Properties

Label 65.2.o.a
Level $65$
Weight $2$
Character orbit 65.o
Analytic conductor $0.519$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 65 = 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 65.o (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.519027613138\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Defining polynomial: \(x^{20} + 26 x^{18} + 279 x^{16} + 1604 x^{14} + 5353 x^{12} + 10466 x^{10} + 11441 x^{8} + 6176 x^{6} + 1263 x^{4} + 78 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} + 26 x^{18} + 279 x^{16} + 1604 x^{14} + 5353 x^{12} + 10466 x^{10} + 11441 x^{8} + 6176 x^{6} + 1263 x^{4} + 78 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -20 \nu^{18} - 389 \nu^{16} - 2695 \nu^{14} - 7125 \nu^{12} + 1214 \nu^{10} + 39860 \nu^{8} + 68102 \nu^{6} + 46015 \nu^{4} + 16571 \nu^{2} + 2996 \nu - 409 \)\()/5992\)
\(\beta_{2}\)\(=\)\((\)\( 20 \nu^{18} + 389 \nu^{16} + 2695 \nu^{14} + 7125 \nu^{12} - 1214 \nu^{10} - 39860 \nu^{8} - 68102 \nu^{6} - 46015 \nu^{4} - 16571 \nu^{2} + 2996 \nu + 409 \)\()/5992\)
\(\beta_{3}\)\(=\)\((\)\( -69 \nu^{18} - 1647 \nu^{16} - 15798 \nu^{14} - 78322 \nu^{12} - 214723 \nu^{10} - 324081 \nu^{8} - 257858 \nu^{6} - 105030 \nu^{4} - 19961 \nu^{2} - 539 \)\()/1712\)
\(\beta_{4}\)\(=\)\((\)\(321 \nu^{19} - 483 \nu^{18} + 9951 \nu^{17} - 11529 \nu^{16} + 127330 \nu^{15} - 110586 \nu^{14} + 869910 \nu^{13} - 548254 \nu^{12} + 3425391 \nu^{11} - 1503061 \nu^{10} + 7807469 \nu^{9} - 2268567 \nu^{8} + 9753906 \nu^{7} - 1805006 \nu^{6} + 5800042 \nu^{5} - 735210 \nu^{4} + 1183313 \nu^{3} - 139727 \nu^{2} + 39483 \nu - 3773\)\()/23968\)
\(\beta_{5}\)\(=\)\((\)\(700 \nu^{19} + 617 \nu^{18} + 17360 \nu^{17} + 16607 \nu^{16} + 174468 \nu^{15} + 185192 \nu^{14} + 912240 \nu^{13} + 1108682 \nu^{12} + 2625448 \nu^{11} + 3846937 \nu^{10} + 3934784 \nu^{9} + 7752327 \nu^{8} + 2192820 \nu^{7} + 8524592 \nu^{6} - 974624 \nu^{5} + 4331072 \nu^{4} - 1268316 \nu^{3} + 669321 \nu^{2} - 170688 \nu + 26961\)\()/11984\)
\(\beta_{6}\)\(=\)\((\)\( -409 \nu^{19} - 10614 \nu^{17} - 113722 \nu^{15} - 653341 \nu^{13} - 2182252 \nu^{11} - 4281808 \nu^{9} - 4719229 \nu^{7} - 2594086 \nu^{5} - 562582 \nu^{3} - 48473 \nu - 2996 \)\()/5992\)
\(\beta_{7}\)\(=\)\((\)\(1635 \nu^{19} - 309 \nu^{18} + 41725 \nu^{17} - 7171 \nu^{16} + 436590 \nu^{15} - 66542 \nu^{14} + 2423698 \nu^{13} - 319614 \nu^{12} + 7689981 \nu^{11} - 870831 \nu^{10} + 13910023 \nu^{9} - 1440917 \nu^{8} + 13309334 \nu^{7} - 1613590 \nu^{6} + 5445918 \nu^{5} - 1272982 \nu^{4} + 481307 \nu^{3} - 501629 \nu^{2} + 71073 \nu - 30699\)\()/23968\)
\(\beta_{8}\)\(=\)\((\)\(700 \nu^{19} - 617 \nu^{18} + 17360 \nu^{17} - 16607 \nu^{16} + 174468 \nu^{15} - 185192 \nu^{14} + 912240 \nu^{13} - 1108682 \nu^{12} + 2625448 \nu^{11} - 3846937 \nu^{10} + 3934784 \nu^{9} - 7752327 \nu^{8} + 2192820 \nu^{7} - 8524592 \nu^{6} - 974624 \nu^{5} - 4331072 \nu^{4} - 1268316 \nu^{3} - 669321 \nu^{2} - 170688 \nu - 26961\)\()/11984\)
\(\beta_{9}\)\(=\)\((\)\(-419 \nu^{19} + 4593 \nu^{18} - 12681 \nu^{17} + 118807 \nu^{16} - 158886 \nu^{15} + 1266230 \nu^{14} - 1064734 \nu^{13} + 7212374 \nu^{12} - 4108073 \nu^{11} + 23752795 \nu^{10} - 9110155 \nu^{9} + 45500681 \nu^{8} - 10820986 \nu^{7} + 47979462 \nu^{6} - 5625126 \nu^{5} + 23917926 \nu^{4} - 518719 \nu^{3} + 3775361 \nu^{2} + 159919 \nu + 83703\)\()/23968\)
\(\beta_{10}\)\(=\)\((\)\(-3773 \nu^{19} - 565 \nu^{18} - 97615 \nu^{17} - 11551 \nu^{16} - 1041138 \nu^{15} - 83062 \nu^{14} - 5941306 \nu^{13} - 198098 \nu^{12} - 19648615 \nu^{11} + 414413 \nu^{10} - 37985157 \nu^{9} + 3110895 \nu^{8} - 40898326 \nu^{7} + 5871486 \nu^{6} - 21497042 \nu^{5} + 4145562 \nu^{4} - 4030089 \nu^{3} + 861543 \nu^{2} - 166551 \nu + 81509\)\()/23968\)
\(\beta_{11}\)\(=\)\((\)\(3773 \nu^{19} + 321 \nu^{18} + 97615 \nu^{17} + 9951 \nu^{16} + 1041138 \nu^{15} + 127330 \nu^{14} + 5941306 \nu^{13} + 869910 \nu^{12} + 19648615 \nu^{11} + 3425391 \nu^{10} + 37985157 \nu^{9} + 7807469 \nu^{8} + 40898326 \nu^{7} + 9753906 \nu^{6} + 21497042 \nu^{5} + 5800042 \nu^{4} + 4030089 \nu^{3} + 1183313 \nu^{2} + 154567 \nu + 39483\)\()/23968\)
\(\beta_{12}\)\(=\)\((\)\(3773 \nu^{19} - 321 \nu^{18} + 97615 \nu^{17} - 9951 \nu^{16} + 1041138 \nu^{15} - 127330 \nu^{14} + 5941306 \nu^{13} - 869910 \nu^{12} + 19648615 \nu^{11} - 3425391 \nu^{10} + 37985157 \nu^{9} - 7807469 \nu^{8} + 40898326 \nu^{7} - 9753906 \nu^{6} + 21497042 \nu^{5} - 5800042 \nu^{4} + 4030089 \nu^{3} - 1183313 \nu^{2} + 154567 \nu - 39483\)\()/23968\)
\(\beta_{13}\)\(=\)\((\)\(-2753 \nu^{19} - 867 \nu^{18} - 71035 \nu^{17} - 22593 \nu^{16} - 754642 \nu^{15} - 243222 \nu^{14} - 4279914 \nu^{13} - 1404094 \nu^{12} - 14010639 \nu^{11} - 4705845 \nu^{10} - 26598933 \nu^{9} - 9214959 \nu^{8} - 27637370 \nu^{7} - 9971998 \nu^{6} - 13384022 \nu^{5} - 5100714 \nu^{4} - 1923401 \nu^{3} - 798859 \nu^{2} - 7127 \nu - 15221\)\()/11984\)
\(\beta_{14}\)\(=\)\((\)\(2753 \nu^{19} - 867 \nu^{18} + 71035 \nu^{17} - 22593 \nu^{16} + 754642 \nu^{15} - 243222 \nu^{14} + 4279914 \nu^{13} - 1404094 \nu^{12} + 14010639 \nu^{11} - 4705845 \nu^{10} + 26598933 \nu^{9} - 9214959 \nu^{8} + 27637370 \nu^{7} - 9971998 \nu^{6} + 13384022 \nu^{5} - 5100714 \nu^{4} + 1923401 \nu^{3} - 798859 \nu^{2} + 7127 \nu - 15221\)\()/11984\)
\(\beta_{15}\)\(=\)\((\)\(-6965 \nu^{19} + 1863 \nu^{18} - 180971 \nu^{17} + 49605 \nu^{16} - 1940134 \nu^{15} + 545958 \nu^{14} - 11139534 \nu^{13} + 3217222 \nu^{12} - 37111487 \nu^{11} + 10951069 \nu^{10} - 72399061 \nu^{9} + 21535831 \nu^{8} - 78924566 \nu^{7} + 22849098 \nu^{6} - 42433790 \nu^{5} + 10804742 \nu^{4} - 8585045 \nu^{3} + 1254563 \nu^{2} - 504763 \nu + 1285\)\()/23968\)
\(\beta_{16}\)\(=\)\((\)\(8297 \nu^{19} - 175 \nu^{18} + 216915 \nu^{17} - 2093 \nu^{16} + 2342074 \nu^{15} + 9562 \nu^{14} + 13551722 \nu^{13} + 273770 \nu^{12} + 45488043 \nu^{11} + 1753171 \nu^{10} + 89206401 \nu^{9} + 5197801 \nu^{8} + 97040830 \nu^{7} + 7462350 \nu^{6} + 50921018 \nu^{5} + 4439554 \nu^{4} + 9271077 \nu^{3} + 591213 \nu^{2} + 419203 \nu + 23947\)\()/23968\)
\(\beta_{17}\)\(=\)\((\)\(-4979 \nu^{19} - 129835 \nu^{17} - 1398012 \nu^{15} - 8068552 \nu^{13} - 27041293 \nu^{11} - 53105203 \nu^{9} - 58320582 \nu^{7} - 31671424 \nu^{5} - 6571905 \nu^{3} + 5992 \nu^{2} - 381235 \nu + 11984\)\()/11984\)
\(\beta_{18}\)\(=\)\((\)\(10417 \nu^{19} - 483 \nu^{18} + 268635 \nu^{17} - 11529 \nu^{16} + 2853942 \nu^{15} - 110586 \nu^{14} + 16210930 \nu^{13} - 548254 \nu^{12} + 53334711 \nu^{11} - 1503061 \nu^{10} + 102576749 \nu^{9} - 2268567 \nu^{8} + 110068986 \nu^{7} - 1805006 \nu^{6} + 58130790 \nu^{5} - 735210 \nu^{4} + 11443805 \nu^{3} - 127743 \nu^{2} + 679767 \nu + 20195\)\()/23968\)
\(\beta_{19}\)\(=\)\((\)\(-16083 \nu^{19} - 483 \nu^{18} - 419809 \nu^{17} - 11529 \nu^{16} - 4525598 \nu^{15} - 110586 \nu^{14} - 26151990 \nu^{13} - 548254 \nu^{12} - 87742737 \nu^{11} - 1503061 \nu^{10} - 172338195 \nu^{9} - 2268567 \nu^{8} - 188614114 \nu^{7} - 1805006 \nu^{6} - 100694134 \nu^{5} - 735210 \nu^{4} - 19274543 \nu^{3} - 139727 \nu^{2} - 820129 \nu - 15757\)\()/23968\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{2} + \beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{12} + \beta_{10} - \beta_{3} + \beta_{1} - 2\)
\(\nu^{3}\)\(=\)\(\beta_{19} + 2 \beta_{18} - \beta_{13} - 3 \beta_{12} - 2 \beta_{11} - 2 \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{3} - 6 \beta_{2} - 7 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{19} - \beta_{18} + \beta_{17} + 2 \beta_{16} - 2 \beta_{15} + 3 \beta_{14} - 6 \beta_{12} - 2 \beta_{11} - 7 \beta_{10} + \beta_{9} - 3 \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + 7 \beta_{3} - \beta_{2} - 7 \beta_{1} + 8\)
\(\nu^{5}\)\(=\)\(-9 \beta_{19} - 17 \beta_{18} - \beta_{17} - 3 \beta_{14} + 11 \beta_{13} + 26 \beta_{12} + 17 \beta_{11} + 17 \beta_{10} + 8 \beta_{9} + 10 \beta_{8} + 8 \beta_{7} + 5 \beta_{6} - 6 \beta_{5} - 18 \beta_{4} + 9 \beta_{3} + 37 \beta_{2} + 46 \beta_{1} - 2\)
\(\nu^{6}\)\(=\)\(-13 \beta_{19} + 13 \beta_{18} - 13 \beta_{17} - 26 \beta_{16} + 26 \beta_{15} - 32 \beta_{14} + 4 \beta_{13} + 38 \beta_{12} + 25 \beta_{11} + 53 \beta_{10} - 10 \beta_{9} + 37 \beta_{8} + 10 \beta_{7} - 13 \beta_{6} - 11 \beta_{5} + 16 \beta_{4} - 53 \beta_{3} + 18 \beta_{2} + 51 \beta_{1} - 39\)
\(\nu^{7}\)\(=\)\(66 \beta_{19} + 122 \beta_{18} + 14 \beta_{17} + 37 \beta_{14} - 89 \beta_{13} - 189 \beta_{12} - 121 \beta_{11} - 120 \beta_{10} - 52 \beta_{9} - 76 \beta_{8} - 52 \beta_{7} - 24 \beta_{6} + 28 \beta_{5} + 138 \beta_{4} - 69 \beta_{3} - 236 \beta_{2} - 304 \beta_{1} + 21\)
\(\nu^{8}\)\(=\)\(120 \beta_{19} - 116 \beta_{18} + 116 \beta_{17} + 232 \beta_{16} - 240 \beta_{15} + 260 \beta_{14} - 56 \beta_{13} - 252 \beta_{12} - 229 \beta_{11} - 397 \beta_{10} + 76 \beta_{9} - 332 \beta_{8} - 84 \beta_{7} + 112 \beta_{6} + 100 \beta_{5} - 152 \beta_{4} + 398 \beta_{3} - 185 \beta_{2} - 368 \beta_{1} + 213\)
\(\nu^{9}\)\(=\)\(-462 \beta_{19} - 840 \beta_{18} - 136 \beta_{17} - 332 \beta_{14} + 657 \beta_{13} + 1306 \beta_{12} + 818 \beta_{11} + 813 \beta_{10} + 325 \beta_{9} + 537 \beta_{8} + 325 \beta_{7} + 122 \beta_{6} - 113 \beta_{5} - 1011 \beta_{4} + 506 \beta_{3} + 1544 \beta_{2} + 2032 \beta_{1} - 170\)
\(\nu^{10}\)\(=\)\(-974 \beta_{19} + 914 \beta_{18} - 914 \beta_{17} - 1828 \beta_{16} + 1948 \beta_{15} - 1941 \beta_{14} + 546 \beta_{13} + 1707 \beta_{12} + 1862 \beta_{11} + 2910 \beta_{10} - 539 \beta_{9} + 2648 \beta_{8} + 659 \beta_{7} - 854 \beta_{6} - 820 \beta_{5} + 1229 \beta_{4} - 2933 \beta_{3} + 1578 \beta_{2} + 2621 \beta_{1} - 1255\)
\(\nu^{11}\)\(=\)\(3210 \beta_{19} + 5733 \beta_{18} + 1135 \beta_{17} + 2648 \beta_{14} - 4690 \beta_{13} - 8890 \beta_{12} - 5456 \beta_{11} - 5476 \beta_{10} - 2042 \beta_{9} - 3716 \beta_{8} - 2042 \beta_{7} - 676 \beta_{6} + 368 \beta_{5} + 7277 \beta_{4} - 3655 \beta_{3} - 10296 \beta_{2} - 13730 \beta_{1} + 1267\)
\(\nu^{12}\)\(=\)\(7438 \beta_{19} - 6832 \beta_{18} + 6832 \beta_{17} + 13664 \beta_{16} - 14876 \beta_{15} + 14032 \beta_{14} - 4610 \beta_{13} - 11688 \beta_{12} - 14278 \beta_{11} - 20988 \beta_{10} + 3766 \beta_{9} - 19984 \beta_{8} - 4978 \beta_{7} + 6226 \beta_{6} + 6320 \beta_{5} - 9292 \beta_{4} + 21288 \beta_{3} - 12378 \beta_{2} - 18508 \beta_{1} + 7797\)
\(\nu^{13}\)\(=\)\(-22356 \beta_{19} - 39188 \beta_{18} - 8780 \beta_{17} - 19984 \beta_{14} + 33058 \beta_{13} + 60422 \beta_{12} + 36438 \beta_{11} + 37058 \beta_{10} + 13074 \beta_{9} + 25620 \beta_{8} + 13074 \beta_{7} + 4096 \beta_{6} - 528 \beta_{5} - 51934 \beta_{4} + 26218 \beta_{3} + 69645 \beta_{2} + 93629 \beta_{1} - 9130\)
\(\nu^{14}\)\(=\)\(-54982 \beta_{19} + 49778 \beta_{18} - 49778 \beta_{17} - 99556 \beta_{16} + 109964 \beta_{15} - 100084 \beta_{14} + 36202 \beta_{13} + 80555 \beta_{12} + 105990 \beta_{11} + 149815 \beta_{10} - 26322 \beta_{9} + 146494 \beta_{8} + 36730 \beta_{7} - 44574 \beta_{6} - 46938 \beta_{5} + 68030 \beta_{4} - 152877 \beta_{3} + 92972 \beta_{2} + 130077 \beta_{1} - 50290\)
\(\nu^{15}\)\(=\)\(156271 \beta_{19} + 269080 \beta_{18} + 65190 \beta_{17} + 146494 \beta_{14} - 231975 \beta_{13} - 412141 \beta_{12} - 245006 \beta_{11} - 252616 \beta_{10} - 85481 \beta_{9} - 176993 \beta_{8} - 85481 \beta_{7} - 26707 \beta_{6} - 6031 \beta_{5} + 368826 \beta_{4} - 187213 \beta_{3} - 476096 \beta_{2} - 643231 \beta_{1} + 64782\)
\(\nu^{16}\)\(=\)\(398897 \beta_{19} - 357747 \beta_{18} + 357747 \beta_{17} + 715494 \beta_{16} - 797794 \beta_{15} + 709463 \beta_{14} - 272912 \beta_{13} - 557750 \beta_{12} - 771768 \beta_{11} - 1062637 \beta_{10} + 184581 \beta_{9} - 1056441 \beta_{8} - 266881 \beta_{7} + 316597 \beta_{6} + 340947 \beta_{5} - 489763 \beta_{4} + 1090029 \beta_{3} - 681429 \beta_{2} - 912121 \beta_{1} + 333034\)
\(\nu^{17}\)\(=\)\(-1095413 \beta_{19} - 1856511 \beta_{18} - 472963 \beta_{17} - 1056441 \beta_{14} + 1625819 \beta_{13} + 2825568 \beta_{12} + 1660831 \beta_{11} + 1734115 \beta_{10} + 569378 \beta_{9} + 1226922 \beta_{8} + 569378 \beta_{7} + 182897 \beta_{6} + 88166 \beta_{5} - 2610994 \beta_{4} + 1332033 \beta_{3} + 3279739 \beta_{2} + 4444476 \beta_{1} - 456258\)
\(\nu^{18}\)\(=\)\(-2861437 \beta_{19} + 2550751 \beta_{18} - 2550751 \beta_{17} - 5101502 \beta_{16} + 5722874 \beta_{15} - 5013336 \beta_{14} + 2007386 \beta_{13} + 3875098 \beta_{12} + 5552425 \beta_{11} + 7508303 \beta_{10} - 1297848 \beta_{9} + 7545155 \beta_{8} + 1919220 \beta_{7} - 2240065 \beta_{6} - 2443653 \beta_{5} + 3492968 \beta_{4} - 7734751 \beta_{3} + 4921700 \beta_{2} + 6390257 \beta_{1} - 2246355\)
\(\nu^{19}\)\(=\)\(7691530 \beta_{19} + 12862874 \beta_{18} + 3385870 \beta_{17} + 7545155 \beta_{14} - 11394317 \beta_{13} - 19469181 \beta_{12} - 11344809 \beta_{11} - 11973534 \beta_{10} - 3849162 \beta_{9} - 8532880 \beta_{8} - 3849162 \beta_{7} - 1287312 \beta_{6} - 834556 \beta_{5} + 18442656 \beta_{4} - 9449577 \beta_{3} - 22718962 \beta_{2} - 30843334 \beta_{1} + 3202109\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/65\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(41\)
\(\chi(n)\) \(-\beta_{11} - \beta_{12}\) \(-\beta_{11}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
2.64975i
1.83163i
0.493902i
0.274809i
1.51805i
2.08794i
1.58474i
0.131303i
1.02262i
2.25081i
2.64975i
1.83163i
0.493902i
0.274809i
1.51805i
2.08794i
1.58474i
0.131303i
1.02262i
2.25081i
−1.32488 2.29475i −1.25278 0.335680i −2.51060 + 4.34849i −1.30391 1.81654i 0.889471 + 3.31955i 0.0972962 + 0.0561740i 8.00544 −1.14131 0.658935i −2.44100 + 5.39885i
2.2 −0.915816 1.58624i 1.91432 + 0.512942i −0.677439 + 1.17336i 1.45480 + 1.69810i −0.939520 3.50634i −3.06478 1.76945i −1.18163 0.803451 + 0.463873i 1.36126 3.86282i
2.3 −0.246951 0.427732i 0.908353 + 0.243392i 0.878030 1.52079i −2.21791 + 0.284413i −0.120212 0.448637i 3.18307 + 1.83775i −1.85513 −1.83221 1.05783i 0.669366 + 0.878433i
2.4 −0.137404 0.237991i −2.28256 0.611610i 0.962240 1.66665i 1.69883 1.45395i 0.168076 + 0.627267i −0.334376 0.193052i −1.07848 2.23793 + 1.29207i −0.579454 0.204528i
2.5 0.759023 + 1.31467i −0.653367 0.175069i −0.152233 + 0.263675i 0.600231 + 2.15400i −0.265763 0.991842i −2.24723 1.29744i 2.57390 −2.20184 1.27123i −2.37621 + 2.42404i
32.1 −1.04397 + 1.80821i 0.713171 + 2.66159i −1.17974 2.04338i −0.194361 2.22760i −5.55724 1.48906i 2.52122 1.45563i 0.750585 −3.97738 + 2.29634i 4.23088 + 1.97411i
32.2 −0.792369 + 1.37242i −0.0510678 0.190588i −0.255697 0.442881i 0.0672627 + 2.23506i 0.302032 + 0.0809291i −0.474866 + 0.274164i −2.35905 2.56436 1.48053i −3.12074 1.67868i
32.3 0.0656513 0.113711i −0.0890070 0.332179i 0.991380 + 1.71712i 0.813169 2.08297i −0.0436159 0.0116869i −2.40874 + 1.39069i 0.522947 2.49566 1.44087i −0.183472 0.229216i
32.4 0.511309 0.885613i −0.721300 2.69193i 0.477126 + 0.826407i −1.69584 + 1.45744i −2.75281 0.737614i 0.834479 0.481787i 3.02107 −4.12812 + 2.38337i 0.423625 + 2.24706i
32.5 1.12540 1.94926i 0.514229 + 1.91913i −1.53307 2.65535i −2.22228 0.247944i 4.31958 + 1.15743i −1.10607 + 0.638592i −2.39966 −0.820542 + 0.473740i −2.98427 + 4.05275i
33.1 −1.32488 + 2.29475i −1.25278 + 0.335680i −2.51060 4.34849i −1.30391 + 1.81654i 0.889471 3.31955i 0.0972962 0.0561740i 8.00544 −1.14131 + 0.658935i −2.44100 5.39885i
33.2 −0.915816 + 1.58624i 1.91432 0.512942i −0.677439 1.17336i 1.45480 1.69810i −0.939520 + 3.50634i −3.06478 + 1.76945i −1.18163 0.803451 0.463873i 1.36126 + 3.86282i
33.3 −0.246951 + 0.427732i 0.908353 0.243392i 0.878030 + 1.52079i −2.21791 0.284413i −0.120212 + 0.448637i 3.18307 1.83775i −1.85513 −1.83221 + 1.05783i 0.669366 0.878433i
33.4 −0.137404 + 0.237991i −2.28256 + 0.611610i 0.962240 + 1.66665i 1.69883 + 1.45395i 0.168076 0.627267i −0.334376 + 0.193052i −1.07848 2.23793 1.29207i −0.579454 + 0.204528i
33.5 0.759023 1.31467i −0.653367 + 0.175069i −0.152233 0.263675i 0.600231 2.15400i −0.265763 + 0.991842i −2.24723 + 1.29744i 2.57390 −2.20184 + 1.27123i −2.37621 2.42404i
63.1 −1.04397 1.80821i 0.713171 2.66159i −1.17974 + 2.04338i −0.194361 + 2.22760i −5.55724 + 1.48906i 2.52122 + 1.45563i 0.750585 −3.97738 2.29634i 4.23088 1.97411i
63.2 −0.792369 1.37242i −0.0510678 + 0.190588i −0.255697 + 0.442881i 0.0672627 2.23506i 0.302032 0.0809291i −0.474866 0.274164i −2.35905 2.56436 + 1.48053i −3.12074 + 1.67868i
63.3 0.0656513 + 0.113711i −0.0890070 + 0.332179i 0.991380 1.71712i 0.813169 + 2.08297i −0.0436159 + 0.0116869i −2.40874 1.39069i 0.522947 2.49566 + 1.44087i −0.183472 + 0.229216i
63.4 0.511309 + 0.885613i −0.721300 + 2.69193i 0.477126 0.826407i −1.69584 1.45744i −2.75281 + 0.737614i 0.834479 + 0.481787i 3.02107 −4.12812 2.38337i 0.423625 2.24706i
63.5 1.12540 + 1.94926i 0.514229 1.91913i −1.53307 + 2.65535i −2.22228 + 0.247944i 4.31958 1.15743i −1.10607 0.638592i −2.39966 −0.820542 0.473740i −2.98427 4.05275i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 63.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.o even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 65.2.o.a 20
3.b odd 2 1 585.2.cf.a 20
5.b even 2 1 325.2.s.b 20
5.c odd 4 1 65.2.t.a yes 20
5.c odd 4 1 325.2.x.b 20
13.b even 2 1 845.2.o.g 20
13.c even 3 1 845.2.k.e 20
13.c even 3 1 845.2.o.e 20
13.d odd 4 1 845.2.t.e 20
13.d odd 4 1 845.2.t.f 20
13.e even 6 1 845.2.k.d 20
13.e even 6 1 845.2.o.f 20
13.f odd 12 1 65.2.t.a yes 20
13.f odd 12 1 845.2.f.d 20
13.f odd 12 1 845.2.f.e 20
13.f odd 12 1 845.2.t.g 20
15.e even 4 1 585.2.dp.a 20
39.k even 12 1 585.2.dp.a 20
65.f even 4 1 845.2.o.f 20
65.h odd 4 1 845.2.t.g 20
65.k even 4 1 845.2.o.e 20
65.o even 12 1 inner 65.2.o.a 20
65.o even 12 1 845.2.k.e 20
65.q odd 12 1 845.2.f.e 20
65.q odd 12 1 845.2.t.f 20
65.r odd 12 1 845.2.f.d 20
65.r odd 12 1 845.2.t.e 20
65.s odd 12 1 325.2.x.b 20
65.t even 12 1 325.2.s.b 20
65.t even 12 1 845.2.k.d 20
65.t even 12 1 845.2.o.g 20
195.bn odd 12 1 585.2.cf.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.o.a 20 1.a even 1 1 trivial
65.2.o.a 20 65.o even 12 1 inner
65.2.t.a yes 20 5.c odd 4 1
65.2.t.a yes 20 13.f odd 12 1
325.2.s.b 20 5.b even 2 1
325.2.s.b 20 65.t even 12 1
325.2.x.b 20 5.c odd 4 1
325.2.x.b 20 65.s odd 12 1
585.2.cf.a 20 3.b odd 2 1
585.2.cf.a 20 195.bn odd 12 1
585.2.dp.a 20 15.e even 4 1
585.2.dp.a 20 39.k even 12 1
845.2.f.d 20 13.f odd 12 1
845.2.f.d 20 65.r odd 12 1
845.2.f.e 20 13.f odd 12 1
845.2.f.e 20 65.q odd 12 1
845.2.k.d 20 13.e even 6 1
845.2.k.d 20 65.t even 12 1
845.2.k.e 20 13.c even 3 1
845.2.k.e 20 65.o even 12 1
845.2.o.e 20 13.c even 3 1
845.2.o.e 20 65.k even 4 1
845.2.o.f 20 13.e even 6 1
845.2.o.f 20 65.f even 4 1
845.2.o.g 20 13.b even 2 1
845.2.o.g 20 65.t even 12 1
845.2.t.e 20 13.d odd 4 1
845.2.t.e 20 65.r odd 12 1
845.2.t.f 20 13.d odd 4 1
845.2.t.f 20 65.q odd 12 1
845.2.t.g 20 13.f odd 12 1
845.2.t.g 20 65.h odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(65, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T + 41 T^{2} + 178 T^{3} + 1214 T^{4} + 2208 T^{5} + 4733 T^{6} + 3252 T^{7} + 6569 T^{8} + 4404 T^{9} + 5468 T^{10} + 2864 T^{11} + 2677 T^{12} + 1258 T^{13} + 893 T^{14} + 322 T^{15} + 170 T^{16} + 48 T^{17} + 21 T^{18} + 4 T^{19} + T^{20} \)
$3$ \( 16 + 96 T + 708 T^{2} + 1948 T^{3} + 4097 T^{4} + 2690 T^{5} - 8584 T^{6} - 11256 T^{7} + 5374 T^{8} + 8882 T^{9} + 832 T^{10} + 418 T^{11} + 595 T^{12} - 618 T^{13} - 188 T^{14} + 2 T^{15} - 2 T^{16} + 20 T^{17} + 8 T^{18} + 2 T^{19} + T^{20} \)
$5$ \( 9765625 + 11718750 T + 10546875 T^{2} + 8906250 T^{3} + 6140625 T^{4} + 3887500 T^{5} + 2295000 T^{6} + 1264500 T^{7} + 649950 T^{8} + 316320 T^{9} + 146954 T^{10} + 63264 T^{11} + 25998 T^{12} + 10116 T^{13} + 3672 T^{14} + 1244 T^{15} + 393 T^{16} + 114 T^{17} + 27 T^{18} + 6 T^{19} + T^{20} \)
$7$ \( 64 - 480 T - 1128 T^{2} + 17460 T^{3} + 94417 T^{4} + 166374 T^{5} + 54704 T^{6} - 153888 T^{7} - 73546 T^{8} + 175722 T^{9} + 198624 T^{10} + 46566 T^{11} - 24333 T^{12} - 10674 T^{13} + 2792 T^{14} + 1950 T^{15} + 54 T^{16} - 120 T^{17} - 8 T^{18} + 6 T^{19} + T^{20} \)
$11$ \( 256 + 15616 T + 147920 T^{2} - 3781400 T^{3} + 14856881 T^{4} - 2759960 T^{5} + 33036500 T^{6} - 15386688 T^{7} + 2684514 T^{8} - 1623624 T^{9} - 1360616 T^{10} + 30704 T^{11} + 266651 T^{12} + 196248 T^{13} + 81176 T^{14} + 23048 T^{15} + 5362 T^{16} + 976 T^{17} + 140 T^{18} + 16 T^{19} + T^{20} \)
$13$ \( 137858491849 - 21208998746 T + 12235960815 T^{2} - 1631461442 T^{3} + 699887305 T^{4} - 350500592 T^{5} + 32102564 T^{6} + 456976 T^{7} - 1624090 T^{8} + 970580 T^{9} + 342346 T^{10} + 74660 T^{11} - 9610 T^{12} + 208 T^{13} + 1124 T^{14} - 944 T^{15} + 145 T^{16} - 26 T^{17} + 15 T^{18} - 2 T^{19} + T^{20} \)
$17$ \( 1168561 - 7415660 T + 174841793 T^{2} - 5758158 T^{3} + 44759067 T^{4} + 26251528 T^{5} - 5096966 T^{6} + 2745932 T^{7} - 121343 T^{8} - 1056080 T^{9} + 96031 T^{10} - 38730 T^{11} - 2399 T^{12} + 17836 T^{13} - 446 T^{14} + 444 T^{15} + 107 T^{16} - 232 T^{17} - 7 T^{18} + 10 T^{19} + T^{20} \)
$19$ \( 1583721616 + 3820734368 T + 1167831572 T^{2} + 463061396 T^{3} + 3184775849 T^{4} - 2773365368 T^{5} + 1830432156 T^{6} - 1254413560 T^{7} + 577282318 T^{8} - 205617120 T^{9} + 65790820 T^{10} - 15024876 T^{11} + 1989583 T^{12} + 74704 T^{13} - 116976 T^{14} + 25808 T^{15} - 1690 T^{16} - 440 T^{17} + 152 T^{18} - 20 T^{19} + T^{20} \)
$23$ \( 144 - 1296 T + 2916 T^{2} - 600 T^{3} - 21887 T^{4} + 187570 T^{5} + 260036 T^{6} + 474444 T^{7} + 4070922 T^{8} + 2935886 T^{9} + 2860352 T^{10} + 2226574 T^{11} + 1100975 T^{12} + 380770 T^{13} + 87332 T^{14} + 10138 T^{15} - 326 T^{16} - 344 T^{17} - 52 T^{18} + 2 T^{19} + T^{20} \)
$29$ \( 206213167449 - 1375693542936 T + 3313045860099 T^{2} - 1693561403784 T^{3} - 352134815045 T^{4} + 421814740512 T^{5} + 49257519182 T^{6} - 87906944184 T^{7} + 13003839581 T^{8} + 4237769376 T^{9} - 990563695 T^{10} - 156526392 T^{11} + 52332613 T^{12} + 1675704 T^{13} - 1236386 T^{14} - 14640 T^{15} + 20819 T^{16} - 173 T^{18} + T^{20} \)
$31$ \( 2166784 + 14602240 T + 49203200 T^{2} - 45220864 T^{3} + 317008128 T^{4} + 1365233664 T^{5} + 2473766912 T^{6} + 2175176448 T^{7} + 1058401664 T^{8} + 187005312 T^{9} + 8238080 T^{10} - 315200 T^{11} + 3495856 T^{12} + 416544 T^{13} + 5408 T^{14} - 12992 T^{15} + 6072 T^{16} - 104 T^{17} + T^{20} \)
$37$ \( 4508182449 - 23327492490 T + 49998796215 T^{2} - 50518059150 T^{3} + 19210271947 T^{4} + 5364502944 T^{5} - 2412054490 T^{6} - 9147396588 T^{7} + 11463108577 T^{8} - 6758027358 T^{9} + 2417790097 T^{10} - 546547590 T^{11} + 72292689 T^{12} - 3632340 T^{13} - 163690 T^{14} - 82248 T^{15} + 43003 T^{16} - 7854 T^{17} + 775 T^{18} - 42 T^{19} + T^{20} \)
$41$ \( 3748255729 - 2192518076 T - 3998259115 T^{2} + 21282088462 T^{3} - 38294393429 T^{4} - 45649423224 T^{5} + 114452844834 T^{6} - 20978724348 T^{7} + 6465808597 T^{8} - 344134940 T^{9} - 181905193 T^{10} - 11505094 T^{11} - 5999131 T^{12} + 1385824 T^{13} + 318594 T^{14} + 7852 T^{15} + 5227 T^{16} - 1188 T^{17} + 53 T^{18} - 10 T^{19} + T^{20} \)
$43$ \( 1370772640000 - 18203832560000 T + 57172378499600 T^{2} + 58430182109120 T^{3} + 12962355503081 T^{4} - 826155173166 T^{5} - 373100342360 T^{6} - 181959452144 T^{7} - 25100897594 T^{8} + 1062248466 T^{9} + 1284829516 T^{10} + 499552270 T^{11} + 114310435 T^{12} + 20569910 T^{13} + 3269428 T^{14} + 414538 T^{15} + 46734 T^{16} + 4500 T^{17} + 332 T^{18} + 22 T^{19} + T^{20} \)
$47$ \( 807469056 + 11345362944 T^{2} + 44521738240 T^{4} + 54003212288 T^{6} + 17345956864 T^{8} + 2242437632 T^{10} + 133025920 T^{12} + 3738752 T^{14} + 52960 T^{16} + 368 T^{18} + T^{20} \)
$53$ \( 2978634160384 + 1638501455872 T + 450657394688 T^{2} + 451854140672 T^{3} + 938935450496 T^{4} + 662260749312 T^{5} + 256514576512 T^{6} + 53765212096 T^{7} + 8410884016 T^{8} + 2137889376 T^{9} + 822713344 T^{10} + 169978000 T^{11} + 19301176 T^{12} + 1286456 T^{13} + 475072 T^{14} + 100768 T^{15} + 10713 T^{16} + 78 T^{17} + 50 T^{18} + 10 T^{19} + T^{20} \)
$59$ \( 33856 + 28392672 T + 6360946824 T^{2} - 9278523364 T^{3} + 8028346497 T^{4} - 6999681172 T^{5} + 4362869144 T^{6} - 1471550856 T^{7} + 112358362 T^{8} + 234546884 T^{9} - 132826648 T^{10} + 26431368 T^{11} + 5027435 T^{12} - 5604924 T^{13} + 1943264 T^{14} - 363348 T^{15} + 43194 T^{16} - 4136 T^{17} + 320 T^{18} - 16 T^{19} + T^{20} \)
$61$ \( 826457355409 + 152931933728 T + 714855732255 T^{2} + 167561667840 T^{3} + 464934395195 T^{4} + 101940472768 T^{5} + 118218961526 T^{6} + 20885521424 T^{7} + 20383774133 T^{8} + 3839755696 T^{9} + 1784573189 T^{10} + 385162944 T^{11} + 121471973 T^{12} + 21057264 T^{13} + 3732310 T^{14} + 391232 T^{15} + 44619 T^{16} + 3184 T^{17} + 319 T^{18} + 16 T^{19} + T^{20} \)
$67$ \( 15478905336976 - 87299958482384 T + 620793296665580 T^{2} + 855544406309724 T^{3} + 710415918999441 T^{4} + 375296027359966 T^{5} + 149347427286784 T^{6} + 45533093686732 T^{7} + 11395073107926 T^{8} + 2376642817662 T^{9} + 433822790124 T^{10} + 69587684462 T^{11} + 9990258843 T^{12} + 1247036682 T^{13} + 133985072 T^{14} + 11877582 T^{15} + 858582 T^{16} + 48068 T^{17} + 2044 T^{18} + 58 T^{19} + T^{20} \)
$71$ \( 112163628352758544 + 124423487088552896 T + 74662563174729188 T^{2} + 31199257022438180 T^{3} + 9682913168509481 T^{4} + 2301254681694640 T^{5} + 419603164556892 T^{6} + 55401972752540 T^{7} + 4475729325790 T^{8} - 28579680612 T^{9} - 68391405476 T^{10} - 9818440548 T^{11} - 554987105 T^{12} + 39729340 T^{13} + 10077960 T^{14} + 615752 T^{15} - 10042 T^{16} - 5924 T^{17} - 256 T^{18} + 16 T^{19} + T^{20} \)
$73$ \( ( 253113232 - 309584640 T + 145220576 T^{2} - 29843280 T^{3} + 1070440 T^{4} + 651712 T^{5} - 115932 T^{6} + 5932 T^{7} + 249 T^{8} - 36 T^{9} + T^{10} )^{2} \)
$79$ \( 7586504765095936 + 5385471164325888 T^{2} + 1241871864168704 T^{4} + 134326970823168 T^{6} + 7680125227520 T^{8} + 243260666240 T^{10} + 4371796960 T^{12} + 45175136 T^{14} + 264160 T^{16} + 808 T^{18} + T^{20} \)
$83$ \( 11512611864576 + 13783820697600 T^{2} + 5842988808448 T^{4} + 1204294956032 T^{6} + 135217417088 T^{8} + 8634255104 T^{10} + 316478512 T^{12} + 6614080 T^{14} + 76568 T^{16} + 448 T^{18} + T^{20} \)
$89$ \( 329648222500 + 1177174003500 T + 2273701009050 T^{2} + 5083577201910 T^{3} + 8474675429921 T^{4} + 6925519621868 T^{5} + 3497464766988 T^{6} + 1358361684772 T^{7} + 446547141742 T^{8} + 120073964504 T^{9} + 25613241926 T^{10} + 4417984798 T^{11} + 621940219 T^{12} + 69631212 T^{13} + 6066956 T^{14} + 351592 T^{15} - 498 T^{16} - 1876 T^{17} - 150 T^{18} - 6 T^{19} + T^{20} \)
$97$ \( 28761567213630736 - 55405419075392720 T + 143232059180324060 T^{2} + 79826789169984204 T^{3} + 36437734918301265 T^{4} + 8992046654782666 T^{5} + 2040319607503288 T^{6} + 320016998462884 T^{7} + 53902773927654 T^{8} + 6669078921834 T^{9} + 926859773148 T^{10} + 85742329586 T^{11} + 9510255507 T^{12} + 663004590 T^{13} + 67111568 T^{14} + 3595530 T^{15} + 288774 T^{16} + 10988 T^{17} + 796 T^{18} + 22 T^{19} + T^{20} \)
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