Properties

Label 1155.2.c.f
Level 1155
Weight 2
Character orbit 1155.c
Analytic conductor 9.223
Analytic rank 0
Dimension 20
CM No
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1155.c (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(9.22272143346\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20}\mathstrut +\mathstrut \) \(33\) \(x^{18}\mathstrut +\mathstrut \) \(456\) \(x^{16}\mathstrut +\mathstrut \) \(3426\) \(x^{14}\mathstrut +\mathstrut \) \(15210\) \(x^{12}\mathstrut +\mathstrut \) \(40640\) \(x^{10}\mathstrut +\mathstrut \) \(63865\) \(x^{8}\mathstrut +\mathstrut \) \(55281\) \(x^{6}\mathstrut +\mathstrut \) \(22984\) \(x^{4}\mathstrut +\mathstrut \) \(3428\) \(x^{2}\mathstrut +\mathstrut \) \(4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 3 \)
\(\beta_{3}\)\(=\)\((\)\( -932 \nu^{18} - 29599 \nu^{16} - 388661 \nu^{14} - 2720792 \nu^{12} - 10903803 \nu^{10} - 24948019 \nu^{8} - 30691092 \nu^{6} - 17834990 \nu^{4} - 3622164 \nu^{2} - 10112 \)\()/171356\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(21445\) \(\nu^{19}\mathstrut -\mathstrut \) \(701609\) \(\nu^{17}\mathstrut -\mathstrut \) \(9437765\) \(\nu^{15}\mathstrut -\mathstrut \) \(67184034\) \(\nu^{13}\mathstrut -\mathstrut \) \(271650511\) \(\nu^{11}\mathstrut -\mathstrut \) \(625930321\) \(\nu^{9}\mathstrut -\mathstrut \) \(801263181\) \(\nu^{7}\mathstrut -\mathstrut \) \(580092520\) \(\nu^{5}\mathstrut -\mathstrut \) \(263850094\) \(\nu^{3}\mathstrut -\mathstrut \) \(52934572\) \(\nu\)\()/3255764\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(17223\) \(\nu^{19}\mathstrut -\mathstrut \) \(39658\) \(\nu^{18}\mathstrut -\mathstrut \) \(539302\) \(\nu^{17}\mathstrut -\mathstrut \) \(1304803\) \(\nu^{16}\mathstrut -\mathstrut \) \(7239623\) \(\nu^{15}\mathstrut -\mathstrut \) \(17627746\) \(\nu^{14}\mathstrut -\mathstrut \) \(55067041\) \(\nu^{13}\mathstrut -\mathstrut \) \(125789839\) \(\nu^{12}\mathstrut -\mathstrut \) \(262962518\) \(\nu^{11}\mathstrut -\mathstrut \) \(508016561\) \(\nu^{10}\mathstrut -\mathstrut \) \(813882325\) \(\nu^{9}\mathstrut -\mathstrut \) \(1157693438\) \(\nu^{8}\mathstrut -\mathstrut \) \(1595603488\) \(\nu^{7}\mathstrut -\mathstrut \) \(1413013347\) \(\nu^{6}\mathstrut -\mathstrut \) \(1831293380\) \(\nu^{5}\mathstrut -\mathstrut \) \(842531572\) \(\nu^{4}\mathstrut -\mathstrut \) \(1064523572\) \(\nu^{3}\mathstrut -\mathstrut \) \(207237480\) \(\nu^{2}\mathstrut -\mathstrut \) \(233638456\) \(\nu\mathstrut -\mathstrut \) \(12649508\)\()/6511528\)
\(\beta_{6}\)\(=\)\((\)\( 28357 \nu^{18} + 807316 \nu^{16} + 9309882 \nu^{14} + 55840854 \nu^{12} + 186103148 \nu^{10} + 341926714 \nu^{8} + 328984885 \nu^{6} + 163946632 \nu^{4} + 46688632 \nu^{2} + 1859432 \)\()/3255764\)
\(\beta_{7}\)\(=\)\((\)\(11377\) \(\nu^{19}\mathstrut -\mathstrut \) \(77206\) \(\nu^{18}\mathstrut +\mathstrut \) \(402088\) \(\nu^{17}\mathstrut -\mathstrut \) \(2366551\) \(\nu^{16}\mathstrut +\mathstrut \) \(6106853\) \(\nu^{15}\mathstrut -\mathstrut \) \(29875558\) \(\nu^{14}\mathstrut +\mathstrut \) \(52047565\) \(\nu^{13}\mathstrut -\mathstrut \) \(200454623\) \(\nu^{12}\mathstrut +\mathstrut \) \(271905052\) \(\nu^{11}\mathstrut -\mathstrut \) \(769151417\) \(\nu^{10}\mathstrut +\mathstrut \) \(890119595\) \(\nu^{9}\mathstrut -\mathstrut \) \(1690185486\) \(\nu^{8}\mathstrut +\mathstrut \) \(1787344850\) \(\nu^{7}\mathstrut -\mathstrut \) \(2021456739\) \(\nu^{6}\mathstrut +\mathstrut \) \(2060015128\) \(\nu^{5}\mathstrut -\mathstrut \) \(1184482080\) \(\nu^{4}\mathstrut +\mathstrut \) \(1186256172\) \(\nu^{3}\mathstrut -\mathstrut \) \(280996784\) \(\nu^{2}\mathstrut +\mathstrut \) \(242033984\) \(\nu\mathstrut -\mathstrut \) \(12324404\)\()/6511528\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(40638\) \(\nu^{19}\mathstrut +\mathstrut \) \(2923\) \(\nu^{18}\mathstrut -\mathstrut \) \(1359828\) \(\nu^{17}\mathstrut +\mathstrut \) \(68607\) \(\nu^{16}\mathstrut -\mathstrut \) \(19061802\) \(\nu^{15}\mathstrut +\mathstrut \) \(566385\) \(\nu^{14}\mathstrut -\mathstrut \) \(145349694\) \(\nu^{13}\mathstrut +\mathstrut \) \(1509738\) \(\nu^{12}\mathstrut -\mathstrut \) \(655436372\) \(\nu^{11}\mathstrut -\mathstrut \) \(4471267\) \(\nu^{10}\mathstrut -\mathstrut \) \(1782095748\) \(\nu^{9}\mathstrut -\mathstrut \) \(38118635\) \(\nu^{8}\mathstrut -\mathstrut \) \(2861591894\) \(\nu^{7}\mathstrut -\mathstrut \) \(95870681\) \(\nu^{6}\mathstrut -\mathstrut \) \(2550730974\) \(\nu^{5}\mathstrut -\mathstrut \) \(114360874\) \(\nu^{4}\mathstrut -\mathstrut \) \(1104999046\) \(\nu^{3}\mathstrut -\mathstrut \) \(60866300\) \(\nu^{2}\mathstrut -\mathstrut \) \(176186716\) \(\nu\mathstrut -\mathstrut \) \(4197764\)\()/3255764\)
\(\beta_{9}\)\(=\)\((\)\(40638\) \(\nu^{19}\mathstrut +\mathstrut \) \(2923\) \(\nu^{18}\mathstrut +\mathstrut \) \(1359828\) \(\nu^{17}\mathstrut +\mathstrut \) \(68607\) \(\nu^{16}\mathstrut +\mathstrut \) \(19061802\) \(\nu^{15}\mathstrut +\mathstrut \) \(566385\) \(\nu^{14}\mathstrut +\mathstrut \) \(145349694\) \(\nu^{13}\mathstrut +\mathstrut \) \(1509738\) \(\nu^{12}\mathstrut +\mathstrut \) \(655436372\) \(\nu^{11}\mathstrut -\mathstrut \) \(4471267\) \(\nu^{10}\mathstrut +\mathstrut \) \(1782095748\) \(\nu^{9}\mathstrut -\mathstrut \) \(38118635\) \(\nu^{8}\mathstrut +\mathstrut \) \(2861591894\) \(\nu^{7}\mathstrut -\mathstrut \) \(95870681\) \(\nu^{6}\mathstrut +\mathstrut \) \(2550730974\) \(\nu^{5}\mathstrut -\mathstrut \) \(114360874\) \(\nu^{4}\mathstrut +\mathstrut \) \(1104999046\) \(\nu^{3}\mathstrut -\mathstrut \) \(60866300\) \(\nu^{2}\mathstrut +\mathstrut \) \(176186716\) \(\nu\mathstrut -\mathstrut \) \(4197764\)\()/3255764\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(11377\) \(\nu^{19}\mathstrut -\mathstrut \) \(77206\) \(\nu^{18}\mathstrut -\mathstrut \) \(402088\) \(\nu^{17}\mathstrut -\mathstrut \) \(2366551\) \(\nu^{16}\mathstrut -\mathstrut \) \(6106853\) \(\nu^{15}\mathstrut -\mathstrut \) \(29875558\) \(\nu^{14}\mathstrut -\mathstrut \) \(52047565\) \(\nu^{13}\mathstrut -\mathstrut \) \(200454623\) \(\nu^{12}\mathstrut -\mathstrut \) \(271905052\) \(\nu^{11}\mathstrut -\mathstrut \) \(769151417\) \(\nu^{10}\mathstrut -\mathstrut \) \(890119595\) \(\nu^{9}\mathstrut -\mathstrut \) \(1690185486\) \(\nu^{8}\mathstrut -\mathstrut \) \(1787344850\) \(\nu^{7}\mathstrut -\mathstrut \) \(2021456739\) \(\nu^{6}\mathstrut -\mathstrut \) \(2060015128\) \(\nu^{5}\mathstrut -\mathstrut \) \(1184482080\) \(\nu^{4}\mathstrut -\mathstrut \) \(1186256172\) \(\nu^{3}\mathstrut -\mathstrut \) \(280996784\) \(\nu^{2}\mathstrut -\mathstrut \) \(242033984\) \(\nu\mathstrut -\mathstrut \) \(12324404\)\()/6511528\)
\(\beta_{11}\)\(=\)\((\)\( 2528 \nu^{19} + 82492 \nu^{17} + 1123169 \nu^{15} + 8272267 \nu^{13} + 35730088 \nu^{11} + 91834117 \nu^{9} + 136502701 \nu^{7} + 109059276 \nu^{5} + 40268562 \nu^{3} + 5043820 \nu \)\()/171356\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(3503\) \(\nu^{18}\mathstrut -\mathstrut \) \(111664\) \(\nu^{16}\mathstrut -\mathstrut \) \(1471065\) \(\nu^{14}\mathstrut -\mathstrut \) \(10332043\) \(\nu^{12}\mathstrut -\mathstrut \) \(41590554\) \(\nu^{10}\mathstrut -\mathstrut \) \(95909007\) \(\nu^{8}\mathstrut -\mathstrut \) \(119768558\) \(\nu^{6}\mathstrut -\mathstrut \) \(71376026\) \(\nu^{4}\mathstrut -\mathstrut \) \(14754680\) \(\nu^{2}\mathstrut +\mathstrut \) \(467236\)\()/171356\)
\(\beta_{13}\)\(=\)\((\)\(158482\) \(\nu^{19}\mathstrut -\mathstrut \) \(20801\) \(\nu^{18}\mathstrut +\mathstrut \) \(5086207\) \(\nu^{17}\mathstrut -\mathstrut \) \(781727\) \(\nu^{16}\mathstrut +\mathstrut \) \(67999162\) \(\nu^{15}\mathstrut -\mathstrut \) \(11937193\) \(\nu^{14}\mathstrut +\mathstrut \) \(491154011\) \(\nu^{13}\mathstrut -\mathstrut \) \(95841406\) \(\nu^{12}\mathstrut +\mathstrut \) \(2080024161\) \(\nu^{11}\mathstrut -\mathstrut \) \(436700849\) \(\nu^{10}\mathstrut +\mathstrut \) \(5254376982\) \(\nu^{9}\mathstrut -\mathstrut \) \(1136990015\) \(\nu^{8}\mathstrut +\mathstrut \) \(7744640527\) \(\nu^{7}\mathstrut -\mathstrut \) \(1622824553\) \(\nu^{6}\mathstrut +\mathstrut \) \(6285944028\) \(\nu^{5}\mathstrut -\mathstrut \) \(1153488952\) \(\nu^{4}\mathstrut +\mathstrut \) \(2490994876\) \(\nu^{3}\mathstrut -\mathstrut \) \(324766228\) \(\nu^{2}\mathstrut +\mathstrut \) \(364697836\) \(\nu\mathstrut -\mathstrut \) \(8350020\)\()/6511528\)
\(\beta_{14}\)\(=\)\((\)\(-\)\(158482\) \(\nu^{19}\mathstrut -\mathstrut \) \(20801\) \(\nu^{18}\mathstrut -\mathstrut \) \(5086207\) \(\nu^{17}\mathstrut -\mathstrut \) \(781727\) \(\nu^{16}\mathstrut -\mathstrut \) \(67999162\) \(\nu^{15}\mathstrut -\mathstrut \) \(11937193\) \(\nu^{14}\mathstrut -\mathstrut \) \(491154011\) \(\nu^{13}\mathstrut -\mathstrut \) \(95841406\) \(\nu^{12}\mathstrut -\mathstrut \) \(2080024161\) \(\nu^{11}\mathstrut -\mathstrut \) \(436700849\) \(\nu^{10}\mathstrut -\mathstrut \) \(5254376982\) \(\nu^{9}\mathstrut -\mathstrut \) \(1136990015\) \(\nu^{8}\mathstrut -\mathstrut \) \(7744640527\) \(\nu^{7}\mathstrut -\mathstrut \) \(1622824553\) \(\nu^{6}\mathstrut -\mathstrut \) \(6285944028\) \(\nu^{5}\mathstrut -\mathstrut \) \(1153488952\) \(\nu^{4}\mathstrut -\mathstrut \) \(2490994876\) \(\nu^{3}\mathstrut -\mathstrut \) \(324766228\) \(\nu^{2}\mathstrut -\mathstrut \) \(364697836\) \(\nu\mathstrut -\mathstrut \) \(8350020\)\()/6511528\)
\(\beta_{15}\)\(=\)\((\)\(-\)\(192793\) \(\nu^{19}\mathstrut +\mathstrut \) \(21163\) \(\nu^{18}\mathstrut -\mathstrut \) \(6452077\) \(\nu^{17}\mathstrut +\mathstrut \) \(777136\) \(\nu^{16}\mathstrut -\mathstrut \) \(90636213\) \(\nu^{15}\mathstrut +\mathstrut \) \(12008451\) \(\nu^{14}\mathstrut -\mathstrut \) \(694260760\) \(\nu^{13}\mathstrut +\mathstrut \) \(101264279\) \(\nu^{12}\mathstrut -\mathstrut \) \(3152696619\) \(\nu^{11}\mathstrut +\mathstrut \) \(504609204\) \(\nu^{10}\mathstrut -\mathstrut \) \(8644761455\) \(\nu^{9}\mathstrut +\mathstrut \) \(1498448529\) \(\nu^{8}\mathstrut -\mathstrut \) \(13969193167\) \(\nu^{7}\mathstrut +\mathstrut \) \(2525053938\) \(\nu^{6}\mathstrut -\mathstrut \) \(12401839144\) \(\nu^{5}\mathstrut +\mathstrut \) \(2109983528\) \(\nu^{4}\mathstrut -\mathstrut \) \(5229989096\) \(\nu^{3}\mathstrut +\mathstrut \) \(617774332\) \(\nu^{2}\mathstrut -\mathstrut \) \(769879164\) \(\nu\mathstrut +\mathstrut \) \(12865832\)\()/6511528\)
\(\beta_{16}\)\(=\)\((\)\(-\)\(110783\) \(\nu^{19}\mathstrut -\mathstrut \) \(3732736\) \(\nu^{17}\mathstrut -\mathstrut \) \(52853791\) \(\nu^{15}\mathstrut -\mathstrut \) \(408736717\) \(\nu^{13}\mathstrut -\mathstrut \) \(1878395562\) \(\nu^{11}\mathstrut -\mathstrut \) \(5231616809\) \(\nu^{9}\mathstrut -\mathstrut \) \(8634683824\) \(\nu^{7}\mathstrut -\mathstrut \) \(7886314178\) \(\nu^{5}\mathstrut -\mathstrut \) \(3430562652\) \(\nu^{3}\mathstrut -\mathstrut \) \(500823852\) \(\nu\)\()/3255764\)
\(\beta_{17}\)\(=\)\((\)\( 6652 \nu^{19} + 217877 \nu^{17} + 2980846 \nu^{15} + 22096009 \nu^{13} + 96286461 \nu^{11} + 250554332 \nu^{9} + 378817011 \nu^{7} + 309342838 \nu^{5} + 117183522 \nu^{3} + 15121348 \nu \)\()/171356\)
\(\beta_{18}\)\(=\)\((\)\(-\)\(6652\) \(\nu^{19}\mathstrut -\mathstrut \) \(217877\) \(\nu^{17}\mathstrut -\mathstrut \) \(2980846\) \(\nu^{15}\mathstrut -\mathstrut \) \(22096009\) \(\nu^{13}\mathstrut -\mathstrut \) \(96286461\) \(\nu^{11}\mathstrut -\mathstrut \) \(250554332\) \(\nu^{9}\mathstrut -\mathstrut \) \(378817011\) \(\nu^{7}\mathstrut -\mathstrut \) \(309342838\) \(\nu^{5}\mathstrut -\mathstrut \) \(117012166\) \(\nu^{3}\mathstrut -\mathstrut \) \(14264568\) \(\nu\)\()/171356\)
\(\beta_{19}\)\(=\)\((\)\(-\)\(8594\) \(\nu^{19}\mathstrut -\mathstrut \) \(286263\) \(\nu^{17}\mathstrut -\mathstrut \) \(3995054\) \(\nu^{15}\mathstrut -\mathstrut \) \(30327549\) \(\nu^{13}\mathstrut -\mathstrut \) \(136061779\) \(\nu^{11}\mathstrut -\mathstrut \) \(367228218\) \(\nu^{9}\mathstrut -\mathstrut \) \(581983849\) \(\nu^{7}\mathstrut -\mathstrut \) \(506157206\) \(\nu^{5}\mathstrut -\mathstrut \) \(211022344\) \(\nu^{3}\mathstrut -\mathstrut \) \(32364064\) \(\nu\)\()/171356\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut -\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{18}\mathstrut +\mathstrut \) \(\beta_{17}\mathstrut -\mathstrut \) \(5\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{12}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(15\)
\(\nu^{5}\)\(=\)\(-\)\(9\) \(\beta_{18}\mathstrut -\mathstrut \) \(10\) \(\beta_{17}\mathstrut +\mathstrut \) \(\beta_{16}\mathstrut +\mathstrut \) \(2\) \(\beta_{11}\mathstrut +\mathstrut \) \(2\) \(\beta_{9}\mathstrut -\mathstrut \) \(2\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(28\) \(\beta_{1}\)
\(\nu^{6}\)\(=\)\(-\)\(\beta_{19}\mathstrut +\mathstrut \) \(3\) \(\beta_{18}\mathstrut -\mathstrut \) \(\beta_{16}\mathstrut -\mathstrut \) \(\beta_{15}\mathstrut -\mathstrut \) \(3\) \(\beta_{14}\mathstrut -\mathstrut \) \(10\) \(\beta_{12}\mathstrut +\mathstrut \) \(9\) \(\beta_{10}\mathstrut -\mathstrut \) \(3\) \(\beta_{9}\mathstrut +\mathstrut \) \(2\) \(\beta_{8}\mathstrut +\mathstrut \) \(9\) \(\beta_{7}\mathstrut +\mathstrut \) \(10\) \(\beta_{6}\mathstrut +\mathstrut \) \(3\) \(\beta_{5}\mathstrut +\mathstrut \) \(12\) \(\beta_{3}\mathstrut +\mathstrut \) \(48\) \(\beta_{2}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut -\mathstrut \) \(85\)
\(\nu^{7}\)\(=\)\(-\)\(3\) \(\beta_{19}\mathstrut +\mathstrut \) \(71\) \(\beta_{18}\mathstrut +\mathstrut \) \(83\) \(\beta_{17}\mathstrut -\mathstrut \) \(13\) \(\beta_{16}\mathstrut -\mathstrut \) \(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{13}\mathstrut -\mathstrut \) \(32\) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(29\) \(\beta_{9}\mathstrut +\mathstrut \) \(29\) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(14\) \(\beta_{4}\mathstrut -\mathstrut \) \(165\) \(\beta_{1}\)
\(\nu^{8}\)\(=\)\(14\) \(\beta_{19}\mathstrut -\mathstrut \) \(42\) \(\beta_{18}\mathstrut +\mathstrut \) \(14\) \(\beta_{16}\mathstrut +\mathstrut \) \(14\) \(\beta_{15}\mathstrut +\mathstrut \) \(44\) \(\beta_{14}\mathstrut +\mathstrut \) \(2\) \(\beta_{13}\mathstrut +\mathstrut \) \(81\) \(\beta_{12}\mathstrut -\mathstrut \) \(67\) \(\beta_{10}\mathstrut +\mathstrut \) \(37\) \(\beta_{9}\mathstrut -\mathstrut \) \(33\) \(\beta_{8}\mathstrut -\mathstrut \) \(67\) \(\beta_{7}\mathstrut -\mathstrut \) \(81\) \(\beta_{6}\mathstrut -\mathstrut \) \(42\) \(\beta_{5}\mathstrut -\mathstrut \) \(113\) \(\beta_{3}\mathstrut -\mathstrut \) \(331\) \(\beta_{2}\mathstrut +\mathstrut \) \(14\) \(\beta_{1}\mathstrut +\mathstrut \) \(513\)
\(\nu^{9}\)\(=\)\(49\) \(\beta_{19}\mathstrut -\mathstrut \) \(545\) \(\beta_{18}\mathstrut -\mathstrut \) \(648\) \(\beta_{17}\mathstrut +\mathstrut \) \(130\) \(\beta_{16}\mathstrut +\mathstrut \) \(19\) \(\beta_{14}\mathstrut -\mathstrut \) \(19\) \(\beta_{13}\mathstrut +\mathstrut \) \(350\) \(\beta_{11}\mathstrut -\mathstrut \) \(23\) \(\beta_{10}\mathstrut +\mathstrut \) \(300\) \(\beta_{9}\mathstrut -\mathstrut \) \(300\) \(\beta_{8}\mathstrut +\mathstrut \) \(23\) \(\beta_{7}\mathstrut +\mathstrut \) \(141\) \(\beta_{4}\mathstrut +\mathstrut \) \(1006\) \(\beta_{1}\)
\(\nu^{10}\)\(=\)\(-\)\(140\) \(\beta_{19}\mathstrut +\mathstrut \) \(421\) \(\beta_{18}\mathstrut -\mathstrut \) \(140\) \(\beta_{16}\mathstrut -\mathstrut \) \(141\) \(\beta_{15}\mathstrut -\mathstrut \) \(464\) \(\beta_{14}\mathstrut -\mathstrut \) \(43\) \(\beta_{13}\mathstrut -\mathstrut \) \(610\) \(\beta_{12}\mathstrut +\mathstrut \) \(478\) \(\beta_{10}\mathstrut -\mathstrut \) \(330\) \(\beta_{9}\mathstrut +\mathstrut \) \(371\) \(\beta_{8}\mathstrut +\mathstrut \) \(478\) \(\beta_{7}\mathstrut +\mathstrut \) \(626\) \(\beta_{6}\mathstrut +\mathstrut \) \(421\) \(\beta_{5}\mathstrut +\mathstrut \) \(960\) \(\beta_{3}\mathstrut +\mathstrut \) \(2302\) \(\beta_{2}\mathstrut -\mathstrut \) \(141\) \(\beta_{1}\mathstrut -\mathstrut \) \(3235\)
\(\nu^{11}\)\(=\)\(-\)\(555\) \(\beta_{19}\mathstrut +\mathstrut \) \(4157\) \(\beta_{18}\mathstrut +\mathstrut \) \(4919\) \(\beta_{17}\mathstrut -\mathstrut \) \(1181\) \(\beta_{16}\mathstrut -\mathstrut \) \(239\) \(\beta_{14}\mathstrut +\mathstrut \) \(239\) \(\beta_{13}\mathstrut -\mathstrut \) \(3268\) \(\beta_{11}\mathstrut +\mathstrut \) \(317\) \(\beta_{10}\mathstrut -\mathstrut \) \(2732\) \(\beta_{9}\mathstrut +\mathstrut \) \(2732\) \(\beta_{8}\mathstrut -\mathstrut \) \(317\) \(\beta_{7}\mathstrut -\mathstrut \) \(1258\) \(\beta_{4}\mathstrut -\mathstrut \) \(6301\) \(\beta_{1}\)
\(\nu^{12}\)\(=\)\(1235\) \(\beta_{19}\mathstrut -\mathstrut \) \(3728\) \(\beta_{18}\mathstrut +\mathstrut \) \(1235\) \(\beta_{16}\mathstrut +\mathstrut \) \(1258\) \(\beta_{15}\mathstrut +\mathstrut \) \(4307\) \(\beta_{14}\mathstrut +\mathstrut \) \(579\) \(\beta_{13}\mathstrut +\mathstrut \) \(4441\) \(\beta_{12}\mathstrut -\mathstrut \) \(3386\) \(\beta_{10}\mathstrut +\mathstrut \) \(2617\) \(\beta_{9}\mathstrut -\mathstrut \) \(3581\) \(\beta_{8}\mathstrut -\mathstrut \) \(3386\) \(\beta_{7}\mathstrut -\mathstrut \) \(4783\) \(\beta_{6}\mathstrut -\mathstrut \) \(3728\) \(\beta_{5}\mathstrut -\mathstrut \) \(7691\) \(\beta_{3}\mathstrut -\mathstrut \) \(16157\) \(\beta_{2}\mathstrut +\mathstrut \) \(1258\) \(\beta_{1}\mathstrut +\mathstrut \) \(21117\)
\(\nu^{13}\)\(=\)\(5418\) \(\beta_{19}\mathstrut -\mathstrut \) \(31673\) \(\beta_{18}\mathstrut -\mathstrut \) \(36800\) \(\beta_{17}\mathstrut +\mathstrut \) \(10201\) \(\beta_{16}\mathstrut +\mathstrut \) \(2508\) \(\beta_{14}\mathstrut -\mathstrut \) \(2508\) \(\beta_{13}\mathstrut +\mathstrut \) \(28062\) \(\beta_{11}\mathstrut -\mathstrut \) \(3504\) \(\beta_{10}\mathstrut +\mathstrut \) \(23371\) \(\beta_{9}\mathstrut -\mathstrut \) \(23371\) \(\beta_{8}\mathstrut +\mathstrut \) \(3504\) \(\beta_{7}\mathstrut +\mathstrut \) \(10585\) \(\beta_{4}\mathstrut +\mathstrut \) \(40376\) \(\beta_{1}\)
\(\nu^{14}\)\(=\)\(-\)\(10278\) \(\beta_{19}\mathstrut +\mathstrut \) \(31141\) \(\beta_{18}\mathstrut -\mathstrut \) \(10278\) \(\beta_{16}\mathstrut -\mathstrut \) \(10585\) \(\beta_{15}\mathstrut -\mathstrut \) \(37478\) \(\beta_{14}\mathstrut -\mathstrut \) \(6337\) \(\beta_{13}\mathstrut -\mathstrut \) \(31766\) \(\beta_{12}\mathstrut +\mathstrut \) \(24072\) \(\beta_{10}\mathstrut -\mathstrut \) \(19711\) \(\beta_{9}\mathstrut +\mathstrut \) \(31986\) \(\beta_{8}\mathstrut +\mathstrut \) \(24072\) \(\beta_{7}\mathstrut +\mathstrut \) \(36456\) \(\beta_{6}\mathstrut +\mathstrut \) \(31141\) \(\beta_{5}\mathstrut +\mathstrut \) \(59386\) \(\beta_{3}\mathstrut +\mathstrut \) \(114418\) \(\beta_{2}\mathstrut -\mathstrut \) \(10585\) \(\beta_{1}\mathstrut -\mathstrut \) \(141781\)
\(\nu^{15}\)\(=\)\(-\)\(48908\) \(\beta_{19}\mathstrut +\mathstrut \) \(241331\) \(\beta_{18}\mathstrut +\mathstrut \) \(273167\) \(\beta_{17}\mathstrut -\mathstrut \) \(85364\) \(\beta_{16}\mathstrut -\mathstrut \) \(23814\) \(\beta_{14}\mathstrut +\mathstrut \) \(23814\) \(\beta_{13}\mathstrut -\mathstrut \) \(229332\) \(\beta_{11}\mathstrut +\mathstrut \) \(34382\) \(\beta_{10}\mathstrut -\mathstrut \) \(192948\) \(\beta_{9}\mathstrut +\mathstrut \) \(192948\) \(\beta_{8}\mathstrut -\mathstrut \) \(34382\) \(\beta_{7}\mathstrut -\mathstrut \) \(86166\) \(\beta_{4}\mathstrut -\mathstrut \) \(263895\) \(\beta_{1}\)
\(\nu^{16}\)\(=\)\(82968\) \(\beta_{19}\mathstrut -\mathstrut \) \(252102\) \(\beta_{18}\mathstrut +\mathstrut \) \(82968\) \(\beta_{16}\mathstrut +\mathstrut \) \(86166\) \(\beta_{15}\mathstrut +\mathstrut \) \(313974\) \(\beta_{14}\mathstrut +\mathstrut \) \(61872\) \(\beta_{13}\mathstrut +\mathstrut \) \(225061\) \(\beta_{12}\mathstrut -\mathstrut \) \(172285\) \(\beta_{10}\mathstrut +\mathstrut \) \(144998\) \(\beta_{9}\mathstrut -\mathstrut \) \(273040\) \(\beta_{8}\mathstrut -\mathstrut \) \(172285\) \(\beta_{7}\mathstrut -\mathstrut \) \(277787\) \(\beta_{6}\mathstrut -\mathstrut \) \(252102\) \(\beta_{5}\mathstrut -\mathstrut \) \(447691\) \(\beta_{3}\mathstrut -\mathstrut \) \(816931\) \(\beta_{2}\mathstrut +\mathstrut \) \(86166\) \(\beta_{1}\mathstrut +\mathstrut \) \(974063\)
\(\nu^{17}\)\(=\)\(421078\) \(\beta_{19}\mathstrut -\mathstrut \) \(1838815\) \(\beta_{18}\mathstrut -\mathstrut \) \(2019572\) \(\beta_{17}\mathstrut +\mathstrut \) \(698865\) \(\beta_{16}\mathstrut +\mathstrut \) \(212606\) \(\beta_{14}\mathstrut -\mathstrut \) \(212606\) \(\beta_{13}\mathstrut +\mathstrut \) \(1816978\) \(\beta_{11}\mathstrut -\mathstrut \) \(313816\) \(\beta_{10}\mathstrut +\mathstrut \) \(1557768\) \(\beta_{9}\mathstrut -\mathstrut \) \(1557768\) \(\beta_{8}\mathstrut +\mathstrut \) \(313816\) \(\beta_{7}\mathstrut +\mathstrut \) \(687073\) \(\beta_{4}\mathstrut +\mathstrut \) \(1754916\) \(\beta_{1}\)
\(\nu^{18}\)\(=\)\(-\)\(658089\) \(\beta_{19}\mathstrut +\mathstrut \) \(2003251\) \(\beta_{18}\mathstrut -\mathstrut \) \(658089\) \(\beta_{16}\mathstrut -\mathstrut \) \(687073\) \(\beta_{15}\mathstrut -\mathstrut \) \(2566315\) \(\beta_{14}\mathstrut -\mathstrut \) \(563064\) \(\beta_{13}\mathstrut -\mathstrut \) \(1586702\) \(\beta_{12}\mathstrut +\mathstrut \) \(1241773\) \(\beta_{10}\mathstrut -\mathstrut \) \(1055751\) \(\beta_{9}\mathstrut +\mathstrut \) \(2263678\) \(\beta_{8}\mathstrut +\mathstrut \) \(1241773\) \(\beta_{7}\mathstrut +\mathstrut \) \(2116602\) \(\beta_{6}\mathstrut +\mathstrut \) \(2003251\) \(\beta_{5}\mathstrut +\mathstrut \) \(3322576\) \(\beta_{3}\mathstrut +\mathstrut \) \(5875164\) \(\beta_{2}\mathstrut -\mathstrut \) \(687073\) \(\beta_{1}\mathstrut -\mathstrut \) \(6817561\)
\(\nu^{19}\)\(=\)\(-\)\(3513815\) \(\beta_{19}\mathstrut +\mathstrut \) \(14006619\) \(\beta_{18}\mathstrut +\mathstrut \) \(14904295\) \(\beta_{17}\mathstrut -\mathstrut \) \(5630417\) \(\beta_{16}\mathstrut -\mathstrut \) \(1822329\) \(\beta_{14}\mathstrut +\mathstrut \) \(1822329\) \(\beta_{13}\mathstrut -\mathstrut \) \(14109884\) \(\beta_{11}\mathstrut +\mathstrut \) \(2731717\) \(\beta_{10}\mathstrut -\mathstrut \) \(12387861\) \(\beta_{9}\mathstrut +\mathstrut \) \(12387861\) \(\beta_{8}\mathstrut -\mathstrut \) \(2731717\) \(\beta_{7}\mathstrut -\mathstrut \) \(5403154\) \(\beta_{4}\mathstrut -\mathstrut \) \(11848557\) \(\beta_{1}\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1155\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(232\) \(386\) \(661\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
694.1
2.74150i
2.55795i
2.32980i
2.28323i
1.93467i
1.40171i
1.28094i
0.792050i
0.568210i
0.0342944i
0.0342944i
0.568210i
0.792050i
1.28094i
1.40171i
1.93467i
2.28323i
2.32980i
2.55795i
2.74150i
2.74150i 1.00000i −5.51584 −0.739422 2.11027i 2.74150 1.00000i 9.63869i −1.00000 −5.78532 + 2.02713i
694.2 2.55795i 1.00000i −4.54312 0.0738813 + 2.23485i 2.55795 1.00000i 6.50518i −1.00000 5.71663 0.188985i
694.3 2.32980i 1.00000i −3.42795 1.29334 + 1.82408i −2.32980 1.00000i 3.32682i −1.00000 4.24974 3.01322i
694.4 2.28323i 1.00000i −3.21315 −2.23363 0.104355i −2.28323 1.00000i 2.76990i −1.00000 −0.238267 + 5.09990i
694.5 1.93467i 1.00000i −1.74296 1.72871 1.41829i 1.93467 1.00000i 0.497289i −1.00000 −2.74393 3.34450i
694.6 1.40171i 1.00000i 0.0352144 −2.22487 + 0.223473i 1.40171 1.00000i 2.85278i −1.00000 0.313244 + 3.11862i
694.7 1.28094i 1.00000i 0.359182 2.05729 0.876110i −1.28094 1.00000i 3.02198i −1.00000 −1.12225 2.63527i
694.8 0.792050i 1.00000i 1.37266 −0.199975 + 2.22711i 0.792050 1.00000i 2.67131i −1.00000 1.76398 + 0.158390i
694.9 0.568210i 1.00000i 1.67714 1.08688 1.95415i −0.568210 1.00000i 2.08939i −1.00000 −1.11037 0.617578i
694.10 0.0342944i 1.00000i 1.99882 −1.84220 1.26739i 0.0342944 1.00000i 0.137137i −1.00000 −0.0434646 + 0.0631773i
694.11 0.0342944i 1.00000i 1.99882 −1.84220 + 1.26739i 0.0342944 1.00000i 0.137137i −1.00000 −0.0434646 0.0631773i
694.12 0.568210i 1.00000i 1.67714 1.08688 + 1.95415i −0.568210 1.00000i 2.08939i −1.00000 −1.11037 + 0.617578i
694.13 0.792050i 1.00000i 1.37266 −0.199975 2.22711i 0.792050 1.00000i 2.67131i −1.00000 1.76398 0.158390i
694.14 1.28094i 1.00000i 0.359182 2.05729 + 0.876110i −1.28094 1.00000i 3.02198i −1.00000 −1.12225 + 2.63527i
694.15 1.40171i 1.00000i 0.0352144 −2.22487 0.223473i 1.40171 1.00000i 2.85278i −1.00000 0.313244 3.11862i
694.16 1.93467i 1.00000i −1.74296 1.72871 + 1.41829i 1.93467 1.00000i 0.497289i −1.00000 −2.74393 + 3.34450i
694.17 2.28323i 1.00000i −3.21315 −2.23363 + 0.104355i −2.28323 1.00000i 2.76990i −1.00000 −0.238267 5.09990i
694.18 2.32980i 1.00000i −3.42795 1.29334 1.82408i −2.32980 1.00000i 3.32682i −1.00000 4.24974 + 3.01322i
694.19 2.55795i 1.00000i −4.54312 0.0738813 2.23485i 2.55795 1.00000i 6.50518i −1.00000 5.71663 + 0.188985i
694.20 2.74150i 1.00000i −5.51584 −0.739422 + 2.11027i 2.74150 1.00000i 9.63869i −1.00000 −5.78532 2.02713i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 694.20
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
5.b Even 1 yes

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1155, [\chi])\):

\(T_{2}^{20} + \cdots\)
\(T_{13}^{20} + \cdots\)