Properties

Label 230.3.f.a
Level $230$
Weight $3$
Character orbit 230.f
Analytic conductor $6.267$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 230.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.26704608029\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 52 x^{17} + 1020 x^{16} - 1316 x^{15} + 1352 x^{14} - 18724 x^{13} + 250686 x^{12} - 439644 x^{11} + 460536 x^{10} - 1833716 x^{9} + 16970128 x^{8} - 36894604 x^{7} + 40544632 x^{6} - 4380588 x^{5} + 34721 x^{4} - 4419956 x^{3} + 9945800 x^{2} + 1329080 x + 88804\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{8} ) q^{2} + \beta_{1} q^{3} + 2 \beta_{8} q^{4} -\beta_{7} q^{5} + ( -\beta_{1} + \beta_{2} ) q^{6} -\beta_{12} q^{7} + ( 2 - 2 \beta_{8} ) q^{8} + ( -\beta_{1} - \beta_{2} - \beta_{8} - \beta_{10} + \beta_{13} + \beta_{14} ) q^{9} +O(q^{10})\) \( q + ( -1 - \beta_{8} ) q^{2} + \beta_{1} q^{3} + 2 \beta_{8} q^{4} -\beta_{7} q^{5} + ( -\beta_{1} + \beta_{2} ) q^{6} -\beta_{12} q^{7} + ( 2 - 2 \beta_{8} ) q^{8} + ( -\beta_{1} - \beta_{2} - \beta_{8} - \beta_{10} + \beta_{13} + \beta_{14} ) q^{9} + ( \beta_{7} - \beta_{15} ) q^{10} + ( 4 - \beta_{6} + \beta_{9} - \beta_{12} + \beta_{15} - \beta_{16} + \beta_{18} + \beta_{19} ) q^{11} -2 \beta_{2} q^{12} + ( 1 + \beta_{1} + \beta_{2} - \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{8} + 2 \beta_{10} + \beta_{12} - \beta_{13} + \beta_{14} - \beta_{17} - \beta_{19} ) q^{13} + ( \beta_{9} + \beta_{12} ) q^{14} + ( -2 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{6} + 2 \beta_{8} + \beta_{9} + \beta_{10} - \beta_{13} - \beta_{14} + \beta_{16} + \beta_{19} ) q^{15} -4 q^{16} + ( -2 + \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{8} + \beta_{9} - \beta_{10} - 2 \beta_{11} + \beta_{13} - \beta_{14} + \beta_{18} + 2 \beta_{19} ) q^{17} + ( -1 + 2 \beta_{1} - \beta_{4} + \beta_{8} + 2 \beta_{10} - \beta_{13} ) q^{18} + ( 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{8} - 2 \beta_{13} - \beta_{17} - \beta_{18} + \beta_{19} ) q^{19} + 2 \beta_{15} q^{20} + ( 2 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{6} - \beta_{7} + \beta_{9} - \beta_{12} + 2 \beta_{15} - 2 \beta_{16} + \beta_{18} + \beta_{19} ) q^{21} + ( -4 - \beta_{3} + \beta_{6} - \beta_{7} - 4 \beta_{8} + 2 \beta_{12} - \beta_{15} + \beta_{16} - \beta_{17} - 2 \beta_{19} ) q^{22} -\beta_{10} q^{23} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{24} + ( 5 - 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{10} - \beta_{11} + \beta_{12} - \beta_{14} + \beta_{15} - 2 \beta_{16} - \beta_{17} + \beta_{18} ) q^{25} + ( -2 - 2 \beta_{1} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{9} - \beta_{10} + 2 \beta_{11} - \beta_{12} - 3 \beta_{14} - \beta_{18} + \beta_{19} ) q^{26} + ( 7 - \beta_{1} + \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + \beta_{6} + 7 \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{11} - 3 \beta_{13} - 4 \beta_{14} + 2 \beta_{15} - \beta_{17} - \beta_{18} + 2 \beta_{19} ) q^{27} -2 \beta_{9} q^{28} + ( \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{5} - \beta_{7} - 11 \beta_{8} + \beta_{10} + \beta_{11} + \beta_{12} - 4 \beta_{13} - \beta_{15} - \beta_{16} - 2 \beta_{17} - 2 \beta_{18} + \beta_{19} ) q^{29} + ( 4 + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{6} - \beta_{9} - 2 \beta_{10} + \beta_{12} - \beta_{16} - \beta_{17} + \beta_{18} - \beta_{19} ) q^{30} + ( -13 - 2 \beta_{2} + 4 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} - 4 \beta_{14} + 2 \beta_{19} ) q^{31} + ( 4 + 4 \beta_{8} ) q^{32} + ( -6 + 7 \beta_{1} + \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} + 6 \beta_{8} + 5 \beta_{9} + 9 \beta_{10} + \beta_{12} - 3 \beta_{13} + \beta_{14} - 2 \beta_{15} + \beta_{16} - 2 \beta_{17} - \beta_{19} ) q^{33} + ( 2 \beta_{2} + 2 \beta_{5} + 4 \beta_{8} - \beta_{9} + 2 \beta_{11} + \beta_{12} - 2 \beta_{13} + 2 \beta_{14} + \beta_{18} - 3 \beta_{19} ) q^{34} + ( -2 + 4 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{9} - 2 \beta_{12} + \beta_{13} - 3 \beta_{14} - \beta_{16} - \beta_{17} + \beta_{18} + 3 \beta_{19} ) q^{35} + ( 2 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{10} - 2 \beta_{14} ) q^{36} + ( 4 + 4 \beta_{2} - 2 \beta_{3} - \beta_{6} + 4 \beta_{8} + \beta_{12} - 2 \beta_{14} - 2 \beta_{15} + \beta_{17} ) q^{37} + ( 4 - 4 \beta_{1} + 2 \beta_{4} + \beta_{6} - 4 \beta_{8} + 2 \beta_{13} + \beta_{17} + 2 \beta_{18} ) q^{38} + ( \beta_{2} - \beta_{3} + \beta_{5} - \beta_{7} - 14 \beta_{8} + \beta_{9} - 3 \beta_{10} + \beta_{11} + 2 \beta_{12} - \beta_{13} + 4 \beta_{14} + \beta_{15} + \beta_{16} - 4 \beta_{17} + \beta_{18} - 2 \beta_{19} ) q^{39} + ( -2 \beta_{7} - 2 \beta_{15} ) q^{40} + ( 5 + \beta_{1} + \beta_{2} - 2 \beta_{3} - 6 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} + 3 \beta_{9} - \beta_{10} - 2 \beta_{11} - \beta_{12} + \beta_{14} + 3 \beta_{18} + \beta_{19} ) q^{41} + ( -2 + 4 \beta_{2} - 3 \beta_{3} + \beta_{6} - \beta_{7} - 2 \beta_{8} + 2 \beta_{12} - 3 \beta_{15} + \beta_{16} - \beta_{17} - 2 \beta_{19} ) q^{42} + ( 8 - \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - 8 \beta_{8} - \beta_{9} + 5 \beta_{10} + \beta_{12} - \beta_{13} + \beta_{14} + 2 \beta_{15} - 2 \beta_{16} + 2 \beta_{17} - \beta_{19} ) q^{43} + ( 2 \beta_{3} + 2 \beta_{7} + 8 \beta_{8} - 2 \beta_{9} - 2 \beta_{12} + 2 \beta_{17} - 2 \beta_{18} + 2 \beta_{19} ) q^{44} + ( -2 \beta_{1} - 6 \beta_{2} - \beta_{3} - 4 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - \beta_{7} - 6 \beta_{8} + 3 \beta_{9} - 2 \beta_{11} + 4 \beta_{13} + 6 \beta_{14} + 2 \beta_{16} + 4 \beta_{18} - 4 \beta_{19} ) q^{45} + ( \beta_{10} + \beta_{14} ) q^{46} + ( 2 + 4 \beta_{3} + \beta_{4} + \beta_{6} + 2 \beta_{8} + \beta_{12} - \beta_{13} - 3 \beta_{14} + 4 \beta_{15} - \beta_{17} + 3 \beta_{19} ) q^{47} -4 \beta_{1} q^{48} + ( -3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + \beta_{5} + 3 \beta_{7} + 4 \beta_{8} - 3 \beta_{9} - 7 \beta_{10} + \beta_{11} - 2 \beta_{12} + \beta_{13} + 8 \beta_{14} + 3 \beta_{15} + 3 \beta_{16} + 2 \beta_{17} + \beta_{18} - 2 \beta_{19} ) q^{49} + ( -5 + \beta_{1} - 3 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - 5 \beta_{8} - \beta_{9} + \beta_{11} - \beta_{12} - 2 \beta_{13} + 2 \beta_{14} - \beta_{15} + 3 \beta_{16} + \beta_{17} - \beta_{18} - \beta_{19} ) q^{50} + ( 6 - 2 \beta_{1} + 4 \beta_{2} - \beta_{3} - 6 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} + \beta_{7} + 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{15} + 2 \beta_{16} + \beta_{18} - \beta_{19} ) q^{51} + ( 2 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{6} + 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - 4 \beta_{11} + 2 \beta_{13} + 4 \beta_{14} + 2 \beta_{17} + 2 \beta_{18} ) q^{52} + ( -16 + 7 \beta_{1} - \beta_{2} + 4 \beta_{3} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} + 16 \beta_{8} + 2 \beta_{9} + 5 \beta_{10} - \beta_{12} + \beta_{13} - \beta_{14} - 4 \beta_{15} - \beta_{16} - 2 \beta_{17} - 2 \beta_{18} + \beta_{19} ) q^{53} + ( -2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{7} - 14 \beta_{8} + \beta_{9} - 5 \beta_{10} - 2 \beta_{11} - \beta_{12} + 6 \beta_{13} + 3 \beta_{14} - 2 \beta_{15} - 2 \beta_{16} + 2 \beta_{17} + 3 \beta_{18} - \beta_{19} ) q^{54} + ( 1 - 4 \beta_{1} - 3 \beta_{2} + 6 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} - 11 \beta_{8} + \beta_{9} - 2 \beta_{10} + 3 \beta_{12} + 3 \beta_{13} - 9 \beta_{14} - 3 \beta_{16} + \beta_{17} ) q^{55} + ( 2 \beta_{9} - 2 \beta_{12} ) q^{56} + ( -18 - \beta_{1} - 11 \beta_{2} - 2 \beta_{3} - 5 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} - 18 \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{11} - 3 \beta_{12} + 5 \beta_{13} + 7 \beta_{14} - 2 \beta_{15} - 2 \beta_{16} + 2 \beta_{17} - \beta_{18} - 2 \beta_{19} ) q^{57} + ( -11 - 3 \beta_{1} - \beta_{2} + 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 11 \beta_{8} - \beta_{9} - \beta_{10} - \beta_{12} + 4 \beta_{13} - \beta_{14} + 2 \beta_{16} + 2 \beta_{17} + 3 \beta_{18} + \beta_{19} ) q^{58} + ( 3 \beta_{1} + \beta_{2} - 2 \beta_{5} + 21 \beta_{8} - 2 \beta_{9} - 5 \beta_{10} - 2 \beta_{11} - 4 \beta_{12} + 3 \beta_{13} + 3 \beta_{14} - 2 \beta_{15} - 2 \beta_{16} + 4 \beta_{17} - 2 \beta_{18} + 4 \beta_{19} ) q^{59} + ( -4 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{8} + 2 \beta_{10} - 2 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} + 2 \beta_{17} - 2 \beta_{18} ) q^{60} + ( 10 - 2 \beta_{1} + 5 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} + \beta_{6} - 5 \beta_{7} - 4 \beta_{9} + 2 \beta_{11} + 2 \beta_{12} - 2 \beta_{14} - \beta_{15} + \beta_{16} - 5 \beta_{18} - 3 \beta_{19} ) q^{61} + ( 13 + 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{4} - 4 \beta_{6} + 13 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - 4 \beta_{11} - 2 \beta_{12} + 4 \beta_{13} + 6 \beta_{14} + 4 \beta_{17} + 2 \beta_{18} - 2 \beta_{19} ) q^{62} + ( 10 + \beta_{1} + \beta_{2} + \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 10 \beta_{8} - 2 \beta_{9} + 9 \beta_{10} + \beta_{12} - 3 \beta_{13} + \beta_{14} - \beta_{15} + 2 \beta_{16} - 2 \beta_{17} - 2 \beta_{18} - \beta_{19} ) q^{63} -8 \beta_{8} q^{64} + ( 2 - 4 \beta_{4} - 6 \beta_{6} + \beta_{7} + 2 \beta_{8} + 3 \beta_{9} + 4 \beta_{10} - 6 \beta_{11} - 4 \beta_{12} - \beta_{15} + 3 \beta_{16} - 2 \beta_{18} + 2 \beta_{19} ) q^{65} + ( 12 - 8 \beta_{1} + 6 \beta_{2} - \beta_{3} + 6 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} + \beta_{7} - 6 \beta_{9} - 8 \beta_{10} + 2 \beta_{11} + 4 \beta_{12} - 10 \beta_{14} + 3 \beta_{15} - 3 \beta_{16} - \beta_{18} + \beta_{19} ) q^{66} + ( 8 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} + \beta_{6} + \beta_{12} - 4 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} - \beta_{17} - 2 \beta_{19} ) q^{67} + ( 4 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 4 \beta_{5} - 4 \beta_{8} + 2 \beta_{10} - 2 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} - 4 \beta_{18} + 2 \beta_{19} ) q^{68} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{5} + \beta_{7} - \beta_{9} + \beta_{10} + \beta_{11} - \beta_{13} - \beta_{18} ) q^{69} + ( 1 - \beta_{1} + 7 \beta_{2} + \beta_{3} - 4 \beta_{4} + 3 \beta_{5} - \beta_{6} + \beta_{7} + 3 \beta_{8} - 3 \beta_{10} - 3 \beta_{11} + 4 \beta_{12} + 2 \beta_{13} + 3 \beta_{14} - \beta_{15} + 3 \beta_{16} + 3 \beta_{17} + 2 \beta_{18} - 4 \beta_{19} ) q^{70} + ( 1 - 3 \beta_{1} + 4 \beta_{2} + \beta_{3} - 6 \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} - 4 \beta_{9} - \beta_{10} - \beta_{11} + 5 \beta_{12} + 5 \beta_{15} - 5 \beta_{16} - 4 \beta_{18} - 5 \beta_{19} ) q^{71} + ( -2 - 4 \beta_{2} - 2 \beta_{4} - 2 \beta_{8} + 2 \beta_{13} + 4 \beta_{14} ) q^{72} + ( 2 - 12 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} + 10 \beta_{4} - 6 \beta_{5} + 8 \beta_{6} + 4 \beta_{7} - 2 \beta_{8} - 7 \beta_{9} - 5 \beta_{10} - 3 \beta_{12} + 10 \beta_{13} - 3 \beta_{14} + 4 \beta_{15} + 4 \beta_{16} + 8 \beta_{17} - 3 \beta_{18} + 3 \beta_{19} ) q^{73} + ( -4 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{7} - 8 \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{12} + 2 \beta_{14} + 2 \beta_{15} + 2 \beta_{16} - 2 \beta_{17} ) q^{74} + ( -5 - 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + \beta_{4} + 2 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} + 37 \beta_{8} - 5 \beta_{9} + 4 \beta_{10} + \beta_{11} + 4 \beta_{12} - 3 \beta_{13} - 8 \beta_{14} + \beta_{15} - 2 \beta_{16} + 3 \beta_{17} - 4 \beta_{18} - \beta_{19} ) q^{75} + ( -8 + 4 \beta_{1} - 4 \beta_{2} - 4 \beta_{4} - 2 \beta_{6} - 2 \beta_{18} - 2 \beta_{19} ) q^{76} + ( 13 - 3 \beta_{1} - 17 \beta_{2} + 6 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} + 4 \beta_{7} + 13 \beta_{8} - 3 \beta_{9} + 3 \beta_{10} + 6 \beta_{11} - 9 \beta_{12} - 2 \beta_{13} + 5 \beta_{14} + 6 \beta_{15} - 4 \beta_{16} + 2 \beta_{17} - 3 \beta_{18} - \beta_{19} ) q^{77} + ( -14 - \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} + 4 \beta_{6} + 14 \beta_{8} - 3 \beta_{9} + 7 \beta_{10} - \beta_{12} + \beta_{13} - \beta_{14} - 2 \beta_{15} + 4 \beta_{17} - 3 \beta_{18} + \beta_{19} ) q^{78} + ( 2 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} - 4 \beta_{5} + 5 \beta_{7} + 18 \beta_{8} + 2 \beta_{9} - 8 \beta_{10} - 4 \beta_{11} - 2 \beta_{12} + 4 \beta_{13} + 4 \beta_{14} - \beta_{15} - \beta_{16} + 3 \beta_{17} + 3 \beta_{18} + \beta_{19} ) q^{79} + 4 \beta_{7} q^{80} + ( -13 + 8 \beta_{1} - 7 \beta_{2} - 5 \beta_{3} - 4 \beta_{4} + \beta_{5} - 2 \beta_{6} + 5 \beta_{7} + 4 \beta_{9} + 5 \beta_{10} - \beta_{11} - 3 \beta_{12} + 6 \beta_{14} - 7 \beta_{15} + 7 \beta_{16} - 6 \beta_{18} - 7 \beta_{19} ) q^{81} + ( -5 - 2 \beta_{1} + 2 \beta_{3} + 6 \beta_{4} + 4 \beta_{6} - 2 \beta_{7} - 5 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + 4 \beta_{11} + 4 \beta_{12} - 6 \beta_{13} + 2 \beta_{15} + 2 \beta_{16} - 4 \beta_{17} - 2 \beta_{18} - 4 \beta_{19} ) q^{82} + ( 8 - \beta_{1} + 3 \beta_{2} - 8 \beta_{3} - 3 \beta_{4} + 6 \beta_{5} - 3 \beta_{6} - 8 \beta_{8} - 8 \beta_{9} + \beta_{10} + 3 \beta_{12} - 3 \beta_{13} + 3 \beta_{14} + 8 \beta_{15} - 3 \beta_{17} - 3 \beta_{19} ) q^{83} + ( -4 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} + 4 \beta_{7} + 4 \beta_{8} - 2 \beta_{9} - 2 \beta_{12} + 2 \beta_{15} + 2 \beta_{16} + 2 \beta_{17} - 2 \beta_{18} + 2 \beta_{19} ) q^{84} + ( -13 - 11 \beta_{1} - \beta_{2} + 5 \beta_{4} - 5 \beta_{5} + 6 \beta_{6} + 3 \beta_{7} + 8 \beta_{8} - 9 \beta_{9} + 6 \beta_{11} - 5 \beta_{12} + 2 \beta_{13} - 13 \beta_{14} - 4 \beta_{15} + 9 \beta_{16} + \beta_{17} - 4 \beta_{18} - 2 \beta_{19} ) q^{85} + ( -16 - 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} - 4 \beta_{6} - 4 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} - 6 \beta_{14} - 4 \beta_{15} + 4 \beta_{16} - \beta_{18} + \beta_{19} ) q^{86} + ( -12 - 2 \beta_{1} + 6 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} - \beta_{6} - 12 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + 4 \beta_{11} - \beta_{12} + 3 \beta_{13} + 7 \beta_{14} - 4 \beta_{15} + \beta_{17} - 2 \beta_{18} - 3 \beta_{19} ) q^{87} + ( 8 - 2 \beta_{3} - 2 \beta_{6} - 2 \beta_{7} - 8 \beta_{8} + 4 \beta_{9} + 2 \beta_{15} - 2 \beta_{16} - 2 \beta_{17} + 4 \beta_{18} ) q^{88} + ( -6 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} + 4 \beta_{7} + 22 \beta_{8} - 2 \beta_{10} - 4 \beta_{13} + 2 \beta_{14} + 4 \beta_{15} + 4 \beta_{16} + 9 \beta_{17} - 8 \beta_{18} + 8 \beta_{19} ) q^{89} + ( -6 + 8 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} + 3 \beta_{6} + \beta_{7} + 6 \beta_{8} - 3 \beta_{9} + 6 \beta_{10} + 4 \beta_{11} + 3 \beta_{12} - 8 \beta_{13} - 6 \beta_{14} - \beta_{15} - \beta_{16} - 3 \beta_{17} - 8 \beta_{18} ) q^{90} + ( -26 - 4 \beta_{1} - 2 \beta_{3} + 16 \beta_{4} - 4 \beta_{5} + 7 \beta_{6} + 2 \beta_{7} - 2 \beta_{9} - 2 \beta_{10} + 4 \beta_{11} - 2 \beta_{12} - 6 \beta_{14} - 9 \beta_{15} + 9 \beta_{16} - \beta_{18} + 3 \beta_{19} ) q^{91} -2 \beta_{14} q^{92} + ( 8 - 17 \beta_{1} - 2 \beta_{2} - 4 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} - 8 \beta_{8} - 4 \beta_{9} + 4 \beta_{10} - 2 \beta_{12} - 2 \beta_{14} + 4 \beta_{16} + 4 \beta_{17} - 8 \beta_{18} + 2 \beta_{19} ) q^{93} + ( -4 \beta_{3} - 4 \beta_{7} - 4 \beta_{8} - \beta_{9} - 3 \beta_{10} - \beta_{12} + 2 \beta_{13} + 3 \beta_{14} - 4 \beta_{15} - 4 \beta_{16} + 2 \beta_{17} + 3 \beta_{18} - 3 \beta_{19} ) q^{94} + ( 2 + 12 \beta_{1} + 10 \beta_{2} - \beta_{3} - \beta_{4} + 3 \beta_{5} - 4 \beta_{6} - \beta_{7} + 6 \beta_{8} + \beta_{9} + 5 \beta_{10} + 3 \beta_{11} - 2 \beta_{12} - 8 \beta_{13} - 10 \beta_{14} + 5 \beta_{15} - 5 \beta_{16} - 2 \beta_{17} + 5 \beta_{18} ) q^{95} + ( 4 \beta_{1} - 4 \beta_{2} ) q^{96} + ( -12 + \beta_{1} + 11 \beta_{2} - \beta_{3} - 7 \beta_{4} - \beta_{6} - 2 \beta_{7} - 12 \beta_{8} + \beta_{9} - \beta_{10} - 2 \beta_{11} + 3 \beta_{12} + 7 \beta_{13} + 9 \beta_{14} - \beta_{15} + 2 \beta_{16} + \beta_{17} + \beta_{18} - 4 \beta_{19} ) q^{97} + ( 4 + 5 \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 6 \beta_{7} - 4 \beta_{8} + 5 \beta_{9} + 15 \beta_{10} - \beta_{12} - \beta_{13} - \beta_{14} - 6 \beta_{16} - 2 \beta_{17} - 3 \beta_{18} + \beta_{19} ) q^{98} + ( -12 \beta_{1} - 16 \beta_{2} - 3 \beta_{3} - 4 \beta_{5} - 3 \beta_{7} - 28 \beta_{8} + 3 \beta_{9} - 16 \beta_{10} - 4 \beta_{11} - \beta_{12} + 12 \beta_{13} + 12 \beta_{14} - 9 \beta_{15} - 9 \beta_{16} - 2 \beta_{17} + 8 \beta_{18} - 4 \beta_{19} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 20 q^{2} + 4 q^{5} + 8 q^{7} + 40 q^{8} + O(q^{10}) \) \( 20 q - 20 q^{2} + 4 q^{5} + 8 q^{7} + 40 q^{8} + 4 q^{10} + 56 q^{11} - 4 q^{13} - 48 q^{15} - 80 q^{16} - 72 q^{17} - 28 q^{18} - 16 q^{20} + 8 q^{21} - 56 q^{22} + 36 q^{25} + 8 q^{26} + 156 q^{27} - 16 q^{28} + 84 q^{30} - 212 q^{31} + 80 q^{32} - 100 q^{33} + 56 q^{36} + 72 q^{37} + 88 q^{38} + 24 q^{40} - 12 q^{41} - 8 q^{42} + 120 q^{43} - 32 q^{45} + 8 q^{47} - 28 q^{50} + 64 q^{51} - 8 q^{52} - 244 q^{53} + 68 q^{55} + 32 q^{56} - 384 q^{57} - 188 q^{58} - 72 q^{60} + 328 q^{61} + 212 q^{62} + 172 q^{63} + 20 q^{65} + 200 q^{66} + 56 q^{67} + 144 q^{68} - 28 q^{70} - 92 q^{71} - 56 q^{72} + 144 q^{73} - 124 q^{75} - 176 q^{76} + 292 q^{77} - 208 q^{78} - 16 q^{80} - 84 q^{81} + 12 q^{82} - 72 q^{83} - 20 q^{85} - 240 q^{86} - 208 q^{87} + 112 q^{88} - 56 q^{90} - 192 q^{91} + 256 q^{93} - 96 q^{95} - 276 q^{97} + 104 q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 52 x^{17} + 1020 x^{16} - 1316 x^{15} + 1352 x^{14} - 18724 x^{13} + 250686 x^{12} - 439644 x^{11} + 460536 x^{10} - 1833716 x^{9} + 16970128 x^{8} - 36894604 x^{7} + 40544632 x^{6} - 4380588 x^{5} + 34721 x^{4} - 4419956 x^{3} + 9945800 x^{2} + 1329080 x + 88804\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(\)\(88\!\cdots\!61\)\( \nu^{19} + \)\(24\!\cdots\!33\)\( \nu^{18} + \)\(36\!\cdots\!62\)\( \nu^{17} - \)\(45\!\cdots\!43\)\( \nu^{16} + \)\(88\!\cdots\!41\)\( \nu^{15} - \)\(93\!\cdots\!91\)\( \nu^{14} + \)\(12\!\cdots\!42\)\( \nu^{13} - \)\(16\!\cdots\!07\)\( \nu^{12} + \)\(21\!\cdots\!11\)\( \nu^{11} - \)\(33\!\cdots\!73\)\( \nu^{10} + \)\(39\!\cdots\!14\)\( \nu^{9} - \)\(16\!\cdots\!49\)\( \nu^{8} + \)\(14\!\cdots\!27\)\( \nu^{7} - \)\(29\!\cdots\!73\)\( \nu^{6} + \)\(33\!\cdots\!18\)\( \nu^{5} - \)\(76\!\cdots\!89\)\( \nu^{4} + \)\(16\!\cdots\!68\)\( \nu^{3} - \)\(56\!\cdots\!76\)\( \nu^{2} + \)\(78\!\cdots\!84\)\( \nu + \)\(10\!\cdots\!08\)\(\)\()/ \)\(87\!\cdots\!16\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(18\!\cdots\!24\)\( \nu^{19} - \)\(47\!\cdots\!04\)\( \nu^{18} + \)\(62\!\cdots\!74\)\( \nu^{17} - \)\(94\!\cdots\!67\)\( \nu^{16} + \)\(21\!\cdots\!57\)\( \nu^{15} - \)\(73\!\cdots\!41\)\( \nu^{14} + \)\(93\!\cdots\!59\)\( \nu^{13} - \)\(41\!\cdots\!70\)\( \nu^{12} + \)\(54\!\cdots\!84\)\( \nu^{11} - \)\(20\!\cdots\!00\)\( \nu^{10} + \)\(30\!\cdots\!54\)\( \nu^{9} - \)\(58\!\cdots\!73\)\( \nu^{8} + \)\(40\!\cdots\!95\)\( \nu^{7} - \)\(14\!\cdots\!51\)\( \nu^{6} + \)\(26\!\cdots\!89\)\( \nu^{5} - \)\(22\!\cdots\!06\)\( \nu^{4} + \)\(60\!\cdots\!40\)\( \nu^{3} - \)\(20\!\cdots\!84\)\( \nu^{2} + \)\(48\!\cdots\!84\)\( \nu - \)\(22\!\cdots\!44\)\(\)\()/ \)\(51\!\cdots\!20\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(65\!\cdots\!41\)\( \nu^{19} + \)\(53\!\cdots\!78\)\( \nu^{18} + \)\(12\!\cdots\!43\)\( \nu^{17} - \)\(34\!\cdots\!08\)\( \nu^{16} + \)\(63\!\cdots\!82\)\( \nu^{15} - \)\(32\!\cdots\!96\)\( \nu^{14} + \)\(32\!\cdots\!46\)\( \nu^{13} - \)\(11\!\cdots\!02\)\( \nu^{12} + \)\(15\!\cdots\!01\)\( \nu^{11} - \)\(15\!\cdots\!46\)\( \nu^{10} + \)\(10\!\cdots\!05\)\( \nu^{9} - \)\(10\!\cdots\!44\)\( \nu^{8} + \)\(10\!\cdots\!02\)\( \nu^{7} - \)\(15\!\cdots\!92\)\( \nu^{6} + \)\(93\!\cdots\!86\)\( \nu^{5} + \)\(10\!\cdots\!18\)\( \nu^{4} + \)\(49\!\cdots\!90\)\( \nu^{3} - \)\(34\!\cdots\!04\)\( \nu^{2} - \)\(48\!\cdots\!72\)\( \nu - \)\(27\!\cdots\!80\)\(\)\()/ \)\(80\!\cdots\!72\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(30\!\cdots\!81\)\( \nu^{19} + \)\(60\!\cdots\!32\)\( \nu^{18} - \)\(21\!\cdots\!03\)\( \nu^{17} - \)\(15\!\cdots\!77\)\( \nu^{16} + \)\(31\!\cdots\!37\)\( \nu^{15} - \)\(33\!\cdots\!58\)\( \nu^{14} + \)\(11\!\cdots\!07\)\( \nu^{13} - \)\(53\!\cdots\!19\)\( \nu^{12} + \)\(76\!\cdots\!19\)\( \nu^{11} - \)\(11\!\cdots\!48\)\( \nu^{10} + \)\(59\!\cdots\!07\)\( \nu^{9} - \)\(42\!\cdots\!51\)\( \nu^{8} + \)\(50\!\cdots\!67\)\( \nu^{7} - \)\(10\!\cdots\!18\)\( \nu^{6} + \)\(64\!\cdots\!97\)\( \nu^{5} + \)\(10\!\cdots\!07\)\( \nu^{4} - \)\(97\!\cdots\!04\)\( \nu^{3} - \)\(16\!\cdots\!28\)\( \nu^{2} + \)\(35\!\cdots\!32\)\( \nu + \)\(97\!\cdots\!60\)\(\)\()/ \)\(35\!\cdots\!40\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(28\!\cdots\!89\)\( \nu^{19} + \)\(47\!\cdots\!96\)\( \nu^{18} - \)\(13\!\cdots\!65\)\( \nu^{17} + \)\(15\!\cdots\!11\)\( \nu^{16} - \)\(29\!\cdots\!01\)\( \nu^{15} + \)\(39\!\cdots\!30\)\( \nu^{14} - \)\(54\!\cdots\!95\)\( \nu^{13} + \)\(56\!\cdots\!69\)\( \nu^{12} - \)\(72\!\cdots\!27\)\( \nu^{11} + \)\(13\!\cdots\!88\)\( \nu^{10} - \)\(17\!\cdots\!75\)\( \nu^{9} + \)\(61\!\cdots\!97\)\( \nu^{8} - \)\(49\!\cdots\!87\)\( \nu^{7} + \)\(10\!\cdots\!30\)\( \nu^{6} - \)\(14\!\cdots\!05\)\( \nu^{5} + \)\(83\!\cdots\!35\)\( \nu^{4} - \)\(72\!\cdots\!36\)\( \nu^{3} + \)\(20\!\cdots\!96\)\( \nu^{2} + \)\(32\!\cdots\!40\)\( \nu + \)\(15\!\cdots\!68\)\(\)\()/ \)\(24\!\cdots\!60\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(62\!\cdots\!91\)\( \nu^{19} + \)\(45\!\cdots\!93\)\( \nu^{18} - \)\(60\!\cdots\!16\)\( \nu^{17} - \)\(32\!\cdots\!60\)\( \nu^{16} + \)\(61\!\cdots\!90\)\( \nu^{15} - \)\(35\!\cdots\!86\)\( \nu^{14} + \)\(18\!\cdots\!69\)\( \nu^{13} - \)\(11\!\cdots\!27\)\( \nu^{12} + \)\(14\!\cdots\!05\)\( \nu^{11} - \)\(15\!\cdots\!29\)\( \nu^{10} + \)\(74\!\cdots\!44\)\( \nu^{9} - \)\(92\!\cdots\!64\)\( \nu^{8} + \)\(97\!\cdots\!66\)\( \nu^{7} - \)\(15\!\cdots\!86\)\( \nu^{6} + \)\(77\!\cdots\!59\)\( \nu^{5} + \)\(16\!\cdots\!59\)\( \nu^{4} - \)\(27\!\cdots\!12\)\( \nu^{3} - \)\(31\!\cdots\!92\)\( \nu^{2} + \)\(47\!\cdots\!64\)\( \nu + \)\(28\!\cdots\!52\)\(\)\()/ \)\(51\!\cdots\!20\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(17\!\cdots\!23\)\( \nu^{19} - \)\(13\!\cdots\!89\)\( \nu^{18} - \)\(36\!\cdots\!17\)\( \nu^{17} - \)\(93\!\cdots\!34\)\( \nu^{16} + \)\(18\!\cdots\!67\)\( \nu^{15} - \)\(24\!\cdots\!77\)\( \nu^{14} + \)\(25\!\cdots\!55\)\( \nu^{13} - \)\(33\!\cdots\!10\)\( \nu^{12} + \)\(45\!\cdots\!21\)\( \nu^{11} - \)\(82\!\cdots\!51\)\( \nu^{10} + \)\(87\!\cdots\!05\)\( \nu^{9} - \)\(33\!\cdots\!54\)\( \nu^{8} + \)\(30\!\cdots\!45\)\( \nu^{7} - \)\(68\!\cdots\!15\)\( \nu^{6} + \)\(77\!\cdots\!13\)\( \nu^{5} - \)\(12\!\cdots\!06\)\( \nu^{4} + \)\(12\!\cdots\!44\)\( \nu^{3} - \)\(10\!\cdots\!20\)\( \nu^{2} + \)\(18\!\cdots\!24\)\( \nu + \)\(12\!\cdots\!24\)\(\)\()/ \)\(13\!\cdots\!84\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(12\!\cdots\!71\)\( \nu^{19} - \)\(68\!\cdots\!40\)\( \nu^{18} + \)\(49\!\cdots\!18\)\( \nu^{17} - \)\(65\!\cdots\!11\)\( \nu^{16} + \)\(12\!\cdots\!91\)\( \nu^{15} - \)\(17\!\cdots\!92\)\( \nu^{14} + \)\(18\!\cdots\!48\)\( \nu^{13} - \)\(23\!\cdots\!23\)\( \nu^{12} + \)\(31\!\cdots\!17\)\( \nu^{11} - \)\(57\!\cdots\!56\)\( \nu^{10} + \)\(62\!\cdots\!38\)\( \nu^{9} - \)\(23\!\cdots\!05\)\( \nu^{8} + \)\(21\!\cdots\!29\)\( \nu^{7} - \)\(47\!\cdots\!72\)\( \nu^{6} + \)\(54\!\cdots\!28\)\( \nu^{5} - \)\(82\!\cdots\!77\)\( \nu^{4} + \)\(77\!\cdots\!52\)\( \nu^{3} - \)\(76\!\cdots\!80\)\( \nu^{2} + \)\(17\!\cdots\!68\)\( \nu - \)\(65\!\cdots\!24\)\(\)\()/ \)\(35\!\cdots\!40\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(97\!\cdots\!51\)\( \nu^{19} + \)\(11\!\cdots\!71\)\( \nu^{18} - \)\(37\!\cdots\!21\)\( \nu^{17} + \)\(50\!\cdots\!34\)\( \nu^{16} - \)\(10\!\cdots\!57\)\( \nu^{15} + \)\(14\!\cdots\!97\)\( \nu^{14} - \)\(15\!\cdots\!41\)\( \nu^{13} + \)\(18\!\cdots\!94\)\( \nu^{12} - \)\(24\!\cdots\!73\)\( \nu^{11} + \)\(46\!\cdots\!45\)\( \nu^{10} - \)\(51\!\cdots\!35\)\( \nu^{9} + \)\(18\!\cdots\!38\)\( \nu^{8} - \)\(16\!\cdots\!39\)\( \nu^{7} + \)\(38\!\cdots\!15\)\( \nu^{6} - \)\(44\!\cdots\!15\)\( \nu^{5} + \)\(95\!\cdots\!46\)\( \nu^{4} - \)\(92\!\cdots\!80\)\( \nu^{3} + \)\(45\!\cdots\!92\)\( \nu^{2} - \)\(11\!\cdots\!80\)\( \nu + \)\(39\!\cdots\!28\)\(\)\()/ \)\(23\!\cdots\!56\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-\)\(15\!\cdots\!99\)\( \nu^{19} + \)\(12\!\cdots\!05\)\( \nu^{18} + \)\(78\!\cdots\!63\)\( \nu^{17} + \)\(78\!\cdots\!34\)\( \nu^{16} - \)\(15\!\cdots\!99\)\( \nu^{15} + \)\(21\!\cdots\!83\)\( \nu^{14} - \)\(21\!\cdots\!27\)\( \nu^{13} + \)\(28\!\cdots\!52\)\( \nu^{12} - \)\(38\!\cdots\!13\)\( \nu^{11} + \)\(69\!\cdots\!99\)\( \nu^{10} - \)\(73\!\cdots\!47\)\( \nu^{9} + \)\(28\!\cdots\!90\)\( \nu^{8} - \)\(25\!\cdots\!21\)\( \nu^{7} + \)\(57\!\cdots\!73\)\( \nu^{6} - \)\(64\!\cdots\!57\)\( \nu^{5} + \)\(97\!\cdots\!48\)\( \nu^{4} + \)\(16\!\cdots\!12\)\( \nu^{3} + \)\(29\!\cdots\!80\)\( \nu^{2} - \)\(11\!\cdots\!12\)\( \nu - \)\(47\!\cdots\!84\)\(\)\()/ \)\(35\!\cdots\!40\)\( \)
\(\beta_{12}\)\(=\)\((\)\(\)\(16\!\cdots\!24\)\( \nu^{19} + \)\(59\!\cdots\!08\)\( \nu^{18} + \)\(20\!\cdots\!93\)\( \nu^{17} - \)\(88\!\cdots\!93\)\( \nu^{16} + \)\(17\!\cdots\!38\)\( \nu^{15} - \)\(21\!\cdots\!12\)\( \nu^{14} + \)\(24\!\cdots\!53\)\( \nu^{13} - \)\(32\!\cdots\!61\)\( \nu^{12} + \)\(42\!\cdots\!36\)\( \nu^{11} - \)\(73\!\cdots\!52\)\( \nu^{10} + \)\(80\!\cdots\!23\)\( \nu^{9} - \)\(31\!\cdots\!79\)\( \nu^{8} + \)\(28\!\cdots\!58\)\( \nu^{7} - \)\(62\!\cdots\!72\)\( \nu^{6} + \)\(70\!\cdots\!03\)\( \nu^{5} - \)\(13\!\cdots\!27\)\( \nu^{4} + \)\(88\!\cdots\!84\)\( \nu^{3} - \)\(10\!\cdots\!92\)\( \nu^{2} + \)\(16\!\cdots\!28\)\( \nu + \)\(22\!\cdots\!00\)\(\)\()/ \)\(35\!\cdots\!40\)\( \)
\(\beta_{13}\)\(=\)\((\)\(\)\(57\!\cdots\!61\)\( \nu^{19} - \)\(85\!\cdots\!27\)\( \nu^{18} - \)\(89\!\cdots\!75\)\( \nu^{17} - \)\(29\!\cdots\!47\)\( \nu^{16} + \)\(59\!\cdots\!10\)\( \nu^{15} - \)\(83\!\cdots\!99\)\( \nu^{14} + \)\(80\!\cdots\!74\)\( \nu^{13} - \)\(10\!\cdots\!53\)\( \nu^{12} + \)\(14\!\cdots\!85\)\( \nu^{11} - \)\(27\!\cdots\!65\)\( \nu^{10} + \)\(28\!\cdots\!79\)\( \nu^{9} - \)\(10\!\cdots\!13\)\( \nu^{8} + \)\(98\!\cdots\!70\)\( \nu^{7} - \)\(22\!\cdots\!05\)\( \nu^{6} + \)\(25\!\cdots\!74\)\( \nu^{5} - \)\(40\!\cdots\!27\)\( \nu^{4} - \)\(52\!\cdots\!30\)\( \nu^{3} - \)\(44\!\cdots\!56\)\( \nu^{2} + \)\(55\!\cdots\!60\)\( \nu + \)\(36\!\cdots\!00\)\(\)\()/ \)\(11\!\cdots\!28\)\( \)
\(\beta_{14}\)\(=\)\((\)\(\)\(12\!\cdots\!73\)\( \nu^{19} + \)\(55\!\cdots\!43\)\( \nu^{18} + \)\(84\!\cdots\!41\)\( \nu^{17} - \)\(62\!\cdots\!70\)\( \nu^{16} + \)\(12\!\cdots\!09\)\( \nu^{15} - \)\(15\!\cdots\!91\)\( \nu^{14} + \)\(16\!\cdots\!83\)\( \nu^{13} - \)\(22\!\cdots\!62\)\( \nu^{12} + \)\(30\!\cdots\!23\)\( \nu^{11} - \)\(51\!\cdots\!83\)\( \nu^{10} + \)\(55\!\cdots\!31\)\( \nu^{9} - \)\(22\!\cdots\!30\)\( \nu^{8} + \)\(20\!\cdots\!03\)\( \nu^{7} - \)\(43\!\cdots\!93\)\( \nu^{6} + \)\(48\!\cdots\!45\)\( \nu^{5} - \)\(61\!\cdots\!14\)\( \nu^{4} + \)\(28\!\cdots\!28\)\( \nu^{3} - \)\(57\!\cdots\!56\)\( \nu^{2} + \)\(11\!\cdots\!60\)\( \nu + \)\(15\!\cdots\!16\)\(\)\()/ \)\(23\!\cdots\!56\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-\)\(25\!\cdots\!11\)\( \nu^{19} + \)\(14\!\cdots\!97\)\( \nu^{18} - \)\(55\!\cdots\!39\)\( \nu^{17} + \)\(13\!\cdots\!15\)\( \nu^{16} - \)\(26\!\cdots\!10\)\( \nu^{15} + \)\(35\!\cdots\!76\)\( \nu^{14} - \)\(37\!\cdots\!44\)\( \nu^{13} + \)\(48\!\cdots\!02\)\( \nu^{12} - \)\(64\!\cdots\!85\)\( \nu^{11} + \)\(11\!\cdots\!99\)\( \nu^{10} - \)\(12\!\cdots\!09\)\( \nu^{9} + \)\(47\!\cdots\!49\)\( \nu^{8} - \)\(43\!\cdots\!26\)\( \nu^{7} + \)\(97\!\cdots\!56\)\( \nu^{6} - \)\(11\!\cdots\!44\)\( \nu^{5} + \)\(17\!\cdots\!66\)\( \nu^{4} + \)\(92\!\cdots\!92\)\( \nu^{3} + \)\(78\!\cdots\!72\)\( \nu^{2} - \)\(26\!\cdots\!84\)\( \nu - \)\(14\!\cdots\!92\)\(\)\()/ \)\(35\!\cdots\!40\)\( \)
\(\beta_{16}\)\(=\)\((\)\(-\)\(26\!\cdots\!94\)\( \nu^{19} + \)\(14\!\cdots\!84\)\( \nu^{18} - \)\(10\!\cdots\!89\)\( \nu^{17} + \)\(13\!\cdots\!22\)\( \nu^{16} - \)\(27\!\cdots\!97\)\( \nu^{15} + \)\(36\!\cdots\!61\)\( \nu^{14} - \)\(38\!\cdots\!24\)\( \nu^{13} + \)\(50\!\cdots\!85\)\( \nu^{12} - \)\(66\!\cdots\!54\)\( \nu^{11} + \)\(12\!\cdots\!80\)\( \nu^{10} - \)\(13\!\cdots\!99\)\( \nu^{9} + \)\(49\!\cdots\!78\)\( \nu^{8} - \)\(45\!\cdots\!35\)\( \nu^{7} + \)\(10\!\cdots\!51\)\( \nu^{6} - \)\(11\!\cdots\!64\)\( \nu^{5} + \)\(19\!\cdots\!91\)\( \nu^{4} - \)\(11\!\cdots\!20\)\( \nu^{3} + \)\(84\!\cdots\!04\)\( \nu^{2} - \)\(26\!\cdots\!84\)\( \nu - \)\(19\!\cdots\!16\)\(\)\()/ \)\(35\!\cdots\!40\)\( \)
\(\beta_{17}\)\(=\)\((\)\(-\)\(36\!\cdots\!85\)\( \nu^{19} + \)\(47\!\cdots\!91\)\( \nu^{18} + \)\(36\!\cdots\!27\)\( \nu^{17} + \)\(18\!\cdots\!92\)\( \nu^{16} - \)\(37\!\cdots\!57\)\( \nu^{15} + \)\(52\!\cdots\!97\)\( \nu^{14} - \)\(51\!\cdots\!43\)\( \nu^{13} + \)\(67\!\cdots\!62\)\( \nu^{12} - \)\(91\!\cdots\!19\)\( \nu^{11} + \)\(16\!\cdots\!49\)\( \nu^{10} - \)\(17\!\cdots\!63\)\( \nu^{9} + \)\(67\!\cdots\!52\)\( \nu^{8} - \)\(61\!\cdots\!71\)\( \nu^{7} + \)\(14\!\cdots\!47\)\( \nu^{6} - \)\(15\!\cdots\!33\)\( \nu^{5} + \)\(24\!\cdots\!62\)\( \nu^{4} + \)\(71\!\cdots\!32\)\( \nu^{3} + \)\(16\!\cdots\!36\)\( \nu^{2} - \)\(33\!\cdots\!88\)\( \nu - \)\(21\!\cdots\!68\)\(\)\()/ \)\(35\!\cdots\!40\)\( \)
\(\beta_{18}\)\(=\)\((\)\(-\)\(16\!\cdots\!78\)\( \nu^{19} + \)\(21\!\cdots\!00\)\( \nu^{18} - \)\(15\!\cdots\!09\)\( \nu^{17} + \)\(83\!\cdots\!68\)\( \nu^{16} - \)\(16\!\cdots\!28\)\( \nu^{15} + \)\(23\!\cdots\!46\)\( \nu^{14} - \)\(24\!\cdots\!99\)\( \nu^{13} + \)\(30\!\cdots\!04\)\( \nu^{12} - \)\(40\!\cdots\!46\)\( \nu^{11} + \)\(76\!\cdots\!68\)\( \nu^{10} - \)\(83\!\cdots\!79\)\( \nu^{9} + \)\(30\!\cdots\!20\)\( \nu^{8} - \)\(27\!\cdots\!72\)\( \nu^{7} + \)\(63\!\cdots\!86\)\( \nu^{6} - \)\(73\!\cdots\!29\)\( \nu^{5} + \)\(16\!\cdots\!16\)\( \nu^{4} - \)\(15\!\cdots\!56\)\( \nu^{3} + \)\(72\!\cdots\!60\)\( \nu^{2} - \)\(16\!\cdots\!44\)\( \nu + \)\(63\!\cdots\!92\)\(\)\()/ \)\(11\!\cdots\!80\)\( \)
\(\beta_{19}\)\(=\)\((\)\(\)\(23\!\cdots\!81\)\( \nu^{19} - \)\(26\!\cdots\!88\)\( \nu^{18} - \)\(42\!\cdots\!48\)\( \nu^{17} - \)\(12\!\cdots\!42\)\( \nu^{16} + \)\(24\!\cdots\!07\)\( \nu^{15} - \)\(31\!\cdots\!58\)\( \nu^{14} + \)\(32\!\cdots\!62\)\( \nu^{13} - \)\(44\!\cdots\!54\)\( \nu^{12} + \)\(59\!\cdots\!39\)\( \nu^{11} - \)\(10\!\cdots\!68\)\( \nu^{10} + \)\(10\!\cdots\!72\)\( \nu^{9} - \)\(43\!\cdots\!46\)\( \nu^{8} + \)\(40\!\cdots\!57\)\( \nu^{7} - \)\(87\!\cdots\!58\)\( \nu^{6} + \)\(96\!\cdots\!82\)\( \nu^{5} - \)\(98\!\cdots\!18\)\( \nu^{4} - \)\(18\!\cdots\!84\)\( \nu^{3} - \)\(10\!\cdots\!28\)\( \nu^{2} + \)\(23\!\cdots\!32\)\( \nu + \)\(31\!\cdots\!60\)\(\)\()/ \)\(17\!\cdots\!40\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{14} + \beta_{13} - \beta_{10} - 10 \beta_{8} - \beta_{2} - \beta_{1}\)
\(\nu^{3}\)\(=\)\(2 \beta_{19} - \beta_{18} - \beta_{17} + 2 \beta_{15} - 4 \beta_{14} - 3 \beta_{13} + 2 \beta_{11} + \beta_{10} - \beta_{9} + 7 \beta_{8} + \beta_{6} + 3 \beta_{4} + 2 \beta_{3} + 19 \beta_{2} - \beta_{1} + 7\)
\(\nu^{4}\)\(=\)\(-7 \beta_{19} - 6 \beta_{18} + 7 \beta_{16} - 7 \beta_{15} + 33 \beta_{14} - 3 \beta_{12} - \beta_{11} + 32 \beta_{10} + 4 \beta_{9} + 5 \beta_{7} - 2 \beta_{6} + \beta_{5} - 31 \beta_{4} - 5 \beta_{3} - 34 \beta_{2} + 35 \beta_{1} - 202\)
\(\nu^{5}\)\(=\)\(30 \beta_{19} + 84 \beta_{18} + 28 \beta_{17} - 88 \beta_{16} + 10 \beta_{15} - 30 \beta_{14} + 117 \beta_{13} - 30 \beta_{12} - 180 \beta_{10} - 12 \beta_{9} - 321 \beta_{8} - 88 \beta_{7} + 28 \beta_{6} - 60 \beta_{5} + 117 \beta_{4} - 10 \beta_{3} - 30 \beta_{2} - 451 \beta_{1} + 321\)
\(\nu^{6}\)\(=\)\(255 \beta_{19} - 349 \beta_{18} - 110 \beta_{17} + 244 \beta_{16} + 244 \beta_{15} - 976 \beta_{14} - 1013 \beta_{13} + 181 \beta_{12} + 94 \beta_{11} + 1070 \beta_{10} + 87 \beta_{9} + 4956 \beta_{8} + 340 \beta_{7} + 94 \beta_{5} + 340 \beta_{3} + 1210 \beta_{2} + 1116 \beta_{1}\)
\(\nu^{7}\)\(=\)\(-2937 \beta_{19} + 866 \beta_{18} + 787 \beta_{17} + 470 \beta_{16} - 3066 \beta_{15} + 6737 \beta_{14} + 3993 \beta_{13} - 657 \beta_{12} - 1732 \beta_{11} - 866 \beta_{10} + 866 \beta_{9} - 12085 \beta_{8} - 470 \beta_{7} - 787 \beta_{6} - 3993 \beta_{4} - 3066 \beta_{3} - 12318 \beta_{2} + 866 \beta_{1} - 12085\)
\(\nu^{8}\)\(=\)\(13270 \beta_{19} + 8973 \beta_{18} - 12905 \beta_{16} + 12905 \beta_{15} - 34504 \beta_{14} + 2132 \beta_{12} + 4297 \beta_{11} - 30207 \beta_{10} - 6429 \beta_{9} - 9111 \beta_{7} + 4288 \beta_{6} - 4297 \beta_{5} + 33069 \beta_{4} + 9111 \beta_{3} + 36883 \beta_{2} - 41180 \beta_{1} + 137518\)
\(\nu^{9}\)\(=\)\(-25919 \beta_{19} - 97647 \beta_{18} - 23234 \beta_{17} + 101044 \beta_{16} - 17220 \beta_{15} + 25919 \beta_{14} - 132039 \beta_{13} + 25919 \beta_{12} + 234181 \beta_{10} + 26181 \beta_{9} + 424345 \beta_{8} + 101044 \beta_{7} - 23234 \beta_{6} + 51838 \beta_{5} - 132039 \beta_{4} + 17220 \beta_{3} + 25919 \beta_{2} + 365770 \beta_{1} - 424345\)
\(\nu^{10}\)\(=\)\(-299718 \beta_{19} + 461512 \beta_{18} + 150960 \beta_{17} - 313546 \beta_{16} - 313546 \beta_{15} + 952399 \beta_{14} + 1076827 \beta_{13} - 214942 \beta_{12} - 161794 \beta_{11} - 1114193 \beta_{10} - 53148 \beta_{9} - 4108936 \beta_{8} - 449438 \beta_{7} - 161794 \beta_{5} - 449438 \beta_{3} - 1379653 \beta_{2} - 1217859 \beta_{1}\)
\(\nu^{11}\)\(=\)\(3196192 \beta_{19} - 801439 \beta_{18} - 712957 \beta_{17} - 585768 \beta_{16} + 3285114 \beta_{15} - 7869462 \beta_{14} - 4322653 \beta_{13} + 931906 \beta_{12} + 1602878 \beta_{11} + 801439 \beta_{10} - 801439 \beta_{9} + 14393681 \beta_{8} + 585768 \beta_{7} + 712957 \beta_{6} + 4322653 \beta_{4} + 3285114 \beta_{3} + 11363497 \beta_{2} - 801439 \beta_{1} + 14393681\)
\(\nu^{12}\)\(=\)\(-15480507 \beta_{19} - 9838398 \beta_{18} + 15092379 \beta_{16} - 15092379 \beta_{15} + 36070713 \beta_{14} - 1416823 \beta_{12} - 5642109 \beta_{11} + 30428604 \beta_{10} + 7058932 \beta_{9} + 10458009 \beta_{7} - 5106386 \beta_{6} + 5642109 \beta_{5} - 35037671 \beta_{4} - 10458009 \beta_{3} - 40054042 \beta_{2} + 45696151 \beta_{1} - 127905718\)
\(\nu^{13}\)\(=\)\(25322218 \beta_{19} + 104126788 \beta_{18} + 22434272 \beta_{17} - 106565104 \beta_{16} + 19389946 \beta_{15} - 25322218 \beta_{14} + 141006745 \beta_{13} - 25322218 \beta_{12} - 260204332 \beta_{10} - 31619740 \beta_{9} - 479184449 \beta_{8} - 106565104 \beta_{7} + 22434272 \beta_{6} - 50644436 \beta_{5} + 141006745 \beta_{4} - 19389946 \beta_{3} - 25322218 \beta_{2} - 361307767 \beta_{1} + 479184449\)
\(\nu^{14}\)\(=\)\(321015807 \beta_{19} - 511029377 \beta_{18} - 169583230 \beta_{17} + 344231680 \beta_{16} + 344231680 \beta_{15} - 980118624 \beta_{14} - 1139972065 \beta_{13} + 230498189 \beta_{12} + 190013570 \beta_{11} + 1170132194 \beta_{10} + 40484619 \beta_{9} + 4068512004 \beta_{8} + 498537008 \beta_{7} + 190013570 \beta_{5} + 498537008 \beta_{3} + 1502433522 \beta_{2} + 1312419952 \beta_{1}\)
\(\nu^{15}\)\(=\)\(-3388147513 \beta_{19} + 810535394 \beta_{18} + 716587555 \beta_{17} + 635027174 \beta_{16} - 3458682090 \beta_{15} + 8535876497 \beta_{14} + 4593422385 \beta_{13} - 1049782377 \beta_{12} - 1621070788 \beta_{11} - 810535394 \beta_{10} + 810535394 \beta_{9} - 15789324785 \beta_{8} - 635027174 \beta_{7} - 716587555 \beta_{6} - 4593422385 \beta_{4} - 3458682090 \beta_{3} - 11622197754 \beta_{2} + 810535394 \beta_{1} - 15789324785\)
\(\nu^{16}\)\(=\)\(16745667574 \beta_{19} + 10453501581 \beta_{18} - 16341965065 \beta_{16} + 16341965065 \beta_{15} - 38010411804 \beta_{14} + 1219565940 \beta_{12} + 6292165993 \beta_{11} - 31718245811 \beta_{10} - 7511731933 \beta_{9} - 11263703055 \beta_{7} + 5579838872 \beta_{6} - 6292165993 \beta_{5} + 37093207193 \beta_{4} + 11263703055 \beta_{3} + 42888005275 \beta_{2} - 49180171268 \beta_{1} + 130865658022\)
\(\nu^{17}\)\(=\)\(-26138406939 \beta_{19} - 110226681007 \beta_{18} - 23084709966 \beta_{17} + 112357677276 \beta_{16} - 20710794876 \beta_{15} + 26138406939 \beta_{14} - 149554028679 \beta_{13} + 26138406939 \beta_{12} + 278918451453 \beta_{10} + 34498233909 \beta_{9} + 517330362901 \beta_{8} + 112357677276 \beta_{7} - 23084709966 \beta_{6} + 52276813878 \beta_{5} - 149554028679 \beta_{4} + 20710794876 \beta_{3} + 26138406939 \beta_{2} + 376027097254 \beta_{1} - 517330362901\)
\(\nu^{18}\)\(=\)\(-340205444706 \beta_{19} + 546816069288 \beta_{18} + 182691867016 \beta_{17} - 367548747726 \beta_{16} - 367548747726 \beta_{15} + 1029134050323 \beta_{14} + 1207077685191 \beta_{13} - 244621531306 \beta_{12} - 206610624582 \beta_{11} - 1235744674905 \beta_{10} - 38010906724 \beta_{9} - 4233489741208 \beta_{8} - 533733867362 \beta_{7} - 206610624582 \beta_{5} - 533733867362 \beta_{3} - 1605734864641 \beta_{2} - 1399124240059 \beta_{1}\)
\(\nu^{19}\)\(=\)\(3586311908268 \beta_{19} - 846442183307 \beta_{18} - 747173987357 \beta_{17} - 674413682784 \beta_{16} + 3652652232538 \beta_{15} - 9096045995658 \beta_{14} - 4868152493421 \beta_{13} + 1128127714686 \beta_{12} + 1692884366614 \beta_{11} + 846442183307 \beta_{10} - 846442183307 \beta_{9} + 16897845076885 \beta_{8} + 674413682784 \beta_{7} + 747173987357 \beta_{6} + 4868152493421 \beta_{4} + 3652652232538 \beta_{3} + 12201497633521 \beta_{2} - 846442183307 \beta_{1} + 16897845076885\)

Character values

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

\(n\) \(47\) \(51\)
\(\chi(n)\) \(\beta_{8}\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
−4.03398 + 4.03398i
−2.63948 + 2.63948i
−2.41818 + 2.41818i
−0.454110 + 0.454110i
−0.0649756 + 0.0649756i
0.608645 0.608645i
1.21198 1.21198i
1.94658 1.94658i
2.91022 2.91022i
2.93330 2.93330i
−4.03398 4.03398i
−2.63948 2.63948i
−2.41818 2.41818i
−0.454110 0.454110i
−0.0649756 0.0649756i
0.608645 + 0.608645i
1.21198 + 1.21198i
1.94658 + 1.94658i
2.91022 + 2.91022i
2.93330 + 2.93330i
−1.00000 1.00000i −4.03398 + 4.03398i 2.00000i 4.54274 2.08891i 8.06796 −1.88621 1.88621i 2.00000 2.00000i 23.5460i −6.63165 2.45382i
47.2 −1.00000 1.00000i −2.63948 + 2.63948i 2.00000i 1.86266 + 4.64010i 5.27895 5.81530 + 5.81530i 2.00000 2.00000i 4.93366i 2.77744 6.50275i
47.3 −1.00000 1.00000i −2.41818 + 2.41818i 2.00000i −4.87518 + 1.11024i 4.83636 3.71756 + 3.71756i 2.00000 2.00000i 2.69521i 5.98542 + 3.76494i
47.4 −1.00000 1.00000i −0.454110 + 0.454110i 2.00000i 4.61282 1.92922i 0.908220 −8.21299 8.21299i 2.00000 2.00000i 8.58757i −6.54204 2.68360i
47.5 −1.00000 1.00000i −0.0649756 + 0.0649756i 2.00000i −4.95703 0.654141i 0.129951 −3.31044 3.31044i 2.00000 2.00000i 8.99156i 4.30288 + 5.61117i
47.6 −1.00000 1.00000i 0.608645 0.608645i 2.00000i 4.88434 + 1.06922i −1.21729 3.48181 + 3.48181i 2.00000 2.00000i 8.25910i −3.81512 5.95356i
47.7 −1.00000 1.00000i 1.21198 1.21198i 2.00000i −0.802406 + 4.93519i −2.42397 −5.44701 5.44701i 2.00000 2.00000i 6.06220i 5.73760 4.13279i
47.8 −1.00000 1.00000i 1.94658 1.94658i 2.00000i 0.283956 4.99193i −3.89316 9.76397 + 9.76397i 2.00000 2.00000i 1.42166i −5.27589 + 4.70797i
47.9 −1.00000 1.00000i 2.91022 2.91022i 2.00000i −3.94243 3.07527i −5.82043 −1.42062 1.42062i 2.00000 2.00000i 7.93871i 0.867158 + 7.01769i
47.10 −1.00000 1.00000i 2.93330 2.93330i 2.00000i 0.390531 + 4.98473i −5.86660 1.49862 + 1.49862i 2.00000 2.00000i 8.20851i 4.59419 5.37526i
93.1 −1.00000 + 1.00000i −4.03398 4.03398i 2.00000i 4.54274 + 2.08891i 8.06796 −1.88621 + 1.88621i 2.00000 + 2.00000i 23.5460i −6.63165 + 2.45382i
93.2 −1.00000 + 1.00000i −2.63948 2.63948i 2.00000i 1.86266 4.64010i 5.27895 5.81530 5.81530i 2.00000 + 2.00000i 4.93366i 2.77744 + 6.50275i
93.3 −1.00000 + 1.00000i −2.41818 2.41818i 2.00000i −4.87518 1.11024i 4.83636 3.71756 3.71756i 2.00000 + 2.00000i 2.69521i 5.98542 3.76494i
93.4 −1.00000 + 1.00000i −0.454110 0.454110i 2.00000i 4.61282 + 1.92922i 0.908220 −8.21299 + 8.21299i 2.00000 + 2.00000i 8.58757i −6.54204 + 2.68360i
93.5 −1.00000 + 1.00000i −0.0649756 0.0649756i 2.00000i −4.95703 + 0.654141i 0.129951 −3.31044 + 3.31044i 2.00000 + 2.00000i 8.99156i 4.30288 5.61117i
93.6 −1.00000 + 1.00000i 0.608645 + 0.608645i 2.00000i 4.88434 1.06922i −1.21729 3.48181 3.48181i 2.00000 + 2.00000i 8.25910i −3.81512 + 5.95356i
93.7 −1.00000 + 1.00000i 1.21198 + 1.21198i 2.00000i −0.802406 4.93519i −2.42397 −5.44701 + 5.44701i 2.00000 + 2.00000i 6.06220i 5.73760 + 4.13279i
93.8 −1.00000 + 1.00000i 1.94658 + 1.94658i 2.00000i 0.283956 + 4.99193i −3.89316 9.76397 9.76397i 2.00000 + 2.00000i 1.42166i −5.27589 4.70797i
93.9 −1.00000 + 1.00000i 2.91022 + 2.91022i 2.00000i −3.94243 + 3.07527i −5.82043 −1.42062 + 1.42062i 2.00000 + 2.00000i 7.93871i 0.867158 7.01769i
93.10 −1.00000 + 1.00000i 2.93330 + 2.93330i 2.00000i 0.390531 4.98473i −5.86660 1.49862 1.49862i 2.00000 + 2.00000i 8.20851i 4.59419 + 5.37526i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 93.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 230.3.f.a 20
5.c odd 4 1 inner 230.3.f.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.3.f.a 20 1.a even 1 1 trivial
230.3.f.a 20 5.c odd 4 1 inner

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 2 + 2 T + T^{2} )^{10} \)
$3$ \( 88804 + 1329080 T + 9945800 T^{2} - 4419956 T^{3} + 34721 T^{4} - 4380588 T^{5} + 40544632 T^{6} - 36894604 T^{7} + 16970128 T^{8} - 1833716 T^{9} + 460536 T^{10} - 439644 T^{11} + 250686 T^{12} - 18724 T^{13} + 1352 T^{14} - 1316 T^{15} + 1020 T^{16} - 52 T^{17} + T^{20} \)
$5$ \( 95367431640625 - 15258789062500 T - 1525878906250 T^{2} + 732421875000 T^{3} - 344970703125 T^{4} + 34882812500 T^{5} + 12825000000 T^{6} - 2186062500 T^{7} + 333775000 T^{8} + 27535500 T^{9} - 29495500 T^{10} + 1101420 T^{11} + 534040 T^{12} - 139908 T^{13} + 32832 T^{14} + 3572 T^{15} - 1413 T^{16} + 120 T^{17} - 10 T^{18} - 4 T^{19} + T^{20} \)
$7$ \( 195628000677264 + 67226832267840 T + 11551124995200 T^{2} - 4771880205024 T^{3} + 11358258723616 T^{4} + 3433257455040 T^{5} + 567361404192 T^{6} - 209020366672 T^{7} + 66805292296 T^{8} + 8678129056 T^{9} + 1321578464 T^{10} - 480645928 T^{11} + 107626716 T^{12} + 4730912 T^{13} + 479752 T^{14} - 152348 T^{15} + 27517 T^{16} + 532 T^{17} + 32 T^{18} - 8 T^{19} + T^{20} \)
$11$ \( ( -1481201960 - 1024786496 T + 8591372 T^{2} + 75696072 T^{3} + 1067500 T^{4} - 1858096 T^{5} + 30636 T^{6} + 13484 T^{7} - 388 T^{8} - 28 T^{9} + T^{10} )^{2} \)
$13$ \( 46778503687027787536 + 66463618814009857984 T + 47216266849927096448 T^{2} - 28796382463620954400 T^{3} + 8242488617783342801 T^{4} - 378490708283117292 T^{5} + 5994575045101384 T^{6} - 1081142143902492 T^{7} + 1088324631454056 T^{8} - 29382816599044 T^{9} + 232151157624 T^{10} + 57972962252 T^{11} + 33280843982 T^{12} - 407377956 T^{13} + 2686392 T^{14} + 1144364 T^{15} + 335608 T^{16} - 1260 T^{17} + 8 T^{18} + 4 T^{19} + T^{20} \)
$17$ \( \)\(13\!\cdots\!00\)\( + \)\(33\!\cdots\!00\)\( T + \)\(41\!\cdots\!00\)\( T^{2} + \)\(26\!\cdots\!00\)\( T^{3} + \)\(92\!\cdots\!00\)\( T^{4} + \)\(12\!\cdots\!60\)\( T^{5} + 8632667502575375648 T^{6} + 235408099091834816 T^{7} + 50253648133953276 T^{8} + 5793848995300192 T^{9} + 356530868604672 T^{10} + 9591580024832 T^{11} + 395923927324 T^{12} + 31934939648 T^{13} + 1909725192 T^{14} + 52410324 T^{15} + 1091369 T^{16} + 43932 T^{17} + 2592 T^{18} + 72 T^{19} + T^{20} \)
$19$ \( \)\(10\!\cdots\!44\)\( + \)\(88\!\cdots\!76\)\( T^{2} + \)\(30\!\cdots\!12\)\( T^{4} + 5765121822676730464 T^{6} + 62726897758142864 T^{8} + 412252437335072 T^{10} + 1640199958236 T^{12} + 3873089960 T^{14} + 5209368 T^{16} + 3628 T^{18} + T^{20} \)
$23$ \( ( 529 + T^{4} )^{5} \)
$29$ \( \)\(13\!\cdots\!25\)\( + \)\(96\!\cdots\!66\)\( T^{2} + \)\(13\!\cdots\!05\)\( T^{4} + \)\(82\!\cdots\!48\)\( T^{6} + 28568621908873074498 T^{8} + 60172441369515724 T^{10} + 79271696392578 T^{12} + 65218713688 T^{14} + 32334205 T^{16} + 8786 T^{18} + T^{20} \)
$31$ \( ( -1910123254695 + 1772036568042 T - 304150628147 T^{2} - 13202190216 T^{3} + 3435481442 T^{4} + 120347580 T^{5} - 5176206 T^{6} - 223880 T^{7} + 773 T^{8} + 106 T^{9} + T^{10} )^{2} \)
$37$ \( \)\(65\!\cdots\!04\)\( + \)\(84\!\cdots\!36\)\( T + \)\(54\!\cdots\!12\)\( T^{2} - \)\(16\!\cdots\!28\)\( T^{3} + \)\(84\!\cdots\!08\)\( T^{4} - 59428625594291160000 T^{5} + 5604378004991694624 T^{6} - 3441698733278361104 T^{7} + 1389314972256657352 T^{8} - 153380444805071584 T^{9} + 9092953009403168 T^{10} - 261654757304920 T^{11} + 5958139665612 T^{12} - 263291243696 T^{13} + 15779761544 T^{14} - 441926876 T^{15} + 6533293 T^{16} - 42316 T^{17} + 2592 T^{18} - 72 T^{19} + T^{20} \)
$41$ \( ( 621850944556593 + 108006335475462 T + 1283442504253 T^{2} - 473320515864 T^{3} - 15593013598 T^{4} + 421054756 T^{5} + 18442306 T^{6} - 117016 T^{7} - 7507 T^{8} + 6 T^{9} + T^{10} )^{2} \)
$43$ \( \)\(13\!\cdots\!00\)\( - \)\(89\!\cdots\!00\)\( T + \)\(29\!\cdots\!00\)\( T^{2} - \)\(51\!\cdots\!80\)\( T^{3} + \)\(72\!\cdots\!24\)\( T^{4} - \)\(44\!\cdots\!72\)\( T^{5} + \)\(14\!\cdots\!08\)\( T^{6} - \)\(24\!\cdots\!40\)\( T^{7} + \)\(96\!\cdots\!52\)\( T^{8} - 50120655129797548608 T^{9} + 1660322689474800160 T^{10} - 27244467822730736 T^{11} + 348698971541900 T^{12} - 8235165427184 T^{13} + 270888900800 T^{14} - 4443089400 T^{15} + 40093980 T^{16} - 229760 T^{17} + 7200 T^{18} - 120 T^{19} + T^{20} \)
$47$ \( \)\(16\!\cdots\!00\)\( - \)\(73\!\cdots\!00\)\( T + \)\(16\!\cdots\!00\)\( T^{2} + \)\(13\!\cdots\!20\)\( T^{3} + \)\(60\!\cdots\!81\)\( T^{4} - \)\(33\!\cdots\!48\)\( T^{5} + \)\(10\!\cdots\!52\)\( T^{6} + 95193701998438760948 T^{7} + 4025784299699135136 T^{8} + 2190010362102652 T^{9} + 18106515941597464 T^{10} + 1665903501880500 T^{11} + 75410027448862 T^{12} + 900175691644 T^{13} + 2284530152 T^{14} + 136702300 T^{15} + 25340780 T^{16} + 91532 T^{17} + 32 T^{18} - 8 T^{19} + T^{20} \)
$53$ \( \)\(94\!\cdots\!36\)\( + \)\(61\!\cdots\!52\)\( T + \)\(20\!\cdots\!32\)\( T^{2} - \)\(49\!\cdots\!64\)\( T^{3} + \)\(31\!\cdots\!12\)\( T^{4} + \)\(10\!\cdots\!16\)\( T^{5} + \)\(18\!\cdots\!36\)\( T^{6} - \)\(16\!\cdots\!72\)\( T^{7} + \)\(20\!\cdots\!92\)\( T^{8} + 67282012253709884800 T^{9} + 996624973888543360 T^{10} + 3628488508400000 T^{11} + 572380047437552 T^{12} + 17976880111824 T^{13} + 279329642304 T^{14} + 2129602992 T^{15} + 61464457 T^{16} + 1782908 T^{17} + 29768 T^{18} + 244 T^{19} + T^{20} \)
$59$ \( \)\(87\!\cdots\!00\)\( + \)\(17\!\cdots\!40\)\( T^{2} + \)\(57\!\cdots\!04\)\( T^{4} + \)\(38\!\cdots\!24\)\( T^{6} + \)\(10\!\cdots\!40\)\( T^{8} + 1573516217383617216 T^{10} + 1294679419414792 T^{12} + 614020973488 T^{14} + 161145593 T^{16} + 20802 T^{18} + T^{20} \)
$61$ \( ( -38679925234168 + 23507776447920 T + 4270310693116 T^{2} - 1017774209624 T^{3} + 20584897008 T^{4} + 2522239496 T^{5} - 126604438 T^{6} + 1937972 T^{7} - 2934 T^{8} - 164 T^{9} + T^{10} )^{2} \)
$67$ \( \)\(45\!\cdots\!00\)\( + \)\(67\!\cdots\!00\)\( T + \)\(50\!\cdots\!00\)\( T^{2} - \)\(55\!\cdots\!20\)\( T^{3} + \)\(95\!\cdots\!16\)\( T^{4} + \)\(17\!\cdots\!76\)\( T^{5} + \)\(16\!\cdots\!68\)\( T^{6} - \)\(33\!\cdots\!20\)\( T^{7} + 5990306469992412488 T^{8} + 761254545017310560 T^{9} + 94687519777585056 T^{10} - 2596584783885816 T^{11} + 34414645641964 T^{12} + 631728043200 T^{13} + 153047558280 T^{14} - 5295332332 T^{15} + 91544525 T^{16} - 10076 T^{17} + 1568 T^{18} - 56 T^{19} + T^{20} \)
$71$ \( ( 244806049860225129 + 26462138966588814 T + 425377632786329 T^{2} - 28765379322000 T^{3} - 703660036618 T^{4} + 8999697764 T^{5} + 240698230 T^{6} - 1103152 T^{7} - 27599 T^{8} + 46 T^{9} + T^{10} )^{2} \)
$73$ \( \)\(70\!\cdots\!76\)\( - \)\(68\!\cdots\!92\)\( T + \)\(33\!\cdots\!32\)\( T^{2} + \)\(21\!\cdots\!00\)\( T^{3} + \)\(54\!\cdots\!89\)\( T^{4} - \)\(39\!\cdots\!92\)\( T^{5} + \)\(12\!\cdots\!16\)\( T^{6} - \)\(17\!\cdots\!52\)\( T^{7} + \)\(15\!\cdots\!88\)\( T^{8} - \)\(17\!\cdots\!92\)\( T^{9} + \)\(57\!\cdots\!56\)\( T^{10} - 8137291309317621492 T^{11} + 60050734238782862 T^{12} - 148348463028316 T^{13} + 5272289243912 T^{14} - 74514446556 T^{15} + 529282508 T^{16} - 244172 T^{17} + 10368 T^{18} - 144 T^{19} + T^{20} \)
$79$ \( \)\(19\!\cdots\!24\)\( + \)\(21\!\cdots\!00\)\( T^{2} + \)\(85\!\cdots\!16\)\( T^{4} + \)\(14\!\cdots\!12\)\( T^{6} + \)\(13\!\cdots\!52\)\( T^{8} + 69615472453936153600 T^{10} + 22522946417389980 T^{12} + 4510238122984 T^{14} + 543330808 T^{16} + 35948 T^{18} + T^{20} \)
$83$ \( \)\(74\!\cdots\!04\)\( + \)\(57\!\cdots\!04\)\( T + \)\(21\!\cdots\!52\)\( T^{2} + \)\(98\!\cdots\!60\)\( T^{3} + \)\(21\!\cdots\!84\)\( T^{4} - \)\(17\!\cdots\!40\)\( T^{5} + \)\(79\!\cdots\!68\)\( T^{6} + \)\(21\!\cdots\!60\)\( T^{7} + \)\(30\!\cdots\!00\)\( T^{8} - \)\(16\!\cdots\!36\)\( T^{9} + \)\(15\!\cdots\!36\)\( T^{10} + 4641436687516667184 T^{11} + 67817516123568732 T^{12} - 15000157707336 T^{13} + 1058744154312 T^{14} + 31694727404 T^{15} + 475870649 T^{16} - 142692 T^{17} + 2592 T^{18} + 72 T^{19} + T^{20} \)
$89$ \( \)\(13\!\cdots\!64\)\( + \)\(18\!\cdots\!64\)\( T^{2} + \)\(82\!\cdots\!72\)\( T^{4} + \)\(18\!\cdots\!72\)\( T^{6} + \)\(25\!\cdots\!80\)\( T^{8} + \)\(22\!\cdots\!64\)\( T^{10} + 12825623877814462044 T^{12} + 488447034783160 T^{14} + 11806232824 T^{16} + 164180 T^{18} + T^{20} \)
$97$ \( \)\(17\!\cdots\!76\)\( - \)\(64\!\cdots\!08\)\( T + \)\(12\!\cdots\!32\)\( T^{2} - \)\(10\!\cdots\!72\)\( T^{3} + \)\(43\!\cdots\!16\)\( T^{4} - \)\(42\!\cdots\!72\)\( T^{5} + \)\(19\!\cdots\!56\)\( T^{6} - \)\(14\!\cdots\!32\)\( T^{7} + \)\(31\!\cdots\!68\)\( T^{8} + \)\(66\!\cdots\!88\)\( T^{9} + \)\(20\!\cdots\!20\)\( T^{10} + 237487702270127056 T^{11} + 5083978143773048 T^{12} + 345064261887472 T^{13} + 7293366478304 T^{14} + 46180424952 T^{15} + 190402196 T^{16} + 1897160 T^{17} + 38088 T^{18} + 276 T^{19} + T^{20} \)
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