Properties

Label 230.2.g.b
Level $230$
Weight $2$
Character orbit 230.g
Analytic conductor $1.837$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 230.g (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.83655924649\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(2\) over \(\Q(\zeta_{11})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 5 x^{19} + 12 x^{18} + 16 x^{17} - 49 x^{16} + 59 x^{15} - 197 x^{14} + 42 x^{13} + 1625 x^{12} - 5910 x^{11} + 14651 x^{10} - 22501 x^{9} + 26003 x^{8} - 19607 x^{7} + 13040 x^{6} + 348 x^{5} + 7193 x^{4} + 10771 x^{3} + 8781 x^{2} + 3105 x + 529\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 5 x^{19} + 12 x^{18} + 16 x^{17} - 49 x^{16} + 59 x^{15} - 197 x^{14} + 42 x^{13} + 1625 x^{12} - 5910 x^{11} + 14651 x^{10} - 22501 x^{9} + 26003 x^{8} - 19607 x^{7} + 13040 x^{6} + 348 x^{5} + 7193 x^{4} + 10771 x^{3} + 8781 x^{2} + 3105 x + 529\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(\)\(25\!\cdots\!20\)\( \nu^{19} - \)\(41\!\cdots\!21\)\( \nu^{18} - \)\(20\!\cdots\!17\)\( \nu^{17} + \)\(18\!\cdots\!40\)\( \nu^{16} - \)\(10\!\cdots\!26\)\( \nu^{15} - \)\(32\!\cdots\!70\)\( \nu^{14} + \)\(35\!\cdots\!75\)\( \nu^{13} - \)\(22\!\cdots\!69\)\( \nu^{12} + \)\(65\!\cdots\!21\)\( \nu^{11} - \)\(25\!\cdots\!57\)\( \nu^{10} - \)\(25\!\cdots\!53\)\( \nu^{9} + \)\(11\!\cdots\!66\)\( \nu^{8} - \)\(27\!\cdots\!67\)\( \nu^{7} + \)\(43\!\cdots\!01\)\( \nu^{6} - \)\(50\!\cdots\!11\)\( \nu^{5} + \)\(51\!\cdots\!81\)\( \nu^{4} - \)\(37\!\cdots\!07\)\( \nu^{3} + \)\(24\!\cdots\!78\)\( \nu^{2} + \)\(39\!\cdots\!12\)\( \nu + \)\(57\!\cdots\!44\)\(\)\()/ \)\(14\!\cdots\!87\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(68\!\cdots\!51\)\( \nu^{19} - \)\(75\!\cdots\!16\)\( \nu^{18} + \)\(54\!\cdots\!21\)\( \nu^{17} - \)\(18\!\cdots\!19\)\( \nu^{16} - \)\(34\!\cdots\!59\)\( \nu^{15} + \)\(59\!\cdots\!75\)\( \nu^{14} - \)\(93\!\cdots\!55\)\( \nu^{13} + \)\(27\!\cdots\!44\)\( \nu^{12} - \)\(32\!\cdots\!48\)\( \nu^{11} - \)\(12\!\cdots\!71\)\( \nu^{10} + \)\(67\!\cdots\!90\)\( \nu^{9} - \)\(19\!\cdots\!14\)\( \nu^{8} + \)\(35\!\cdots\!44\)\( \nu^{7} - \)\(48\!\cdots\!49\)\( \nu^{6} + \)\(46\!\cdots\!84\)\( \nu^{5} - \)\(35\!\cdots\!22\)\( \nu^{4} + \)\(13\!\cdots\!54\)\( \nu^{3} - \)\(11\!\cdots\!65\)\( \nu^{2} - \)\(69\!\cdots\!20\)\( \nu - \)\(28\!\cdots\!27\)\(\)\()/ \)\(14\!\cdots\!87\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(10\!\cdots\!36\)\( \nu^{19} - \)\(57\!\cdots\!00\)\( \nu^{18} + \)\(13\!\cdots\!53\)\( \nu^{17} + \)\(19\!\cdots\!93\)\( \nu^{16} - \)\(72\!\cdots\!04\)\( \nu^{15} + \)\(74\!\cdots\!50\)\( \nu^{14} - \)\(18\!\cdots\!22\)\( \nu^{13} + \)\(99\!\cdots\!37\)\( \nu^{12} + \)\(19\!\cdots\!69\)\( \nu^{11} - \)\(70\!\cdots\!81\)\( \nu^{10} + \)\(16\!\cdots\!93\)\( \nu^{9} - \)\(22\!\cdots\!83\)\( \nu^{8} + \)\(16\!\cdots\!42\)\( \nu^{7} + \)\(56\!\cdots\!15\)\( \nu^{6} - \)\(29\!\cdots\!61\)\( \nu^{5} + \)\(51\!\cdots\!39\)\( \nu^{4} - \)\(43\!\cdots\!33\)\( \nu^{3} + \)\(49\!\cdots\!63\)\( \nu^{2} - \)\(14\!\cdots\!62\)\( \nu - \)\(61\!\cdots\!32\)\(\)\()/ \)\(14\!\cdots\!87\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(64\!\cdots\!88\)\( \nu^{19} - \)\(38\!\cdots\!51\)\( \nu^{18} + \)\(11\!\cdots\!38\)\( \nu^{17} + \)\(16\!\cdots\!32\)\( \nu^{16} - \)\(38\!\cdots\!27\)\( \nu^{15} + \)\(71\!\cdots\!28\)\( \nu^{14} - \)\(17\!\cdots\!29\)\( \nu^{13} + \)\(17\!\cdots\!43\)\( \nu^{12} + \)\(96\!\cdots\!06\)\( \nu^{11} - \)\(48\!\cdots\!56\)\( \nu^{10} + \)\(13\!\cdots\!14\)\( \nu^{9} - \)\(25\!\cdots\!33\)\( \nu^{8} + \)\(34\!\cdots\!08\)\( \nu^{7} - \)\(35\!\cdots\!23\)\( \nu^{6} + \)\(29\!\cdots\!95\)\( \nu^{5} - \)\(15\!\cdots\!05\)\( \nu^{4} + \)\(93\!\cdots\!61\)\( \nu^{3} + \)\(26\!\cdots\!33\)\( \nu^{2} + \)\(43\!\cdots\!83\)\( \nu - \)\(88\!\cdots\!56\)\(\)\()/ \)\(14\!\cdots\!87\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(87\!\cdots\!15\)\( \nu^{19} - \)\(51\!\cdots\!36\)\( \nu^{18} + \)\(14\!\cdots\!78\)\( \nu^{17} + \)\(26\!\cdots\!55\)\( \nu^{16} - \)\(49\!\cdots\!87\)\( \nu^{15} + \)\(92\!\cdots\!13\)\( \nu^{14} - \)\(23\!\cdots\!09\)\( \nu^{13} + \)\(22\!\cdots\!54\)\( \nu^{12} + \)\(12\!\cdots\!89\)\( \nu^{11} - \)\(63\!\cdots\!42\)\( \nu^{10} + \)\(17\!\cdots\!33\)\( \nu^{9} - \)\(33\!\cdots\!22\)\( \nu^{8} + \)\(47\!\cdots\!25\)\( \nu^{7} - \)\(49\!\cdots\!75\)\( \nu^{6} + \)\(41\!\cdots\!03\)\( \nu^{5} - \)\(23\!\cdots\!17\)\( \nu^{4} + \)\(14\!\cdots\!78\)\( \nu^{3} + \)\(28\!\cdots\!84\)\( \nu^{2} + \)\(13\!\cdots\!27\)\( \nu + \)\(23\!\cdots\!08\)\(\)\()/ \)\(14\!\cdots\!87\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(16\!\cdots\!91\)\( \nu^{19} - \)\(84\!\cdots\!91\)\( \nu^{18} + \)\(20\!\cdots\!63\)\( \nu^{17} + \)\(24\!\cdots\!44\)\( \nu^{16} - \)\(85\!\cdots\!57\)\( \nu^{15} + \)\(10\!\cdots\!95\)\( \nu^{14} - \)\(33\!\cdots\!06\)\( \nu^{13} + \)\(10\!\cdots\!36\)\( \nu^{12} + \)\(26\!\cdots\!24\)\( \nu^{11} - \)\(10\!\cdots\!53\)\( \nu^{10} + \)\(25\!\cdots\!51\)\( \nu^{9} - \)\(39\!\cdots\!94\)\( \nu^{8} + \)\(45\!\cdots\!50\)\( \nu^{7} - \)\(34\!\cdots\!09\)\( \nu^{6} + \)\(21\!\cdots\!79\)\( \nu^{5} + \)\(26\!\cdots\!01\)\( \nu^{4} + \)\(75\!\cdots\!45\)\( \nu^{3} + \)\(19\!\cdots\!08\)\( \nu^{2} + \)\(10\!\cdots\!21\)\( \nu + \)\(24\!\cdots\!54\)\(\)\()/ \)\(14\!\cdots\!87\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(16\!\cdots\!64\)\( \nu^{19} - \)\(77\!\cdots\!32\)\( \nu^{18} + \)\(16\!\cdots\!17\)\( \nu^{17} + \)\(37\!\cdots\!62\)\( \nu^{16} - \)\(80\!\cdots\!04\)\( \nu^{15} + \)\(59\!\cdots\!49\)\( \nu^{14} - \)\(25\!\cdots\!80\)\( \nu^{13} - \)\(10\!\cdots\!41\)\( \nu^{12} + \)\(28\!\cdots\!43\)\( \nu^{11} - \)\(89\!\cdots\!34\)\( \nu^{10} + \)\(19\!\cdots\!08\)\( \nu^{9} - \)\(24\!\cdots\!50\)\( \nu^{8} + \)\(18\!\cdots\!59\)\( \nu^{7} + \)\(22\!\cdots\!60\)\( \nu^{6} - \)\(14\!\cdots\!63\)\( \nu^{5} + \)\(30\!\cdots\!67\)\( \nu^{4} - \)\(34\!\cdots\!53\)\( \nu^{3} + \)\(27\!\cdots\!05\)\( \nu^{2} + \)\(17\!\cdots\!17\)\( \nu + \)\(56\!\cdots\!03\)\(\)\()/ \)\(14\!\cdots\!87\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(76\!\cdots\!98\)\( \nu^{19} + \)\(40\!\cdots\!47\)\( \nu^{18} - \)\(10\!\cdots\!32\)\( \nu^{17} - \)\(97\!\cdots\!42\)\( \nu^{16} + \)\(40\!\cdots\!99\)\( \nu^{15} - \)\(56\!\cdots\!71\)\( \nu^{14} + \)\(16\!\cdots\!39\)\( \nu^{13} - \)\(69\!\cdots\!16\)\( \nu^{12} - \)\(12\!\cdots\!17\)\( \nu^{11} + \)\(48\!\cdots\!82\)\( \nu^{10} - \)\(12\!\cdots\!33\)\( \nu^{9} + \)\(20\!\cdots\!95\)\( \nu^{8} - \)\(24\!\cdots\!05\)\( \nu^{7} + \)\(19\!\cdots\!16\)\( \nu^{6} - \)\(13\!\cdots\!79\)\( \nu^{5} + \)\(84\!\cdots\!92\)\( \nu^{4} - \)\(43\!\cdots\!30\)\( \nu^{3} - \)\(76\!\cdots\!28\)\( \nu^{2} - \)\(46\!\cdots\!65\)\( \nu - \)\(94\!\cdots\!93\)\(\)\()/ \)\(61\!\cdots\!69\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(21\!\cdots\!27\)\( \nu^{19} + \)\(11\!\cdots\!32\)\( \nu^{18} - \)\(27\!\cdots\!56\)\( \nu^{17} - \)\(32\!\cdots\!05\)\( \nu^{16} + \)\(10\!\cdots\!22\)\( \nu^{15} - \)\(13\!\cdots\!60\)\( \nu^{14} + \)\(44\!\cdots\!71\)\( \nu^{13} - \)\(12\!\cdots\!87\)\( \nu^{12} - \)\(35\!\cdots\!28\)\( \nu^{11} + \)\(13\!\cdots\!17\)\( \nu^{10} - \)\(33\!\cdots\!34\)\( \nu^{9} + \)\(51\!\cdots\!72\)\( \nu^{8} - \)\(60\!\cdots\!18\)\( \nu^{7} + \)\(47\!\cdots\!37\)\( \nu^{6} - \)\(32\!\cdots\!33\)\( \nu^{5} + \)\(16\!\cdots\!50\)\( \nu^{4} - \)\(15\!\cdots\!84\)\( \nu^{3} - \)\(22\!\cdots\!27\)\( \nu^{2} - \)\(16\!\cdots\!68\)\( \nu - \)\(39\!\cdots\!82\)\(\)\()/ \)\(14\!\cdots\!87\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(22\!\cdots\!84\)\( \nu^{19} - \)\(12\!\cdots\!05\)\( \nu^{18} + \)\(33\!\cdots\!10\)\( \nu^{17} + \)\(19\!\cdots\!03\)\( \nu^{16} - \)\(12\!\cdots\!00\)\( \nu^{15} + \)\(19\!\cdots\!04\)\( \nu^{14} - \)\(52\!\cdots\!75\)\( \nu^{13} + \)\(35\!\cdots\!59\)\( \nu^{12} + \)\(35\!\cdots\!88\)\( \nu^{11} - \)\(15\!\cdots\!81\)\( \nu^{10} + \)\(40\!\cdots\!43\)\( \nu^{9} - \)\(70\!\cdots\!81\)\( \nu^{8} + \)\(89\!\cdots\!90\)\( \nu^{7} - \)\(81\!\cdots\!46\)\( \nu^{6} + \)\(59\!\cdots\!19\)\( \nu^{5} - \)\(20\!\cdots\!24\)\( \nu^{4} + \)\(19\!\cdots\!99\)\( \nu^{3} + \)\(15\!\cdots\!82\)\( \nu^{2} + \)\(84\!\cdots\!34\)\( \nu - \)\(32\!\cdots\!98\)\(\)\()/ \)\(14\!\cdots\!87\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(22\!\cdots\!71\)\( \nu^{19} + \)\(11\!\cdots\!05\)\( \nu^{18} - \)\(29\!\cdots\!55\)\( \nu^{17} - \)\(32\!\cdots\!69\)\( \nu^{16} + \)\(11\!\cdots\!81\)\( \nu^{15} - \)\(15\!\cdots\!21\)\( \nu^{14} + \)\(46\!\cdots\!16\)\( \nu^{13} - \)\(15\!\cdots\!13\)\( \nu^{12} - \)\(36\!\cdots\!46\)\( \nu^{11} + \)\(14\!\cdots\!45\)\( \nu^{10} - \)\(35\!\cdots\!47\)\( \nu^{9} + \)\(55\!\cdots\!95\)\( \nu^{8} - \)\(65\!\cdots\!96\)\( \nu^{7} + \)\(49\!\cdots\!63\)\( \nu^{6} - \)\(29\!\cdots\!04\)\( \nu^{5} - \)\(42\!\cdots\!36\)\( \nu^{4} - \)\(10\!\cdots\!18\)\( \nu^{3} - \)\(25\!\cdots\!83\)\( \nu^{2} - \)\(14\!\cdots\!76\)\( \nu - \)\(58\!\cdots\!01\)\(\)\()/ \)\(14\!\cdots\!87\)\( \)
\(\beta_{13}\)\(=\)\((\)\(\)\(28\!\cdots\!65\)\( \nu^{19} - \)\(15\!\cdots\!46\)\( \nu^{18} + \)\(39\!\cdots\!36\)\( \nu^{17} + \)\(33\!\cdots\!63\)\( \nu^{16} - \)\(15\!\cdots\!34\)\( \nu^{15} + \)\(22\!\cdots\!67\)\( \nu^{14} - \)\(62\!\cdots\!11\)\( \nu^{13} + \)\(30\!\cdots\!08\)\( \nu^{12} + \)\(46\!\cdots\!56\)\( \nu^{11} - \)\(18\!\cdots\!70\)\( \nu^{10} + \)\(47\!\cdots\!07\)\( \nu^{9} - \)\(78\!\cdots\!92\)\( \nu^{8} + \)\(96\!\cdots\!59\)\( \nu^{7} - \)\(79\!\cdots\!77\)\( \nu^{6} + \)\(52\!\cdots\!21\)\( \nu^{5} - \)\(46\!\cdots\!79\)\( \nu^{4} + \)\(13\!\cdots\!00\)\( \nu^{3} + \)\(29\!\cdots\!25\)\( \nu^{2} + \)\(13\!\cdots\!60\)\( \nu + \)\(30\!\cdots\!95\)\(\)\()/ \)\(14\!\cdots\!87\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(29\!\cdots\!15\)\( \nu^{19} + \)\(14\!\cdots\!15\)\( \nu^{18} - \)\(34\!\cdots\!73\)\( \nu^{17} - \)\(48\!\cdots\!59\)\( \nu^{16} + \)\(15\!\cdots\!61\)\( \nu^{15} - \)\(17\!\cdots\!03\)\( \nu^{14} + \)\(55\!\cdots\!40\)\( \nu^{13} - \)\(94\!\cdots\!03\)\( \nu^{12} - \)\(48\!\cdots\!42\)\( \nu^{11} + \)\(17\!\cdots\!55\)\( \nu^{10} - \)\(42\!\cdots\!80\)\( \nu^{9} + \)\(63\!\cdots\!13\)\( \nu^{8} - \)\(69\!\cdots\!83\)\( \nu^{7} + \)\(45\!\cdots\!85\)\( \nu^{6} - \)\(22\!\cdots\!29\)\( \nu^{5} - \)\(14\!\cdots\!24\)\( \nu^{4} - \)\(11\!\cdots\!11\)\( \nu^{3} - \)\(33\!\cdots\!33\)\( \nu^{2} - \)\(22\!\cdots\!21\)\( \nu - \)\(61\!\cdots\!61\)\(\)\()/ \)\(14\!\cdots\!87\)\( \)
\(\beta_{15}\)\(=\)\((\)\(\)\(40\!\cdots\!38\)\( \nu^{19} - \)\(22\!\cdots\!38\)\( \nu^{18} + \)\(59\!\cdots\!35\)\( \nu^{17} + \)\(38\!\cdots\!91\)\( \nu^{16} - \)\(22\!\cdots\!11\)\( \nu^{15} + \)\(34\!\cdots\!83\)\( \nu^{14} - \)\(95\!\cdots\!07\)\( \nu^{13} + \)\(61\!\cdots\!53\)\( \nu^{12} + \)\(64\!\cdots\!62\)\( \nu^{11} - \)\(27\!\cdots\!10\)\( \nu^{10} + \)\(72\!\cdots\!64\)\( \nu^{9} - \)\(12\!\cdots\!24\)\( \nu^{8} + \)\(16\!\cdots\!65\)\( \nu^{7} - \)\(15\!\cdots\!99\)\( \nu^{6} + \)\(11\!\cdots\!59\)\( \nu^{5} - \)\(44\!\cdots\!18\)\( \nu^{4} + \)\(42\!\cdots\!49\)\( \nu^{3} + \)\(30\!\cdots\!65\)\( \nu^{2} + \)\(17\!\cdots\!74\)\( \nu + \)\(34\!\cdots\!17\)\(\)\()/ \)\(14\!\cdots\!87\)\( \)
\(\beta_{16}\)\(=\)\((\)\(\)\(17\!\cdots\!17\)\( \nu^{19} - \)\(97\!\cdots\!83\)\( \nu^{18} + \)\(25\!\cdots\!51\)\( \nu^{17} + \)\(18\!\cdots\!40\)\( \nu^{16} - \)\(97\!\cdots\!75\)\( \nu^{15} + \)\(14\!\cdots\!02\)\( \nu^{14} - \)\(40\!\cdots\!20\)\( \nu^{13} + \)\(24\!\cdots\!53\)\( \nu^{12} + \)\(28\!\cdots\!09\)\( \nu^{11} - \)\(11\!\cdots\!87\)\( \nu^{10} + \)\(31\!\cdots\!49\)\( \nu^{9} - \)\(52\!\cdots\!50\)\( \nu^{8} + \)\(66\!\cdots\!46\)\( \nu^{7} - \)\(59\!\cdots\!24\)\( \nu^{6} + \)\(43\!\cdots\!96\)\( \nu^{5} - \)\(12\!\cdots\!63\)\( \nu^{4} + \)\(13\!\cdots\!73\)\( \nu^{3} + \)\(14\!\cdots\!77\)\( \nu^{2} + \)\(80\!\cdots\!49\)\( \nu + \)\(94\!\cdots\!20\)\(\)\()/ \)\(61\!\cdots\!69\)\( \)
\(\beta_{17}\)\(=\)\((\)\(-\)\(56\!\cdots\!79\)\( \nu^{19} + \)\(28\!\cdots\!87\)\( \nu^{18} - \)\(66\!\cdots\!63\)\( \nu^{17} - \)\(97\!\cdots\!10\)\( \nu^{16} + \)\(28\!\cdots\!50\)\( \nu^{15} - \)\(31\!\cdots\!56\)\( \nu^{14} + \)\(10\!\cdots\!10\)\( \nu^{13} - \)\(12\!\cdots\!29\)\( \nu^{12} - \)\(93\!\cdots\!42\)\( \nu^{11} + \)\(33\!\cdots\!61\)\( \nu^{10} - \)\(80\!\cdots\!38\)\( \nu^{9} + \)\(11\!\cdots\!53\)\( \nu^{8} - \)\(12\!\cdots\!75\)\( \nu^{7} + \)\(83\!\cdots\!39\)\( \nu^{6} - \)\(41\!\cdots\!95\)\( \nu^{5} - \)\(31\!\cdots\!45\)\( \nu^{4} - \)\(20\!\cdots\!66\)\( \nu^{3} - \)\(71\!\cdots\!59\)\( \nu^{2} - \)\(47\!\cdots\!59\)\( \nu - \)\(13\!\cdots\!22\)\(\)\()/ \)\(14\!\cdots\!87\)\( \)
\(\beta_{18}\)\(=\)\((\)\(-\)\(72\!\cdots\!74\)\( \nu^{19} + \)\(39\!\cdots\!18\)\( \nu^{18} - \)\(10\!\cdots\!49\)\( \nu^{17} - \)\(70\!\cdots\!30\)\( \nu^{16} + \)\(39\!\cdots\!31\)\( \nu^{15} - \)\(60\!\cdots\!31\)\( \nu^{14} + \)\(16\!\cdots\!85\)\( \nu^{13} - \)\(10\!\cdots\!85\)\( \nu^{12} - \)\(11\!\cdots\!75\)\( \nu^{11} + \)\(48\!\cdots\!54\)\( \nu^{10} - \)\(12\!\cdots\!60\)\( \nu^{9} + \)\(21\!\cdots\!14\)\( \nu^{8} - \)\(28\!\cdots\!18\)\( \nu^{7} + \)\(25\!\cdots\!47\)\( \nu^{6} - \)\(18\!\cdots\!68\)\( \nu^{5} + \)\(59\!\cdots\!34\)\( \nu^{4} - \)\(61\!\cdots\!18\)\( \nu^{3} - \)\(58\!\cdots\!58\)\( \nu^{2} - \)\(33\!\cdots\!47\)\( \nu - \)\(38\!\cdots\!34\)\(\)\()/ \)\(14\!\cdots\!87\)\( \)
\(\beta_{19}\)\(=\)\((\)\(-\)\(14\!\cdots\!30\)\( \nu^{19} + \)\(79\!\cdots\!64\)\( \nu^{18} - \)\(20\!\cdots\!94\)\( \nu^{17} - \)\(17\!\cdots\!82\)\( \nu^{16} + \)\(80\!\cdots\!63\)\( \nu^{15} - \)\(11\!\cdots\!05\)\( \nu^{14} + \)\(32\!\cdots\!96\)\( \nu^{13} - \)\(16\!\cdots\!52\)\( \nu^{12} - \)\(23\!\cdots\!87\)\( \nu^{11} + \)\(96\!\cdots\!28\)\( \nu^{10} - \)\(24\!\cdots\!05\)\( \nu^{9} + \)\(41\!\cdots\!43\)\( \nu^{8} - \)\(50\!\cdots\!87\)\( \nu^{7} + \)\(42\!\cdots\!34\)\( \nu^{6} - \)\(29\!\cdots\!16\)\( \nu^{5} + \)\(56\!\cdots\!77\)\( \nu^{4} - \)\(99\!\cdots\!29\)\( \nu^{3} - \)\(13\!\cdots\!99\)\( \nu^{2} - \)\(77\!\cdots\!12\)\( \nu - \)\(16\!\cdots\!18\)\(\)\()/ \)\(14\!\cdots\!87\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{18} - \beta_{17} + \beta_{16} + \beta_{15} - \beta_{13} + \beta_{12} + \beta_{11} - \beta_{9} - \beta_{8} - \beta_{7} - 2 \beta_{6} - \beta_{5} + 3 \beta_{4} + \beta_{3} - \beta_{2} + \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(3 \beta_{19} + \beta_{18} - 3 \beta_{17} + 6 \beta_{16} + 3 \beta_{15} - 3 \beta_{14} + 5 \beta_{13} + 5 \beta_{10} + 2 \beta_{9} - 3 \beta_{8} - 3 \beta_{7} + \beta_{6} - 2 \beta_{5} - \beta_{4} - 2 \beta_{3} - 5 \beta_{2} - 2\)
\(\nu^{4}\)\(=\)\(18 \beta_{19} - 2 \beta_{18} + 25 \beta_{16} + 12 \beta_{15} - 27 \beta_{14} + 18 \beta_{13} - 15 \beta_{12} - 4 \beta_{11} + 15 \beta_{9} - 4 \beta_{8} - 15 \beta_{7} - 4 \beta_{6} + 13 \beta_{5} - 4 \beta_{3} - 5 \beta_{1} - 19\)
\(\nu^{5}\)\(=\)\(29 \beta_{19} - 79 \beta_{18} + 52 \beta_{17} - 70 \beta_{16} - 27 \beta_{15} - 32 \beta_{14} + 15 \beta_{13} - 62 \beta_{12} + 20 \beta_{11} - 80 \beta_{10} + 79 \beta_{9} + 80 \beta_{8} + 42 \beta_{6} + 39 \beta_{5} - 5 \beta_{4} - 5 \beta_{3} + 52 \beta_{2} - 79 \beta_{1} - 62\)
\(\nu^{6}\)\(=\)\(-56 \beta_{19} - 239 \beta_{18} + 282 \beta_{17} - 369 \beta_{16} - 239 \beta_{15} + 37 \beta_{14} - 210 \beta_{13} - 166 \beta_{12} - 19 \beta_{11} - 356 \beta_{10} + 56 \beta_{9} + 359 \beta_{8} + 221 \beta_{7} + 123 \beta_{6} + 226 \beta_{5} + 60 \beta_{3} + 222 \beta_{2} - 178 \beta_{1} - 89\)
\(\nu^{7}\)\(=\)\(-1081 \beta_{19} - 361 \beta_{18} + 535 \beta_{17} - 1933 \beta_{16} - 1081 \beta_{15} + 1400 \beta_{14} - 1420 \beta_{13} + 361 \beta_{12} + 14 \beta_{11} - 854 \beta_{10} - 535 \beta_{9} + 640 \beta_{8} + 1038 \beta_{7} - 45 \beta_{5} + 105 \beta_{4} + 339 \beta_{3} + 503 \beta_{2} - 45 \beta_{1} + 852\)
\(\nu^{8}\)\(=\)\(-3798 \beta_{19} + 2438 \beta_{18} - 1058 \beta_{17} - 1773 \beta_{16} - 1058 \beta_{15} + 4464 \beta_{14} - 3798 \beta_{13} + 3798 \beta_{12} - 810 \beta_{11} + 1868 \beta_{10} - 4325 \beta_{9} - 1922 \beta_{8} + 2438 \beta_{7} - 1876 \beta_{6} - 2797 \beta_{5} + 413 \beta_{4} + 1221 \beta_{3} - 1001 \beta_{2} + 3267 \beta_{1} + 4464\)
\(\nu^{9}\)\(=\)\(-5135 \beta_{19} + 16473 \beta_{18} - 15916 \beta_{17} + 15193 \beta_{16} + 8789 \beta_{15} + 6870 \beta_{14} + 420 \beta_{13} + 15916 \beta_{12} - 1210 \beta_{11} + 21605 \beta_{10} - 12724 \beta_{9} - 21471 \beta_{8} - 5135 \beta_{7} - 10069 \beta_{6} - 15916 \beta_{5} + 1735 \beta_{4} - 12724 \beta_{2} + 14481 \beta_{1} + 14706\)
\(\nu^{10}\)\(=\)\(36734 \beta_{19} + 50703 \beta_{18} - 59591 \beta_{17} + 118069 \beta_{16} + 66327 \beta_{15} - 47053 \beta_{14} + 67366 \beta_{13} + 17683 \beta_{12} - 4813 \beta_{11} + 87212 \beta_{10} - 77404 \beta_{8} - 59591 \beta_{7} - 19274 \beta_{6} - 33307 \beta_{5} - 4813 \beta_{4} - 17813 \beta_{3} - 50703 \beta_{2} + 33020 \beta_{1} - 9938\)
\(\nu^{11}\)\(=\)\(256857 \beta_{19} - 76135 \beta_{17} + 344581 \beta_{16} + 196934 \beta_{15} - 318507 \beta_{14} + 318507 \beta_{13} - 138918 \beta_{12} + 22073 \beta_{11} + 101338 \beta_{10} + 196934 \beta_{9} - 76135 \beta_{8} - 240976 \beta_{7} + 37850 \beta_{6} + 59923 \beta_{5} - 25203 \beta_{4} - 87724 \beta_{3} - 59923 \beta_{2} - 76135 \beta_{1} - 232313\)
\(\nu^{12}\)\(=\)\(778490 \beta_{19} - 778490 \beta_{18} + 560354 \beta_{17} - 972564 \beta_{14} + 654831 \beta_{13} - 1019617 \beta_{12} + 123659 \beta_{11} - 777936 \beta_{10} + 1019617 \beta_{9} + 830113 \beta_{8} - 282060 \beta_{7} + 560908 \beta_{6} + 842414 \beta_{5} - 133529 \beta_{4} - 194074 \beta_{3} + 459263 \beta_{2} - 863156 \beta_{1} - 1153146\)
\(\nu^{13}\)\(=\)\(-3979257 \beta_{18} + 3979257 \beta_{17} - 5344858 \beta_{16} - 3042920 \beta_{15} - 1672508 \beta_{13} - 3042920 \beta_{12} + 397588 \beta_{11} - 5651765 \beta_{10} + 2164279 \beta_{9} + 5344858 \beta_{8} + 2164279 \beta_{7} + 2111735 \beta_{6} + 3414324 \beta_{5} - 117728 \beta_{4} + 397588 \beta_{3} + 3317425 \beta_{2} - 3317425 \beta_{1} - 2111735\)
\(\nu^{14}\)\(=\)\(-11985926 \beta_{19} - 8599271 \beta_{18} + 11985926 \beta_{17} - 27688350 \beta_{16} - 15700939 \beta_{15} + 14954050 \beta_{14} - 18419890 \beta_{13} - 16918620 \beta_{10} - 4413673 \beta_{9} + 14954050 \beta_{8} + 15700939 \beta_{7} + 2687957 \beta_{6} + 4413673 \beta_{5} + 1027056 \beta_{4} + 4932694 \beta_{3} + 9944941 \beta_{2} - 3715013 \beta_{1} + 6433964\)
\(\nu^{15}\)\(=\)\(-61492387 \beta_{19} + 17642007 \beta_{18} - 58907654 \beta_{16} - 33525393 \beta_{15} + 76549661 \beta_{14} - 69535225 \beta_{13} + 46978173 \beta_{12} - 6084356 \beta_{11} - 56720847 \beta_{9} - 6084356 \beta_{8} + 46978173 \beta_{7} - 19834963 \beta_{6} - 29336166 \beta_{5} + 8042838 \beta_{4} + 19558179 \beta_{3} + 32156221 \beta_{1} + 66536352\)
\(\nu^{16}\)\(=\)\(-132340428 \beta_{19} + 221906105 \beta_{18} - 184749586 \beta_{17} + 120271015 \beta_{16} + 68506823 \beta_{15} + 165278556 \beta_{14} - 72491788 \beta_{13} + 242001342 \beta_{12} - 30705338 \beta_{11} + 261063925 \beta_{10} - 221906105 \beta_{9} - 261063925 \beta_{8} - 140998611 \beta_{6} - 220942488 \beta_{5} + 23474981 \beta_{4} + 23474981 \beta_{3} - 153399282 \beta_{2} + 221906105 \beta_{1} + 242001342\)
\(\nu^{17}\)\(=\)\(271144644 \beta_{19} + 873784229 \beta_{18} - 949293604 \beta_{17} + 1539365880 \beta_{16} + 873784229 \beta_{15} - 338975821 \beta_{14} + 698859189 \beta_{13} + 518087766 \beta_{12} - 67831177 \beta_{11} + 1343730611 \beta_{10} - 271144644 \beta_{9} - 1250285953 \beta_{8} - 724971372 \beta_{7} - 424471848 \beta_{6} - 678148960 \beta_{5} - 178167737 \beta_{3} - 789232410 \beta_{2} + 649461997 \beta_{1} + 217095417\)
\(\nu^{18}\)\(=\)\(3426020339 \beta_{19} + 1059630164 \beta_{18} - 2039347103 \beta_{17} + 6028541248 \beta_{16} + 3426020339 \beta_{15} - 4266560602 \beta_{14} + 4710719625 \beta_{13} - 1059630164 \beta_{12} + 140416675 \beta_{11} + 2879887366 \beta_{10} + 2039347103 \beta_{9} - 2403439748 \beta_{8} - 3731718622 \beta_{7} - 97039871 \beta_{5} - 364092645 \beta_{4} - 1284699286 \beta_{3} - 1692371519 \beta_{2} - 97039871 \beta_{1} - 2602520909\)
\(\nu^{19}\)\(=\)\(13471243579 \beta_{19} - 7998279224 \beta_{18} + 4175877645 \beta_{17} + 7359415616 \beta_{16} + 4175877645 \beta_{15} - 16796005904 \beta_{14} + 13471243579 \beta_{13} - 13471243579 \beta_{12} + 1716522174 \beta_{11} - 5892399819 \beta_{10} + 14648648527 \beta_{9} + 6927879328 \beta_{8} - 7998279224 \beta_{7} + 6543364251 \beta_{6} + 10008173667 \beta_{5} - 1886451261 \beta_{4} - 3850103282 \beta_{3} + 3463069912 \beta_{2} - 10472770882 \beta_{1} - 16796005904\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

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

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
0.588453 1.28853i
−0.905632 + 1.98306i
2.57205 + 2.96831i
−0.271308 0.313106i
−0.138474 + 0.963109i
0.212636 1.47891i
−0.138474 0.963109i
0.212636 + 1.47891i
2.57205 2.96831i
−0.271308 + 0.313106i
−2.49760 + 0.733362i
1.92732 0.565914i
−0.319826 0.205540i
1.33237 + 0.856263i
0.588453 + 1.28853i
−0.905632 1.98306i
−0.319826 + 0.205540i
1.33237 0.856263i
−2.49760 0.733362i
1.92732 + 0.565914i
0.142315 0.989821i −1.10832 2.42687i −0.959493 0.281733i −0.654861 0.755750i −2.55990 + 0.751655i −0.986597 0.634047i −0.415415 + 0.909632i −2.69677 + 3.11224i −0.841254 + 0.540641i
31.2 0.142315 0.989821i 1.40549 + 3.07760i −0.959493 0.281733i −0.654861 0.755750i 3.24630 0.953198i 2.97617 + 1.91267i −0.415415 + 0.909632i −5.53162 + 6.38383i −0.841254 + 0.540641i
41.1 0.959493 + 0.281733i −0.880272 + 1.01589i 0.841254 + 0.540641i −0.142315 + 0.989821i −1.13082 + 0.726736i 0.394126 + 0.863015i 0.654861 + 0.755750i 0.169795 + 1.18095i −0.415415 + 0.909632i
41.2 0.959493 + 0.281733i 1.48208 1.71041i 0.841254 + 0.540641i −0.142315 + 0.989821i 1.90392 1.22358i −0.119262 0.261148i 0.654861 + 0.755750i −0.302001 2.10046i −0.415415 + 0.909632i
71.1 −0.841254 0.540641i −0.420808 2.92679i 0.415415 + 0.909632i −0.959493 0.281733i −1.22833 + 2.68968i 2.73530 3.15671i 0.142315 0.989821i −5.51052 + 1.61803i 0.654861 + 0.755750i
71.2 −0.841254 0.540641i 0.0390478 + 0.271583i 0.415415 + 0.909632i −0.959493 0.281733i 0.113980 0.249581i −0.365070 + 0.421313i 0.142315 0.989821i 2.80625 0.823988i 0.654861 + 0.755750i
81.1 −0.841254 + 0.540641i −0.420808 + 2.92679i 0.415415 0.909632i −0.959493 + 0.281733i −1.22833 2.68968i 2.73530 + 3.15671i 0.142315 + 0.989821i −5.51052 1.61803i 0.654861 0.755750i
81.2 −0.841254 + 0.540641i 0.0390478 0.271583i 0.415415 0.909632i −0.959493 + 0.281733i 0.113980 + 0.249581i −0.365070 0.421313i 0.142315 + 0.989821i 2.80625 + 0.823988i 0.654861 0.755750i
101.1 0.959493 0.281733i −0.880272 1.01589i 0.841254 0.540641i −0.142315 0.989821i −1.13082 0.726736i 0.394126 0.863015i 0.654861 0.755750i 0.169795 1.18095i −0.415415 0.909632i
101.2 0.959493 0.281733i 1.48208 + 1.71041i 0.841254 0.540641i −0.142315 0.989821i 1.90392 + 1.22358i −0.119262 + 0.261148i 0.654861 0.755750i −0.302001 + 2.10046i −0.415415 0.909632i
121.1 −0.415415 + 0.909632i −1.50807 0.442808i −0.654861 0.755750i 0.841254 0.540641i 1.02927 1.18784i −0.440465 + 3.06350i 0.959493 0.281733i −0.445574 0.286353i 0.142315 + 0.989821i
121.2 −0.415415 + 0.909632i −0.248602 0.0729960i −0.654861 0.755750i 0.841254 0.540641i 0.169672 0.195812i 0.663794 4.61679i 0.959493 0.281733i −2.46729 1.58563i 0.142315 + 0.989821i
131.1 0.654861 0.755750i −1.71555 + 1.10252i −0.142315 0.989821i 0.415415 + 0.909632i −0.290219 + 2.01852i 3.10381 0.911362i −0.841254 0.540641i 0.481321 1.05395i 0.959493 + 0.281733i
131.2 0.654861 0.755750i 1.45499 0.935068i −0.142315 0.989821i 0.415415 + 0.909632i 0.246141 1.71195i 1.53819 0.451652i −0.841254 0.540641i −0.00358844 + 0.00785758i 0.959493 + 0.281733i
141.1 0.142315 + 0.989821i −1.10832 + 2.42687i −0.959493 + 0.281733i −0.654861 + 0.755750i −2.55990 0.751655i −0.986597 + 0.634047i −0.415415 0.909632i −2.69677 3.11224i −0.841254 0.540641i
141.2 0.142315 + 0.989821i 1.40549 3.07760i −0.959493 + 0.281733i −0.654861 + 0.755750i 3.24630 + 0.953198i 2.97617 1.91267i −0.415415 0.909632i −5.53162 6.38383i −0.841254 0.540641i
151.1 0.654861 + 0.755750i −1.71555 1.10252i −0.142315 + 0.989821i 0.415415 0.909632i −0.290219 2.01852i 3.10381 + 0.911362i −0.841254 + 0.540641i 0.481321 + 1.05395i 0.959493 0.281733i
151.2 0.654861 + 0.755750i 1.45499 + 0.935068i −0.142315 + 0.989821i 0.415415 0.909632i 0.246141 + 1.71195i 1.53819 + 0.451652i −0.841254 + 0.540641i −0.00358844 0.00785758i 0.959493 0.281733i
211.1 −0.415415 0.909632i −1.50807 + 0.442808i −0.654861 + 0.755750i 0.841254 + 0.540641i 1.02927 + 1.18784i −0.440465 3.06350i 0.959493 + 0.281733i −0.445574 + 0.286353i 0.142315 0.989821i
211.2 −0.415415 0.909632i −0.248602 + 0.0729960i −0.654861 + 0.755750i 0.841254 + 0.540641i 0.169672 + 0.195812i 0.663794 + 4.61679i 0.959493 + 0.281733i −2.46729 + 1.58563i 0.142315 0.989821i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 211.2
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 230.2.g.b 20
23.c even 11 1 inner 230.2.g.b 20
23.c even 11 1 5290.2.a.bj 10
23.d odd 22 1 5290.2.a.bi 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.g.b 20 1.a even 1 1 trivial
230.2.g.b 20 23.c even 11 1 inner
5290.2.a.bi 10 23.d odd 22 1
5290.2.a.bj 10 23.c even 11 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{20} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(230, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
$3$ \( 1024 + 8192 T + 32768 T^{2} + 126720 T^{3} + 366464 T^{4} + 431808 T^{5} + 252880 T^{6} + 67672 T^{7} + 31164 T^{8} + 46692 T^{9} + 34605 T^{10} + 15634 T^{11} + 7173 T^{12} + 3432 T^{13} + 1695 T^{14} + 701 T^{15} + 252 T^{16} + 66 T^{17} + 21 T^{18} + 3 T^{19} + T^{20} \)
$5$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
$7$ \( 38809 + 126277 T + 536876 T^{2} + 148808 T^{3} + 209230 T^{4} - 785936 T^{5} + 1709082 T^{6} - 1623479 T^{7} + 183019 T^{8} + 188999 T^{9} + 477641 T^{10} - 734599 T^{11} + 526637 T^{12} - 263428 T^{13} + 102312 T^{14} - 31386 T^{15} + 7556 T^{16} - 1408 T^{17} + 197 T^{18} - 19 T^{19} + T^{20} \)
$11$ \( 2374681 - 11682321 T + 20709116 T^{2} - 306162 T^{3} - 784055 T^{4} + 11498674 T^{5} + 1481982 T^{6} + 2003537 T^{7} + 4030360 T^{8} + 453856 T^{9} + 361008 T^{10} + 129010 T^{11} + 29129 T^{12} + 20470 T^{13} + 1922 T^{14} + 198 T^{15} + 334 T^{16} - 96 T^{17} + 40 T^{18} - 7 T^{19} + T^{20} \)
$13$ \( 13416820561 - 15019921601 T + 11207619488 T^{2} - 3736702926 T^{3} + 884371465 T^{4} - 10527833 T^{5} + 249165720 T^{6} - 99344817 T^{7} + 68348193 T^{8} - 6231580 T^{9} + 4091724 T^{10} - 160564 T^{11} + 321003 T^{12} - 41877 T^{13} + 5567 T^{14} + 3495 T^{15} - 793 T^{16} + 22 T^{17} + 21 T^{18} - 8 T^{19} + T^{20} \)
$17$ \( 605553664 - 420501504 T + 1364533248 T^{2} - 2719784832 T^{3} + 3327405824 T^{4} - 2179861856 T^{5} + 1486084128 T^{6} - 351604200 T^{7} + 113158108 T^{8} - 51645542 T^{9} + 8319607 T^{10} - 2028395 T^{11} + 930412 T^{12} - 108685 T^{13} + 10664 T^{14} - 7231 T^{15} + 1132 T^{16} - 53 T^{17} + 25 T^{18} - 8 T^{19} + T^{20} \)
$19$ \( 392951329 + 2509849499 T + 12605613026 T^{2} + 27998138276 T^{3} + 39337595636 T^{4} + 31572147805 T^{5} + 20835816469 T^{6} + 11554668737 T^{7} + 3581241073 T^{8} + 482090916 T^{9} + 138639280 T^{10} + 39666195 T^{11} + 4305053 T^{12} + 360736 T^{13} + 158468 T^{14} + 19547 T^{15} + 753 T^{16} + 191 T^{17} + 8 T^{18} + T^{19} + T^{20} \)
$23$ \( 41426511213649 + 16210373953167 T + 3680616308207 T^{2} - 388150100958 T^{3} - 280972117322 T^{4} - 52642849397 T^{5} + 6884928123 T^{6} + 4150723382 T^{7} + 564954014 T^{8} - 91002812 T^{9} - 42379117 T^{10} - 3956644 T^{11} + 1067966 T^{12} + 341146 T^{13} + 24603 T^{14} - 8179 T^{15} - 1898 T^{16} - 114 T^{17} + 47 T^{18} + 9 T^{19} + T^{20} \)
$29$ \( 3969466491904 - 302040563200 T + 2396242269184 T^{2} - 327084014720 T^{3} + 302579385280 T^{4} + 59628029248 T^{5} - 9797783200 T^{6} - 3147021736 T^{7} + 640894788 T^{8} + 8566556 T^{9} - 34275803 T^{10} + 5783813 T^{11} + 2761985 T^{12} - 715472 T^{13} + 82592 T^{14} + 20783 T^{15} - 939 T^{16} - 538 T^{17} + 181 T^{18} - 20 T^{19} + T^{20} \)
$31$ \( 1024 + 50176 T + 1254400 T^{2} + 14122496 T^{3} + 72118912 T^{4} + 152469184 T^{5} + 169714240 T^{6} + 109002424 T^{7} + 71814772 T^{8} + 36799614 T^{9} + 16566087 T^{10} + 6369919 T^{11} + 2241675 T^{12} + 661548 T^{13} + 160647 T^{14} + 36895 T^{15} + 7455 T^{16} + 1119 T^{17} + 158 T^{18} + 17 T^{19} + T^{20} \)
$37$ \( 7650363233041 - 12842446216894 T + 6056001856018 T^{2} + 3097167307726 T^{3} + 806596658179 T^{4} - 833717374188 T^{5} + 748643051448 T^{6} - 292019413121 T^{7} + 104077363944 T^{8} - 24098340307 T^{9} + 4757994681 T^{10} - 577835551 T^{11} + 48134142 T^{12} + 1584269 T^{13} - 495914 T^{14} + 22258 T^{15} - 2455 T^{16} + 90 T^{17} + 38 T^{18} - 6 T^{19} + T^{20} \)
$41$ \( 1804635361 - 100073213877 T + 1611592784236 T^{2} - 3463013179703 T^{3} + 2696794264161 T^{4} - 1366581573393 T^{5} + 1032877827084 T^{6} - 196318849793 T^{7} + 172162095906 T^{8} - 24087657794 T^{9} + 4958829446 T^{10} - 351700185 T^{11} + 84127686 T^{12} - 4945292 T^{13} + 1071909 T^{14} - 46208 T^{15} + 10314 T^{16} + 66 T^{17} + 46 T^{18} + 2 T^{19} + T^{20} \)
$43$ \( 23760372736 - 1476600867840 T + 31969782646784 T^{2} + 33438861109760 T^{3} + 19127551711232 T^{4} + 7310999117440 T^{5} + 1999354052848 T^{6} + 395012274456 T^{7} + 59271492876 T^{8} + 8658355072 T^{9} + 1678266501 T^{10} + 283531984 T^{11} + 16159843 T^{12} - 5680715 T^{13} - 1305917 T^{14} - 59044 T^{15} + 18614 T^{16} + 2962 T^{17} + 372 T^{18} + 18 T^{19} + T^{20} \)
$47$ \( ( 130098649 - 9202654 T - 38102297 T^{2} + 8039739 T^{3} + 1243129 T^{4} - 360930 T^{5} - 5893 T^{6} + 5031 T^{7} - 141 T^{8} - 21 T^{9} + T^{10} )^{2} \)
$53$ \( 7447441 + 107495310 T + 889965334 T^{2} + 1568453218 T^{3} + 2601865130 T^{4} + 1042708470 T^{5} + 638295465 T^{6} + 365550580 T^{7} + 371653127 T^{8} + 79460323 T^{9} + 71096442 T^{10} - 14948702 T^{11} + 9849637 T^{12} - 3673419 T^{13} + 446593 T^{14} - 23090 T^{15} + 1161 T^{16} + 177 T^{17} + 78 T^{18} - 19 T^{19} + T^{20} \)
$59$ \( 27292338778849 - 81912483477870 T + 86999990331053 T^{2} - 60214786711490 T^{3} + 64232139968705 T^{4} - 40351308357437 T^{5} + 18636721120910 T^{6} - 3569172259347 T^{7} + 811409899206 T^{8} - 52989069284 T^{9} + 9535673201 T^{10} - 24764761 T^{11} + 119099819 T^{12} - 7945054 T^{13} - 268482 T^{14} + 186219 T^{15} - 12520 T^{16} - 42 T^{17} + 248 T^{18} - 25 T^{19} + T^{20} \)
$61$ \( 2253405457122304 + 2831513170643968 T + 711971601406464 T^{2} - 714687263754752 T^{3} + 132174583068736 T^{4} + 489260564075584 T^{5} + 355052381901728 T^{6} + 142069040515616 T^{7} + 36918826551276 T^{8} + 6956855386674 T^{9} + 1024770399861 T^{10} + 121990095186 T^{11} + 12316750619 T^{12} + 1093149301 T^{13} + 89829004 T^{14} + 6898492 T^{15} + 483951 T^{16} + 29167 T^{17} + 1398 T^{18} + 49 T^{19} + T^{20} \)
$67$ \( 246200702534656 + 230566656409600 T + 121960287006464 T^{2} + 55220985758336 T^{3} + 12794232183808 T^{4} + 1029313640800 T^{5} + 645433250736 T^{6} - 142353555944 T^{7} + 7055154712 T^{8} - 2723792848 T^{9} + 888595697 T^{10} - 122588815 T^{11} + 16105247 T^{12} - 2267437 T^{13} + 598659 T^{14} - 78711 T^{15} + 7583 T^{16} - 949 T^{17} + 191 T^{18} - 19 T^{19} + T^{20} \)
$71$ \( 95950331339776 - 235912521435648 T + 305646800542720 T^{2} - 195664758584448 T^{3} + 61518961753472 T^{4} + 5406760978080 T^{5} - 6291358868128 T^{6} + 1150277485992 T^{7} + 108646647476 T^{8} - 44217346056 T^{9} + 23488118125 T^{10} - 2678112963 T^{11} + 623227474 T^{12} - 27979337 T^{13} + 4307201 T^{14} + 160391 T^{15} - 6783 T^{16} + 2434 T^{17} - 59 T^{18} - 7 T^{19} + T^{20} \)
$73$ \( 859836921496576 + 1264966442085888 T + 1060412615909120 T^{2} + 631480429812992 T^{3} + 267756039870336 T^{4} + 80352207406976 T^{5} + 17048189939312 T^{6} + 2154345152144 T^{7} + 17428821680 T^{8} - 48354946056 T^{9} - 6573931871 T^{10} - 404959833 T^{11} + 107446426 T^{12} + 9026709 T^{13} + 3160448 T^{14} + 513305 T^{15} + 53412 T^{16} + 1707 T^{17} + 71 T^{18} + 10 T^{19} + T^{20} \)
$79$ \( 9084561155797451776 + 9048933320140178432 T + 4123486068743251968 T^{2} + 1029251469152841088 T^{3} + 206468751655754688 T^{4} + 27535026774016960 T^{5} + 2792053848491104 T^{6} + 298072826139000 T^{7} + 37799186854408 T^{8} + 5479113244880 T^{9} + 722881717513 T^{10} + 75572585297 T^{11} + 6909701888 T^{12} + 583534919 T^{13} + 47337058 T^{14} + 3483205 T^{15} + 215205 T^{16} + 12067 T^{17} + 681 T^{18} + 33 T^{19} + T^{20} \)
$83$ \( 160435495936 + 3769747092992 T + 22560840909568 T^{2} - 19479408779008 T^{3} + 27078397206912 T^{4} + 1577595335392 T^{5} - 151256362384 T^{6} + 3751405393088 T^{7} + 1960385419244 T^{8} + 506152903952 T^{9} + 108292497741 T^{10} + 20875934628 T^{11} + 3404360471 T^{12} + 447518876 T^{13} + 49994883 T^{14} + 4661273 T^{15} + 360322 T^{16} + 21790 T^{17} + 1079 T^{18} + 41 T^{19} + T^{20} \)
$89$ \( 20576368474290409 - 71303777523356521 T + 118183040024174686 T^{2} - 113275280496174874 T^{3} + 67664515869731539 T^{4} - 24699316511966283 T^{5} + 6002779451496267 T^{6} - 961489031275762 T^{7} + 122090262567603 T^{8} - 12644852336996 T^{9} + 1124589940419 T^{10} - 98846115687 T^{11} + 9667986757 T^{12} - 1082173543 T^{13} + 115282266 T^{14} - 9933088 T^{15} + 696927 T^{16} - 39523 T^{17} + 1727 T^{18} - 55 T^{19} + T^{20} \)
$97$ \( 286047133533184 + 275423649894400 T + 546194756251136 T^{2} + 609517097500032 T^{3} + 331410155455488 T^{4} + 61921415767520 T^{5} + 3380272671840 T^{6} - 663390232312 T^{7} - 296781093420 T^{8} + 18266228114 T^{9} + 14740230493 T^{10} - 99975454 T^{11} + 237558340 T^{12} - 102260319 T^{13} + 9113908 T^{14} - 287783 T^{15} + 35114 T^{16} - 3075 T^{17} + 207 T^{18} + 9 T^{19} + T^{20} \)
show more
show less