Properties

Label 930.4.d.c
Level $930$
Weight $4$
Character orbit 930.d
Analytic conductor $54.872$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 930.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(54.8717763053\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Defining polynomial: \(x^{20} + 2763 x^{18} + 2652899 x^{16} + 1161420105 x^{14} + 247831438280 x^{12} + 26461073949176 x^{10} + 1433368340491408 x^{8} + 37536884921183444 x^{6} + 414170668954972900 x^{4} + 1987578737873500000 x^{2} + 3460065015625000000\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + 2 \beta_{8} q^{2} -3 \beta_{8} q^{3} -4 q^{4} -\beta_{16} q^{5} + 6 q^{6} + ( \beta_{7} - \beta_{8} ) q^{7} -8 \beta_{8} q^{8} -9 q^{9} +O(q^{10})\) \( q + 2 \beta_{8} q^{2} -3 \beta_{8} q^{3} -4 q^{4} -\beta_{16} q^{5} + 6 q^{6} + ( \beta_{7} - \beta_{8} ) q^{7} -8 \beta_{8} q^{8} -9 q^{9} + 2 \beta_{10} q^{10} + ( -5 + \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{10} + \beta_{11} + \beta_{15} + \beta_{16} ) q^{11} + 12 \beta_{8} q^{12} + ( \beta_{7} + 10 \beta_{8} - \beta_{9} + \beta_{14} + \beta_{15} - \beta_{16} + \beta_{17} - 2 \beta_{18} + \beta_{19} ) q^{13} + ( 2 - 2 \beta_{1} ) q^{14} -3 \beta_{10} q^{15} + 16 q^{16} + ( -\beta_{7} - 19 \beta_{8} - \beta_{9} + 2 \beta_{10} + 2 \beta_{11} - \beta_{12} + \beta_{13} + \beta_{15} - \beta_{16} - 2 \beta_{17} + 2 \beta_{18} + 2 \beta_{19} ) q^{17} -18 \beta_{8} q^{18} + ( 18 + \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + 4 \beta_{5} - \beta_{12} - \beta_{13} + 2 \beta_{15} + 2 \beta_{16} ) q^{19} + 4 \beta_{16} q^{20} + ( -3 + 3 \beta_{1} ) q^{21} + ( -10 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} + 4 \beta_{14} + 2 \beta_{15} - 2 \beta_{16} + 2 \beta_{17} - 2 \beta_{18} ) q^{22} + ( 2 \beta_{7} + 33 \beta_{8} - \beta_{9} + \beta_{12} - \beta_{13} + 5 \beta_{14} + 2 \beta_{15} - 2 \beta_{16} + \beta_{17} + \beta_{18} + \beta_{19} ) q^{23} -24 q^{24} + ( -2 + 2 \beta_{1} - 5 \beta_{2} + \beta_{3} - \beta_{5} - 4 \beta_{6} + \beta_{7} - 11 \beta_{8} + 2 \beta_{9} + 3 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} + \beta_{13} - 3 \beta_{14} + \beta_{15} + 3 \beta_{16} + 3 \beta_{17} ) q^{25} + ( -20 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{10} - 2 \beta_{11} ) q^{26} + 27 \beta_{8} q^{27} + ( -4 \beta_{7} + 4 \beta_{8} ) q^{28} + ( 23 + 4 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} - 6 \beta_{5} - 6 \beta_{6} - \beta_{10} + \beta_{11} - 3 \beta_{15} - 3 \beta_{16} ) q^{29} -6 \beta_{16} q^{30} -31 q^{31} + 32 \beta_{8} q^{32} + ( 15 \beta_{8} + 3 \beta_{9} + 3 \beta_{10} + 3 \beta_{11} - 6 \beta_{14} - 3 \beta_{15} + 3 \beta_{16} - 3 \beta_{17} + 3 \beta_{18} ) q^{33} + ( 38 + 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} + 2 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} + 2 \beta_{13} + 4 \beta_{15} + 4 \beta_{16} ) q^{34} + ( 19 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} - 5 \beta_{7} + 20 \beta_{8} + 5 \beta_{9} - 2 \beta_{10} - 3 \beta_{11} + \beta_{12} - 4 \beta_{13} - 2 \beta_{14} - 3 \beta_{15} + 5 \beta_{16} - 5 \beta_{17} + 2 \beta_{18} - 4 \beta_{19} ) q^{35} + 36 q^{36} + ( -6 \beta_{7} - 49 \beta_{8} - 4 \beta_{9} + \beta_{10} + \beta_{11} + 6 \beta_{12} - 6 \beta_{13} - \beta_{15} + \beta_{16} + 3 \beta_{17} - 6 \beta_{18} ) q^{37} + ( 2 \beta_{7} + 36 \beta_{8} - 4 \beta_{9} - 4 \beta_{10} - 4 \beta_{11} - 2 \beta_{12} + 2 \beta_{13} - 8 \beta_{17} + 2 \beta_{18} - 2 \beta_{19} ) q^{38} + ( 30 + 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + 6 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} - 3 \beta_{10} + 3 \beta_{11} ) q^{39} -8 \beta_{10} q^{40} + ( -41 - 5 \beta_{1} - 8 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} + \beta_{6} + 3 \beta_{10} - 3 \beta_{11} - 7 \beta_{15} - 7 \beta_{16} ) q^{41} + ( 6 \beta_{7} - 6 \beta_{8} ) q^{42} + ( -6 \beta_{7} - 3 \beta_{8} - 19 \beta_{9} - 5 \beta_{10} - 5 \beta_{11} + 3 \beta_{12} - 3 \beta_{13} + 3 \beta_{15} - 3 \beta_{16} - 10 \beta_{17} + 2 \beta_{18} + \beta_{19} ) q^{43} + ( 20 - 4 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} + 8 \beta_{6} + 4 \beta_{10} - 4 \beta_{11} - 4 \beta_{15} - 4 \beta_{16} ) q^{44} + 9 \beta_{16} q^{45} + ( -66 - 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 10 \beta_{6} + 4 \beta_{10} - 4 \beta_{11} - 2 \beta_{12} - 2 \beta_{13} ) q^{46} + ( 5 \beta_{7} - 63 \beta_{8} - 3 \beta_{9} - 18 \beta_{10} - 18 \beta_{11} + 5 \beta_{12} - 5 \beta_{13} + 7 \beta_{14} - 3 \beta_{15} + 3 \beta_{16} - 9 \beta_{17} - 12 \beta_{18} ) q^{47} -48 \beta_{8} q^{48} + ( 61 + \beta_{1} + 19 \beta_{2} + 4 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} + 12 \beta_{6} - 4 \beta_{10} + 4 \beta_{11} - 18 \beta_{15} - 18 \beta_{16} ) q^{49} + ( 22 - 2 \beta_{1} + 4 \beta_{3} + 6 \beta_{5} - 6 \beta_{6} + 4 \beta_{7} - 4 \beta_{8} - 2 \beta_{9} - 6 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} + 4 \beta_{13} + 8 \beta_{14} + 4 \beta_{15} + 6 \beta_{16} + 2 \beta_{17} - 10 \beta_{19} ) q^{50} + ( -57 - 3 \beta_{1} + 6 \beta_{2} + 3 \beta_{3} - 6 \beta_{4} + 6 \beta_{5} - 3 \beta_{10} + 3 \beta_{11} - 3 \beta_{12} - 3 \beta_{13} - 6 \beta_{15} - 6 \beta_{16} ) q^{51} + ( -4 \beta_{7} - 40 \beta_{8} + 4 \beta_{9} - 4 \beta_{14} - 4 \beta_{15} + 4 \beta_{16} - 4 \beta_{17} + 8 \beta_{18} - 4 \beta_{19} ) q^{52} + ( 4 \beta_{7} + 71 \beta_{8} + 19 \beta_{9} + 7 \beta_{10} + 7 \beta_{11} - 12 \beta_{12} + 12 \beta_{13} - 6 \beta_{14} - 11 \beta_{15} + 11 \beta_{16} + 2 \beta_{17} + 7 \beta_{18} + 9 \beta_{19} ) q^{53} -54 q^{54} + ( -70 - 13 \beta_{1} + 7 \beta_{2} + 8 \beta_{3} + 3 \beta_{4} - 8 \beta_{5} + 2 \beta_{6} - 9 \beta_{7} + 13 \beta_{8} - 7 \beta_{9} - 8 \beta_{10} - 8 \beta_{11} + 8 \beta_{12} - 4 \beta_{13} - 10 \beta_{14} - 14 \beta_{15} + 4 \beta_{16} - 13 \beta_{17} + 4 \beta_{18} - \beta_{19} ) q^{55} + ( -8 + 8 \beta_{1} ) q^{56} + ( -3 \beta_{7} - 54 \beta_{8} + 6 \beta_{9} + 6 \beta_{10} + 6 \beta_{11} + 3 \beta_{12} - 3 \beta_{13} + 12 \beta_{17} - 3 \beta_{18} + 3 \beta_{19} ) q^{57} + ( 8 \beta_{7} + 46 \beta_{8} + 8 \beta_{9} + 6 \beta_{10} + 6 \beta_{11} + 12 \beta_{14} + 2 \beta_{15} - 2 \beta_{16} + 12 \beta_{17} + 4 \beta_{18} - 6 \beta_{19} ) q^{58} + ( 171 - 2 \beta_{1} - 13 \beta_{2} + 12 \beta_{3} + 24 \beta_{4} - 15 \beta_{5} - 13 \beta_{6} + 9 \beta_{10} - 9 \beta_{11} + 12 \beta_{12} + 12 \beta_{13} + 12 \beta_{15} + 12 \beta_{16} ) q^{59} + 12 \beta_{10} q^{60} + ( -136 + 16 \beta_{1} - \beta_{2} - 20 \beta_{3} - 17 \beta_{4} + 2 \beta_{5} + 11 \beta_{6} + 19 \beta_{10} - 19 \beta_{11} - 8 \beta_{12} - 8 \beta_{13} - 17 \beta_{15} - 17 \beta_{16} ) q^{61} -62 \beta_{8} q^{62} + ( -9 \beta_{7} + 9 \beta_{8} ) q^{63} -64 q^{64} + ( -1 + 4 \beta_{1} + 5 \beta_{2} - 2 \beta_{4} + 5 \beta_{5} - 15 \beta_{6} - 25 \beta_{7} + 51 \beta_{8} - \beta_{9} + 6 \beta_{10} - 2 \beta_{11} - \beta_{12} - 11 \beta_{13} - 6 \beta_{14} + 3 \beta_{15} + 7 \beta_{16} + \beta_{17} - \beta_{18} + 5 \beta_{19} ) q^{65} + ( -30 + 6 \beta_{3} + 6 \beta_{4} - 6 \beta_{5} - 12 \beta_{6} - 6 \beta_{10} + 6 \beta_{11} + 6 \beta_{15} + 6 \beta_{16} ) q^{66} + ( 19 \beta_{7} - 76 \beta_{8} + 13 \beta_{9} - \beta_{10} - \beta_{11} - 15 \beta_{12} + 15 \beta_{13} + 22 \beta_{14} - 8 \beta_{15} + 8 \beta_{16} + 36 \beta_{17} - 8 \beta_{18} - 7 \beta_{19} ) q^{67} + ( 4 \beta_{7} + 76 \beta_{8} + 4 \beta_{9} - 8 \beta_{10} - 8 \beta_{11} + 4 \beta_{12} - 4 \beta_{13} - 4 \beta_{15} + 4 \beta_{16} + 8 \beta_{17} - 8 \beta_{18} - 8 \beta_{19} ) q^{68} + ( 99 + 6 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} - 15 \beta_{6} - 6 \beta_{10} + 6 \beta_{11} + 3 \beta_{12} + 3 \beta_{13} ) q^{69} + ( -40 + 10 \beta_{1} + 8 \beta_{2} + 10 \beta_{3} + 4 \beta_{4} - 10 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} + 38 \beta_{8} + 4 \beta_{9} - 10 \beta_{10} + 6 \beta_{11} - 8 \beta_{12} - 2 \beta_{13} + 2 \beta_{14} - 6 \beta_{15} - 4 \beta_{16} - 4 \beta_{17} + 2 \beta_{18} - 4 \beta_{19} ) q^{70} + ( -140 - 44 \beta_{1} + 15 \beta_{2} + 33 \beta_{3} + \beta_{4} + 29 \beta_{5} + 28 \beta_{6} + 13 \beta_{10} - 13 \beta_{11} + 21 \beta_{12} + 21 \beta_{13} + 4 \beta_{15} + 4 \beta_{16} ) q^{71} + 72 \beta_{8} q^{72} + ( 37 \beta_{7} - 15 \beta_{8} + 14 \beta_{9} - 5 \beta_{10} - 5 \beta_{11} - 9 \beta_{12} + 9 \beta_{13} - 3 \beta_{14} - 20 \beta_{15} + 20 \beta_{16} + \beta_{17} + 16 \beta_{18} - 16 \beta_{19} ) q^{73} + ( 98 + 12 \beta_{1} - 8 \beta_{3} - 12 \beta_{4} + 6 \beta_{5} - 2 \beta_{10} + 2 \beta_{11} - 12 \beta_{12} - 12 \beta_{13} + 2 \beta_{15} + 2 \beta_{16} ) q^{74} + ( -33 + 3 \beta_{1} - 6 \beta_{3} - 9 \beta_{5} + 9 \beta_{6} - 6 \beta_{7} + 6 \beta_{8} + 3 \beta_{9} + 9 \beta_{10} + 3 \beta_{11} - 3 \beta_{12} - 6 \beta_{13} - 12 \beta_{14} - 6 \beta_{15} - 9 \beta_{16} - 3 \beta_{17} + 15 \beta_{19} ) q^{75} + ( -72 - 4 \beta_{1} + 4 \beta_{2} - 8 \beta_{3} + 4 \beta_{4} - 16 \beta_{5} + 4 \beta_{12} + 4 \beta_{13} - 8 \beta_{15} - 8 \beta_{16} ) q^{76} + ( 9 \beta_{7} - 154 \beta_{8} - 2 \beta_{10} - 2 \beta_{11} - 10 \beta_{12} + 10 \beta_{13} - 7 \beta_{14} + 7 \beta_{15} - 7 \beta_{16} + 15 \beta_{17} - \beta_{18} + 17 \beta_{19} ) q^{77} + ( 6 \beta_{7} + 60 \beta_{8} - 6 \beta_{9} + 6 \beta_{14} + 6 \beta_{15} - 6 \beta_{16} + 6 \beta_{17} - 12 \beta_{18} + 6 \beta_{19} ) q^{78} + ( 218 + 19 \beta_{1} + 4 \beta_{2} - 18 \beta_{3} - 5 \beta_{4} - 3 \beta_{5} - 16 \beta_{6} + 7 \beta_{10} - 7 \beta_{11} - 22 \beta_{12} - 22 \beta_{13} - 22 \beta_{15} - 22 \beta_{16} ) q^{79} -16 \beta_{16} q^{80} + 81 q^{81} + ( -10 \beta_{7} - 82 \beta_{8} + 6 \beta_{9} + 14 \beta_{10} + 14 \beta_{11} - 2 \beta_{14} - 6 \beta_{15} + 6 \beta_{16} - 6 \beta_{17} - 6 \beta_{18} - 16 \beta_{19} ) q^{82} + ( 13 \beta_{7} + 113 \beta_{8} + \beta_{9} - 21 \beta_{10} - 21 \beta_{11} + 17 \beta_{12} - 17 \beta_{13} - 20 \beta_{14} - 6 \beta_{15} + 6 \beta_{16} - 19 \beta_{17} + 5 \beta_{18} + 11 \beta_{19} ) q^{83} + ( 12 - 12 \beta_{1} ) q^{84} + ( 30 - 20 \beta_{1} - \beta_{2} + 20 \beta_{3} + 32 \beta_{4} + 10 \beta_{5} + 3 \beta_{6} + \beta_{7} + 179 \beta_{8} - 3 \beta_{9} - 10 \beta_{10} - 2 \beta_{11} + 11 \beta_{12} + 29 \beta_{13} + 6 \beta_{14} + 7 \beta_{15} + 21 \beta_{16} + 13 \beta_{17} - 4 \beta_{18} - 7 \beta_{19} ) q^{85} + ( 6 + 12 \beta_{1} - 2 \beta_{2} - 38 \beta_{3} + 4 \beta_{4} - 20 \beta_{5} + 6 \beta_{10} - 6 \beta_{11} - 6 \beta_{12} - 6 \beta_{13} - 10 \beta_{15} - 10 \beta_{16} ) q^{86} + ( -12 \beta_{7} - 69 \beta_{8} - 12 \beta_{9} - 9 \beta_{10} - 9 \beta_{11} - 18 \beta_{14} - 3 \beta_{15} + 3 \beta_{16} - 18 \beta_{17} - 6 \beta_{18} + 9 \beta_{19} ) q^{87} + ( 40 \beta_{8} + 8 \beta_{9} + 8 \beta_{10} + 8 \beta_{11} - 16 \beta_{14} - 8 \beta_{15} + 8 \beta_{16} - 8 \beta_{17} + 8 \beta_{18} ) q^{88} + ( 68 - 14 \beta_{1} - \beta_{2} + \beta_{3} - 3 \beta_{4} + 22 \beta_{5} - 11 \beta_{6} + 12 \beta_{10} - 12 \beta_{11} + 18 \beta_{12} + 18 \beta_{13} + 8 \beta_{15} + 8 \beta_{16} ) q^{89} -18 \beta_{10} q^{90} + ( -286 + 10 \beta_{1} + 14 \beta_{2} - 4 \beta_{3} - 21 \beta_{4} + 4 \beta_{5} + 10 \beta_{6} - \beta_{12} - \beta_{13} - 50 \beta_{15} - 50 \beta_{16} ) q^{91} + ( -8 \beta_{7} - 132 \beta_{8} + 4 \beta_{9} - 4 \beta_{12} + 4 \beta_{13} - 20 \beta_{14} - 8 \beta_{15} + 8 \beta_{16} - 4 \beta_{17} - 4 \beta_{18} - 4 \beta_{19} ) q^{92} + 93 \beta_{8} q^{93} + ( 126 - 10 \beta_{1} - 6 \beta_{3} - 24 \beta_{4} - 18 \beta_{5} + 14 \beta_{6} - 6 \beta_{10} + 6 \beta_{11} - 10 \beta_{12} - 10 \beta_{13} - 36 \beta_{15} - 36 \beta_{16} ) q^{94} + ( -69 - 23 \beta_{1} + 3 \beta_{2} - \beta_{3} - 18 \beta_{4} + 11 \beta_{5} + 25 \beta_{6} + 21 \beta_{7} - 5 \beta_{8} + 36 \beta_{9} + 33 \beta_{10} + 12 \beta_{11} - 17 \beta_{12} + 26 \beta_{13} + 24 \beta_{14} - 34 \beta_{15} - 34 \beta_{16} + 39 \beta_{17} + 21 \beta_{18} + \beta_{19} ) q^{95} + 96 q^{96} + ( -31 \beta_{7} - 297 \beta_{8} - 11 \beta_{9} + 6 \beta_{10} + 6 \beta_{11} + 3 \beta_{12} - 3 \beta_{13} - 14 \beta_{14} - 26 \beta_{15} + 26 \beta_{16} - 9 \beta_{17} + 18 \beta_{18} + 41 \beta_{19} ) q^{97} + ( 2 \beta_{7} + 122 \beta_{8} - 8 \beta_{9} + 36 \beta_{10} + 36 \beta_{11} - 24 \beta_{14} + 8 \beta_{15} - 8 \beta_{16} + 8 \beta_{17} + 8 \beta_{18} + 38 \beta_{19} ) q^{98} + ( 45 - 9 \beta_{3} - 9 \beta_{4} + 9 \beta_{5} + 18 \beta_{6} + 9 \beta_{10} - 9 \beta_{11} - 9 \beta_{15} - 9 \beta_{16} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q - 80q^{4} - 2q^{5} + 120q^{6} - 180q^{9} + O(q^{10}) \) \( 20q - 80q^{4} - 2q^{5} + 120q^{6} - 180q^{9} + 8q^{10} - 114q^{11} + 52q^{14} - 12q^{15} + 320q^{16} + 370q^{19} + 8q^{20} - 78q^{21} - 480q^{24} - 90q^{25} - 368q^{26} + 368q^{29} - 12q^{30} - 620q^{31} + 712q^{34} + 374q^{35} + 720q^{36} + 552q^{39} - 32q^{40} - 872q^{41} + 456q^{44} + 18q^{45} - 1236q^{46} + 1334q^{49} + 416q^{50} - 1068q^{51} - 1080q^{54} - 1290q^{55} - 208q^{56} + 3228q^{59} + 48q^{60} - 2604q^{61} - 1280q^{64} + 44q^{65} - 684q^{66} + 1854q^{69} - 852q^{70} - 2290q^{71} + 2008q^{74} - 624q^{75} - 1480q^{76} + 4342q^{79} - 32q^{80} + 1620q^{81} + 312q^{84} + 500q^{85} - 4q^{86} + 1390q^{89} - 72q^{90} - 5744q^{91} + 2608q^{94} - 1136q^{95} + 1920q^{96} + 1026q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} + 2763 x^{18} + 2652899 x^{16} + 1161420105 x^{14} + 247831438280 x^{12} + 26461073949176 x^{10} + 1433368340491408 x^{8} + 37536884921183444 x^{6} + 414170668954972900 x^{4} + 1987578737873500000 x^{2} + 3460065015625000000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-54931471300923625262851678917 \nu^{18} - 151355724231870876258193118110421 \nu^{16} - 144573245074660506648919709820669883 \nu^{14} - 62700498856685305656582358591476463535 \nu^{12} - 13141061810554499633837291992357023300260 \nu^{10} - 1355791024107463781262453358894242818867392 \nu^{8} - 68873935652143522096650132524964672359254136 \nu^{6} - 1576070803725895166160914980074950303892732148 \nu^{4} - 11994743884174515355097759763379045126123010300 \nu^{2} - 28078626625186327718933935904659857519505240000\)\()/ \)\(10\!\cdots\!00\)\( \)
\(\beta_{2}\)\(=\)\((\)\(\)\(36\!\cdots\!79\)\( \nu^{18} + \)\(99\!\cdots\!77\)\( \nu^{16} + \)\(95\!\cdots\!21\)\( \nu^{14} + \)\(41\!\cdots\!95\)\( \nu^{12} + \)\(86\!\cdots\!20\)\( \nu^{10} + \)\(89\!\cdots\!04\)\( \nu^{8} + \)\(45\!\cdots\!32\)\( \nu^{6} + \)\(10\!\cdots\!76\)\( \nu^{4} + \)\(79\!\cdots\!00\)\( \nu^{2} + \)\(18\!\cdots\!00\)\(\)\()/ \)\(18\!\cdots\!00\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(38\!\cdots\!83\)\( \nu^{18} - \)\(10\!\cdots\!29\)\( \nu^{16} - \)\(10\!\cdots\!17\)\( \nu^{14} - \)\(44\!\cdots\!15\)\( \nu^{12} - \)\(93\!\cdots\!40\)\( \nu^{10} - \)\(96\!\cdots\!08\)\( \nu^{8} - \)\(49\!\cdots\!64\)\( \nu^{6} - \)\(11\!\cdots\!52\)\( \nu^{4} - \)\(88\!\cdots\!00\)\( \nu^{2} - \)\(21\!\cdots\!00\)\(\)\()/ \)\(11\!\cdots\!00\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(96\!\cdots\!73\)\( \nu^{18} - \)\(26\!\cdots\!59\)\( \nu^{16} - \)\(25\!\cdots\!47\)\( \nu^{14} - \)\(11\!\cdots\!85\)\( \nu^{12} - \)\(23\!\cdots\!20\)\( \nu^{10} - \)\(24\!\cdots\!68\)\( \nu^{8} - \)\(12\!\cdots\!24\)\( \nu^{6} - \)\(28\!\cdots\!52\)\( \nu^{4} - \)\(23\!\cdots\!00\)\( \nu^{2} - \)\(60\!\cdots\!00\)\(\)\()/ \)\(22\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(31\!\cdots\!71\)\( \nu^{18} - \)\(85\!\cdots\!48\)\( \nu^{16} - \)\(82\!\cdots\!29\)\( \nu^{14} - \)\(35\!\cdots\!80\)\( \nu^{12} - \)\(74\!\cdots\!80\)\( \nu^{10} - \)\(77\!\cdots\!96\)\( \nu^{8} - \)\(39\!\cdots\!68\)\( \nu^{6} - \)\(92\!\cdots\!24\)\( \nu^{4} - \)\(72\!\cdots\!00\)\( \nu^{2} - \)\(17\!\cdots\!00\)\(\)\()/ \)\(69\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(33\!\cdots\!72\)\( \nu^{18} - \)\(92\!\cdots\!11\)\( \nu^{16} - \)\(88\!\cdots\!28\)\( \nu^{14} - \)\(38\!\cdots\!85\)\( \nu^{12} - \)\(81\!\cdots\!60\)\( \nu^{10} - \)\(84\!\cdots\!72\)\( \nu^{8} - \)\(42\!\cdots\!76\)\( \nu^{6} - \)\(99\!\cdots\!68\)\( \nu^{4} - \)\(76\!\cdots\!00\)\( \nu^{2} - \)\(18\!\cdots\!00\)\(\)\()/ \)\(69\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(15\!\cdots\!89\)\( \nu^{19} - \)\(41\!\cdots\!07\)\( \nu^{17} - \)\(39\!\cdots\!11\)\( \nu^{15} - \)\(17\!\cdots\!45\)\( \nu^{13} - \)\(36\!\cdots\!20\)\( \nu^{11} - \)\(37\!\cdots\!64\)\( \nu^{9} - \)\(19\!\cdots\!12\)\( \nu^{7} - \)\(44\!\cdots\!16\)\( \nu^{5} - \)\(33\!\cdots\!00\)\( \nu^{3} - \)\(76\!\cdots\!00\)\( \nu\)\()/ \)\(19\!\cdots\!00\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(15\!\cdots\!89\)\( \nu^{19} - \)\(41\!\cdots\!07\)\( \nu^{17} - \)\(39\!\cdots\!11\)\( \nu^{15} - \)\(17\!\cdots\!45\)\( \nu^{13} - \)\(36\!\cdots\!20\)\( \nu^{11} - \)\(37\!\cdots\!64\)\( \nu^{9} - \)\(19\!\cdots\!12\)\( \nu^{7} - \)\(44\!\cdots\!16\)\( \nu^{5} - \)\(33\!\cdots\!00\)\( \nu^{3} - \)\(78\!\cdots\!00\)\( \nu\)\()/ \)\(19\!\cdots\!00\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(12\!\cdots\!67\)\( \nu^{19} - \)\(34\!\cdots\!21\)\( \nu^{17} - \)\(33\!\cdots\!33\)\( \nu^{15} - \)\(14\!\cdots\!35\)\( \nu^{13} - \)\(30\!\cdots\!60\)\( \nu^{11} - \)\(31\!\cdots\!92\)\( \nu^{9} - \)\(15\!\cdots\!36\)\( \nu^{7} - \)\(35\!\cdots\!48\)\( \nu^{5} - \)\(25\!\cdots\!00\)\( \nu^{3} - \)\(57\!\cdots\!00\)\( \nu\)\()/ \)\(41\!\cdots\!00\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(11\!\cdots\!87\)\( \nu^{19} - \)\(32\!\cdots\!50\)\( \nu^{18} - \)\(31\!\cdots\!81\)\( \nu^{17} - \)\(88\!\cdots\!50\)\( \nu^{16} - \)\(29\!\cdots\!13\)\( \nu^{15} - \)\(85\!\cdots\!50\)\( \nu^{14} - \)\(12\!\cdots\!35\)\( \nu^{13} - \)\(37\!\cdots\!50\)\( \nu^{12} - \)\(27\!\cdots\!60\)\( \nu^{11} - \)\(78\!\cdots\!00\)\( \nu^{10} - \)\(28\!\cdots\!12\)\( \nu^{9} - \)\(81\!\cdots\!00\)\( \nu^{8} - \)\(14\!\cdots\!96\)\( \nu^{7} - \)\(42\!\cdots\!00\)\( \nu^{6} - \)\(33\!\cdots\!28\)\( \nu^{5} - \)\(10\!\cdots\!00\)\( \nu^{4} - \)\(26\!\cdots\!00\)\( \nu^{3} - \)\(83\!\cdots\!00\)\( \nu^{2} - \)\(67\!\cdots\!00\)\( \nu - \)\(21\!\cdots\!00\)\(\)\()/ \)\(28\!\cdots\!00\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-\)\(11\!\cdots\!87\)\( \nu^{19} + \)\(32\!\cdots\!50\)\( \nu^{18} - \)\(31\!\cdots\!81\)\( \nu^{17} + \)\(88\!\cdots\!50\)\( \nu^{16} - \)\(29\!\cdots\!13\)\( \nu^{15} + \)\(85\!\cdots\!50\)\( \nu^{14} - \)\(12\!\cdots\!35\)\( \nu^{13} + \)\(37\!\cdots\!50\)\( \nu^{12} - \)\(27\!\cdots\!60\)\( \nu^{11} + \)\(78\!\cdots\!00\)\( \nu^{10} - \)\(28\!\cdots\!12\)\( \nu^{9} + \)\(81\!\cdots\!00\)\( \nu^{8} - \)\(14\!\cdots\!96\)\( \nu^{7} + \)\(42\!\cdots\!00\)\( \nu^{6} - \)\(33\!\cdots\!28\)\( \nu^{5} + \)\(10\!\cdots\!00\)\( \nu^{4} - \)\(26\!\cdots\!00\)\( \nu^{3} + \)\(83\!\cdots\!00\)\( \nu^{2} - \)\(67\!\cdots\!00\)\( \nu + \)\(21\!\cdots\!00\)\(\)\()/ \)\(28\!\cdots\!00\)\( \)
\(\beta_{12}\)\(=\)\((\)\(\)\(40\!\cdots\!49\)\( \nu^{19} + \)\(27\!\cdots\!50\)\( \nu^{18} + \)\(11\!\cdots\!87\)\( \nu^{17} + \)\(76\!\cdots\!50\)\( \nu^{16} + \)\(10\!\cdots\!51\)\( \nu^{15} + \)\(73\!\cdots\!50\)\( \nu^{14} + \)\(46\!\cdots\!45\)\( \nu^{13} + \)\(31\!\cdots\!50\)\( \nu^{12} + \)\(97\!\cdots\!20\)\( \nu^{11} + \)\(67\!\cdots\!00\)\( \nu^{10} + \)\(10\!\cdots\!24\)\( \nu^{9} + \)\(69\!\cdots\!00\)\( \nu^{8} + \)\(52\!\cdots\!92\)\( \nu^{7} + \)\(35\!\cdots\!00\)\( \nu^{6} + \)\(12\!\cdots\!56\)\( \nu^{5} + \)\(83\!\cdots\!00\)\( \nu^{4} + \)\(10\!\cdots\!00\)\( \nu^{3} + \)\(65\!\cdots\!00\)\( \nu^{2} + \)\(28\!\cdots\!00\)\( \nu + \)\(16\!\cdots\!00\)\(\)\()/ \)\(82\!\cdots\!00\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(40\!\cdots\!49\)\( \nu^{19} + \)\(27\!\cdots\!50\)\( \nu^{18} - \)\(11\!\cdots\!87\)\( \nu^{17} + \)\(76\!\cdots\!50\)\( \nu^{16} - \)\(10\!\cdots\!51\)\( \nu^{15} + \)\(73\!\cdots\!50\)\( \nu^{14} - \)\(46\!\cdots\!45\)\( \nu^{13} + \)\(31\!\cdots\!50\)\( \nu^{12} - \)\(97\!\cdots\!20\)\( \nu^{11} + \)\(67\!\cdots\!00\)\( \nu^{10} - \)\(10\!\cdots\!24\)\( \nu^{9} + \)\(69\!\cdots\!00\)\( \nu^{8} - \)\(52\!\cdots\!92\)\( \nu^{7} + \)\(35\!\cdots\!00\)\( \nu^{6} - \)\(12\!\cdots\!56\)\( \nu^{5} + \)\(83\!\cdots\!00\)\( \nu^{4} - \)\(10\!\cdots\!00\)\( \nu^{3} + \)\(65\!\cdots\!00\)\( \nu^{2} - \)\(28\!\cdots\!00\)\( \nu + \)\(16\!\cdots\!00\)\(\)\()/ \)\(82\!\cdots\!00\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(11\!\cdots\!77\)\( \nu^{19} - \)\(30\!\cdots\!51\)\( \nu^{17} - \)\(29\!\cdots\!23\)\( \nu^{15} - \)\(12\!\cdots\!85\)\( \nu^{13} - \)\(26\!\cdots\!60\)\( \nu^{11} - \)\(27\!\cdots\!52\)\( \nu^{9} - \)\(14\!\cdots\!16\)\( \nu^{7} - \)\(32\!\cdots\!88\)\( \nu^{5} - \)\(23\!\cdots\!00\)\( \nu^{3} - \)\(54\!\cdots\!00\)\( \nu\)\()/ \)\(10\!\cdots\!00\)\( \)
\(\beta_{15}\)\(=\)\((\)\(\)\(10\!\cdots\!59\)\( \nu^{19} + \)\(19\!\cdots\!50\)\( \nu^{18} + \)\(30\!\cdots\!17\)\( \nu^{17} + \)\(55\!\cdots\!50\)\( \nu^{16} + \)\(28\!\cdots\!41\)\( \nu^{15} + \)\(52\!\cdots\!50\)\( \nu^{14} + \)\(12\!\cdots\!95\)\( \nu^{13} + \)\(22\!\cdots\!50\)\( \nu^{12} + \)\(26\!\cdots\!20\)\( \nu^{11} + \)\(48\!\cdots\!00\)\( \nu^{10} + \)\(27\!\cdots\!84\)\( \nu^{9} + \)\(50\!\cdots\!00\)\( \nu^{8} + \)\(14\!\cdots\!72\)\( \nu^{7} + \)\(26\!\cdots\!00\)\( \nu^{6} + \)\(32\!\cdots\!96\)\( \nu^{5} + \)\(62\!\cdots\!00\)\( \nu^{4} + \)\(25\!\cdots\!00\)\( \nu^{3} + \)\(55\!\cdots\!00\)\( \nu^{2} + \)\(63\!\cdots\!00\)\( \nu + \)\(15\!\cdots\!00\)\(\)\()/ \)\(82\!\cdots\!00\)\( \)
\(\beta_{16}\)\(=\)\((\)\(-\)\(10\!\cdots\!59\)\( \nu^{19} + \)\(19\!\cdots\!50\)\( \nu^{18} - \)\(30\!\cdots\!17\)\( \nu^{17} + \)\(55\!\cdots\!50\)\( \nu^{16} - \)\(28\!\cdots\!41\)\( \nu^{15} + \)\(52\!\cdots\!50\)\( \nu^{14} - \)\(12\!\cdots\!95\)\( \nu^{13} + \)\(22\!\cdots\!50\)\( \nu^{12} - \)\(26\!\cdots\!20\)\( \nu^{11} + \)\(48\!\cdots\!00\)\( \nu^{10} - \)\(27\!\cdots\!84\)\( \nu^{9} + \)\(50\!\cdots\!00\)\( \nu^{8} - \)\(14\!\cdots\!72\)\( \nu^{7} + \)\(26\!\cdots\!00\)\( \nu^{6} - \)\(32\!\cdots\!96\)\( \nu^{5} + \)\(62\!\cdots\!00\)\( \nu^{4} - \)\(25\!\cdots\!00\)\( \nu^{3} + \)\(55\!\cdots\!00\)\( \nu^{2} - \)\(63\!\cdots\!00\)\( \nu + \)\(15\!\cdots\!00\)\(\)\()/ \)\(82\!\cdots\!00\)\( \)
\(\beta_{17}\)\(=\)\((\)\(\)\(25\!\cdots\!19\)\( \nu^{19} + \)\(70\!\cdots\!97\)\( \nu^{17} + \)\(67\!\cdots\!81\)\( \nu^{15} + \)\(29\!\cdots\!95\)\( \nu^{13} + \)\(61\!\cdots\!20\)\( \nu^{11} + \)\(64\!\cdots\!44\)\( \nu^{9} + \)\(33\!\cdots\!52\)\( \nu^{7} + \)\(76\!\cdots\!36\)\( \nu^{5} + \)\(60\!\cdots\!00\)\( \nu^{3} + \)\(14\!\cdots\!00\)\( \nu\)\()/ \)\(14\!\cdots\!00\)\( \)
\(\beta_{18}\)\(=\)\((\)\(\)\(69\!\cdots\!31\)\( \nu^{19} + \)\(19\!\cdots\!53\)\( \nu^{17} + \)\(18\!\cdots\!69\)\( \nu^{15} + \)\(79\!\cdots\!55\)\( \nu^{13} + \)\(16\!\cdots\!80\)\( \nu^{11} + \)\(17\!\cdots\!56\)\( \nu^{9} + \)\(89\!\cdots\!48\)\( \nu^{7} + \)\(20\!\cdots\!64\)\( \nu^{5} + \)\(16\!\cdots\!00\)\( \nu^{3} + \)\(42\!\cdots\!00\)\( \nu\)\()/ \)\(28\!\cdots\!00\)\( \)
\(\beta_{19}\)\(=\)\((\)\(-\)\(21\!\cdots\!83\)\( \nu^{19} - \)\(58\!\cdots\!29\)\( \nu^{17} - \)\(55\!\cdots\!17\)\( \nu^{15} - \)\(24\!\cdots\!15\)\( \nu^{13} - \)\(50\!\cdots\!40\)\( \nu^{11} - \)\(52\!\cdots\!08\)\( \nu^{9} - \)\(26\!\cdots\!64\)\( \nu^{7} - \)\(61\!\cdots\!52\)\( \nu^{5} - \)\(47\!\cdots\!00\)\( \nu^{3} - \)\(11\!\cdots\!00\)\( \nu\)\()/ \)\(82\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(-\beta_{8} + \beta_{7}\)
\(\nu^{2}\)\(=\)\(-18 \beta_{16} - 18 \beta_{15} + 4 \beta_{11} - 4 \beta_{10} + 12 \beta_{6} - 4 \beta_{5} - 4 \beta_{4} + 4 \beta_{3} + 19 \beta_{2} + \beta_{1} - 282\)
\(\nu^{3}\)\(=\)\(-235 \beta_{19} - 148 \beta_{18} - 241 \beta_{17} + 27 \beta_{16} - 27 \beta_{15} + 131 \beta_{14} - 132 \beta_{13} + 132 \beta_{12} - 516 \beta_{11} - 516 \beta_{10} + 11 \beta_{9} + 2707 \beta_{8} - 824 \beta_{7}\)
\(\nu^{4}\)\(=\)\(24476 \beta_{16} + 24476 \beta_{15} + 2043 \beta_{13} + 2043 \beta_{12} - 4354 \beta_{11} + 4354 \beta_{10} - 11662 \beta_{6} + 7870 \beta_{5} + 5183 \beta_{4} - 4690 \beta_{3} - 18731 \beta_{2} - 10713 \beta_{1} + 235134\)
\(\nu^{5}\)\(=\)\(445876 \beta_{19} + 191736 \beta_{18} + 322326 \beta_{17} - 90191 \beta_{16} + 90191 \beta_{15} - 255883 \beta_{14} + 143190 \beta_{13} - 143190 \beta_{12} + 787797 \beta_{11} + 787797 \beta_{10} - 96976 \beta_{9} - 5124630 \beta_{8} + 809791 \beta_{7}\)
\(\nu^{6}\)\(=\)\(-31357260 \beta_{16} - 31357260 \beta_{15} - 3624288 \beta_{13} - 3624288 \beta_{12} + 4948816 \beta_{11} - 4948816 \beta_{10} + 12959648 \beta_{6} - 11047117 \beta_{5} - 6830680 \beta_{4} + 5568472 \beta_{3} + 21437018 \beta_{2} + 19617333 \beta_{1} - 259162202\)
\(\nu^{7}\)\(=\)\(-659861794 \beta_{19} - 249280818 \beta_{18} - 419918317 \beta_{17} + 144540755 \beta_{16} - 144540755 \beta_{15} + 384535340 \beta_{14} - 167899446 \beta_{13} + 167899446 \beta_{12} - 1081695691 \beta_{11} - 1081695691 \beta_{10} + 163113612 \beta_{9} + 7711975250 \beta_{8} - 932167435 \beta_{7}\)
\(\nu^{8}\)\(=\)\(40729598390 \beta_{16} + 40729598390 \beta_{15} + 5268444486 \beta_{13} + 5268444486 \beta_{12} - 6074121852 \beta_{11} + 6074121852 \beta_{10} - 15906824746 \beta_{6} + 14863546929 \beta_{5} + 9032130460 \beta_{4} - 6901015644 \beta_{3} - 26665624906 \beta_{2} - 28853960045 \beta_{1} + 318245219188\)
\(\nu^{9}\)\(=\)\(908554023504 \beta_{19} + 328215539042 \beta_{18} + 550284205115 \beta_{17} - 202917784331 \beta_{16} + 202917784331 \beta_{15} - 533541185588 \beta_{14} + 210546061512 \beta_{13} - 210546061512 \beta_{12} + 1448940273009 \beta_{11} + 1448940273009 \beta_{10} - 230636379260 \beta_{9} - 10698992843298 \beta_{8} + 1162332928479 \beta_{7}\)
\(\nu^{10}\)\(=\)\(-53416120942924 \beta_{16} - 53416120942924 \beta_{15} - 7215915884820 \beta_{13} - 7215915884820 \beta_{12} + 7782172208542 \beta_{11} - 7782172208542 \beta_{10} + 20405139570076 \beta_{6} - 19775334081337 \beta_{5} - 11937723766752 \beta_{4} + 8855621988802 \beta_{3} + 34397715866392 \beta_{2} + 39665366454691 \beta_{1} - 408486270446774\)
\(\nu^{11}\)\(=\)\(-1219428692187922 \beta_{19} - 433523440562956 \beta_{18} - 724371154543925 \beta_{17} + 273929960600883 \beta_{16} - 273929960600883 \beta_{15} + 718585150317084 \beta_{14} - 272320565022516 \beta_{13} + 272320565022516 \beta_{12} - 1924799099745427 \beta_{11} - 1924799099745427 \beta_{10} + 311726695881100 \beta_{9} + 14404111689257502 \beta_{8} - 1500534812806415 \beta_{7}\)
\(\nu^{12}\)\(=\)\(70366199982986716 \beta_{16} + 70366199982986716 \beta_{15} + 9669834831711570 \beta_{13} + 9669834831711570 \beta_{12} - 10159571865442428 \beta_{11} + 10159571865442428 \beta_{10} - 26649284568499494 \beta_{6} + 26204690688167043 \beta_{5} + 15773199937077488 \beta_{4} - 11563467035445078 \beta_{3} - 45029152527383728 \beta_{2} - 53214481787307851 \beta_{1} + 533685307788546938\)
\(\nu^{13}\)\(=\)\(1621087649719957806 \beta_{19} + 572959751009264372 \beta_{18} + 955633504653997333 \beta_{17} - 364843708344964835 \beta_{16} + 364843708344964835 \beta_{15} - 956709988575360220 \beta_{14} + 356777600983911654 \beta_{13} - 356777600983911654 \beta_{12} + 2548899992002388849 \beta_{11} + 2548899992002388849 \beta_{10} - 415271668062166008 \beta_{9} - 19172276040055159282 \beta_{8} + 1964732255302319967 \beta_{7}\)
\(\nu^{14}\)\(=\)\(-92868872236010541476 \beta_{16} - 92868872236010541476 \beta_{15} - 12849242774126640744 \beta_{13} - 12849242774126640744 \beta_{12} + 13363263785550646406 \beta_{11} - 13363263785550646406 \beta_{10} + 35053419824715235958 \beta_{6} - 34668756111209704919 \beta_{5} - 20840545288449276588 \beta_{4} + 15210326174852324474 \beta_{3} + 59288184607243621172 \beta_{2} + 70735007405024898947 \beta_{1} - 702131843748773045646\)
\(\nu^{15}\)\(=\)\(-2147183321676617160026 \beta_{19} - 757272190577418236324 \beta_{18} - 1261960866401092014997 \beta_{17} + 483552501935417506983 \beta_{16} - 483552501935417506983 \beta_{15} + 1268003336764180926384 \beta_{14} - 469860482805388536972 \beta_{13} + 469860482805388536972 \beta_{12} - 3371171857236134338463 \beta_{11} - 3371171857236134338463 \beta_{10} + 550405122142983865412 \beta_{9} + 25406863443842773211186 \beta_{8} - 2587009578341443530519 \beta_{7}\)
\(\nu^{16}\)\(=\)\(122662094642718960760268 \beta_{16} + 122662094642718960760268 \beta_{15} + 17017361575118010273906 \beta_{13} + 17017361575118010273906 \beta_{12} - 17628252200092553916116 \beta_{11} + 17628252200092553916116 \beta_{10} - 46238126562384206190678 \beta_{6} + 45836383165501406351663 \beta_{5} + 27537453826454452250184 \beta_{4} - 20065003131371141296958 \beta_{3} - 78238184907968287227100 \beta_{2} - 93689194832978842670115 \beta_{1} + 926260989582136880043754\)
\(\nu^{17}\)\(=\)\(2840016836659035423793610 \beta_{19} + 1000826388446279076977796 \beta_{18} + 1667178701579723573612557 \beta_{17} - 639716858565176998201339 \beta_{16} + 639716858565176998201339 \beta_{15} - 1677602580422482883086632 \beta_{14} + 620070719944709128898094 \beta_{13} - 620070719944709128898094 \beta_{12} + 4456494002932163932584877 \beta_{11} + 4456494002932163932584877 \beta_{10} - 728159855310845111818252 \beta_{9} - 33611552717802194383088638 \beta_{8} + 3413895818699330745264907 \beta_{7}\)
\(\nu^{18}\)\(=\)\(-162063868209435186942043724 \beta_{16} - 162063868209435186942043724 \beta_{15} - 22507926943445165007242796 \beta_{13} - 22507926943445165007242796 \beta_{12} + 23280131879617539525369294 \beta_{11} - 23280131879617539525369294 \beta_{10} + 61059509589742475365047702 \beta_{6} - 60584687966660575302351291 \beta_{5} - 36388293335350867119611632 \beta_{4} + 26498287128243153746278986 \beta_{3} + 103335036081302263782655928 \beta_{2} + 123920248741045270331513995 \beta_{1} - 1223229700933212006981886686\)
\(\nu^{19}\)\(=\)\(-3754369184130634238513176722 \beta_{19} - 1322657670828458251782937684 \beta_{18} - 2202901264967708350075566401 \beta_{17} + 845735709957977068143170095 \beta_{16} - 845735709957977068143170095 \beta_{15} + 2217961812416442356898642500 \beta_{14} - 818978009137341871723720788 \beta_{13} + 818978009137341871723720788 \beta_{12} - 5890051853388504248388608623 \beta_{11} - 5890051853388504248388608623 \beta_{10} + 962658391008771682599091236 \beta_{9} + 44436396653639805026656522498 \beta_{8} - 4508969361145760508507499223 \beta_{7}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/930\mathbb{Z}\right)^\times\).

\(n\) \(187\) \(311\) \(871\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
559.1
22.8518i
36.3525i
7.62904i
2.24994i
2.45273i
7.05659i
2.56665i
9.07744i
22.4343i
14.4198i
22.8518i
36.3525i
7.62904i
2.24994i
2.45273i
7.05659i
2.56665i
9.07744i
22.4343i
14.4198i
2.00000i 3.00000i −4.00000 −11.0046 + 1.97458i 6.00000 22.8518i 8.00000i −9.00000 3.94915 + 22.0092i
559.2 2.00000i 3.00000i −4.00000 −10.3430 4.24534i 6.00000 36.3525i 8.00000i −9.00000 −8.49067 + 20.6859i
559.3 2.00000i 3.00000i −4.00000 −7.42511 + 8.35869i 6.00000 7.62904i 8.00000i −9.00000 16.7174 + 14.8502i
559.4 2.00000i 3.00000i −4.00000 −6.23400 9.28102i 6.00000 2.24994i 8.00000i −9.00000 −18.5620 + 12.4680i
559.5 2.00000i 3.00000i −4.00000 1.65380 + 11.0573i 6.00000 2.45273i 8.00000i −9.00000 22.1147 3.30759i
559.6 2.00000i 3.00000i −4.00000 2.19955 10.9618i 6.00000 7.05659i 8.00000i −9.00000 −21.9237 4.39910i
559.7 2.00000i 3.00000i −4.00000 3.61266 + 10.5806i 6.00000 2.56665i 8.00000i −9.00000 21.1612 7.22532i
559.8 2.00000i 3.00000i −4.00000 4.77279 10.1104i 6.00000 9.07744i 8.00000i −9.00000 −20.2208 9.54558i
559.9 2.00000i 3.00000i −4.00000 10.6574 + 3.37922i 6.00000 22.4343i 8.00000i −9.00000 6.75844 21.3149i
559.10 2.00000i 3.00000i −4.00000 11.1104 + 1.24819i 6.00000 14.4198i 8.00000i −9.00000 2.49638 22.2209i
559.11 2.00000i 3.00000i −4.00000 −11.0046 1.97458i 6.00000 22.8518i 8.00000i −9.00000 3.94915 22.0092i
559.12 2.00000i 3.00000i −4.00000 −10.3430 + 4.24534i 6.00000 36.3525i 8.00000i −9.00000 −8.49067 20.6859i
559.13 2.00000i 3.00000i −4.00000 −7.42511 8.35869i 6.00000 7.62904i 8.00000i −9.00000 16.7174 14.8502i
559.14 2.00000i 3.00000i −4.00000 −6.23400 + 9.28102i 6.00000 2.24994i 8.00000i −9.00000 −18.5620 12.4680i
559.15 2.00000i 3.00000i −4.00000 1.65380 11.0573i 6.00000 2.45273i 8.00000i −9.00000 22.1147 + 3.30759i
559.16 2.00000i 3.00000i −4.00000 2.19955 + 10.9618i 6.00000 7.05659i 8.00000i −9.00000 −21.9237 + 4.39910i
559.17 2.00000i 3.00000i −4.00000 3.61266 10.5806i 6.00000 2.56665i 8.00000i −9.00000 21.1612 + 7.22532i
559.18 2.00000i 3.00000i −4.00000 4.77279 + 10.1104i 6.00000 9.07744i 8.00000i −9.00000 −20.2208 + 9.54558i
559.19 2.00000i 3.00000i −4.00000 10.6574 3.37922i 6.00000 22.4343i 8.00000i −9.00000 6.75844 + 21.3149i
559.20 2.00000i 3.00000i −4.00000 11.1104 1.24819i 6.00000 14.4198i 8.00000i −9.00000 2.49638 + 22.2209i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 559.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.4.d.c 20
5.b even 2 1 inner 930.4.d.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.4.d.c 20 1.a even 1 1 trivial
930.4.d.c 20 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(26\!\cdots\!76\)\( T_{7}^{10} + \)\(14\!\cdots\!08\)\( T_{7}^{8} + \)\(37\!\cdots\!44\)\( T_{7}^{6} + \)\(41\!\cdots\!00\)\( T_{7}^{4} + \)\(19\!\cdots\!00\)\( T_{7}^{2} + \)\(34\!\cdots\!00\)\( \)">\(T_{7}^{20} + \cdots\) acting on \(S_{4}^{\mathrm{new}}(930, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 + T^{2} )^{10} \)
$3$ \( ( 9 + T^{2} )^{10} \)
$5$ \( \)\(93\!\cdots\!25\)\( + 14901161193847656250 T + 2801418304443359375 T^{2} + 353813171386718750 T^{3} - 47847747802734375 T^{4} - 8918579101562500 T^{5} - 537836914062500 T^{6} - 6531054687500 T^{7} - 766441406250 T^{8} + 293650000000 T^{9} + 85737106250 T^{10} + 2349200000 T^{11} - 49052250 T^{12} - 3343900 T^{13} - 2202980 T^{14} - 292244 T^{15} - 12543 T^{16} + 742 T^{17} + 47 T^{18} + 2 T^{19} + T^{20} \)
$7$ \( 3460065015625000000 + 1987578737873500000 T^{2} + 414170668954972900 T^{4} + 37536884921183444 T^{6} + 1433368340491408 T^{8} + 26461073949176 T^{10} + 247831438280 T^{12} + 1161420105 T^{14} + 2652899 T^{16} + 2763 T^{18} + T^{20} \)
$11$ \( ( -156520544587776 + 9214841973504 T + 1672610820928 T^{2} - 91276492992 T^{3} - 4375988600 T^{4} + 209615064 T^{5} + 5425636 T^{6} - 184608 T^{7} - 3564 T^{8} + 57 T^{9} + T^{10} )^{2} \)
$13$ \( \)\(20\!\cdots\!00\)\( + \)\(74\!\cdots\!64\)\( T^{2} + \)\(79\!\cdots\!88\)\( T^{4} + \)\(31\!\cdots\!36\)\( T^{6} + \)\(56\!\cdots\!80\)\( T^{8} + 5436376962189509828 T^{10} + 3042627312787552 T^{12} + 1021954754480 T^{14} + 203057904 T^{16} + 21992 T^{18} + T^{20} \)
$17$ \( \)\(39\!\cdots\!36\)\( + \)\(99\!\cdots\!60\)\( T^{2} + \)\(30\!\cdots\!68\)\( T^{4} + \)\(40\!\cdots\!56\)\( T^{6} + \)\(29\!\cdots\!24\)\( T^{8} + \)\(13\!\cdots\!92\)\( T^{10} + 39643092221437680 T^{12} + 7188337677008 T^{14} + 777036316 T^{16} + 44584 T^{18} + T^{20} \)
$19$ \( ( -2305398529727667968 + 236427323830682624 T - 7942893128034640 T^{2} + 55381030853744 T^{3} + 2252043397000 T^{4} - 41253340264 T^{5} - 51821736 T^{6} + 5182581 T^{7} - 18957 T^{8} - 185 T^{9} + T^{10} )^{2} \)
$23$ \( \)\(75\!\cdots\!64\)\( + \)\(37\!\cdots\!04\)\( T^{2} + \)\(71\!\cdots\!12\)\( T^{4} + \)\(66\!\cdots\!84\)\( T^{6} + \)\(33\!\cdots\!24\)\( T^{8} + \)\(93\!\cdots\!24\)\( T^{10} + 1482376999732803536 T^{12} + 130323419478884 T^{14} + 5987299188 T^{16} + 127929 T^{18} + T^{20} \)
$29$ \( ( 14458251377096669056 - 460949649136330240 T - 30458010273716080 T^{2} + 840148019447576 T^{3} + 6109616324428 T^{4} - 245159962374 T^{5} + 240261062 T^{6} + 20565436 T^{7} - 87306 T^{8} - 184 T^{9} + T^{10} )^{2} \)
$31$ \( ( 31 + T )^{20} \)
$37$ \( \)\(98\!\cdots\!44\)\( + \)\(74\!\cdots\!36\)\( T^{2} + \)\(81\!\cdots\!32\)\( T^{4} + \)\(34\!\cdots\!96\)\( T^{6} + \)\(70\!\cdots\!84\)\( T^{8} + \)\(79\!\cdots\!36\)\( T^{10} + 50617436644961623736 T^{12} + 1776993119617196 T^{14} + 32663822328 T^{16} + 292236 T^{18} + T^{20} \)
$41$ \( ( -\)\(18\!\cdots\!00\)\( - \)\(31\!\cdots\!40\)\( T + 15342533063306479376 T^{2} - 59978742948652424 T^{3} - 839928912175952 T^{4} + 3821158744428 T^{5} + 18455008460 T^{6} - 71536438 T^{7} - 204063 T^{8} + 436 T^{9} + T^{10} )^{2} \)
$43$ \( \)\(12\!\cdots\!16\)\( + \)\(94\!\cdots\!48\)\( T^{2} + \)\(22\!\cdots\!04\)\( T^{4} + \)\(21\!\cdots\!16\)\( T^{6} + \)\(10\!\cdots\!28\)\( T^{8} + \)\(27\!\cdots\!48\)\( T^{10} + \)\(45\!\cdots\!12\)\( T^{12} + 45651800997030576 T^{14} + 265859777504 T^{16} + 816097 T^{18} + T^{20} \)
$47$ \( \)\(17\!\cdots\!00\)\( + \)\(13\!\cdots\!00\)\( T^{2} + \)\(28\!\cdots\!16\)\( T^{4} + \)\(22\!\cdots\!28\)\( T^{6} + \)\(87\!\cdots\!36\)\( T^{8} + \)\(18\!\cdots\!16\)\( T^{10} + \)\(22\!\cdots\!56\)\( T^{12} + 159634237705781184 T^{14} + 625113173516 T^{16} + 1255372 T^{18} + T^{20} \)
$53$ \( \)\(56\!\cdots\!00\)\( + \)\(61\!\cdots\!00\)\( T^{2} + \)\(59\!\cdots\!00\)\( T^{4} + \)\(24\!\cdots\!16\)\( T^{6} + \)\(55\!\cdots\!12\)\( T^{8} + \)\(75\!\cdots\!48\)\( T^{10} + \)\(61\!\cdots\!64\)\( T^{12} + 312674944790788548 T^{14} + 934635509948 T^{16} + 1505713 T^{18} + T^{20} \)
$59$ \( ( -\)\(14\!\cdots\!00\)\( - \)\(60\!\cdots\!52\)\( T + \)\(44\!\cdots\!64\)\( T^{2} - 37450728820810616826 T^{3} + 109914322712821282 T^{4} + 14843414724822 T^{5} - 709592333480 T^{6} + 1179812064 T^{7} + 85839 T^{8} - 1614 T^{9} + T^{10} )^{2} \)
$61$ \( ( -\)\(54\!\cdots\!96\)\( + \)\(56\!\cdots\!32\)\( T - \)\(60\!\cdots\!12\)\( T^{2} - 2118213229352329152 T^{3} + 25945099941295488 T^{4} + 71354287900816 T^{5} - 207538804224 T^{6} - 745250688 T^{7} - 126072 T^{8} + 1302 T^{9} + T^{10} )^{2} \)
$67$ \( \)\(87\!\cdots\!00\)\( + \)\(14\!\cdots\!00\)\( T^{2} + \)\(80\!\cdots\!16\)\( T^{4} + \)\(14\!\cdots\!92\)\( T^{6} + \)\(29\!\cdots\!40\)\( T^{8} + \)\(24\!\cdots\!24\)\( T^{10} + \)\(10\!\cdots\!80\)\( T^{12} + 2650097772624546224 T^{14} + 3863355335128 T^{16} + 3039720 T^{18} + T^{20} \)
$71$ \( ( -\)\(21\!\cdots\!00\)\( + \)\(12\!\cdots\!20\)\( T - \)\(74\!\cdots\!34\)\( T^{2} - 54274409113296534356 T^{3} + 70967921892530182 T^{4} + 554171128621672 T^{5} + 32241365214 T^{6} - 1840371781 T^{7} - 1288963 T^{8} + 1145 T^{9} + T^{10} )^{2} \)
$73$ \( \)\(17\!\cdots\!76\)\( + \)\(29\!\cdots\!24\)\( T^{2} + \)\(12\!\cdots\!56\)\( T^{4} + \)\(23\!\cdots\!72\)\( T^{6} + \)\(21\!\cdots\!08\)\( T^{8} + \)\(11\!\cdots\!48\)\( T^{10} + \)\(35\!\cdots\!88\)\( T^{12} + 6629635874530797292 T^{14} + 7263516903896 T^{16} + 4227289 T^{18} + T^{20} \)
$79$ \( ( -\)\(72\!\cdots\!00\)\( + \)\(10\!\cdots\!00\)\( T - \)\(41\!\cdots\!60\)\( T^{2} - 29152726546155883088 T^{3} + 468182692606809036 T^{4} - 566670326844352 T^{5} - 1122411778504 T^{6} + 2314138100 T^{7} + 144576 T^{8} - 2171 T^{9} + T^{10} )^{2} \)
$83$ \( \)\(13\!\cdots\!00\)\( + \)\(15\!\cdots\!36\)\( T^{2} + \)\(39\!\cdots\!92\)\( T^{4} + \)\(42\!\cdots\!52\)\( T^{6} + \)\(24\!\cdots\!48\)\( T^{8} + \)\(78\!\cdots\!28\)\( T^{10} + \)\(15\!\cdots\!04\)\( T^{12} + 19786562305340642144 T^{14} + 14808286410180 T^{16} + 6015440 T^{18} + T^{20} \)
$89$ \( ( -\)\(16\!\cdots\!00\)\( - \)\(12\!\cdots\!80\)\( T + \)\(25\!\cdots\!84\)\( T^{2} + \)\(24\!\cdots\!16\)\( T^{3} - 329720124902412074 T^{4} - 1288098156690156 T^{5} + 1800596650988 T^{6} + 1740960560 T^{7} - 2485554 T^{8} - 695 T^{9} + T^{10} )^{2} \)
$97$ \( \)\(34\!\cdots\!00\)\( + \)\(11\!\cdots\!16\)\( T^{2} + \)\(11\!\cdots\!40\)\( T^{4} + \)\(51\!\cdots\!12\)\( T^{6} + \)\(13\!\cdots\!44\)\( T^{8} + \)\(20\!\cdots\!28\)\( T^{10} + \)\(20\!\cdots\!68\)\( T^{12} + \)\(12\!\cdots\!44\)\( T^{14} + 49299400047697 T^{16} + 10738450 T^{18} + T^{20} \)
show more
show less