Properties

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

Related objects

Downloads

Learn more about

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: \(x^{20} - 6 x^{19} + 21 x^{18} - 47 x^{17} + 44 x^{16} + 232 x^{15} - 1084 x^{14} + 1484 x^{13} + 2670 x^{12} - 12826 x^{11} + 18393 x^{10} - 2728 x^{9} - 12654 x^{8} - 6818 x^{7} + 39054 x^{6} + 33738 x^{5} + 67716 x^{4} + 45635 x^{3} + 32892 x^{2} + 9761 x + 1849\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 6 x^{19} + 21 x^{18} - 47 x^{17} + 44 x^{16} + 232 x^{15} - 1084 x^{14} + 1484 x^{13} + 2670 x^{12} - 12826 x^{11} + 18393 x^{10} - 2728 x^{9} - 12654 x^{8} - 6818 x^{7} + 39054 x^{6} + 33738 x^{5} + 67716 x^{4} + 45635 x^{3} + 32892 x^{2} + 9761 x + 1849\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(10\!\cdots\!45\)\( \nu^{19} + \)\(53\!\cdots\!53\)\( \nu^{18} - \)\(34\!\cdots\!46\)\( \nu^{17} + \)\(13\!\cdots\!19\)\( \nu^{16} - \)\(33\!\cdots\!24\)\( \nu^{15} + \)\(46\!\cdots\!28\)\( \nu^{14} + \)\(79\!\cdots\!75\)\( \nu^{13} - \)\(59\!\cdots\!58\)\( \nu^{12} + \)\(12\!\cdots\!76\)\( \nu^{11} + \)\(23\!\cdots\!85\)\( \nu^{10} - \)\(64\!\cdots\!78\)\( \nu^{9} + \)\(15\!\cdots\!40\)\( \nu^{8} - \)\(15\!\cdots\!41\)\( \nu^{7} + \)\(50\!\cdots\!81\)\( \nu^{6} + \)\(25\!\cdots\!60\)\( \nu^{5} + \)\(50\!\cdots\!30\)\( \nu^{4} + \)\(42\!\cdots\!47\)\( \nu^{3} + \)\(40\!\cdots\!29\)\( \nu^{2} + \)\(15\!\cdots\!34\)\( \nu + \)\(12\!\cdots\!48\)\(\)\()/ \)\(12\!\cdots\!27\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(58\!\cdots\!18\)\( \nu^{19} + \)\(16\!\cdots\!81\)\( \nu^{18} - \)\(11\!\cdots\!96\)\( \nu^{17} + \)\(45\!\cdots\!13\)\( \nu^{16} - \)\(11\!\cdots\!88\)\( \nu^{15} + \)\(14\!\cdots\!17\)\( \nu^{14} + \)\(36\!\cdots\!45\)\( \nu^{13} - \)\(22\!\cdots\!57\)\( \nu^{12} + \)\(40\!\cdots\!06\)\( \nu^{11} + \)\(32\!\cdots\!84\)\( \nu^{10} - \)\(26\!\cdots\!24\)\( \nu^{9} + \)\(47\!\cdots\!32\)\( \nu^{8} - \)\(23\!\cdots\!60\)\( \nu^{7} - \)\(17\!\cdots\!39\)\( \nu^{6} - \)\(61\!\cdots\!79\)\( \nu^{5} + \)\(84\!\cdots\!48\)\( \nu^{4} + \)\(42\!\cdots\!64\)\( \nu^{3} + \)\(12\!\cdots\!77\)\( \nu^{2} + \)\(45\!\cdots\!00\)\( \nu + \)\(19\!\cdots\!51\)\(\)\()/ \)\(29\!\cdots\!89\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(44\!\cdots\!88\)\( \nu^{19} + \)\(38\!\cdots\!01\)\( \nu^{18} - \)\(16\!\cdots\!16\)\( \nu^{17} + \)\(43\!\cdots\!96\)\( \nu^{16} - \)\(66\!\cdots\!90\)\( \nu^{15} - \)\(72\!\cdots\!39\)\( \nu^{14} + \)\(78\!\cdots\!92\)\( \nu^{13} - \)\(19\!\cdots\!89\)\( \nu^{12} + \)\(17\!\cdots\!93\)\( \nu^{11} + \)\(97\!\cdots\!86\)\( \nu^{10} - \)\(23\!\cdots\!86\)\( \nu^{9} + \)\(18\!\cdots\!65\)\( \nu^{8} + \)\(13\!\cdots\!45\)\( \nu^{7} - \)\(21\!\cdots\!58\)\( \nu^{6} - \)\(29\!\cdots\!46\)\( \nu^{5} + \)\(40\!\cdots\!11\)\( \nu^{4} + \)\(21\!\cdots\!99\)\( \nu^{3} + \)\(51\!\cdots\!83\)\( \nu^{2} + \)\(33\!\cdots\!39\)\( \nu + \)\(19\!\cdots\!93\)\(\)\()/ \)\(12\!\cdots\!27\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(10\!\cdots\!81\)\( \nu^{19} + \)\(74\!\cdots\!07\)\( \nu^{18} - \)\(29\!\cdots\!28\)\( \nu^{17} + \)\(76\!\cdots\!08\)\( \nu^{16} - \)\(11\!\cdots\!76\)\( \nu^{15} - \)\(15\!\cdots\!65\)\( \nu^{14} + \)\(13\!\cdots\!46\)\( \nu^{13} - \)\(28\!\cdots\!82\)\( \nu^{12} - \)\(23\!\cdots\!05\)\( \nu^{11} + \)\(14\!\cdots\!51\)\( \nu^{10} - \)\(35\!\cdots\!60\)\( \nu^{9} + \)\(36\!\cdots\!97\)\( \nu^{8} - \)\(11\!\cdots\!97\)\( \nu^{7} - \)\(68\!\cdots\!30\)\( \nu^{6} - \)\(12\!\cdots\!80\)\( \nu^{5} - \)\(11\!\cdots\!69\)\( \nu^{4} - \)\(96\!\cdots\!28\)\( \nu^{3} - \)\(44\!\cdots\!18\)\( \nu^{2} - \)\(30\!\cdots\!51\)\( \nu - \)\(45\!\cdots\!35\)\(\)\()/ \)\(29\!\cdots\!89\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(20\!\cdots\!69\)\( \nu^{19} + \)\(13\!\cdots\!79\)\( \nu^{18} - \)\(52\!\cdots\!62\)\( \nu^{17} + \)\(12\!\cdots\!50\)\( \nu^{16} - \)\(16\!\cdots\!57\)\( \nu^{15} - \)\(39\!\cdots\!74\)\( \nu^{14} + \)\(25\!\cdots\!18\)\( \nu^{13} - \)\(48\!\cdots\!99\)\( \nu^{12} - \)\(27\!\cdots\!78\)\( \nu^{11} + \)\(30\!\cdots\!65\)\( \nu^{10} - \)\(59\!\cdots\!77\)\( \nu^{9} + \)\(36\!\cdots\!60\)\( \nu^{8} + \)\(26\!\cdots\!12\)\( \nu^{7} - \)\(20\!\cdots\!86\)\( \nu^{6} - \)\(90\!\cdots\!05\)\( \nu^{5} + \)\(24\!\cdots\!30\)\( \nu^{4} - \)\(10\!\cdots\!01\)\( \nu^{3} - \)\(21\!\cdots\!92\)\( \nu^{2} - \)\(13\!\cdots\!61\)\( \nu + \)\(86\!\cdots\!28\)\(\)\()/ \)\(29\!\cdots\!89\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(22\!\cdots\!52\)\( \nu^{19} + \)\(12\!\cdots\!64\)\( \nu^{18} - \)\(38\!\cdots\!07\)\( \nu^{17} + \)\(73\!\cdots\!08\)\( \nu^{16} - \)\(27\!\cdots\!60\)\( \nu^{15} - \)\(58\!\cdots\!14\)\( \nu^{14} + \)\(20\!\cdots\!84\)\( \nu^{13} - \)\(16\!\cdots\!68\)\( \nu^{12} - \)\(84\!\cdots\!11\)\( \nu^{11} + \)\(25\!\cdots\!42\)\( \nu^{10} - \)\(20\!\cdots\!31\)\( \nu^{9} - \)\(25\!\cdots\!54\)\( \nu^{8} + \)\(31\!\cdots\!23\)\( \nu^{7} + \)\(54\!\cdots\!35\)\( \nu^{6} - \)\(10\!\cdots\!07\)\( \nu^{5} - \)\(14\!\cdots\!72\)\( \nu^{4} - \)\(15\!\cdots\!31\)\( \nu^{3} - \)\(13\!\cdots\!09\)\( \nu^{2} - \)\(79\!\cdots\!27\)\( \nu - \)\(19\!\cdots\!83\)\(\)\()/ \)\(29\!\cdots\!89\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(11\!\cdots\!26\)\( \nu^{19} - \)\(90\!\cdots\!49\)\( \nu^{18} + \)\(38\!\cdots\!69\)\( \nu^{17} - \)\(10\!\cdots\!49\)\( \nu^{16} + \)\(17\!\cdots\!97\)\( \nu^{15} + \)\(13\!\cdots\!46\)\( \nu^{14} - \)\(17\!\cdots\!30\)\( \nu^{13} + \)\(42\!\cdots\!38\)\( \nu^{12} - \)\(11\!\cdots\!96\)\( \nu^{11} - \)\(18\!\cdots\!97\)\( \nu^{10} + \)\(51\!\cdots\!03\)\( \nu^{9} - \)\(55\!\cdots\!21\)\( \nu^{8} + \)\(10\!\cdots\!86\)\( \nu^{7} + \)\(19\!\cdots\!28\)\( \nu^{6} + \)\(38\!\cdots\!00\)\( \nu^{5} - \)\(44\!\cdots\!17\)\( \nu^{4} + \)\(53\!\cdots\!28\)\( \nu^{3} - \)\(41\!\cdots\!96\)\( \nu^{2} + \)\(87\!\cdots\!36\)\( \nu - \)\(11\!\cdots\!71\)\(\)\()/ \)\(12\!\cdots\!27\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(42\!\cdots\!50\)\( \nu^{19} + \)\(25\!\cdots\!00\)\( \nu^{18} - \)\(91\!\cdots\!47\)\( \nu^{17} + \)\(20\!\cdots\!63\)\( \nu^{16} - \)\(20\!\cdots\!01\)\( \nu^{15} - \)\(97\!\cdots\!39\)\( \nu^{14} + \)\(47\!\cdots\!43\)\( \nu^{13} - \)\(67\!\cdots\!30\)\( \nu^{12} - \)\(10\!\cdots\!05\)\( \nu^{11} + \)\(56\!\cdots\!64\)\( \nu^{10} - \)\(83\!\cdots\!56\)\( \nu^{9} + \)\(16\!\cdots\!31\)\( \nu^{8} + \)\(61\!\cdots\!61\)\( \nu^{7} + \)\(15\!\cdots\!78\)\( \nu^{6} - \)\(17\!\cdots\!68\)\( \nu^{5} - \)\(11\!\cdots\!12\)\( \nu^{4} - \)\(25\!\cdots\!26\)\( \nu^{3} - \)\(18\!\cdots\!85\)\( \nu^{2} - \)\(96\!\cdots\!69\)\( \nu - \)\(17\!\cdots\!87\)\(\)\()/ \)\(29\!\cdots\!89\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(20\!\cdots\!96\)\( \nu^{19} + \)\(11\!\cdots\!09\)\( \nu^{18} - \)\(36\!\cdots\!19\)\( \nu^{17} + \)\(71\!\cdots\!46\)\( \nu^{16} - \)\(32\!\cdots\!74\)\( \nu^{15} - \)\(53\!\cdots\!23\)\( \nu^{14} + \)\(20\!\cdots\!82\)\( \nu^{13} - \)\(18\!\cdots\!90\)\( \nu^{12} - \)\(74\!\cdots\!77\)\( \nu^{11} + \)\(24\!\cdots\!42\)\( \nu^{10} - \)\(23\!\cdots\!33\)\( \nu^{9} - \)\(20\!\cdots\!23\)\( \nu^{8} + \)\(41\!\cdots\!64\)\( \nu^{7} + \)\(24\!\cdots\!44\)\( \nu^{6} - \)\(86\!\cdots\!82\)\( \nu^{5} - \)\(10\!\cdots\!63\)\( \nu^{4} - \)\(12\!\cdots\!46\)\( \nu^{3} - \)\(13\!\cdots\!03\)\( \nu^{2} - \)\(75\!\cdots\!88\)\( \nu - \)\(25\!\cdots\!79\)\(\)\()/ \)\(12\!\cdots\!27\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-\)\(47\!\cdots\!51\)\( \nu^{19} + \)\(31\!\cdots\!61\)\( \nu^{18} - \)\(11\!\cdots\!89\)\( \nu^{17} + \)\(27\!\cdots\!71\)\( \nu^{16} - \)\(32\!\cdots\!02\)\( \nu^{15} - \)\(10\!\cdots\!22\)\( \nu^{14} + \)\(57\!\cdots\!78\)\( \nu^{13} - \)\(98\!\cdots\!12\)\( \nu^{12} - \)\(90\!\cdots\!47\)\( \nu^{11} + \)\(68\!\cdots\!95\)\( \nu^{10} - \)\(12\!\cdots\!51\)\( \nu^{9} + \)\(58\!\cdots\!30\)\( \nu^{8} + \)\(63\!\cdots\!72\)\( \nu^{7} - \)\(16\!\cdots\!28\)\( \nu^{6} - \)\(19\!\cdots\!35\)\( \nu^{5} - \)\(60\!\cdots\!16\)\( \nu^{4} - \)\(22\!\cdots\!42\)\( \nu^{3} - \)\(69\!\cdots\!92\)\( \nu^{2} - \)\(52\!\cdots\!99\)\( \nu - \)\(50\!\cdots\!18\)\(\)\()/ \)\(29\!\cdots\!89\)\( \)
\(\beta_{12}\)\(=\)\((\)\(\)\(25\!\cdots\!42\)\( \nu^{19} - \)\(14\!\cdots\!22\)\( \nu^{18} + \)\(51\!\cdots\!44\)\( \nu^{17} - \)\(11\!\cdots\!32\)\( \nu^{16} + \)\(10\!\cdots\!44\)\( \nu^{15} + \)\(59\!\cdots\!48\)\( \nu^{14} - \)\(26\!\cdots\!20\)\( \nu^{13} + \)\(34\!\cdots\!57\)\( \nu^{12} + \)\(70\!\cdots\!28\)\( \nu^{11} - \)\(31\!\cdots\!87\)\( \nu^{10} + \)\(42\!\cdots\!48\)\( \nu^{9} - \)\(30\!\cdots\!22\)\( \nu^{8} - \)\(28\!\cdots\!50\)\( \nu^{7} - \)\(27\!\cdots\!92\)\( \nu^{6} + \)\(10\!\cdots\!52\)\( \nu^{5} + \)\(96\!\cdots\!05\)\( \nu^{4} + \)\(18\!\cdots\!58\)\( \nu^{3} + \)\(12\!\cdots\!23\)\( \nu^{2} + \)\(89\!\cdots\!07\)\( \nu + \)\(14\!\cdots\!55\)\(\)\()/ \)\(12\!\cdots\!27\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(27\!\cdots\!20\)\( \nu^{19} + \)\(17\!\cdots\!10\)\( \nu^{18} - \)\(61\!\cdots\!92\)\( \nu^{17} + \)\(14\!\cdots\!05\)\( \nu^{16} - \)\(15\!\cdots\!80\)\( \nu^{15} - \)\(59\!\cdots\!88\)\( \nu^{14} + \)\(31\!\cdots\!26\)\( \nu^{13} - \)\(48\!\cdots\!67\)\( \nu^{12} - \)\(60\!\cdots\!39\)\( \nu^{11} + \)\(36\!\cdots\!52\)\( \nu^{10} - \)\(59\!\cdots\!86\)\( \nu^{9} + \)\(22\!\cdots\!91\)\( \nu^{8} + \)\(28\!\cdots\!17\)\( \nu^{7} + \)\(11\!\cdots\!90\)\( \nu^{6} - \)\(10\!\cdots\!51\)\( \nu^{5} - \)\(68\!\cdots\!06\)\( \nu^{4} - \)\(17\!\cdots\!29\)\( \nu^{3} - \)\(69\!\cdots\!64\)\( \nu^{2} - \)\(69\!\cdots\!05\)\( \nu - \)\(12\!\cdots\!00\)\(\)\()/ \)\(12\!\cdots\!27\)\( \)
\(\beta_{14}\)\(=\)\((\)\(\)\(34\!\cdots\!55\)\( \nu^{19} - \)\(20\!\cdots\!85\)\( \nu^{18} + \)\(74\!\cdots\!21\)\( \nu^{17} - \)\(17\!\cdots\!99\)\( \nu^{16} + \)\(17\!\cdots\!62\)\( \nu^{15} + \)\(75\!\cdots\!82\)\( \nu^{14} - \)\(37\!\cdots\!91\)\( \nu^{13} + \)\(55\!\cdots\!81\)\( \nu^{12} + \)\(80\!\cdots\!13\)\( \nu^{11} - \)\(43\!\cdots\!50\)\( \nu^{10} + \)\(68\!\cdots\!21\)\( \nu^{9} - \)\(22\!\cdots\!66\)\( \nu^{8} - \)\(30\!\cdots\!34\)\( \nu^{7} - \)\(24\!\cdots\!53\)\( \nu^{6} + \)\(13\!\cdots\!95\)\( \nu^{5} + \)\(94\!\cdots\!03\)\( \nu^{4} + \)\(25\!\cdots\!22\)\( \nu^{3} + \)\(16\!\cdots\!42\)\( \nu^{2} + \)\(15\!\cdots\!13\)\( \nu + \)\(48\!\cdots\!26\)\(\)\()/ \)\(12\!\cdots\!27\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-\)\(37\!\cdots\!04\)\( \nu^{19} + \)\(24\!\cdots\!08\)\( \nu^{18} - \)\(88\!\cdots\!14\)\( \nu^{17} + \)\(20\!\cdots\!76\)\( \nu^{16} - \)\(23\!\cdots\!97\)\( \nu^{15} - \)\(82\!\cdots\!27\)\( \nu^{14} + \)\(44\!\cdots\!79\)\( \nu^{13} - \)\(72\!\cdots\!75\)\( \nu^{12} - \)\(80\!\cdots\!54\)\( \nu^{11} + \)\(52\!\cdots\!19\)\( \nu^{10} - \)\(89\!\cdots\!83\)\( \nu^{9} + \)\(36\!\cdots\!13\)\( \nu^{8} + \)\(50\!\cdots\!10\)\( \nu^{7} - \)\(84\!\cdots\!21\)\( \nu^{6} - \)\(14\!\cdots\!25\)\( \nu^{5} - \)\(64\!\cdots\!84\)\( \nu^{4} - \)\(22\!\cdots\!19\)\( \nu^{3} - \)\(10\!\cdots\!02\)\( \nu^{2} - \)\(78\!\cdots\!44\)\( \nu - \)\(17\!\cdots\!84\)\(\)\()/ \)\(12\!\cdots\!27\)\( \)
\(\beta_{16}\)\(=\)\((\)\(-\)\(44\!\cdots\!81\)\( \nu^{19} + \)\(27\!\cdots\!22\)\( \nu^{18} - \)\(98\!\cdots\!53\)\( \nu^{17} + \)\(22\!\cdots\!08\)\( \nu^{16} - \)\(22\!\cdots\!08\)\( \nu^{15} - \)\(10\!\cdots\!12\)\( \nu^{14} + \)\(50\!\cdots\!06\)\( \nu^{13} - \)\(74\!\cdots\!16\)\( \nu^{12} - \)\(11\!\cdots\!46\)\( \nu^{11} + \)\(60\!\cdots\!79\)\( \nu^{10} - \)\(92\!\cdots\!39\)\( \nu^{9} + \)\(20\!\cdots\!01\)\( \nu^{8} + \)\(67\!\cdots\!96\)\( \nu^{7} + \)\(16\!\cdots\!69\)\( \nu^{6} - \)\(19\!\cdots\!79\)\( \nu^{5} - \)\(10\!\cdots\!77\)\( \nu^{4} - \)\(23\!\cdots\!00\)\( \nu^{3} - \)\(13\!\cdots\!02\)\( \nu^{2} - \)\(87\!\cdots\!65\)\( \nu - \)\(92\!\cdots\!80\)\(\)\()/ \)\(12\!\cdots\!27\)\( \)
\(\beta_{17}\)\(=\)\((\)\(\)\(46\!\cdots\!57\)\( \nu^{19} - \)\(27\!\cdots\!16\)\( \nu^{18} + \)\(96\!\cdots\!14\)\( \nu^{17} - \)\(21\!\cdots\!51\)\( \nu^{16} + \)\(18\!\cdots\!49\)\( \nu^{15} + \)\(11\!\cdots\!08\)\( \nu^{14} - \)\(50\!\cdots\!19\)\( \nu^{13} + \)\(67\!\cdots\!53\)\( \nu^{12} + \)\(13\!\cdots\!41\)\( \nu^{11} - \)\(61\!\cdots\!40\)\( \nu^{10} + \)\(83\!\cdots\!89\)\( \nu^{9} - \)\(11\!\cdots\!64\)\( \nu^{8} - \)\(79\!\cdots\!54\)\( \nu^{7} - \)\(21\!\cdots\!46\)\( \nu^{6} + \)\(18\!\cdots\!55\)\( \nu^{5} + \)\(15\!\cdots\!63\)\( \nu^{4} + \)\(27\!\cdots\!48\)\( \nu^{3} + \)\(19\!\cdots\!43\)\( \nu^{2} + \)\(10\!\cdots\!33\)\( \nu + \)\(25\!\cdots\!77\)\(\)\()/ \)\(12\!\cdots\!27\)\( \)
\(\beta_{18}\)\(=\)\((\)\(-\)\(64\!\cdots\!46\)\( \nu^{19} + \)\(38\!\cdots\!99\)\( \nu^{18} - \)\(13\!\cdots\!68\)\( \nu^{17} + \)\(28\!\cdots\!90\)\( \nu^{16} - \)\(21\!\cdots\!65\)\( \nu^{15} - \)\(16\!\cdots\!86\)\( \nu^{14} + \)\(71\!\cdots\!62\)\( \nu^{13} - \)\(88\!\cdots\!42\)\( \nu^{12} - \)\(20\!\cdots\!49\)\( \nu^{11} + \)\(87\!\cdots\!61\)\( \nu^{10} - \)\(11\!\cdots\!00\)\( \nu^{9} - \)\(22\!\cdots\!60\)\( \nu^{8} + \)\(14\!\cdots\!98\)\( \nu^{7} + \)\(11\!\cdots\!02\)\( \nu^{6} - \)\(28\!\cdots\!21\)\( \nu^{5} - \)\(20\!\cdots\!98\)\( \nu^{4} - \)\(33\!\cdots\!76\)\( \nu^{3} - \)\(28\!\cdots\!42\)\( \nu^{2} - \)\(13\!\cdots\!17\)\( \nu - \)\(53\!\cdots\!64\)\(\)\()/ \)\(12\!\cdots\!27\)\( \)
\(\beta_{19}\)\(=\)\((\)\(-\)\(71\!\cdots\!31\)\( \nu^{19} + \)\(44\!\cdots\!99\)\( \nu^{18} - \)\(15\!\cdots\!86\)\( \nu^{17} + \)\(34\!\cdots\!60\)\( \nu^{16} - \)\(33\!\cdots\!43\)\( \nu^{15} - \)\(16\!\cdots\!58\)\( \nu^{14} + \)\(80\!\cdots\!87\)\( \nu^{13} - \)\(11\!\cdots\!24\)\( \nu^{12} - \)\(19\!\cdots\!93\)\( \nu^{11} + \)\(96\!\cdots\!12\)\( \nu^{10} - \)\(13\!\cdots\!99\)\( \nu^{9} + \)\(19\!\cdots\!73\)\( \nu^{8} + \)\(10\!\cdots\!35\)\( \nu^{7} + \)\(45\!\cdots\!76\)\( \nu^{6} - \)\(31\!\cdots\!33\)\( \nu^{5} - \)\(21\!\cdots\!06\)\( \nu^{4} - \)\(41\!\cdots\!75\)\( \nu^{3} - \)\(20\!\cdots\!39\)\( \nu^{2} - \)\(12\!\cdots\!32\)\( \nu + \)\(68\!\cdots\!21\)\(\)\()/ \)\(12\!\cdots\!27\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{19} - \beta_{18} + \beta_{16} + \beta_{14} + 4 \beta_{13} + \beta_{10} + \beta_{8} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2}\)
\(\nu^{3}\)\(=\)\(-6 \beta_{19} - 5 \beta_{18} - 3 \beta_{17} + 6 \beta_{16} + 3 \beta_{15} + 5 \beta_{14} + 8 \beta_{13} - 5 \beta_{12} + 6 \beta_{11} + \beta_{10} - 6 \beta_{9} + 5 \beta_{8} + 5 \beta_{7} + 6 \beta_{6} - 6 \beta_{5} + 3 \beta_{4} - 5 \beta_{3} - 2 \beta_{2} - 6 \beta_{1} - 3\)
\(\nu^{4}\)\(=\)\(2 \beta_{19} - 5 \beta_{18} + 12 \beta_{17} - 4 \beta_{16} + 12 \beta_{15} + 8 \beta_{14} + \beta_{13} - 10 \beta_{12} + 8 \beta_{11} + 17 \beta_{10} - 8 \beta_{9} - \beta_{8} + 8 \beta_{7} - 2 \beta_{6} + 5 \beta_{4} + 2 \beta_{3} - 16 \beta_{2} - 5 \beta_{1} - 9\)
\(\nu^{5}\)\(=\)\(17 \beta_{19} + 37 \beta_{17} - 10 \beta_{16} + 20 \beta_{15} - 2 \beta_{14} - 25 \beta_{13} + 2 \beta_{12} - 18 \beta_{11} + 38 \beta_{10} + 17 \beta_{9} + 8 \beta_{8} - 5 \beta_{7} - 5 \beta_{6} + 48 \beta_{5} + 23 \beta_{4} + 2 \beta_{3} + 5 \beta_{2} + 18 \beta_{1} - 37\)
\(\nu^{6}\)\(=\)\(-49 \beta_{19} + 17 \beta_{18} - 176 \beta_{17} + 59 \beta_{16} - 252 \beta_{15} - 52 \beta_{14} + 124 \beta_{13} + 87 \beta_{12} - 153 \beta_{10} + 52 \beta_{9} - 39 \beta_{8} - 51 \beta_{7} + 49 \beta_{6} + 17 \beta_{5} - 27 \beta_{4} - 45 \beta_{3} - 22 \beta_{2} - 45 \beta_{1} + 118\)
\(\nu^{7}\)\(=\)\(-149 \beta_{19} + 190 \beta_{18} - 279 \beta_{17} + 126 \beta_{16} - 451 \beta_{15} + 26 \beta_{14} + 149 \beta_{13} - 49 \beta_{12} + 237 \beta_{11} - 541 \beta_{10} - 203 \beta_{9} - 328 \beta_{8} + 149 \beta_{7} + 26 \beta_{6} - 237 \beta_{5} - 464 \beta_{4} - 156 \beta_{2} - 203 \beta_{1} + 541\)
\(\nu^{8}\)\(=\)\(958 \beta_{19} + 505 \beta_{18} + 1726 \beta_{17} - 1261 \beta_{16} + 1221 \beta_{15} + 83 \beta_{14} - 1703 \beta_{13} - 658 \beta_{12} - 505 \beta_{11} + 1281 \beta_{10} - 877 \beta_{8} + 83 \beta_{7} - 955 \beta_{6} + 536 \beta_{5} + 70 \beta_{4} + 536 \beta_{3} + 955 \beta_{1} - 839\)
\(\nu^{9}\)\(=\)\(709 \beta_{19} - 442 \beta_{18} + 481 \beta_{17} - 6 \beta_{16} + 442 \beta_{15} - 709 \beta_{14} - 6 \beta_{13} + 1754 \beta_{12} - 3618 \beta_{11} + 2102 \beta_{10} + 2353 \beta_{9} + 923 \beta_{8} - 1933 \beta_{7} + 2353 \beta_{5} + 2434 \beta_{4} - 442 \beta_{3} + 1660 \beta_{2} + 1933 \beta_{1} - 2434\)
\(\nu^{10}\)\(=\)\(-10774 \beta_{19} - 3634 \beta_{18} - 21494 \beta_{17} + 11124 \beta_{16} - 15352 \beta_{15} - 247 \beta_{14} + 16141 \beta_{13} + 2746 \beta_{12} + 5203 \beta_{11} - 15352 \beta_{10} - 3634 \beta_{9} + 3634 \beta_{8} + 9580 \beta_{6} - 9580 \beta_{5} - 1733 \beta_{4} - 5203 \beta_{3} - 350 \beta_{2} - 10774 \beta_{1} + 9886\)
\(\nu^{11}\)\(=\)\(5608 \beta_{18} + 12201 \beta_{17} - 10691 \beta_{16} + 17284 \beta_{15} + 12201 \beta_{14} - 14769 \beta_{13} - 26970 \beta_{12} + 25721 \beta_{11} - 179 \beta_{10} - 25721 \beta_{9} - 12380 \beta_{8} + 20543 \beta_{7} - 5608 \beta_{6} - 20543 \beta_{5} - 19527 \beta_{4} + 14558 \beta_{3} - 13919 \beta_{2} - 12201 \beta_{1} + 10260\)
\(\nu^{12}\)\(=\)\(112497 \beta_{19} + 14864 \beta_{18} + 219206 \beta_{17} - 130827 \beta_{16} + 195024 \beta_{15} - 185292 \beta_{13} - 42785 \beta_{12} - 79021 \beta_{11} + 204342 \beta_{10} + 55048 \beta_{9} + 3466 \beta_{8} - 14864 \beta_{7} - 79021 \beta_{6} + 112497 \beta_{5} + 82527 \beta_{4} + 55048 \beta_{3} + 42785 \beta_{2} + 117180 \beta_{1} - 170428\)
\(\nu^{13}\)\(=\)\(-93436 \beta_{19} - 93436 \beta_{18} - 297818 \beta_{17} + 174858 \beta_{16} - 278418 \beta_{15} - 122960 \beta_{14} + 276872 \beta_{13} + 278418 \beta_{12} - 190935 \beta_{11} - 122960 \beta_{10} + 258952 \beta_{9} + 195376 \beta_{8} - 190935 \beta_{7} + 151094 \beta_{6} + 122960 \beta_{5} + 195376 \beta_{4} - 151094 \beta_{3} + 148557 \beta_{2} - 25597\)
\(\nu^{14}\)\(=\)\(-1073229 \beta_{19} - 66429 \beta_{18} - 1964912 \beta_{17} + 1073229 \beta_{16} - 1654675 \beta_{15} + 66429 \beta_{14} + 1473638 \beta_{13} + 63989 \beta_{12} + 1073229 \beta_{11} - 2044834 \beta_{10} - 686876 \beta_{9} - 63989 \beta_{8} + 389100 \beta_{7} + 686876 \beta_{6} - 1245479 \beta_{5} - 891683 \beta_{4} - 389100 \beta_{3} - 400409 \beta_{2} - 1245479 \beta_{1} + 1654675\)
\(\nu^{15}\)\(=\)\(2034529 \beta_{19} + 1254266 \beta_{18} + 4981939 \beta_{17} - 2952752 \beta_{16} + 4241676 \beta_{15} + 1020549 \beta_{14} - 4475393 \beta_{13} - 2947410 \beta_{12} + 1020549 \beta_{11} + 2719035 \beta_{10} - 1645888 \beta_{9} - 2027881 \beta_{8} + 1645888 \beta_{7} - 2592128 \beta_{6} - 1254266 \beta_{4} + 2034529 \beta_{3} - 1056894 \beta_{2} + 1254266 \beta_{1} - 1027197\)
\(\nu^{16}\)\(=\)\(9176359 \beta_{19} + 15959038 \beta_{17} - 7906237 \beta_{16} + 13052315 \beta_{15} - 1919605 \beta_{14} - 11683125 \beta_{13} + 1919605 \beta_{12} - 12387178 \beta_{11} + 17517742 \beta_{10} + 9176359 \beta_{9} + 2506766 \beta_{8} - 5760657 \beta_{7} - 5760657 \beta_{6} + 13217003 \beta_{5} + 10260988 \beta_{4} + 1919605 \beta_{3} + 5795561 \beta_{2} + 12387178 \beta_{1} - 15959038\)
\(\nu^{17}\)\(=\)\(-30608973 \beta_{19} - 12926543 \beta_{18} - 67672363 \beta_{17} + 40840915 \beta_{16} - 57696684 \beta_{15} - 8671898 \beta_{14} + 59000465 \beta_{13} + 28946063 \beta_{12} - 46628493 \beta_{10} + 8671898 \beta_{9} + 18485766 \beta_{8} - 13578189 \beta_{7} + 30608973 \beta_{6} - 12926543 \beta_{5} + 2694601 \beta_{4} - 23919845 \beta_{3} + 5559223 \beta_{2} - 23919845 \beta_{1} + 26354328\)
\(\nu^{18}\)\(=\)\(-70366964 \beta_{19} + 9689521 \beta_{18} - 103249795 \beta_{17} + 52884467 \beta_{16} - 78414332 \beta_{15} + 32639289 \beta_{14} + 70366964 \beta_{13} - 50121786 \beta_{12} + 130345291 \beta_{11} - 139105414 \beta_{10} - 109425042 \beta_{9} - 40686657 \beta_{8} + 70366964 \beta_{7} + 32639289 \beta_{6} - 130345291 \beta_{5} - 113173091 \beta_{4} - 65522120 \beta_{2} - 109425042 \beta_{1} + 139105414\)
\(\nu^{19}\)\(=\)\(392001502 \beta_{19} + 117472410 \beta_{18} + 821440121 \beta_{17} - 477358238 \beta_{16} + 703967711 \beta_{15} + 71042035 \beta_{14} - 690908071 \beta_{13} - 264493410 \beta_{12} - 117472410 \beta_{11} + 644477696 \beta_{10} - 148570492 \beta_{8} + 71042035 \beta_{7} - 346236458 \beta_{6} + 251732915 \beta_{5} + 75978965 \beta_{4} + 251732915 \beta_{3} + 346236458 \beta_{1} - 430927863\)

Character values

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

\(n\) \(277\) \(461\) \(511\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{13}\)

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
1.66465 + 1.92111i
−0.205162 0.236769i
2.03264 1.30630i
−0.877779 + 0.564114i
0.909132 1.99072i
−0.266817 + 0.584249i
2.03264 + 1.30630i
−0.877779 0.564114i
−0.462319 + 3.21550i
0.121066 0.842031i
2.29563 + 0.674057i
−2.21104 0.649220i
−0.462319 3.21550i
0.121066 + 0.842031i
0.909132 + 1.99072i
−0.266817 0.584249i
2.29563 0.674057i
−2.21104 + 0.649220i
1.66465 1.92111i
−0.205162 + 0.236769i
0.142315 0.989821i −0.415415 0.909632i −0.959493 0.281733i −0.654861 0.755750i −0.959493 + 0.281733i −0.306823 0.197183i −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 4.30262 + 2.76513i −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 −0.421385 + 2.93080i 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 0.223463 1.55422i 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 −2.28976 0.672335i −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 −1.51668 0.445336i −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 −0.421385 2.93080i 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 0.223463 + 1.55422i 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 −1.63866 3.58816i 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 1.22647 + 2.68560i 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 −1.56737 + 1.80884i 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 0.988122 1.14035i 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 −1.63866 + 3.58816i 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 1.22647 2.68560i 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 −2.28976 + 0.672335i −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 −1.51668 + 0.445336i −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 −1.56737 1.80884i 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 0.988122 + 1.14035i 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 −0.306823 + 0.197183i −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 4.30262 2.76513i −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.e 20
23.c even 11 1 inner 690.2.m.e 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.m.e 20 1.a even 1 1 trivial
690.2.m.e 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

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
$3$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
$5$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
$7$ \( 1893376 + 11030016 T + 26599424 T^{2} + 27123456 T^{3} + 22335808 T^{4} + 19208256 T^{5} + 13691664 T^{6} + 8715712 T^{7} + 5555624 T^{8} + 3032832 T^{9} + 1363385 T^{10} + 508541 T^{11} + 165029 T^{12} + 46245 T^{13} + 10628 T^{14} + 2025 T^{15} + 264 T^{16} + 7 T^{17} + 12 T^{18} + 2 T^{19} + T^{20} \)
$11$ \( 707506801 + 4527841374 T + 9614958956 T^{2} + 7104036583 T^{3} + 3567601698 T^{4} + 1346077342 T^{5} + 541803660 T^{6} + 104071504 T^{7} + 64503123 T^{8} + 4886616 T^{9} + 3116850 T^{10} - 182353 T^{11} + 144919 T^{12} - 28456 T^{13} - 2346 T^{14} - 2437 T^{15} + 608 T^{16} + 131 T^{17} + 45 T^{18} + 4 T^{19} + T^{20} \)
$13$ \( 144264121 + 341833060 T + 368866910 T^{2} + 275172840 T^{3} + 155666016 T^{4} + 54940106 T^{5} + 20933679 T^{6} + 10627138 T^{7} + 2513772 T^{8} - 873972 T^{9} + 38435 T^{10} + 113498 T^{11} - 191 T^{12} - 15058 T^{13} + 6272 T^{14} + 4632 T^{15} + 2464 T^{16} + 623 T^{17} + 122 T^{18} + 13 T^{19} + T^{20} \)
$17$ \( 33860761 - 7169008 T + 694927805 T^{2} + 779457316 T^{3} + 414510910 T^{4} + 102152534 T^{5} + 7717336 T^{6} + 6553382 T^{7} + 5375205 T^{8} + 1235256 T^{9} + 1129162 T^{10} - 72932 T^{11} + 266512 T^{12} - 127652 T^{13} + 33984 T^{14} + 1717 T^{15} - 1586 T^{16} + 257 T^{17} + 25 T^{18} - 6 T^{19} + T^{20} \)
$19$ \( 1024 - 29696 T + 1189632 T^{2} - 1615360 T^{3} + 10084800 T^{4} - 49070176 T^{5} + 106786720 T^{6} - 29167120 T^{7} - 10899516 T^{8} + 38887662 T^{9} + 36117247 T^{10} + 12902043 T^{11} + 2575660 T^{12} + 231073 T^{13} - 4074 T^{14} - 8295 T^{15} - 1694 T^{16} - 43 T^{17} + 15 T^{18} + 4 T^{19} + T^{20} \)
$23$ \( 41426511213649 - 43227663875112 T + 26077558098573 T^{2} - 12148417194896 T^{3} + 4827302304401 T^{4} - 1669072466760 T^{5} + 513041740053 T^{6} - 143126455832 T^{7} + 36600257857 T^{8} - 8591066076 T^{9} + 1861418273 T^{10} - 373524612 T^{11} + 69187633 T^{12} - 11763496 T^{13} + 1833333 T^{14} - 259320 T^{15} + 32609 T^{16} - 3568 T^{17} + 333 T^{18} - 24 T^{19} + T^{20} \)
$29$ \( 179533201 + 924825778 T + 2088800606 T^{2} + 4625191476 T^{3} + 8941338215 T^{4} + 7747079401 T^{5} + 10860749866 T^{6} + 6136090636 T^{7} + 2205714536 T^{8} + 129783467 T^{9} - 29008154 T^{10} + 690734 T^{11} + 9717041 T^{12} + 2957063 T^{13} + 100240 T^{14} - 3097 T^{15} + 9939 T^{16} - 262 T^{17} - 4 T^{18} - 2 T^{19} + T^{20} \)
$31$ \( 16570264694281 - 23516083064548 T + 12611972220445 T^{2} - 7536428014030 T^{3} + 6248095837086 T^{4} - 2408665596211 T^{5} + 345559519762 T^{6} - 96607172273 T^{7} + 30882765951 T^{8} + 254270217 T^{9} + 1407938829 T^{10} - 3059877 T^{11} + 32784203 T^{12} - 2758190 T^{13} + 711248 T^{14} - 62068 T^{15} + 13418 T^{16} - 471 T^{17} + 82 T^{18} - 6 T^{19} + T^{20} \)
$37$ \( 15338839089289 - 83767853304820 T + 110489983303031 T^{2} + 36245138443958 T^{3} + 34302117457725 T^{4} + 9084610749154 T^{5} + 2611963393643 T^{6} + 470365803418 T^{7} + 103227541442 T^{8} + 16320595621 T^{9} + 2792322567 T^{10} + 317652863 T^{11} + 53056315 T^{12} + 6582300 T^{13} + 982781 T^{14} + 85981 T^{15} + 10104 T^{16} + 1014 T^{17} + 150 T^{18} + 20 T^{19} + T^{20} \)
$41$ \( 7415295394816 + 6275992250880 T + 25791835479040 T^{2} + 22999600156544 T^{3} + 28294688524096 T^{4} + 10783835165312 T^{5} + 3052294879808 T^{6} + 445903896504 T^{7} + 6968413884 T^{8} - 11556976200 T^{9} - 1168224463 T^{10} + 215096959 T^{11} + 36126853 T^{12} - 1367003 T^{13} - 268571 T^{14} + 33495 T^{15} + 3166 T^{16} - 583 T^{17} - 8 T^{18} + 11 T^{19} + T^{20} \)
$43$ \( 1157771193925369 + 6526797110183936 T + 21889242286338357 T^{2} - 9356716754131413 T^{3} + 1732662765816992 T^{4} - 137760151709876 T^{5} - 9597524356678 T^{6} + 5746993471320 T^{7} - 938193396641 T^{8} + 45753015814 T^{9} + 15251078217 T^{10} - 4317380551 T^{11} + 583851332 T^{12} - 54659281 T^{13} + 5782928 T^{14} - 567641 T^{15} + 40979 T^{16} - 2810 T^{17} + 241 T^{18} - 4 T^{19} + T^{20} \)
$47$ \( ( 10197947 - 7303085 T - 1587498 T^{2} + 1907351 T^{3} - 149414 T^{4} - 98542 T^{5} + 11626 T^{6} + 1678 T^{7} - 198 T^{8} - 9 T^{9} + T^{10} )^{2} \)
$53$ \( 834348308767744 + 74710702410240 T + 202069573977600 T^{2} - 1405145458560 T^{3} - 9057293719936 T^{4} + 6381494525152 T^{5} + 434131980144 T^{6} - 175144626568 T^{7} + 90687179580 T^{8} + 4549184310 T^{9} + 683680361 T^{10} + 310269916 T^{11} + 37976836 T^{12} + 9111606 T^{13} + 1050878 T^{14} + 9888 T^{15} + 4169 T^{16} + 521 T^{17} + 233 T^{18} + 18 T^{19} + T^{20} \)
$59$ \( 241893954330011281 + 30561006490485664 T - 14138121103184876 T^{2} - 2458878158126656 T^{3} + 1539307106437824 T^{4} + 776632952794793 T^{5} + 148132649202449 T^{6} + 5813567737599 T^{7} - 3448432066892 T^{8} - 868580297281 T^{9} - 78718281862 T^{10} + 6522427425 T^{11} + 3453137054 T^{12} + 642375312 T^{13} + 81660101 T^{14} + 7769339 T^{15} + 588867 T^{16} + 35005 T^{17} + 1632 T^{18} + 52 T^{19} + T^{20} \)
$61$ \( 182035342336 + 613988703232 T + 1587252366080 T^{2} - 7046050585856 T^{3} + 11411033406144 T^{4} - 9163508033760 T^{5} + 4274565141952 T^{6} - 1548166979984 T^{7} + 498914608428 T^{8} - 91008736390 T^{9} + 13719085315 T^{10} - 1046986149 T^{11} + 21141558 T^{12} + 2870477 T^{13} + 58880 T^{14} - 159409 T^{15} + 17021 T^{16} + 325 T^{17} - T^{18} + 7 T^{19} + T^{20} \)
$67$ \( 488977134361 - 1834409849425 T + 1823493365630 T^{2} + 58737577495 T^{3} + 127098646163 T^{4} + 150423809423 T^{5} - 580719343752 T^{6} + 211571918766 T^{7} + 548716793626 T^{8} + 187267985201 T^{9} + 44493610536 T^{10} + 7230970076 T^{11} + 993915154 T^{12} + 101707904 T^{13} + 11067124 T^{14} + 783533 T^{15} + 76004 T^{16} + 2637 T^{17} + 244 T^{18} + 10 T^{19} + T^{20} \)
$71$ \( 46053613236315136 - 7537698995197440 T - 906024258930432 T^{2} - 1491558751143168 T^{3} + 518604391541696 T^{4} - 45143424032352 T^{5} + 19565047891120 T^{6} - 5581902639128 T^{7} + 1453213899472 T^{8} - 241106309070 T^{9} + 28901774837 T^{10} - 1047913020 T^{11} - 12363925 T^{12} + 16576225 T^{13} - 788604 T^{14} + 94256 T^{15} + 5999 T^{16} - 501 T^{17} + 208 T^{18} - 7 T^{19} + T^{20} \)
$73$ \( 2774329959424 + 1732257280000 T + 6036944020480 T^{2} + 4973031548416 T^{3} + 3303771693376 T^{4} + 612476499680 T^{5} - 185282995904 T^{6} - 166034578656 T^{7} - 1149627944 T^{8} + 16086739834 T^{9} + 3144105559 T^{10} - 661391555 T^{11} - 210086463 T^{12} - 4664152 T^{13} + 8024240 T^{14} + 1464266 T^{15} + 162976 T^{16} + 13581 T^{17} + 827 T^{18} + 37 T^{19} + T^{20} \)
$79$ \( 91979474091649 + 3056721166535424 T + 27860513327047843 T^{2} + 5511469526861514 T^{3} + 1960635231735814 T^{4} + 363887056961499 T^{5} + 104582179084448 T^{6} + 11374586083603 T^{7} + 2203525930433 T^{8} + 166593135951 T^{9} + 35413964455 T^{10} + 377833555 T^{11} + 589786388 T^{12} - 26357000 T^{13} + 4632669 T^{14} - 151222 T^{15} + 3636 T^{16} - 1080 T^{17} + 161 T^{18} - 20 T^{19} + T^{20} \)
$83$ \( 723454980958618624 - 1783750007307881472 T + 2318619461385361152 T^{2} - 1289282375656627328 T^{3} + 358805611340444032 T^{4} - 44544834904549920 T^{5} + 3116766428015248 T^{6} - 187516132649704 T^{7} + 30169414822144 T^{8} - 562596054504 T^{9} - 282874089255 T^{10} - 6393037585 T^{11} + 931418040 T^{12} + 142859777 T^{13} + 31510709 T^{14} + 2575139 T^{15} + 179813 T^{16} + 11884 T^{17} + 585 T^{18} + 19 T^{19} + T^{20} \)
$89$ \( 118754673664 + 296886684160 T + 874314605824 T^{2} - 1015585966208 T^{3} + 4085684195648 T^{4} - 5685771379552 T^{5} + 4195190080400 T^{6} - 1793739708552 T^{7} + 469876684948 T^{8} - 51404404590 T^{9} - 4851168179 T^{10} + 2039765970 T^{11} - 248206695 T^{12} + 2774046 T^{13} + 5932872 T^{14} - 981508 T^{15} + 90365 T^{16} - 7183 T^{17} + 649 T^{18} - 33 T^{19} + T^{20} \)
$97$ \( 1784647106533295104 + 613970988611518464 T + 452938879990769152 T^{2} - 38125995478036096 T^{3} + 8919461478909120 T^{4} + 1979604117938496 T^{5} + 201575617738912 T^{6} - 688083424616 T^{7} + 1873446399696 T^{8} + 251455004240 T^{9} - 28148268895 T^{10} - 1049200449 T^{11} + 401832380 T^{12} - 35748591 T^{13} + 2475173 T^{14} + 114615 T^{15} - 1127 T^{16} - 1520 T^{17} + 117 T^{18} - 9 T^{19} + T^{20} \)
show more
show less