(\nu^{16}\)\(=\)\(4372 \beta_{14} - 91168 \beta_{11} + 44296 \beta_{10} - 122136 \beta_{8} + 144436 \beta_{6} - 174128 \beta_{5} - 3748 \beta_{4} - 31296 \beta_{3} - 117424 \beta_{2} + 7977\)
\(\nu^{17}\)\(=\)\((\)\(705041 \beta_{19} + 663942 \beta_{18} - 1159944 \beta_{17} - 1184441 \beta_{16} - 387079 \beta_{15} + 184761 \beta_{13} - 326738 \beta_{12} + 734962 \beta_{9} + 556622 \beta_{7} - 871623 \beta_{1}\)\()/5\)
\(\nu^{18}\)\(=\)\(382341 \beta_{14} + 322700 \beta_{11} - 66062 \beta_{10} + 100520 \beta_{8} + 245460 \beta_{6} - 679359 \beta_{5} + 167860 \beta_{4} - 413338 \beta_{3} - 668900 \beta_{2} + 36220\)
\(\nu^{19}\)\(=\)\((\)\(1878479 \beta_{19} + 3726878 \beta_{18} - 2959921 \beta_{17} + 3582121 \beta_{16} - 1188006 \beta_{15} + 782474 \beta_{13} + 658353 \beta_{12} + 1550053 \beta_{9} - 2428342 \beta_{7} + 755268 \beta_{1}\)\()/5\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/115\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(\beta_{6}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1
−1.76871 1.36809i
1.76871 + 1.36809i
0.0985483 2.23390i
−0.0985483 + 2.23390i
−2.11159 0.735651i
2.11159 + 0.735651i
−2.22384 0.233538i
2.22384 + 0.233538i
1.20735 1.88210i
−1.20735 + 1.88210i
−1.76871 + 1.36809i
1.76871 1.36809i
0.0985483 + 2.23390i
−0.0985483 2.23390i
−2.11159 + 0.735651i
2.11159 0.735651i
−2.22384 + 0.233538i
2.22384 0.233538i
1.20735 + 1.88210i
−1.20735 1.88210i
−1.68447 1.68447i −0.639677 + 0.639677i 3.67489i −1.36809 1.76871i 2.15503 −2.98466 + 2.98466i 2.82130 2.82130i 2.18163i −0.674828 + 5.28385i
22.2 −1.68447 1.68447i −0.639677 + 0.639677i 3.67489i 1.36809 + 1.76871i 2.15503 2.98466 2.98466i 2.82130 2.82130i 2.18163i 0.674828 5.28385i
22.3 −0.819998 0.819998i 1.37823 1.37823i 0.655207i −2.23390 + 0.0985483i −2.26029 1.78496 1.78496i −2.17726 + 2.17726i 0.799034i 1.91260 + 1.75098i
22.4 −0.819998 0.819998i 1.37823 1.37823i 0.655207i 2.23390 0.0985483i −2.26029 −1.78496 + 1.78496i −2.17726 + 2.17726i 0.799034i −1.91260 1.75098i
22.5 −0.459187 0.459187i −1.79404 + 1.79404i 1.57829i −0.735651 2.11159i 1.64760 2.52854 2.52854i −1.64311 + 1.64311i 3.43715i −0.631815 + 1.30742i
22.6 −0.459187 0.459187i −1.79404 + 1.79404i 1.57829i 0.735651 + 2.11159i 1.64760 −2.52854 + 2.52854i −1.64311 + 1.64311i 3.43715i 0.631815 1.30742i
22.7 0.562704 + 0.562704i 0.534388 0.534388i 1.36673i −0.233538 2.22384i 0.601404 −0.567230 + 0.567230i 1.89447 1.89447i 2.42886i 1.11995 1.38278i
22.8 0.562704 + 0.562704i 0.534388 0.534388i 1.36673i 0.233538 + 2.22384i 0.601404 0.567230 0.567230i 1.89447 1.89447i 2.42886i −1.11995 + 1.38278i
22.9 1.40095 + 1.40095i −1.47890 + 1.47890i 1.92534i −1.88210 + 1.20735i −4.14375 2.58659 2.58659i 0.104596 0.104596i 1.37431i −4.32817 0.945306i
22.10 1.40095 + 1.40095i −1.47890 + 1.47890i 1.92534i 1.88210 1.20735i −4.14375 −2.58659 + 2.58659i 0.104596 0.104596i 1.37431i 4.32817 + 0.945306i
68.1 −1.68447 + 1.68447i −0.639677 0.639677i 3.67489i −1.36809 + 1.76871i 2.15503 −2.98466 2.98466i 2.82130 + 2.82130i 2.18163i −0.674828 5.28385i
68.2 −1.68447 + 1.68447i −0.639677 0.639677i 3.67489i 1.36809 1.76871i 2.15503 2.98466 + 2.98466i 2.82130 + 2.82130i 2.18163i 0.674828 + 5.28385i
68.3 −0.819998 + 0.819998i 1.37823 + 1.37823i 0.655207i −2.23390 0.0985483i −2.26029 1.78496 + 1.78496i −2.17726 2.17726i 0.799034i 1.91260 1.75098i
68.4 −0.819998 + 0.819998i 1.37823 + 1.37823i 0.655207i 2.23390 + 0.0985483i −2.26029 −1.78496 1.78496i −2.17726 2.17726i 0.799034i −1.91260 + 1.75098i
68.5 −0.459187 + 0.459187i −1.79404 1.79404i 1.57829i −0.735651 + 2.11159i 1.64760 2.52854 + 2.52854i −1.64311 1.64311i 3.43715i −0.631815 1.30742i
68.6 −0.459187 + 0.459187i −1.79404 1.79404i 1.57829i 0.735651 2.11159i 1.64760 −2.52854 2.52854i −1.64311 1.64311i 3.43715i 0.631815 + 1.30742i
68.7 0.562704 0.562704i 0.534388 + 0.534388i 1.36673i −0.233538 + 2.22384i 0.601404 −0.567230 0.567230i 1.89447 + 1.89447i 2.42886i 1.11995 + 1.38278i
68.8 0.562704 0.562704i 0.534388 + 0.534388i 1.36673i 0.233538 2.22384i 0.601404 0.567230 + 0.567230i 1.89447 + 1.89447i 2.42886i −1.11995 1.38278i
68.9 1.40095 1.40095i −1.47890 1.47890i 1.92534i −1.88210 1.20735i −4.14375 2.58659 + 2.58659i 0.104596 + 0.104596i 1.37431i −4.32817 + 0.945306i
68.10 1.40095 1.40095i −1.47890 1.47890i 1.92534i 1.88210 + 1.20735i −4.14375 −2.58659 2.58659i 0.104596 + 0.104596i 1.37431i 4.32817 0.945306i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 68.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
23.b odd 2 1 inner
115.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 115.2.e.a 20
5.b even 2 1 575.2.e.d 20
5.c odd 4 1 inner 115.2.e.a 20
5.c odd 4 1 575.2.e.d 20
23.b odd 2 1 inner 115.2.e.a 20
115.c odd 2 1 575.2.e.d 20
115.e even 4 1 inner 115.2.e.a 20
115.e even 4 1 575.2.e.d 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.2.e.a 20 1.a even 1 1 trivial
115.2.e.a 20 5.c odd 4 1 inner
115.2.e.a 20 23.b odd 2 1 inner
115.2.e.a 20 115.e even 4 1 inner
575.2.e.d 20 5.b even 2 1
575.2.e.d 20 5.c odd 4 1
575.2.e.d 20 115.c odd 2 1
575.2.e.d 20 115.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(115, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 8 + 12 T + 9 T^{2} - 4 T^{3} + 24 T^{4} + 30 T^{5} + 19 T^{6} - 2 T^{7} + 2 T^{8} + 2 T^{9} + T^{10} )^{2} \)
$3$ \( ( 50 + 10 T + T^{2} + 22 T^{3} + 122 T^{4} + 58 T^{5} + 14 T^{6} + 2 T^{7} + 8 T^{8} + 4 T^{9} + T^{10} )^{2} \)
$5$ \( 9765625 + 2343750 T^{2} + 46875 T^{4} - 50000 T^{6} - 15000 T^{8} - 3500 T^{10} - 600 T^{12} - 80 T^{14} + 3 T^{16} + 6 T^{18} + T^{20} \)
$7$ \( 156250000 + 383500000 T^{4} + 14965000 T^{8} + 165100 T^{12} + 701 T^{16} + T^{20} \)
$11$ \( ( 25000 + 43500 T^{2} + 15300 T^{4} + 1620 T^{6} + 68 T^{8} + T^{10} )^{2} \)
$13$ \( ( 8 - 140 T + 1225 T^{2} - 2246 T^{3} + 2186 T^{4} - 1130 T^{5} + 326 T^{6} - 34 T^{7} + 2 T^{8} - 2 T^{9} + T^{10} )^{2} \)
$17$ \( 156250000 + 306000000 T^{4} + 109537500 T^{8} + 4313100 T^{12} + 4361 T^{16} + T^{20} \)
$19$ \( ( -25000 + 103500 T^{2} - 46600 T^{4} + 4290 T^{6} - 118 T^{8} + T^{10} )^{2} \)
$23$ \( 41426511213649 + 381340450064 T^{3} + 211543285381 T^{4} + 9165352432 T^{5} + 1755162752 T^{6} + 1217284016 T^{7} + 642321322 T^{8} + 51862608 T^{9} + 3256576 T^{10} + 2254896 T^{11} + 1214218 T^{12} + 100048 T^{13} + 6272 T^{14} + 1424 T^{15} + 1429 T^{16} + 112 T^{17} + T^{20} \)
$29$ \( ( 1545049 + 496761 T^{2} + 60950 T^{4} + 3542 T^{6} + 97 T^{8} + T^{10} )^{2} \)
$31$ \( ( 25 + 61 T - 62 T^{2} - 58 T^{3} + T^{4} + T^{5} )^{4} \)
$37$ \( 94933312656250000 + 388907948500000 T^{4} + 337110445000 T^{8} + 116761500 T^{12} + 17821 T^{16} + T^{20} \)
$41$ \( ( -47 + 185 T - 106 T^{2} - 10 T^{3} + 9 T^{4} + T^{5} )^{4} \)
$43$ \( 1562500000000 + 3899142250000 T^{4} + 47646280000 T^{8} + 45572300 T^{12} + 12556 T^{16} + T^{20} \)
$47$ \( ( 315218 - 729686 T + 844561 T^{2} + 20102 T^{3} + 9226 T^{4} - 14102 T^{5} + 8566 T^{6} - 518 T^{7} + 8 T^{8} + 4 T^{9} + T^{10} )^{2} \)
$53$ \( 25000000000000 + 66850000000000 T^{4} + 357806000000 T^{8} + 149130000 T^{12} + 21225 T^{16} + T^{20} \)
$59$ \( ( 53290000 + 15647744 T^{2} + 1083052 T^{4} + 29140 T^{6} + 313 T^{8} + T^{10} )^{2} \)
$61$ \( ( 235225000 + 35043500 T^{2} + 1693000 T^{4} + 35290 T^{6} + 322 T^{8} + T^{10} )^{2} \)
$67$ \( 156250000 + 21568500000 T^{4} + 5713085000 T^{8} + 374693500 T^{12} + 47421 T^{16} + T^{20} \)
$71$ \( ( 13903 - 9525 T + 2104 T^{2} - 120 T^{3} - 11 T^{4} + T^{5} )^{4} \)
$73$ \( ( 1086338 - 1835130 T + 1550025 T^{2} + 177674 T^{3} - 9014 T^{4} + 9510 T^{5} + 11326 T^{6} + 2886 T^{7} + 392 T^{8} + 28 T^{9} + T^{10} )^{2} \)
$79$ \( ( -133225000 + 79441500 T^{2} - 4135600 T^{4} + 69410 T^{6} - 458 T^{8} + T^{10} )^{2} \)
$83$ \( 97656250000 + 5621506000000 T^{4} + 72100857500 T^{8} + 80141500 T^{12} + 24841 T^{16} + T^{20} \)
$89$ \( ( -2175625000 + 181503500 T^{2} - 5197800 T^{4} + 68210 T^{6} - 422 T^{8} + T^{10} )^{2} \)
$97$ \( 74966440000000000 + 790656602250000 T^{4} + 2234390010000 T^{8} + 903193400 T^{12} + 61156 T^{16} + T^{20} \)
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