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Properties

Label 690.2.m.c
Level $690$
Weight $2$
Character orbit 690.m
Analytic conductor $5.510$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 690.m (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.50967773947\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(2\) over \(\Q(\zeta_{11})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 4 x^{19} - 3 x^{18} + 66 x^{17} - 163 x^{16} - 52 x^{15} + 1567 x^{14} - 6182 x^{13} + 17043 x^{12} - 35832 x^{11} + 60906 x^{10} - 87666 x^{9} + 106197 x^{8} - 102542 x^{7} + 92618 x^{6} - 8374 x^{5} + 38352 x^{4} + 60038 x^{3} + 32919 x^{2} + 5681 x + 529\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 4 x^{19} - 3 x^{18} + 66 x^{17} - 163 x^{16} - 52 x^{15} + 1567 x^{14} - 6182 x^{13} + 17043 x^{12} - 35832 x^{11} + 60906 x^{10} - 87666 x^{9} + 106197 x^{8} - 102542 x^{7} + 92618 x^{6} - 8374 x^{5} + 38352 x^{4} + 60038 x^{3} + 32919 x^{2} + 5681 x + 529\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(77\!\cdots\!35\)\( \nu^{19} + \)\(29\!\cdots\!38\)\( \nu^{18} + \)\(22\!\cdots\!82\)\( \nu^{17} - \)\(47\!\cdots\!58\)\( \nu^{16} + \)\(11\!\cdots\!58\)\( \nu^{15} + \)\(16\!\cdots\!16\)\( \nu^{14} - \)\(10\!\cdots\!92\)\( \nu^{13} + \)\(44\!\cdots\!60\)\( \nu^{12} - \)\(13\!\cdots\!90\)\( \nu^{11} + \)\(30\!\cdots\!76\)\( \nu^{10} - \)\(58\!\cdots\!22\)\( \nu^{9} + \)\(96\!\cdots\!75\)\( \nu^{8} - \)\(13\!\cdots\!81\)\( \nu^{7} + \)\(16\!\cdots\!92\)\( \nu^{6} - \)\(18\!\cdots\!32\)\( \nu^{5} + \)\(13\!\cdots\!83\)\( \nu^{4} - \)\(17\!\cdots\!22\)\( \nu^{3} + \)\(20\!\cdots\!65\)\( \nu^{2} - \)\(45\!\cdots\!47\)\( \nu - \)\(13\!\cdots\!32\)\(\)\()/ \)\(73\!\cdots\!31\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(47\!\cdots\!30\)\( \nu^{19} - \)\(21\!\cdots\!91\)\( \nu^{18} - \)\(73\!\cdots\!03\)\( \nu^{17} + \)\(30\!\cdots\!23\)\( \nu^{16} - \)\(96\!\cdots\!05\)\( \nu^{15} + \)\(40\!\cdots\!75\)\( \nu^{14} + \)\(71\!\cdots\!81\)\( \nu^{13} - \)\(33\!\cdots\!49\)\( \nu^{12} + \)\(10\!\cdots\!07\)\( \nu^{11} - \)\(23\!\cdots\!61\)\( \nu^{10} + \)\(42\!\cdots\!49\)\( \nu^{9} - \)\(65\!\cdots\!52\)\( \nu^{8} + \)\(86\!\cdots\!95\)\( \nu^{7} - \)\(94\!\cdots\!70\)\( \nu^{6} + \)\(92\!\cdots\!06\)\( \nu^{5} - \)\(47\!\cdots\!90\)\( \nu^{4} + \)\(37\!\cdots\!35\)\( \nu^{3} + \)\(18\!\cdots\!01\)\( \nu^{2} + \)\(60\!\cdots\!87\)\( \nu - \)\(57\!\cdots\!72\)\(\)\()/ \)\(73\!\cdots\!31\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(55\!\cdots\!96\)\( \nu^{19} + \)\(22\!\cdots\!61\)\( \nu^{18} + \)\(18\!\cdots\!68\)\( \nu^{17} - \)\(37\!\cdots\!54\)\( \nu^{16} + \)\(91\!\cdots\!89\)\( \nu^{15} + \)\(41\!\cdots\!18\)\( \nu^{14} - \)\(91\!\cdots\!61\)\( \nu^{13} + \)\(34\!\cdots\!60\)\( \nu^{12} - \)\(92\!\cdots\!38\)\( \nu^{11} + \)\(18\!\cdots\!95\)\( \nu^{10} - \)\(30\!\cdots\!56\)\( \nu^{9} + \)\(41\!\cdots\!11\)\( \nu^{8} - \)\(45\!\cdots\!56\)\( \nu^{7} + \)\(37\!\cdots\!87\)\( \nu^{6} - \)\(27\!\cdots\!08\)\( \nu^{5} - \)\(19\!\cdots\!83\)\( \nu^{4} - \)\(23\!\cdots\!76\)\( \nu^{3} - \)\(39\!\cdots\!28\)\( \nu^{2} - \)\(17\!\cdots\!74\)\( \nu - \)\(18\!\cdots\!58\)\(\)\()/ \)\(73\!\cdots\!31\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(56\!\cdots\!12\)\( \nu^{19} - \)\(25\!\cdots\!74\)\( \nu^{18} - \)\(48\!\cdots\!08\)\( \nu^{17} + \)\(37\!\cdots\!57\)\( \nu^{16} - \)\(10\!\cdots\!20\)\( \nu^{15} + \)\(23\!\cdots\!30\)\( \nu^{14} + \)\(87\!\cdots\!83\)\( \nu^{13} - \)\(39\!\cdots\!18\)\( \nu^{12} + \)\(11\!\cdots\!61\)\( \nu^{11} - \)\(25\!\cdots\!70\)\( \nu^{10} + \)\(45\!\cdots\!02\)\( \nu^{9} - \)\(69\!\cdots\!74\)\( \nu^{8} + \)\(89\!\cdots\!16\)\( \nu^{7} - \)\(94\!\cdots\!66\)\( \nu^{6} + \)\(88\!\cdots\!00\)\( \nu^{5} - \)\(37\!\cdots\!29\)\( \nu^{4} + \)\(31\!\cdots\!23\)\( \nu^{3} + \)\(27\!\cdots\!79\)\( \nu^{2} + \)\(12\!\cdots\!38\)\( \nu + \)\(20\!\cdots\!88\)\(\)\()/ \)\(73\!\cdots\!31\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(73\!\cdots\!18\)\( \nu^{19} - \)\(33\!\cdots\!19\)\( \nu^{18} - \)\(67\!\cdots\!75\)\( \nu^{17} + \)\(50\!\cdots\!65\)\( \nu^{16} - \)\(14\!\cdots\!93\)\( \nu^{15} + \)\(22\!\cdots\!79\)\( \nu^{14} + \)\(11\!\cdots\!44\)\( \nu^{13} - \)\(51\!\cdots\!90\)\( \nu^{12} + \)\(14\!\cdots\!86\)\( \nu^{11} - \)\(32\!\cdots\!46\)\( \nu^{10} + \)\(58\!\cdots\!31\)\( \nu^{9} - \)\(87\!\cdots\!81\)\( \nu^{8} + \)\(11\!\cdots\!82\)\( \nu^{7} - \)\(11\!\cdots\!18\)\( \nu^{6} + \)\(10\!\cdots\!27\)\( \nu^{5} - \)\(41\!\cdots\!76\)\( \nu^{4} + \)\(33\!\cdots\!61\)\( \nu^{3} + \)\(28\!\cdots\!95\)\( \nu^{2} + \)\(55\!\cdots\!21\)\( \nu - \)\(63\!\cdots\!04\)\(\)\()/ \)\(73\!\cdots\!31\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(82\!\cdots\!88\)\( \nu^{19} - \)\(33\!\cdots\!68\)\( \nu^{18} - \)\(23\!\cdots\!26\)\( \nu^{17} + \)\(54\!\cdots\!08\)\( \nu^{16} - \)\(13\!\cdots\!92\)\( \nu^{15} - \)\(33\!\cdots\!79\)\( \nu^{14} + \)\(12\!\cdots\!73\)\( \nu^{13} - \)\(51\!\cdots\!26\)\( \nu^{12} + \)\(14\!\cdots\!17\)\( \nu^{11} - \)\(30\!\cdots\!56\)\( \nu^{10} + \)\(51\!\cdots\!34\)\( \nu^{9} - \)\(74\!\cdots\!49\)\( \nu^{8} + \)\(89\!\cdots\!33\)\( \nu^{7} - \)\(85\!\cdots\!63\)\( \nu^{6} + \)\(76\!\cdots\!40\)\( \nu^{5} - \)\(33\!\cdots\!26\)\( \nu^{4} + \)\(25\!\cdots\!69\)\( \nu^{3} + \)\(58\!\cdots\!68\)\( \nu^{2} + \)\(21\!\cdots\!46\)\( \nu + \)\(25\!\cdots\!38\)\(\)\()/ \)\(73\!\cdots\!31\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(87\!\cdots\!48\)\( \nu^{19} + \)\(31\!\cdots\!97\)\( \nu^{18} + \)\(39\!\cdots\!68\)\( \nu^{17} - \)\(56\!\cdots\!19\)\( \nu^{16} + \)\(11\!\cdots\!40\)\( \nu^{15} + \)\(10\!\cdots\!71\)\( \nu^{14} - \)\(13\!\cdots\!20\)\( \nu^{13} + \)\(48\!\cdots\!58\)\( \nu^{12} - \)\(12\!\cdots\!37\)\( \nu^{11} + \)\(25\!\cdots\!95\)\( \nu^{10} - \)\(40\!\cdots\!79\)\( \nu^{9} + \)\(54\!\cdots\!33\)\( \nu^{8} - \)\(58\!\cdots\!36\)\( \nu^{7} + \)\(45\!\cdots\!00\)\( \nu^{6} - \)\(33\!\cdots\!35\)\( \nu^{5} - \)\(40\!\cdots\!01\)\( \nu^{4} - \)\(11\!\cdots\!83\)\( \nu^{3} - \)\(78\!\cdots\!28\)\( \nu^{2} - \)\(33\!\cdots\!42\)\( \nu - \)\(10\!\cdots\!60\)\(\)\()/ \)\(73\!\cdots\!31\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(10\!\cdots\!68\)\( \nu^{19} - \)\(39\!\cdots\!42\)\( \nu^{18} - \)\(54\!\cdots\!95\)\( \nu^{17} + \)\(72\!\cdots\!85\)\( \nu^{16} - \)\(14\!\cdots\!61\)\( \nu^{15} - \)\(15\!\cdots\!41\)\( \nu^{14} + \)\(17\!\cdots\!31\)\( \nu^{13} - \)\(60\!\cdots\!95\)\( \nu^{12} + \)\(15\!\cdots\!75\)\( \nu^{11} - \)\(29\!\cdots\!69\)\( \nu^{10} + \)\(43\!\cdots\!47\)\( \nu^{9} - \)\(53\!\cdots\!39\)\( \nu^{8} + \)\(50\!\cdots\!44\)\( \nu^{7} - \)\(25\!\cdots\!61\)\( \nu^{6} + \)\(67\!\cdots\!54\)\( \nu^{5} + \)\(82\!\cdots\!74\)\( \nu^{4} - \)\(59\!\cdots\!54\)\( \nu^{3} + \)\(10\!\cdots\!19\)\( \nu^{2} + \)\(54\!\cdots\!93\)\( \nu + \)\(12\!\cdots\!95\)\(\)\()/ \)\(73\!\cdots\!31\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(11\!\cdots\!76\)\( \nu^{19} - \)\(40\!\cdots\!86\)\( \nu^{18} - \)\(69\!\cdots\!47\)\( \nu^{17} + \)\(78\!\cdots\!41\)\( \nu^{16} - \)\(14\!\cdots\!23\)\( \nu^{15} - \)\(20\!\cdots\!45\)\( \nu^{14} + \)\(18\!\cdots\!71\)\( \nu^{13} - \)\(61\!\cdots\!88\)\( \nu^{12} + \)\(15\!\cdots\!78\)\( \nu^{11} - \)\(27\!\cdots\!46\)\( \nu^{10} + \)\(39\!\cdots\!10\)\( \nu^{9} - \)\(46\!\cdots\!85\)\( \nu^{8} + \)\(39\!\cdots\!91\)\( \nu^{7} - \)\(11\!\cdots\!10\)\( \nu^{6} - \)\(52\!\cdots\!50\)\( \nu^{5} + \)\(96\!\cdots\!03\)\( \nu^{4} + \)\(42\!\cdots\!76\)\( \nu^{3} + \)\(10\!\cdots\!49\)\( \nu^{2} + \)\(67\!\cdots\!39\)\( \nu + \)\(12\!\cdots\!77\)\(\)\()/ \)\(73\!\cdots\!31\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(14\!\cdots\!02\)\( \nu^{19} - \)\(54\!\cdots\!98\)\( \nu^{18} - \)\(63\!\cdots\!90\)\( \nu^{17} + \)\(97\!\cdots\!59\)\( \nu^{16} - \)\(21\!\cdots\!05\)\( \nu^{15} - \)\(15\!\cdots\!51\)\( \nu^{14} + \)\(23\!\cdots\!32\)\( \nu^{13} - \)\(84\!\cdots\!21\)\( \nu^{12} + \)\(22\!\cdots\!00\)\( \nu^{11} - \)\(44\!\cdots\!57\)\( \nu^{10} + \)\(72\!\cdots\!56\)\( \nu^{9} - \)\(98\!\cdots\!91\)\( \nu^{8} + \)\(10\!\cdots\!10\)\( \nu^{7} - \)\(91\!\cdots\!11\)\( \nu^{6} + \)\(76\!\cdots\!77\)\( \nu^{5} + \)\(43\!\cdots\!73\)\( \nu^{4} + \)\(42\!\cdots\!89\)\( \nu^{3} + \)\(10\!\cdots\!89\)\( \nu^{2} + \)\(77\!\cdots\!17\)\( \nu + \)\(11\!\cdots\!84\)\(\)\()/ \)\(73\!\cdots\!31\)\( \)
\(\beta_{12}\)\(=\)\((\)\(\)\(26\!\cdots\!08\)\( \nu^{19} - \)\(10\!\cdots\!67\)\( \nu^{18} - \)\(75\!\cdots\!86\)\( \nu^{17} + \)\(17\!\cdots\!10\)\( \nu^{16} - \)\(42\!\cdots\!62\)\( \nu^{15} - \)\(12\!\cdots\!58\)\( \nu^{14} + \)\(40\!\cdots\!52\)\( \nu^{13} - \)\(16\!\cdots\!48\)\( \nu^{12} + \)\(44\!\cdots\!04\)\( \nu^{11} - \)\(94\!\cdots\!46\)\( \nu^{10} + \)\(16\!\cdots\!24\)\( \nu^{9} - \)\(23\!\cdots\!50\)\( \nu^{8} + \)\(28\!\cdots\!51\)\( \nu^{7} - \)\(28\!\cdots\!17\)\( \nu^{6} + \)\(25\!\cdots\!36\)\( \nu^{5} - \)\(40\!\cdots\!24\)\( \nu^{4} + \)\(11\!\cdots\!99\)\( \nu^{3} + \)\(13\!\cdots\!82\)\( \nu^{2} + \)\(87\!\cdots\!17\)\( \nu + \)\(10\!\cdots\!01\)\(\)\()/ \)\(73\!\cdots\!31\)\( \)
\(\beta_{13}\)\(=\)\((\)\(\)\(31\!\cdots\!52\)\( \nu^{19} - \)\(14\!\cdots\!52\)\( \nu^{18} - \)\(22\!\cdots\!81\)\( \nu^{17} + \)\(21\!\cdots\!35\)\( \nu^{16} - \)\(62\!\cdots\!35\)\( \nu^{15} + \)\(13\!\cdots\!07\)\( \nu^{14} + \)\(49\!\cdots\!61\)\( \nu^{13} - \)\(22\!\cdots\!89\)\( \nu^{12} + \)\(64\!\cdots\!45\)\( \nu^{11} - \)\(14\!\cdots\!93\)\( \nu^{10} + \)\(25\!\cdots\!17\)\( \nu^{9} - \)\(39\!\cdots\!25\)\( \nu^{8} + \)\(49\!\cdots\!43\)\( \nu^{7} - \)\(52\!\cdots\!49\)\( \nu^{6} + \)\(48\!\cdots\!16\)\( \nu^{5} - \)\(18\!\cdots\!06\)\( \nu^{4} + \)\(12\!\cdots\!42\)\( \nu^{3} + \)\(15\!\cdots\!73\)\( \nu^{2} - \)\(16\!\cdots\!45\)\( \nu - \)\(20\!\cdots\!18\)\(\)\()/ \)\(73\!\cdots\!31\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(34\!\cdots\!02\)\( \nu^{19} + \)\(14\!\cdots\!04\)\( \nu^{18} + \)\(81\!\cdots\!45\)\( \nu^{17} - \)\(22\!\cdots\!00\)\( \nu^{16} + \)\(60\!\cdots\!80\)\( \nu^{15} + \)\(88\!\cdots\!15\)\( \nu^{14} - \)\(54\!\cdots\!52\)\( \nu^{13} + \)\(22\!\cdots\!25\)\( \nu^{12} - \)\(62\!\cdots\!46\)\( \nu^{11} + \)\(13\!\cdots\!02\)\( \nu^{10} - \)\(22\!\cdots\!07\)\( \nu^{9} + \)\(33\!\cdots\!88\)\( \nu^{8} - \)\(40\!\cdots\!05\)\( \nu^{7} + \)\(39\!\cdots\!40\)\( \nu^{6} - \)\(35\!\cdots\!23\)\( \nu^{5} + \)\(56\!\cdots\!56\)\( \nu^{4} - \)\(11\!\cdots\!21\)\( \nu^{3} - \)\(20\!\cdots\!00\)\( \nu^{2} - \)\(74\!\cdots\!10\)\( \nu - \)\(18\!\cdots\!88\)\(\)\()/ \)\(73\!\cdots\!31\)\( \)
\(\beta_{15}\)\(=\)\((\)\(\)\(38\!\cdots\!72\)\( \nu^{19} - \)\(15\!\cdots\!00\)\( \nu^{18} - \)\(90\!\cdots\!42\)\( \nu^{17} + \)\(25\!\cdots\!60\)\( \nu^{16} - \)\(66\!\cdots\!93\)\( \nu^{15} - \)\(90\!\cdots\!24\)\( \nu^{14} + \)\(59\!\cdots\!94\)\( \nu^{13} - \)\(24\!\cdots\!87\)\( \nu^{12} + \)\(69\!\cdots\!14\)\( \nu^{11} - \)\(14\!\cdots\!65\)\( \nu^{10} + \)\(25\!\cdots\!02\)\( \nu^{9} - \)\(38\!\cdots\!54\)\( \nu^{8} + \)\(47\!\cdots\!58\)\( \nu^{7} - \)\(48\!\cdots\!40\)\( \nu^{6} + \)\(45\!\cdots\!62\)\( \nu^{5} - \)\(12\!\cdots\!28\)\( \nu^{4} + \)\(18\!\cdots\!73\)\( \nu^{3} + \)\(19\!\cdots\!13\)\( \nu^{2} + \)\(99\!\cdots\!89\)\( \nu + \)\(92\!\cdots\!94\)\(\)\()/ \)\(73\!\cdots\!31\)\( \)
\(\beta_{16}\)\(=\)\((\)\(\)\(47\!\cdots\!22\)\( \nu^{19} - \)\(19\!\cdots\!76\)\( \nu^{18} - \)\(10\!\cdots\!98\)\( \nu^{17} + \)\(31\!\cdots\!78\)\( \nu^{16} - \)\(82\!\cdots\!94\)\( \nu^{15} - \)\(11\!\cdots\!52\)\( \nu^{14} + \)\(74\!\cdots\!53\)\( \nu^{13} - \)\(30\!\cdots\!77\)\( \nu^{12} + \)\(86\!\cdots\!72\)\( \nu^{11} - \)\(18\!\cdots\!21\)\( \nu^{10} + \)\(31\!\cdots\!88\)\( \nu^{9} - \)\(46\!\cdots\!86\)\( \nu^{8} + \)\(57\!\cdots\!83\)\( \nu^{7} - \)\(57\!\cdots\!57\)\( \nu^{6} + \)\(52\!\cdots\!59\)\( \nu^{5} - \)\(11\!\cdots\!68\)\( \nu^{4} + \)\(18\!\cdots\!70\)\( \nu^{3} + \)\(25\!\cdots\!67\)\( \nu^{2} + \)\(97\!\cdots\!50\)\( \nu + \)\(55\!\cdots\!36\)\(\)\()/ \)\(73\!\cdots\!31\)\( \)
\(\beta_{17}\)\(=\)\((\)\(-\)\(47\!\cdots\!34\)\( \nu^{19} + \)\(20\!\cdots\!07\)\( \nu^{18} + \)\(10\!\cdots\!22\)\( \nu^{17} - \)\(31\!\cdots\!99\)\( \nu^{16} + \)\(85\!\cdots\!27\)\( \nu^{15} + \)\(68\!\cdots\!21\)\( \nu^{14} - \)\(75\!\cdots\!03\)\( \nu^{13} + \)\(31\!\cdots\!23\)\( \nu^{12} - \)\(88\!\cdots\!44\)\( \nu^{11} + \)\(19\!\cdots\!31\)\( \nu^{10} - \)\(33\!\cdots\!95\)\( \nu^{9} + \)\(48\!\cdots\!04\)\( \nu^{8} - \)\(60\!\cdots\!41\)\( \nu^{7} + \)\(61\!\cdots\!62\)\( \nu^{6} - \)\(56\!\cdots\!30\)\( \nu^{5} + \)\(15\!\cdots\!17\)\( \nu^{4} - \)\(20\!\cdots\!33\)\( \nu^{3} - \)\(24\!\cdots\!87\)\( \nu^{2} - \)\(10\!\cdots\!80\)\( \nu - \)\(78\!\cdots\!96\)\(\)\()/ \)\(73\!\cdots\!31\)\( \)
\(\beta_{18}\)\(=\)\((\)\(-\)\(66\!\cdots\!69\)\( \nu^{19} + \)\(28\!\cdots\!56\)\( \nu^{18} + \)\(12\!\cdots\!62\)\( \nu^{17} - \)\(44\!\cdots\!08\)\( \nu^{16} + \)\(11\!\cdots\!03\)\( \nu^{15} + \)\(53\!\cdots\!34\)\( \nu^{14} - \)\(10\!\cdots\!02\)\( \nu^{13} + \)\(43\!\cdots\!03\)\( \nu^{12} - \)\(12\!\cdots\!14\)\( \nu^{11} + \)\(26\!\cdots\!17\)\( \nu^{10} - \)\(47\!\cdots\!07\)\( \nu^{9} + \)\(70\!\cdots\!35\)\( \nu^{8} - \)\(88\!\cdots\!64\)\( \nu^{7} + \)\(91\!\cdots\!47\)\( \nu^{6} - \)\(86\!\cdots\!65\)\( \nu^{5} + \)\(29\!\cdots\!28\)\( \nu^{4} - \)\(35\!\cdots\!90\)\( \nu^{3} - \)\(28\!\cdots\!80\)\( \nu^{2} - \)\(14\!\cdots\!76\)\( \nu + \)\(20\!\cdots\!21\)\(\)\()/ \)\(73\!\cdots\!31\)\( \)
\(\beta_{19}\)\(=\)\((\)\(\)\(74\!\cdots\!15\)\( \nu^{19} - \)\(29\!\cdots\!26\)\( \nu^{18} - \)\(23\!\cdots\!01\)\( \nu^{17} + \)\(49\!\cdots\!78\)\( \nu^{16} - \)\(11\!\cdots\!31\)\( \nu^{15} - \)\(46\!\cdots\!38\)\( \nu^{14} + \)\(11\!\cdots\!09\)\( \nu^{13} - \)\(45\!\cdots\!37\)\( \nu^{12} + \)\(12\!\cdots\!08\)\( \nu^{11} - \)\(25\!\cdots\!92\)\( \nu^{10} + \)\(43\!\cdots\!67\)\( \nu^{9} - \)\(61\!\cdots\!64\)\( \nu^{8} + \)\(72\!\cdots\!39\)\( \nu^{7} - \)\(68\!\cdots\!17\)\( \nu^{6} + \)\(59\!\cdots\!57\)\( \nu^{5} + \)\(24\!\cdots\!97\)\( \nu^{4} + \)\(22\!\cdots\!90\)\( \nu^{3} + \)\(46\!\cdots\!68\)\( \nu^{2} + \)\(24\!\cdots\!08\)\( \nu + \)\(17\!\cdots\!51\)\(\)\()/ \)\(73\!\cdots\!31\)\( \)

Display \(\nu^j\) in terms of \(\beta_i\)

\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{19} - \beta_{18} - 2 \beta_{17} - 2 \beta_{16} - \beta_{15} - \beta_{14} - \beta_{13} - 4 \beta_{12} - \beta_{11} + \beta_{10} - 2 \beta_{9} - 2 \beta_{8} + \beta_{5} + 1\)
\(\nu^{3}\)\(=\)\(-2 \beta_{19} + 4 \beta_{17} + 4 \beta_{16} + \beta_{15} - \beta_{14} + 2 \beta_{12} - \beta_{10} + 5 \beta_{9} + 3 \beta_{8} + \beta_{7} - \beta_{3} - 6 \beta_{2} + 2 \beta_{1} - 2\)
\(\nu^{4}\)\(=\)\(7 \beta_{19} - 2 \beta_{18} - 49 \beta_{17} - 34 \beta_{16} - 38 \beta_{15} - 21 \beta_{14} - 13 \beta_{13} - \beta_{12} - 21 \beta_{11} + 18 \beta_{10} - 49 \beta_{9} - 47 \beta_{8} + 2 \beta_{6} + 11 \beta_{5} + 11 \beta_{4} + 7 \beta_{2} - 13 \beta_{1} - 19\)
\(\nu^{5}\)\(=\)\(-71 \beta_{19} + 13 \beta_{18} + 134 \beta_{17} + 66 \beta_{16} + 114 \beta_{15} - 13 \beta_{14} + 48 \beta_{13} + 74 \beta_{12} + 48 \beta_{11} + 37 \beta_{10} + 123 \beta_{9} + 136 \beta_{8} - 13 \beta_{7} - 18 \beta_{6} - 18 \beta_{5} - 24 \beta_{4} - 48 \beta_{2} + 24 \beta_{1} - 2\)
\(\nu^{6}\)\(=\)\(151 \beta_{19} - 4 \beta_{18} - 568 \beta_{17} - 238 \beta_{16} - 505 \beta_{15} - 151 \beta_{13} - 48 \beta_{12} - 94 \beta_{11} - 44 \beta_{10} - 501 \beta_{9} - 572 \beta_{8} + 43 \beta_{7} + 135 \beta_{6} + 43 \beta_{5} + 177 \beta_{4} - 4 \beta_{3} + 177 \beta_{2} - 135 \beta_{1} - 234\)
\(\nu^{7}\)\(=\)\(-702 \beta_{19} + 2126 \beta_{17} + 408 \beta_{16} + 2126 \beta_{15} - 38 \beta_{14} + 877 \beta_{13} + 294 \beta_{12} + 664 \beta_{11} + 702 \beta_{10} + 1767 \beta_{9} + 2268 \beta_{8} - 290 \beta_{7} - 438 \beta_{6} - 158 \beta_{5} - 702 \beta_{4} + 158 \beta_{3} - 438 \beta_{2} + 290 \beta_{1} + 1065\)
\(\nu^{8}\)\(=\)\(2390 \beta_{19} + 132 \beta_{18} - 6136 \beta_{17} - 7012 \beta_{15} + 2086 \beta_{14} - 2444 \beta_{13} - 383 \beta_{12} - 934 \beta_{11} - 3900 \beta_{10} - 4451 \beta_{9} - 7012 \beta_{8} + 1692 \beta_{7} + 2390 \beta_{6} + 2444 \beta_{4} - 613 \beta_{3} + 1692 \beta_{2} - 613 \beta_{1} - 3746\)
\(\nu^{9}\)\(=\)\(-5111 \beta_{19} - 2040 \beta_{18} + 17477 \beta_{17} - 6362 \beta_{16} + 22925 \beta_{15} - 6791 \beta_{14} + 8259 \beta_{13} - 4107 \beta_{12} + 1897 \beta_{11} + 15050 \beta_{10} + 10299 \beta_{9} + 20884 \beta_{8} - 5111 \beta_{7} - 8259 \beta_{6} + 2040 \beta_{5} - 9914 \beta_{4} + 2437 \beta_{3} - 2437 \beta_{2} + 17813\)
\(\nu^{10}\)\(=\)\(14021 \beta_{19} + 9229 \beta_{18} - 34089 \beta_{17} + 49307 \beta_{16} - 64960 \beta_{15} + 38468 \beta_{14} - 24860 \beta_{13} + 20295 \beta_{12} + 2741 \beta_{11} - 71749 \beta_{10} - 6488 \beta_{9} - 49947 \beta_{8} + 24860 \beta_{7} + 28745 \beta_{6} - 11904 \beta_{5} + 28745 \beta_{4} - 14021 \beta_{3} + 9229 \beta_{1} - 60168\)
\(\nu^{11}\)\(=\)\(-39811 \beta_{18} - 23464 \beta_{17} - 258961 \beta_{16} + 124682 \beta_{15} - 159691 \beta_{14} + 43390 \beta_{13} - 121103 \beta_{12} - 63275 \beta_{11} + 258961 \beta_{10} - 119880 \beta_{9} + 43390 \beta_{8} - 84461 \beta_{7} - 103018 \beta_{6} + 67780 \beta_{5} - 84461 \beta_{4} + 43390 \beta_{3} + 39811 \beta_{2} - 67780 \beta_{1} + 194703\)
\(\nu^{12}\)\(=\)\(-194262 \beta_{19} + 194262 \beta_{18} + 590249 \beta_{17} + 1222410 \beta_{16} + 590249 \beta_{14} + 612124 \beta_{12} + 374692 \beta_{11} - 907833 \beta_{10} + 907833 \beta_{9} + 374692 \beta_{8} + 243649 \beta_{7} + 243649 \beta_{6} - 332046 \beta_{5} + 164196 \beta_{4} - 164196 \beta_{3} - 332046 \beta_{2} + 377677 \beta_{1} - 612124\)
\(\nu^{13}\)\(=\)\(1291028 \beta_{19} - 672687 \beta_{18} - 4241226 \beta_{17} - 5090636 \beta_{16} - 1882408 \beta_{15} - 2176925 \beta_{14} - 672687 \beta_{13} - 2413435 \beta_{12} - 1882408 \beta_{11} + 2950198 \beta_{10} - 5090636 \beta_{9} - 3467953 \beta_{8} - 805301 \beta_{7} - 570177 \beta_{6} + 1291028 \beta_{5} + 570177 \beta_{3} + 1689671 \beta_{2} - 1689671 \beta_{1} + 1291028\)
\(\nu^{14}\)\(=\)\(-7306375 \beta_{19} + 2462944 \beta_{18} + 22504950 \beta_{17} + 19715461 \beta_{16} + 14176470 \beta_{15} + 6870095 \beta_{14} + 5252433 \beta_{13} + 9769319 \beta_{12} + 8583611 \beta_{11} - 7717856 \beta_{10} + 24266226 \beta_{9} + 20276664 \beta_{8} + 1335137 \beta_{7} - 5252433 \beta_{5} - 2462944 \beta_{4} - 1335137 \beta_{3} - 8030139 \beta_{2} + 7306375 \beta_{1} - 1185708\)
\(\nu^{15}\)\(=\)\(33609406 \beta_{19} - 8199524 \beta_{18} - 102255210 \beta_{17} - 69885095 \beta_{16} - 76091057 \beta_{15} - 18895885 \beta_{14} - 27803501 \beta_{13} - 35108969 \beta_{12} - 34503462 \beta_{11} + 14678150 \beta_{10} - 102255210 \beta_{9} - 97688596 \beta_{8} + 8199524 \beta_{6} + 18412561 \beta_{5} + 18412561 \beta_{4} + 3215628 \beta_{3} + 33609406 \beta_{2} - 27803501 \beta_{1} - 10696361\)
\(\nu^{16}\)\(=\)\(-140271748 \beta_{19} + 20954963 \beta_{18} + 427756633 \beta_{17} + 223899277 \beta_{16} + 353063932 \beta_{15} + 38521255 \beta_{14} + 129164655 \beta_{13} + 111598363 \beta_{12} + 129164655 \beta_{11} + 7232985 \beta_{10} + 401721400 \beta_{9} + 422676363 \beta_{8} - 20954963 \beta_{7} - 57749347 \beta_{6} - 57749347 \beta_{5} - 98556170 \beta_{4} - 129164655 \beta_{2} + 98556170 \beta_{1} + 100976707\)
\(\nu^{17}\)\(=\)\(546164663 \beta_{19} - 45887093 \beta_{18} - 1632307100 \beta_{17} - 619032881 \beta_{16} - 1487267752 \beta_{15} - 546164663 \beta_{13} - 324068924 \beta_{12} - 446349270 \beta_{11} - 278181831 \beta_{10} - 1441380659 \beta_{9} - 1678194193 \beta_{8} + 165959869 \beta_{7} + 319490924 \beta_{6} + 165959869 \beta_{5} + 460650686 \beta_{4} - 45887093 \beta_{3} + 460650686 \beta_{2} - 319490924 \beta_{1} - 573145788\)
\(\nu^{18}\)\(=\)\(-1926997917 \beta_{19} + 5726696450 \beta_{17} + 1243247780 \beta_{16} + 5726696450 \beta_{15} - 571490131 \beta_{14} + 2094830388 \beta_{13} + 683750137 \beta_{12} + 1355507786 \beta_{11} + 1926997917 \beta_{10} + 4691582511 \beta_{9} + 6113161453 \beta_{8} - 868401336 \beta_{7} - 1470925231 \beta_{6} - 319657389 \beta_{5} - 1926997917 \beta_{4} + 319657389 \beta_{3} - 1470925231 \beta_{2} + 868401336 \beta_{1} + 2764584594\)
\(\nu^{19}\)\(=\)\(6182769136 \beta_{19} + 669331509 \beta_{18} - 18145320837 \beta_{17} - 20325168529 \beta_{15} + 4210658628 \beta_{14} - 7449454423 \beta_{13} - 286561532 \beta_{12} - 3459043292 \beta_{11} - 10173180267 \beta_{10} - 13345662027 \beta_{9} - 20325168529 \beta_{8} + 4071120020 \beta_{7} + 6182769136 \beta_{6} + 7449454423 \beta_{4} - 1774053131 \beta_{3} + 4071120020 \beta_{2} - 1774053131 \beta_{1} - 11962551701\)

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/690\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(461\) \(511\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{16}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
0.180562 − 1.25584i
−0.248715 + 1.72985i
−0.608732 + 1.33294i
0.706969 − 1.54805i
1.74971 + 2.01927i
−0.103821 − 0.119816i
−0.608732 − 1.33294i
0.706969 + 1.54805i
2.17335 + 0.638152i
−3.70312 − 1.08733i
2.21300 + 1.42221i
−0.359204 − 0.230846i
2.17335 − 0.638152i
−3.70312 + 1.08733i
1.74971 − 2.01927i
−0.103821 + 0.119816i
2.21300 − 1.42221i
−0.359204 + 0.230846i
0.180562 + 1.25584i
−0.248715 − 1.72985i
−0.142315 + 0.989821i −0.415415 − 0.909632i −0.959493 − 0.281733i −0.654861 − 0.755750i 0.959493 − 0.281733i −1.32376 − 0.850727i 0.415415 − 0.909632i −0.654861 + 0.755750i 0.841254 − 0.540641i
31.2 −0.142315 + 0.989821i −0.415415 − 0.909632i −0.959493 − 0.281733i −0.654861 − 0.755750i 0.959493 − 0.281733i 1.99973 + 1.28515i 0.415415 − 0.909632i −0.654861 + 0.755750i 0.841254 − 0.540641i
121.1 0.415415 − 0.909632i 0.959493 + 0.281733i −0.654861 − 0.755750i 0.841254 − 0.540641i 0.654861 − 0.755750i −0.668052 + 4.64640i −0.959493 + 0.281733i 0.841254 + 0.540641i −0.142315 − 0.989821i
121.2 0.415415 − 0.909632i 0.959493 + 0.281733i −0.654861 − 0.755750i 0.841254 − 0.540641i 0.654861 − 0.755750i 0.0903198 − 0.628188i −0.959493 + 0.281733i 0.841254 + 0.540641i −0.142315 − 0.989821i
151.1 −0.654861 − 0.755750i −0.841254 − 0.540641i −0.142315 + 0.989821i 0.415415 − 0.909632i 0.142315 + 0.989821i 0.170792 + 0.0501489i 0.841254 − 0.540641i 0.415415 + 0.909632i −0.959493 + 0.281733i
151.2 −0.654861 − 0.755750i −0.841254 − 0.540641i −0.142315 + 0.989821i 0.415415 − 0.909632i 0.142315 + 0.989821i 2.42713 + 0.712670i 0.841254 − 0.540641i 0.415415 + 0.909632i −0.959493 + 0.281733i
211.1 0.415415 + 0.909632i 0.959493 − 0.281733i −0.654861 + 0.755750i 0.841254 + 0.540641i 0.654861 + 0.755750i −0.668052 − 4.64640i −0.959493 − 0.281733i 0.841254 − 0.540641i −0.142315 + 0.989821i
211.2 0.415415 + 0.909632i 0.959493 − 0.281733i −0.654861 + 0.755750i 0.841254 + 0.540641i 0.654861 + 0.755750i 0.0903198 + 0.628188i −0.959493 − 0.281733i 0.841254 − 0.540641i −0.142315 + 0.989821i
271.1 −0.959493 − 0.281733i 0.654861 − 0.755750i 0.841254 + 0.540641i −0.142315 + 0.989821i −0.841254 + 0.540641i −0.838100 − 1.83518i −0.654861 − 0.755750i −0.142315 − 0.989821i 0.415415 − 0.909632i
271.2 −0.959493 − 0.281733i 0.654861 − 0.755750i 0.841254 + 0.540641i −0.142315 + 0.989821i −0.841254 + 0.540641i −0.113936 − 0.249486i −0.654861 − 0.755750i −0.142315 − 0.989821i 0.415415 − 0.909632i
301.1 0.841254 + 0.540641i 0.142315 + 0.989821i 0.415415 + 0.909632i −0.959493 − 0.281733i −0.415415 + 0.909632i −2.29325 + 2.64655i −0.142315 + 0.989821i −0.959493 + 0.281733i −0.654861 − 0.755750i
301.2 0.841254 + 0.540641i 0.142315 + 0.989821i 0.415415 + 0.909632i −0.959493 − 0.281733i −0.415415 + 0.909632i 1.54913 − 1.78779i −0.142315 + 0.989821i −0.959493 + 0.281733i −0.654861 − 0.755750i
331.1 −0.959493 + 0.281733i 0.654861 + 0.755750i 0.841254 − 0.540641i −0.142315 − 0.989821i −0.841254 − 0.540641i −0.838100 + 1.83518i −0.654861 + 0.755750i −0.142315 + 0.989821i 0.415415 + 0.909632i
331.2 −0.959493 + 0.281733i 0.654861 + 0.755750i 0.841254 − 0.540641i −0.142315 − 0.989821i −0.841254 − 0.540641i −0.113936 + 0.249486i −0.654861 + 0.755750i −0.142315 + 0.989821i 0.415415 + 0.909632i
361.1 −0.654861 + 0.755750i −0.841254 + 0.540641i −0.142315 − 0.989821i 0.415415 + 0.909632i 0.142315 − 0.989821i 0.170792 − 0.0501489i 0.841254 + 0.540641i 0.415415 − 0.909632i −0.959493 − 0.281733i
361.2 −0.654861 + 0.755750i −0.841254 + 0.540641i −0.142315 − 0.989821i 0.415415 + 0.909632i 0.142315 − 0.989821i 2.42713 − 0.712670i 0.841254 + 0.540641i 0.415415 − 0.909632i −0.959493 − 0.281733i
541.1 0.841254 − 0.540641i 0.142315 − 0.989821i 0.415415 − 0.909632i −0.959493 + 0.281733i −0.415415 − 0.909632i −2.29325 − 2.64655i −0.142315 − 0.989821i −0.959493 − 0.281733i −0.654861 + 0.755750i
541.2 0.841254 − 0.540641i 0.142315 − 0.989821i 0.415415 − 0.909632i −0.959493 + 0.281733i −0.415415 − 0.909632i 1.54913 + 1.78779i −0.142315 − 0.989821i −0.959493 − 0.281733i −0.654861 + 0.755750i
601.1 −0.142315 − 0.989821i −0.415415 + 0.909632i −0.959493 + 0.281733i −0.654861 + 0.755750i 0.959493 + 0.281733i −1.32376 + 0.850727i 0.415415 + 0.909632i −0.654861 − 0.755750i 0.841254 + 0.540641i
601.2 −0.142315 − 0.989821i −0.415415 + 0.909632i −0.959493 + 0.281733i −0.654861 + 0.755750i 0.959493 + 0.281733i 1.99973 − 1.28515i 0.415415 + 0.909632i −0.654861 − 0.755750i 0.841254 + 0.540641i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 601.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 690.2.m.c ✓ 20
23.c even 11 1 inner 690.2.m.c ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.m.c ✓ 20 1.a even 1 1 trivial
690.2.m.c ✓ 20 23.c even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{7}^{20} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(690, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
$3$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
$5$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
$7$ \( 529 - 4393 T + 9991 T^{2} - 38812 T^{3} + 252482 T^{4} - 184623 T^{5} + 540780 T^{6} - 61583 T^{7} - 30534 T^{8} - 66922 T^{9} + 61478 T^{10} - 17376 T^{11} + 11360 T^{12} - 9544 T^{13} + 3796 T^{14} - 705 T^{15} + 200 T^{16} - 94 T^{17} + 20 T^{18} - 2 T^{19} + T^{20} \)
$11$ \( 4489 + 143246 T + 2669703 T^{2} + 4823295 T^{3} + 24048199 T^{4} + 14541174 T^{5} + 8773829 T^{6} + 6951312 T^{7} + 3431573 T^{8} - 39052 T^{9} - 556975 T^{10} - 344777 T^{11} - 120497 T^{12} + 12877 T^{13} + 45629 T^{14} + 26456 T^{15} + 8813 T^{16} + 1909 T^{17} + 273 T^{18} + 24 T^{19} + T^{20} \)
$13$ \( 529 - 2507 T + 90605 T^{2} + 629679 T^{3} + 34114015 T^{4} + 269541970 T^{5} + 832131170 T^{6} + 266880315 T^{7} + 52831342 T^{8} - 6251797 T^{9} - 4832850 T^{10} - 1350531 T^{11} - 17078 T^{12} + 47466 T^{13} + 26342 T^{14} + 5824 T^{15} + 1004 T^{16} + 197 T^{17} + 23 T^{18} + 4 T^{19} + T^{20} \)
$17$ \( 3500852224 + 17748506624 T + 35072897024 T^{2} + 18489339520 T^{3} + 7030898496 T^{4} - 693780736 T^{5} + 2891896544 T^{6} - 233080632 T^{7} + 7453072 T^{8} + 72225224 T^{9} - 12774719 T^{10} + 1674682 T^{11} + 836880 T^{12} - 530489 T^{13} + 160429 T^{14} - 37521 T^{15} + 7284 T^{16} - 1085 T^{17} + 143 T^{18} - 16 T^{19} + T^{20} \)
$19$ \( 59274601 - 191358645 T + 1560996959 T^{2} - 6313247503 T^{3} + 12381404616 T^{4} - 14196430077 T^{5} + 10983486601 T^{6} - 6289487143 T^{7} + 2980996272 T^{8} - 1246692924 T^{9} + 444571798 T^{10} - 123273945 T^{11} + 25051360 T^{12} - 3592912 T^{13} + 349443 T^{14} - 23941 T^{15} + 2600 T^{16} - 701 T^{17} + 147 T^{18} - 14 T^{19} + T^{20} \)
$23$ \( 41426511213649 + 3602305322926 T + 8849141336753 T^{2} + 313243941124 T^{3} + 873559780989 T^{4} - 32529277522 T^{5} + 51421063591 T^{6} - 7220944162 T^{7} + 2117748345 T^{8} - 570296178 T^{9} + 82913127 T^{10} - 24795486 T^{11} + 4003305 T^{12} - 593486 T^{13} + 183751 T^{14} - 5054 T^{15} + 5901 T^{16} + 92 T^{17} + 113 T^{18} + 2 T^{19} + T^{20} \)
$29$ \( 59067463607296 + 115945225781760 T + 78848647689216 T^{2} - 5722737582336 T^{3} + 5780549739328 T^{4} - 580200928896 T^{5} + 640523124144 T^{6} - 163875134832 T^{7} + 44576316392 T^{8} - 5089766112 T^{9} + 176888217 T^{10} + 114662135 T^{11} - 24116076 T^{12} + 1956362 T^{13} + 210812 T^{14} - 82539 T^{15} + 14963 T^{16} - 1816 T^{17} + 188 T^{18} - 18 T^{19} + T^{20} \)
$31$ \( 152659665224704 - 35760921264640 T + 201218171677952 T^{2} - 112453503978112 T^{3} + 36574051014336 T^{4} - 11204669986400 T^{5} + 3334830274864 T^{6} - 860493193032 T^{7} + 189372033972 T^{8} - 37275450938 T^{9} + 7272197405 T^{10} - 1302121898 T^{11} + 189926803 T^{12} - 21062107 T^{13} + 1900547 T^{14} - 143550 T^{15} + 9196 T^{16} - 836 T^{17} + 176 T^{18} - 22 T^{19} + T^{20} \)
$37$ \( 42213400681 + 210380770345 T + 6953231649958 T^{2} + 9559374314553 T^{3} + 8124675536335 T^{4} + 1044869341747 T^{5} + 266308447623 T^{6} - 236913492746 T^{7} + 19508179108 T^{8} + 2086576619 T^{9} - 3437704909 T^{10} + 286729852 T^{11} + 213538798 T^{12} + 17902800 T^{13} + 915568 T^{14} + 35508 T^{15} + 8175 T^{16} + 447 T^{17} + 132 T^{18} + 16 T^{19} + T^{20} \)
$41$ \( 3543563854121281 - 5773566248326304 T + 4989209671292361 T^{2} - 1850158315434764 T^{3} + 287990193653240 T^{4} + 54176686416506 T^{5} - 29420843177273 T^{6} + 4412569265333 T^{7} + 656649054394 T^{8} - 198980154254 T^{9} + 37894463354 T^{10} - 4100990757 T^{11} + 363229223 T^{12} - 27305970 T^{13} + 2965971 T^{14} - 478847 T^{15} + 68148 T^{16} - 7239 T^{17} + 562 T^{18} - 29 T^{19} + T^{20} \)
$43$ \( 5143909873755136 + 2246763636215296 T - 159056664355072 T^{2} - 234243039084032 T^{3} + 137685940183616 T^{4} + 149053597231968 T^{5} + 57968672172752 T^{6} + 14476666551688 T^{7} + 2976302382592 T^{8} + 546893064674 T^{9} + 86135488219 T^{10} + 11512863835 T^{11} + 1375395021 T^{12} + 144754236 T^{13} + 13832478 T^{14} + 1228513 T^{15} + 97977 T^{16} + 7304 T^{17} + 473 T^{18} + 22 T^{19} + T^{20} \)
$47$ \( ( 1159181 + 866947 T - 12606490 T^{2} - 11161471 T^{3} - 3815776 T^{4} - 598784 T^{5} - 26374 T^{6} + 4944 T^{7} + 812 T^{8} + 47 T^{9} + T^{10} )^{2} \)
$53$ \( 439932686936963089 - 418541241131344244 T + 346383353398075067 T^{2} - 177192375328256193 T^{3} + 61482945148199679 T^{4} - 15142829863438684 T^{5} + 2742491494003593 T^{6} - 371976624379461 T^{7} + 38502742851028 T^{8} - 2982295068332 T^{9} + 128381106699 T^{10} + 9704601248 T^{11} - 2778406562 T^{12} + 265348879 T^{13} - 3473519 T^{14} - 2362448 T^{15} + 359181 T^{16} - 30237 T^{17} + 1667 T^{18} - 58 T^{19} + T^{20} \)
$59$ \( 1050454057882681 - 2213284632706247 T + 3835395702786164 T^{2} - 4187182058952980 T^{3} + 2880347841521556 T^{4} - 1280822387126656 T^{5} + 385215942992992 T^{6} - 78577042025406 T^{7} + 10996914196719 T^{8} - 1117091281547 T^{9} + 118082584366 T^{10} - 17053303761 T^{11} + 2385758870 T^{12} - 292074091 T^{13} + 33197121 T^{14} - 3389617 T^{15} + 302950 T^{16} - 21614 T^{17} + 1155 T^{18} - 45 T^{19} + T^{20} \)
$61$ \( 554879105680384 - 1492269013927424 T + 2329166224538624 T^{2} - 1300707115492480 T^{3} + 630167668691200 T^{4} - 125019463964256 T^{5} + 31494715777088 T^{6} - 4628552041480 T^{7} + 774481276884 T^{8} - 26080076120 T^{9} + 4394103053 T^{10} - 105908257 T^{11} - 245614334 T^{12} + 10765241 T^{13} + 2942278 T^{14} - 145185 T^{15} - 1873 T^{16} + 675 T^{17} - 51 T^{18} - T^{19} + T^{20} \)
$67$ \( 1535935108056064 + 4604162755120640 T + 27250634019937024 T^{2} - 22566789703443840 T^{3} + 5651505832843072 T^{4} - 388233035563136 T^{5} + 27049306122592 T^{6} - 16386349658848 T^{7} + 2775037864372 T^{8} - 364382614940 T^{9} + 65665581729 T^{10} - 8353601898 T^{11} + 950580566 T^{12} - 103826299 T^{13} + 12555498 T^{14} - 1052430 T^{15} + 88691 T^{16} - 6085 T^{17} + 456 T^{18} - 16 T^{19} + T^{20} \)
$71$ \( 59348755456 - 4728769759232 T + 133373238553344 T^{2} - 26158750656000 T^{3} + 37013566222976 T^{4} - 7730437518464 T^{5} + 4650616282192 T^{6} - 2404506874568 T^{7} + 895757550300 T^{8} - 145859817198 T^{9} + 15325882593 T^{10} + 2131062496 T^{11} - 611975417 T^{12} + 84275967 T^{13} + 909764 T^{14} - 1776312 T^{15} + 300761 T^{16} - 28285 T^{17} + 1664 T^{18} - 59 T^{19} + T^{20} \)
$73$ \( 20837416757330944 + 27933834685091840 T + 15069183317576704 T^{2} + 5357616140595968 T^{3} + 1959215714500544 T^{4} + 564425107938560 T^{5} + 124008200100864 T^{6} + 27167314780912 T^{7} + 5085911984268 T^{8} + 645232158354 T^{9} + 77250557119 T^{10} + 9863878641 T^{11} + 1129873385 T^{12} + 103366344 T^{13} + 11348288 T^{14} + 489082 T^{15} + 46000 T^{16} + 3577 T^{17} + 171 T^{18} - 3 T^{19} + T^{20} \)
$79$ \( 23911777870160896 - 773923821410304 T + 39340312019968 T^{2} - 334787984129152 T^{3} + 139007055975424 T^{4} + 43224735083520 T^{5} + 9973710286032 T^{6} + 36281065112 T^{7} - 105085303904 T^{8} - 6345635140 T^{9} + 4670596061 T^{10} + 281447841 T^{11} - 98646397 T^{12} - 13685393 T^{13} + 1715407 T^{14} + 484494 T^{15} + 61536 T^{16} + 6418 T^{17} + 449 T^{18} + 20 T^{19} + T^{20} \)
$83$ \( 10300915926016 - 15432784441856 T + 9817891176704 T^{2} - 256051262720 T^{3} + 845242467968 T^{4} + 1178919324960 T^{5} + 930982028464 T^{6} + 383148687328 T^{7} + 83388978924 T^{8} + 8215295640 T^{9} - 134634523 T^{10} - 141051547 T^{11} - 3474290 T^{12} - 559713 T^{13} + 444909 T^{14} - 18065 T^{15} + 6167 T^{16} - 1210 T^{17} + 123 T^{18} - 13 T^{19} + T^{20} \)
$89$ \( \)\(10\!\cdots\!29\)\( + 15770657295829092059 T - 9703469336441968810 T^{2} - 3458255255263431198 T^{3} - 222038010082364543 T^{4} + 163302476048680739 T^{5} + 71598572030931207 T^{6} + 17159844544903616 T^{7} + 2994042239733987 T^{8} + 413898560972272 T^{9} + 47337962318327 T^{10} + 4591475233591 T^{11} + 384287495435 T^{12} + 28023550053 T^{13} + 1784378432 T^{14} + 98232848 T^{15} + 4569545 T^{16} + 172441 T^{17} + 4959 T^{18} + 97 T^{19} + T^{20} \)
$97$ \( 61671996335104 - 82849497095680 T + 121664127838976 T^{2} - 50636687927936 T^{3} + 26074300000960 T^{4} - 6877018673440 T^{5} + 1582273196080 T^{6} - 388871077320 T^{7} + 69374087468 T^{8} + 3570052458 T^{9} - 3341172253 T^{10} - 50453033 T^{11} + 113929214 T^{12} + 7819207 T^{13} - 1304305 T^{14} - 120491 T^{15} + 17925 T^{16} + 2882 T^{17} + 227 T^{18} + 17 T^{19} + T^{20} \)
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