Properties

Label 483.2.i.h
Level $483$
Weight $2$
Character orbit 483.i
Analytic conductor $3.857$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 483 = 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 483.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.85677441763\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 3 x^{19} + 22 x^{18} - 43 x^{17} + 245 x^{16} - 416 x^{15} + 1707 x^{14} - 2021 x^{13} + 7135 x^{12} - 6640 x^{11} + 20315 x^{10} - 10565 x^{9} + 29358 x^{8} - 9009 x^{7} + 30661 x^{6} - 4026 x^{5} + 15786 x^{4} - 84 x^{3} + 5800 x^{2} + 152 x + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( \beta_{1} + \beta_{4} ) q^{2} -\beta_{9} q^{3} + ( -\beta_{1} - \beta_{8} + \beta_{9} - \beta_{17} ) q^{4} + ( 1 - \beta_{6} + \beta_{9} + \beta_{18} ) q^{5} + \beta_{4} q^{6} + ( \beta_{14} - \beta_{16} ) q^{7} + ( \beta_{3} - 3 \beta_{4} + \beta_{5} - \beta_{7} + \beta_{10} + \beta_{11} + \beta_{12} + \beta_{14} ) q^{8} + ( -1 - \beta_{9} ) q^{9} +O(q^{10})\) \( q + ( \beta_{1} + \beta_{4} ) q^{2} -\beta_{9} q^{3} + ( -\beta_{1} - \beta_{8} + \beta_{9} - \beta_{17} ) q^{4} + ( 1 - \beta_{6} + \beta_{9} + \beta_{18} ) q^{5} + \beta_{4} q^{6} + ( \beta_{14} - \beta_{16} ) q^{7} + ( \beta_{3} - 3 \beta_{4} + \beta_{5} - \beta_{7} + \beta_{10} + \beta_{11} + \beta_{12} + \beta_{14} ) q^{8} + ( -1 - \beta_{9} ) q^{9} + ( \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} - \beta_{12} - \beta_{13} - 2 \beta_{14} + \beta_{15} + \beta_{16} - \beta_{17} + \beta_{19} ) q^{10} + ( -\beta_{1} + \beta_{9} + \beta_{10} - \beta_{11} - \beta_{14} + \beta_{15} + \beta_{16} + \beta_{18} + \beta_{19} ) q^{11} + ( 1 - \beta_{1} + \beta_{2} - \beta_{8} + \beta_{9} - \beta_{17} ) q^{12} + ( \beta_{2} - \beta_{3} + \beta_{4} - \beta_{12} - \beta_{14} ) q^{13} + ( 1 + \beta_{1} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} + \beta_{17} - \beta_{18} - \beta_{19} ) q^{14} + ( 1 - \beta_{6} ) q^{15} + ( -3 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{8} - 3 \beta_{9} - \beta_{11} + \beta_{14} + \beta_{15} + 2 \beta_{17} - \beta_{19} ) q^{16} + ( -\beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{17} ) q^{17} -\beta_{1} q^{18} + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{7} - \beta_{8} - \beta_{11} - 2 \beta_{12} - \beta_{13} - \beta_{16} - \beta_{17} + \beta_{18} ) q^{19} + ( -2 - 2 \beta_{2} - 3 \beta_{4} - \beta_{5} - \beta_{7} + 3 \beta_{12} + 3 \beta_{14} + \beta_{15} - \beta_{16} ) q^{20} -\beta_{16} q^{21} + ( \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{6} + 2 \beta_{7} - \beta_{10} - \beta_{11} - \beta_{12} - \beta_{14} + \beta_{15} - \beta_{16} ) q^{22} + ( -1 - \beta_{9} ) q^{23} + ( \beta_{1} - \beta_{4} + \beta_{5} + \beta_{10} - \beta_{13} + \beta_{18} ) q^{24} + ( \beta_{4} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} - 2 \beta_{12} - \beta_{13} - 3 \beta_{14} + \beta_{15} + \beta_{16} - \beta_{17} + 2 \beta_{19} ) q^{25} + ( -\beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} - \beta_{11} - \beta_{13} - \beta_{14} + \beta_{15} + \beta_{16} ) q^{26} - q^{27} + ( 2 - \beta_{7} + \beta_{10} - \beta_{13} + \beta_{14} + \beta_{16} + \beta_{17} - \beta_{18} ) q^{28} + ( 1 + \beta_{2} + \beta_{3} + \beta_{6} ) q^{29} + ( 1 + \beta_{4} + \beta_{6} + \beta_{7} + \beta_{9} - \beta_{13} - \beta_{14} + \beta_{15} + \beta_{16} - \beta_{17} + \beta_{19} ) q^{30} + ( -\beta_{4} - \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{14} - \beta_{15} - \beta_{17} - \beta_{18} - \beta_{19} ) q^{31} + ( -\beta_{1} + \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} - 3 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} - \beta_{15} - 2 \beta_{16} + 2 \beta_{17} - 2 \beta_{18} - 3 \beta_{19} ) q^{32} + ( 1 - \beta_{1} + \beta_{7} + \beta_{9} + \beta_{10} - \beta_{11} - \beta_{14} + \beta_{15} + \beta_{16} + \beta_{18} + \beta_{19} ) q^{33} + ( \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{7} - 2 \beta_{12} - 2 \beta_{14} ) q^{34} + ( 1 - \beta_{2} + \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{12} - \beta_{14} + \beta_{16} + 2 \beta_{17} + \beta_{18} + \beta_{19} ) q^{35} + ( 1 + \beta_{2} ) q^{36} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{17} - \beta_{18} + 2 \beta_{19} ) q^{37} + ( \beta_{1} + \beta_{4} + \beta_{6} + \beta_{7} + \beta_{9} - 2 \beta_{10} + \beta_{11} + \beta_{12} - 2 \beta_{14} + \beta_{15} - \beta_{16} - 2 \beta_{17} - \beta_{18} ) q^{38} + ( \beta_{8} - \beta_{14} + \beta_{15} + \beta_{16} ) q^{39} + ( -3 - \beta_{2} + \beta_{3} - 4 \beta_{4} - 4 \beta_{6} - 3 \beta_{7} + \beta_{8} - 3 \beta_{9} + \beta_{10} + \beta_{11} + 3 \beta_{12} + 4 \beta_{13} + 2 \beta_{14} - \beta_{15} + 4 \beta_{17} + \beta_{18} - \beta_{19} ) q^{40} + ( -3 - \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + \beta_{5} - \beta_{15} + \beta_{16} ) q^{41} + ( -\beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{11} + \beta_{12} + \beta_{16} - \beta_{18} ) q^{42} + ( 3 - 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{43} + ( -1 + 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} - 3 \beta_{7} + 2 \beta_{8} - \beta_{9} - \beta_{10} + 3 \beta_{11} + 4 \beta_{12} + 2 \beta_{13} + \beta_{14} - \beta_{15} + \beta_{16} + 2 \beta_{17} - \beta_{18} - \beta_{19} ) q^{44} + ( -\beta_{9} - \beta_{18} ) q^{45} -\beta_{1} q^{46} + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} + 2 \beta_{14} - \beta_{18} - 2 \beta_{19} ) q^{47} + ( -3 - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} - \beta_{10} - \beta_{11} + \beta_{15} - \beta_{16} ) q^{48} + ( 2 - \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{9} + 2 \beta_{10} - \beta_{15} + \beta_{16} - \beta_{17} + \beta_{18} ) q^{49} + ( -2 - \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 3 \beta_{12} + 3 \beta_{14} + \beta_{15} - \beta_{16} ) q^{50} + ( -1 + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{13} + \beta_{17} ) q^{51} + ( \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + 3 \beta_{9} - \beta_{10} + 2 \beta_{12} + 2 \beta_{13} + \beta_{14} - \beta_{16} - \beta_{17} + \beta_{18} - \beta_{19} ) q^{52} + ( -\beta_{1} - 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{10} + \beta_{11} - \beta_{13} + 2 \beta_{14} - \beta_{15} + \beta_{17} - \beta_{18} - 2 \beta_{19} ) q^{53} + ( -\beta_{1} - \beta_{4} ) q^{54} + ( -3 + \beta_{2} - 2 \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{10} - \beta_{11} - \beta_{12} - \beta_{14} ) q^{55} + ( 4 + \beta_{1} + 6 \beta_{4} - 3 \beta_{5} + 2 \beta_{7} - \beta_{8} + 2 \beta_{9} - 3 \beta_{11} + \beta_{12} + \beta_{13} + \beta_{15} - \beta_{16} - \beta_{17} + \beta_{18} ) q^{56} + ( \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{7} - \beta_{10} - \beta_{11} - \beta_{12} - \beta_{14} + \beta_{15} - \beta_{16} ) q^{57} + ( 4 \beta_{1} + 5 \beta_{4} + \beta_{7} - \beta_{11} - 2 \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} - \beta_{16} - \beta_{17} + \beta_{18} ) q^{58} + ( \beta_{4} - \beta_{5} - 2 \beta_{9} - \beta_{10} + \beta_{13} - 2 \beta_{17} - \beta_{18} ) q^{59} + ( \beta_{1} - \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + \beta_{10} - \beta_{11} + 2 \beta_{12} + \beta_{13} + 2 \beta_{14} - \beta_{15} - 2 \beta_{16} - \beta_{17} + \beta_{18} ) q^{60} + ( -3 - \beta_{1} - 2 \beta_{4} - \beta_{7} - 3 \beta_{9} - \beta_{10} + \beta_{11} + \beta_{14} - \beta_{15} - \beta_{16} - \beta_{18} - \beta_{19} ) q^{61} + ( -3 - \beta_{2} - \beta_{4} - \beta_{5} + 2 \beta_{7} + \beta_{15} - \beta_{16} ) q^{62} -\beta_{14} q^{63} + ( 6 + 5 \beta_{3} - 5 \beta_{4} + \beta_{5} - 4 \beta_{6} - 4 \beta_{7} + 2 \beta_{10} + 2 \beta_{11} + \beta_{12} + \beta_{14} - 3 \beta_{15} + 3 \beta_{16} ) q^{64} + ( -1 - 2 \beta_{1} + 2 \beta_{2} - \beta_{4} + \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} - 2 \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} + 2 \beta_{14} - \beta_{15} - 2 \beta_{16} - 3 \beta_{17} - 2 \beta_{18} - 2 \beta_{19} ) q^{65} + ( 2 \beta_{1} + \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{10} + \beta_{11} - \beta_{14} - \beta_{16} ) q^{66} + ( 2 \beta_{1} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} + 2 \beta_{14} - \beta_{15} - \beta_{16} + \beta_{17} - \beta_{18} - 2 \beta_{19} ) q^{67} + ( 1 + 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{6} + 2 \beta_{7} + \beta_{9} + \beta_{11} - 2 \beta_{13} - 4 \beta_{14} + 2 \beta_{16} - 4 \beta_{17} - \beta_{18} + 3 \beta_{19} ) q^{68} - q^{69} + ( 5 - 2 \beta_{1} + \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{8} + 3 \beta_{9} - 3 \beta_{10} + \beta_{11} - \beta_{14} + 2 \beta_{15} - 2 \beta_{18} ) q^{70} + ( 2 + 2 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} - 3 \beta_{12} - 3 \beta_{14} ) q^{71} + ( \beta_{1} - \beta_{3} + 2 \beta_{4} + \beta_{7} - \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} + \beta_{18} ) q^{72} + ( \beta_{4} + \beta_{6} + \beta_{7} - \beta_{9} - 3 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} + \beta_{14} - 2 \beta_{15} - \beta_{16} - \beta_{17} - \beta_{19} ) q^{73} + ( -2 \beta_{1} - \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} + \beta_{13} + \beta_{14} - \beta_{16} - \beta_{17} + 3 \beta_{18} + \beta_{19} ) q^{74} + ( 2 - \beta_{2} + \beta_{3} + \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{11} - \beta_{13} - \beta_{14} + \beta_{15} + \beta_{16} - \beta_{17} + 2 \beta_{19} ) q^{75} + ( -4 - 2 \beta_{2} + 2 \beta_{3} - 10 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 6 \beta_{7} + 2 \beta_{10} + 2 \beta_{11} + 4 \beta_{12} + 4 \beta_{14} - \beta_{15} + \beta_{16} ) q^{76} + ( 2 - \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{5} - \beta_{7} + \beta_{8} + 3 \beta_{9} - \beta_{10} + \beta_{16} + \beta_{18} + \beta_{19} ) q^{77} + ( -\beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{10} - \beta_{11} ) q^{78} + ( -1 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - \beta_{9} + 2 \beta_{10} - \beta_{11} - \beta_{12} - 2 \beta_{14} + \beta_{15} + \beta_{16} + \beta_{18} + 3 \beta_{19} ) q^{79} + ( -\beta_{1} - \beta_{4} + \beta_{5} + 3 \beta_{8} - 6 \beta_{9} + \beta_{10} - 2 \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} + 3 \beta_{16} + 5 \beta_{17} - 3 \beta_{18} ) q^{80} + \beta_{9} q^{81} + ( -5 - 3 \beta_{1} - \beta_{2} - 2 \beta_{4} + 3 \beta_{6} + 2 \beta_{7} + \beta_{8} - 5 \beta_{9} - 3 \beta_{12} - 2 \beta_{13} + \beta_{15} - \beta_{16} - \beta_{17} - \beta_{18} + 2 \beta_{19} ) q^{82} + ( -1 + \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{10} + \beta_{11} - 2 \beta_{12} - 2 \beta_{14} ) q^{83} + ( -\beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{9} + \beta_{10} - 2 \beta_{11} + \beta_{14} + \beta_{15} + 2 \beta_{17} ) q^{84} + ( 5 - \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{5} - 3 \beta_{6} - 2 \beta_{12} - 2 \beta_{14} - \beta_{15} + \beta_{16} ) q^{85} + ( 2 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - 3 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{10} - \beta_{11} + \beta_{12} + 2 \beta_{13} - \beta_{15} + 2 \beta_{17} + 2 \beta_{18} - \beta_{19} ) q^{86} + ( \beta_{1} + \beta_{8} - \beta_{9} + 2 \beta_{17} + \beta_{18} ) q^{87} + ( -2 \beta_{1} - \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{10} - \beta_{11} - 4 \beta_{12} - 2 \beta_{13} - \beta_{14} + 3 \beta_{16} + 2 \beta_{17} + 2 \beta_{18} + 2 \beta_{19} ) q^{88} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{8} - \beta_{11} - 2 \beta_{12} - 2 \beta_{15} - 2 \beta_{16} - \beta_{19} ) q^{89} + ( 1 + \beta_{4} - \beta_{5} + \beta_{12} + \beta_{14} ) q^{90} + ( -2 - 2 \beta_{1} - 2 \beta_{3} + \beta_{6} + 2 \beta_{7} + \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} - \beta_{12} - \beta_{13} - 2 \beta_{14} + \beta_{15} + \beta_{17} + \beta_{18} ) q^{91} + ( 1 + \beta_{2} ) q^{92} + ( -1 + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{6} - 2 \beta_{8} - \beta_{9} - \beta_{11} - \beta_{12} + \beta_{14} - \beta_{16} - \beta_{17} - \beta_{18} - \beta_{19} ) q^{93} + ( -3 \beta_{1} - \beta_{4} - \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 3 \beta_{8} + 3 \beta_{9} + 3 \beta_{10} - 2 \beta_{11} + \beta_{13} + 3 \beta_{14} - \beta_{15} + \beta_{16} + \beta_{18} ) q^{94} + ( -\beta_{1} - 2 \beta_{4} + 3 \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} + 4 \beta_{9} + 2 \beta_{10} - 2 \beta_{12} - 3 \beta_{13} + \beta_{14} - 2 \beta_{15} - \beta_{16} - 2 \beta_{17} + \beta_{19} ) q^{95} + ( -\beta_{1} - 5 \beta_{4} - 2 \beta_{6} - 4 \beta_{7} - 2 \beta_{10} + 3 \beta_{11} + 3 \beta_{12} + 2 \beta_{13} + 3 \beta_{14} - 2 \beta_{15} - \beta_{16} + 2 \beta_{17} - 2 \beta_{18} - 3 \beta_{19} ) q^{96} + ( \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{6} + 4 \beta_{7} - \beta_{10} - \beta_{11} - 3 \beta_{12} - 3 \beta_{14} - \beta_{15} + \beta_{16} ) q^{97} + ( 2 + \beta_{1} - 3 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + 5 \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{11} - 4 \beta_{12} - 3 \beta_{13} - 4 \beta_{14} + 2 \beta_{15} - 2 \beta_{17} + 2 \beta_{19} ) q^{98} + ( 1 + \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q - 3q^{2} + 10q^{3} - 15q^{4} + 5q^{5} - 6q^{6} + 18q^{8} - 10q^{9} + O(q^{10}) \) \( 20q - 3q^{2} + 10q^{3} - 15q^{4} + 5q^{5} - 6q^{6} + 18q^{8} - 10q^{9} - 11q^{10} - 8q^{11} + 15q^{12} + 11q^{14} + 10q^{15} - 37q^{16} + 11q^{17} - 3q^{18} - q^{19} - 30q^{20} + 12q^{22} - 10q^{23} + 9q^{24} - 21q^{25} - q^{26} - 20q^{27} + 44q^{28} + 44q^{29} + 11q^{30} + 3q^{31} - 11q^{32} + 8q^{33} - 6q^{34} - 9q^{35} + 30q^{36} + 3q^{37} - 16q^{38} - 39q^{40} - 52q^{41} + 7q^{42} + 54q^{43} - 16q^{44} + 5q^{45} - 3q^{46} - 11q^{47} - 74q^{48} + 22q^{49} + 4q^{50} - 11q^{51} - 29q^{52} - 5q^{53} + 3q^{54} - 36q^{55} + 43q^{56} - 2q^{57} - 16q^{58} + 10q^{59} - 15q^{60} - 22q^{61} - 64q^{62} + 138q^{64} + 11q^{65} + 6q^{66} + 2q^{67} + 21q^{68} - 20q^{69} + 84q^{70} + 54q^{71} - 9q^{72} + 8q^{73} - 14q^{74} + 21q^{75} - 44q^{76} + 8q^{77} - 2q^{78} - 21q^{79} + 53q^{80} - 10q^{81} - 36q^{82} - 24q^{83} + 25q^{84} + 46q^{85} - 18q^{86} + 22q^{87} + 10q^{88} - 6q^{89} + 22q^{90} - 62q^{91} + 30q^{92} - 3q^{93} - 35q^{94} - 44q^{95} + 11q^{96} + 12q^{97} + 2q^{98} + 16q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 3 x^{19} + 22 x^{18} - 43 x^{17} + 245 x^{16} - 416 x^{15} + 1707 x^{14} - 2021 x^{13} + 7135 x^{12} - 6640 x^{11} + 20315 x^{10} - 10565 x^{9} + 29358 x^{8} - 9009 x^{7} + 30661 x^{6} - 4026 x^{5} + 15786 x^{4} - 84 x^{3} + 5800 x^{2} + 152 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(22\!\cdots\!01\)\( \nu^{19} - \)\(72\!\cdots\!30\)\( \nu^{18} + \)\(18\!\cdots\!45\)\( \nu^{17} - \)\(15\!\cdots\!11\)\( \nu^{16} + \)\(27\!\cdots\!00\)\( \nu^{15} - \)\(17\!\cdots\!95\)\( \nu^{14} + \)\(27\!\cdots\!89\)\( \nu^{13} - \)\(11\!\cdots\!00\)\( \nu^{12} + \)\(12\!\cdots\!00\)\( \nu^{11} - \)\(48\!\cdots\!85\)\( \nu^{10} + \)\(39\!\cdots\!75\)\( \nu^{9} - \)\(13\!\cdots\!20\)\( \nu^{8} + \)\(52\!\cdots\!85\)\( \nu^{7} - \)\(16\!\cdots\!65\)\( \nu^{6} + \)\(37\!\cdots\!10\)\( \nu^{5} - \)\(20\!\cdots\!17\)\( \nu^{4} + \)\(10\!\cdots\!60\)\( \nu^{3} - \)\(10\!\cdots\!38\)\( \nu^{2} - \)\(27\!\cdots\!90\)\( \nu - \)\(20\!\cdots\!14\)\(\)\()/ \)\(69\!\cdots\!38\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(22\!\cdots\!60\)\( \nu^{19} - \)\(25\!\cdots\!28\)\( \nu^{18} + \)\(10\!\cdots\!88\)\( \nu^{17} - \)\(49\!\cdots\!89\)\( \nu^{16} + \)\(12\!\cdots\!94\)\( \nu^{15} - \)\(51\!\cdots\!08\)\( \nu^{14} + \)\(10\!\cdots\!12\)\( \nu^{13} - \)\(32\!\cdots\!17\)\( \nu^{12} + \)\(44\!\cdots\!00\)\( \nu^{11} - \)\(12\!\cdots\!04\)\( \nu^{10} + \)\(12\!\cdots\!52\)\( \nu^{9} - \)\(30\!\cdots\!95\)\( \nu^{8} + \)\(13\!\cdots\!12\)\( \nu^{7} - \)\(32\!\cdots\!90\)\( \nu^{6} + \)\(92\!\cdots\!64\)\( \nu^{5} - \)\(42\!\cdots\!98\)\( \nu^{4} + \)\(83\!\cdots\!78\)\( \nu^{3} - \)\(21\!\cdots\!80\)\( \nu^{2} - \)\(57\!\cdots\!88\)\( \nu - \)\(13\!\cdots\!22\)\(\)\()/ \)\(36\!\cdots\!14\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(62\!\cdots\!25\)\( \nu^{19} + \)\(18\!\cdots\!76\)\( \nu^{18} - \)\(12\!\cdots\!20\)\( \nu^{17} + \)\(24\!\cdots\!30\)\( \nu^{16} - \)\(13\!\cdots\!14\)\( \nu^{15} + \)\(23\!\cdots\!00\)\( \nu^{14} - \)\(88\!\cdots\!80\)\( \nu^{13} + \)\(98\!\cdots\!36\)\( \nu^{12} - \)\(32\!\cdots\!75\)\( \nu^{11} + \)\(28\!\cdots\!00\)\( \nu^{10} - \)\(78\!\cdots\!90\)\( \nu^{9} + \)\(25\!\cdots\!50\)\( \nu^{8} - \)\(49\!\cdots\!30\)\( \nu^{7} + \)\(33\!\cdots\!40\)\( \nu^{6} - \)\(23\!\cdots\!60\)\( \nu^{5} - \)\(12\!\cdots\!60\)\( \nu^{4} + \)\(10\!\cdots\!67\)\( \nu^{3} - \)\(96\!\cdots\!60\)\( \nu^{2} - \)\(25\!\cdots\!00\)\( \nu + \)\(18\!\cdots\!90\)\(\)\()/ \)\(69\!\cdots\!38\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(18\!\cdots\!73\)\( \nu^{19} + \)\(43\!\cdots\!07\)\( \nu^{18} - \)\(34\!\cdots\!22\)\( \nu^{17} + \)\(45\!\cdots\!54\)\( \nu^{16} - \)\(35\!\cdots\!77\)\( \nu^{15} + \)\(37\!\cdots\!26\)\( \nu^{14} - \)\(21\!\cdots\!06\)\( \nu^{13} + \)\(81\!\cdots\!86\)\( \nu^{12} - \)\(73\!\cdots\!45\)\( \nu^{11} + \)\(51\!\cdots\!58\)\( \nu^{10} - \)\(15\!\cdots\!36\)\( \nu^{9} - \)\(15\!\cdots\!00\)\( \nu^{8} - \)\(52\!\cdots\!28\)\( \nu^{7} - \)\(29\!\cdots\!55\)\( \nu^{6} - \)\(35\!\cdots\!76\)\( \nu^{5} - \)\(39\!\cdots\!58\)\( \nu^{4} + \)\(16\!\cdots\!30\)\( \nu^{3} - \)\(21\!\cdots\!72\)\( \nu^{2} - \)\(55\!\cdots\!24\)\( \nu - \)\(89\!\cdots\!04\)\(\)\()/ \)\(36\!\cdots\!14\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(20\!\cdots\!92\)\( \nu^{19} - \)\(24\!\cdots\!70\)\( \nu^{18} + \)\(32\!\cdots\!99\)\( \nu^{17} - \)\(56\!\cdots\!02\)\( \nu^{16} + \)\(30\!\cdots\!10\)\( \nu^{15} + \)\(15\!\cdots\!43\)\( \nu^{14} + \)\(15\!\cdots\!43\)\( \nu^{13} + \)\(31\!\cdots\!15\)\( \nu^{12} + \)\(39\!\cdots\!65\)\( \nu^{11} + \)\(18\!\cdots\!69\)\( \nu^{10} + \)\(39\!\cdots\!31\)\( \nu^{9} + \)\(72\!\cdots\!45\)\( \nu^{8} - \)\(15\!\cdots\!79\)\( \nu^{7} + \)\(12\!\cdots\!85\)\( \nu^{6} - \)\(15\!\cdots\!58\)\( \nu^{5} + \)\(13\!\cdots\!23\)\( \nu^{4} - \)\(13\!\cdots\!59\)\( \nu^{3} + \)\(69\!\cdots\!42\)\( \nu^{2} + \)\(18\!\cdots\!38\)\( \nu + \)\(13\!\cdots\!44\)\(\)\()/ \)\(36\!\cdots\!14\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(51\!\cdots\!36\)\( \nu^{19} + \)\(15\!\cdots\!47\)\( \nu^{18} - \)\(10\!\cdots\!42\)\( \nu^{17} + \)\(20\!\cdots\!19\)\( \nu^{16} - \)\(11\!\cdots\!24\)\( \nu^{15} + \)\(19\!\cdots\!38\)\( \nu^{14} - \)\(73\!\cdots\!94\)\( \nu^{13} + \)\(81\!\cdots\!43\)\( \nu^{12} - \)\(26\!\cdots\!30\)\( \nu^{11} + \)\(23\!\cdots\!74\)\( \nu^{10} - \)\(64\!\cdots\!96\)\( \nu^{9} + \)\(18\!\cdots\!55\)\( \nu^{8} - \)\(39\!\cdots\!38\)\( \nu^{7} - \)\(85\!\cdots\!73\)\( \nu^{6} - \)\(18\!\cdots\!16\)\( \nu^{5} - \)\(16\!\cdots\!76\)\( \nu^{4} + \)\(31\!\cdots\!32\)\( \nu^{3} - \)\(11\!\cdots\!20\)\( \nu^{2} - \)\(29\!\cdots\!52\)\( \nu - \)\(12\!\cdots\!86\)\(\)\()/ \)\(36\!\cdots\!14\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(14\!\cdots\!44\)\( \nu^{19} + \)\(43\!\cdots\!98\)\( \nu^{18} - \)\(31\!\cdots\!62\)\( \nu^{17} + \)\(62\!\cdots\!69\)\( \nu^{16} - \)\(34\!\cdots\!06\)\( \nu^{15} + \)\(60\!\cdots\!66\)\( \nu^{14} - \)\(24\!\cdots\!78\)\( \nu^{13} + \)\(29\!\cdots\!10\)\( \nu^{12} - \)\(10\!\cdots\!14\)\( \nu^{11} + \)\(97\!\cdots\!94\)\( \nu^{10} - \)\(28\!\cdots\!80\)\( \nu^{9} + \)\(15\!\cdots\!42\)\( \nu^{8} - \)\(41\!\cdots\!10\)\( \nu^{7} + \)\(14\!\cdots\!44\)\( \nu^{6} - \)\(43\!\cdots\!70\)\( \nu^{5} + \)\(83\!\cdots\!83\)\( \nu^{4} - \)\(22\!\cdots\!04\)\( \nu^{3} + \)\(76\!\cdots\!32\)\( \nu^{2} - \)\(81\!\cdots\!68\)\( \nu + \)\(92\!\cdots\!32\)\(\)\()/ \)\(36\!\cdots\!14\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(90\!\cdots\!45\)\( \nu^{19} - \)\(27\!\cdots\!85\)\( \nu^{18} + \)\(19\!\cdots\!38\)\( \nu^{17} - \)\(38\!\cdots\!95\)\( \nu^{16} + \)\(22\!\cdots\!65\)\( \nu^{15} - \)\(37\!\cdots\!92\)\( \nu^{14} + \)\(15\!\cdots\!15\)\( \nu^{13} - \)\(18\!\cdots\!85\)\( \nu^{12} + \)\(64\!\cdots\!03\)\( \nu^{11} - \)\(59\!\cdots\!50\)\( \nu^{10} + \)\(18\!\cdots\!75\)\( \nu^{9} - \)\(94\!\cdots\!45\)\( \nu^{8} + \)\(26\!\cdots\!10\)\( \nu^{7} - \)\(80\!\cdots\!45\)\( \nu^{6} + \)\(27\!\cdots\!65\)\( \nu^{5} - \)\(35\!\cdots\!50\)\( \nu^{4} + \)\(14\!\cdots\!90\)\( \nu^{3} - \)\(28\!\cdots\!14\)\( \nu^{2} + \)\(52\!\cdots\!20\)\( \nu - \)\(45\!\cdots\!36\)\(\)\()/ \)\(13\!\cdots\!76\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(74\!\cdots\!85\)\( \nu^{19} - \)\(22\!\cdots\!47\)\( \nu^{18} + \)\(16\!\cdots\!72\)\( \nu^{17} - \)\(32\!\cdots\!27\)\( \nu^{16} + \)\(18\!\cdots\!99\)\( \nu^{15} - \)\(31\!\cdots\!94\)\( \nu^{14} + \)\(12\!\cdots\!77\)\( \nu^{13} - \)\(15\!\cdots\!29\)\( \nu^{12} + \)\(53\!\cdots\!19\)\( \nu^{11} - \)\(50\!\cdots\!68\)\( \nu^{10} + \)\(15\!\cdots\!09\)\( \nu^{9} - \)\(81\!\cdots\!13\)\( \nu^{8} + \)\(22\!\cdots\!82\)\( \nu^{7} - \)\(69\!\cdots\!77\)\( \nu^{6} + \)\(23\!\cdots\!89\)\( \nu^{5} - \)\(27\!\cdots\!42\)\( \nu^{4} + \)\(12\!\cdots\!56\)\( \nu^{3} + \)\(34\!\cdots\!98\)\( \nu^{2} + \)\(43\!\cdots\!40\)\( \nu + \)\(40\!\cdots\!04\)\(\)\()/ \)\(73\!\cdots\!28\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-\)\(77\!\cdots\!09\)\( \nu^{19} + \)\(24\!\cdots\!71\)\( \nu^{18} - \)\(17\!\cdots\!22\)\( \nu^{17} + \)\(35\!\cdots\!39\)\( \nu^{16} - \)\(19\!\cdots\!77\)\( \nu^{15} + \)\(34\!\cdots\!28\)\( \nu^{14} - \)\(13\!\cdots\!11\)\( \nu^{13} + \)\(17\!\cdots\!67\)\( \nu^{12} - \)\(56\!\cdots\!89\)\( \nu^{11} + \)\(58\!\cdots\!30\)\( \nu^{10} - \)\(16\!\cdots\!35\)\( \nu^{9} + \)\(99\!\cdots\!63\)\( \nu^{8} - \)\(23\!\cdots\!52\)\( \nu^{7} + \)\(90\!\cdots\!83\)\( \nu^{6} - \)\(23\!\cdots\!49\)\( \nu^{5} + \)\(52\!\cdots\!60\)\( \nu^{4} - \)\(11\!\cdots\!60\)\( \nu^{3} + \)\(91\!\cdots\!82\)\( \nu^{2} - \)\(42\!\cdots\!96\)\( \nu + \)\(28\!\cdots\!96\)\(\)\()/ \)\(73\!\cdots\!28\)\( \)
\(\beta_{12}\)\(=\)\((\)\(\)\(10\!\cdots\!99\)\( \nu^{19} - \)\(32\!\cdots\!85\)\( \nu^{18} + \)\(23\!\cdots\!18\)\( \nu^{17} - \)\(47\!\cdots\!53\)\( \nu^{16} + \)\(26\!\cdots\!53\)\( \nu^{15} - \)\(45\!\cdots\!16\)\( \nu^{14} + \)\(18\!\cdots\!53\)\( \nu^{13} - \)\(22\!\cdots\!25\)\( \nu^{12} + \)\(77\!\cdots\!77\)\( \nu^{11} - \)\(75\!\cdots\!62\)\( \nu^{10} + \)\(22\!\cdots\!37\)\( \nu^{9} - \)\(12\!\cdots\!77\)\( \nu^{8} + \)\(32\!\cdots\!94\)\( \nu^{7} - \)\(11\!\cdots\!97\)\( \nu^{6} + \)\(33\!\cdots\!95\)\( \nu^{5} - \)\(60\!\cdots\!28\)\( \nu^{4} + \)\(17\!\cdots\!04\)\( \nu^{3} - \)\(67\!\cdots\!82\)\( \nu^{2} + \)\(63\!\cdots\!68\)\( \nu - \)\(47\!\cdots\!12\)\(\)\()/ \)\(73\!\cdots\!28\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(55\!\cdots\!40\)\( \nu^{19} + \)\(16\!\cdots\!24\)\( \nu^{18} - \)\(12\!\cdots\!21\)\( \nu^{17} + \)\(24\!\cdots\!41\)\( \nu^{16} - \)\(13\!\cdots\!14\)\( \nu^{15} + \)\(23\!\cdots\!11\)\( \nu^{14} - \)\(95\!\cdots\!77\)\( \nu^{13} + \)\(11\!\cdots\!66\)\( \nu^{12} - \)\(39\!\cdots\!88\)\( \nu^{11} + \)\(39\!\cdots\!23\)\( \nu^{10} - \)\(11\!\cdots\!89\)\( \nu^{9} + \)\(65\!\cdots\!20\)\( \nu^{8} - \)\(16\!\cdots\!58\)\( \nu^{7} + \)\(60\!\cdots\!77\)\( \nu^{6} - \)\(16\!\cdots\!00\)\( \nu^{5} + \)\(33\!\cdots\!07\)\( \nu^{4} - \)\(84\!\cdots\!98\)\( \nu^{3} + \)\(60\!\cdots\!22\)\( \nu^{2} - \)\(29\!\cdots\!06\)\( \nu + \)\(11\!\cdots\!72\)\(\)\()/ \)\(36\!\cdots\!14\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(11\!\cdots\!71\)\( \nu^{19} + \)\(34\!\cdots\!89\)\( \nu^{18} - \)\(25\!\cdots\!24\)\( \nu^{17} + \)\(49\!\cdots\!09\)\( \nu^{16} - \)\(28\!\cdots\!45\)\( \nu^{15} + \)\(48\!\cdots\!22\)\( \nu^{14} - \)\(19\!\cdots\!79\)\( \nu^{13} + \)\(23\!\cdots\!47\)\( \nu^{12} - \)\(81\!\cdots\!27\)\( \nu^{11} + \)\(77\!\cdots\!40\)\( \nu^{10} - \)\(23\!\cdots\!15\)\( \nu^{9} + \)\(12\!\cdots\!27\)\( \nu^{8} - \)\(32\!\cdots\!28\)\( \nu^{7} + \)\(10\!\cdots\!15\)\( \nu^{6} - \)\(34\!\cdots\!63\)\( \nu^{5} + \)\(48\!\cdots\!50\)\( \nu^{4} - \)\(17\!\cdots\!66\)\( \nu^{3} - \)\(19\!\cdots\!54\)\( \nu^{2} - \)\(63\!\cdots\!92\)\( \nu - \)\(21\!\cdots\!28\)\(\)\()/ \)\(73\!\cdots\!28\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-\)\(71\!\cdots\!56\)\( \nu^{19} + \)\(21\!\cdots\!56\)\( \nu^{18} - \)\(15\!\cdots\!68\)\( \nu^{17} + \)\(30\!\cdots\!23\)\( \nu^{16} - \)\(17\!\cdots\!24\)\( \nu^{15} + \)\(29\!\cdots\!49\)\( \nu^{14} - \)\(12\!\cdots\!18\)\( \nu^{13} + \)\(14\!\cdots\!61\)\( \nu^{12} - \)\(51\!\cdots\!23\)\( \nu^{11} + \)\(47\!\cdots\!85\)\( \nu^{10} - \)\(14\!\cdots\!06\)\( \nu^{9} + \)\(76\!\cdots\!03\)\( \nu^{8} - \)\(21\!\cdots\!73\)\( \nu^{7} + \)\(64\!\cdots\!83\)\( \nu^{6} - \)\(22\!\cdots\!82\)\( \nu^{5} + \)\(27\!\cdots\!68\)\( \nu^{4} - \)\(11\!\cdots\!63\)\( \nu^{3} - \)\(50\!\cdots\!78\)\( \nu^{2} - \)\(40\!\cdots\!28\)\( \nu - \)\(83\!\cdots\!36\)\(\)\()/ \)\(36\!\cdots\!14\)\( \)
\(\beta_{16}\)\(=\)\((\)\(-\)\(73\!\cdots\!06\)\( \nu^{19} + \)\(22\!\cdots\!91\)\( \nu^{18} - \)\(16\!\cdots\!57\)\( \nu^{17} + \)\(32\!\cdots\!22\)\( \nu^{16} - \)\(18\!\cdots\!39\)\( \nu^{15} + \)\(31\!\cdots\!36\)\( \nu^{14} - \)\(12\!\cdots\!99\)\( \nu^{13} + \)\(15\!\cdots\!85\)\( \nu^{12} - \)\(52\!\cdots\!88\)\( \nu^{11} + \)\(51\!\cdots\!06\)\( \nu^{10} - \)\(15\!\cdots\!49\)\( \nu^{9} + \)\(83\!\cdots\!23\)\( \nu^{8} - \)\(21\!\cdots\!64\)\( \nu^{7} + \)\(74\!\cdots\!07\)\( \nu^{6} - \)\(22\!\cdots\!44\)\( \nu^{5} + \)\(38\!\cdots\!69\)\( \nu^{4} - \)\(11\!\cdots\!39\)\( \nu^{3} + \)\(47\!\cdots\!96\)\( \nu^{2} - \)\(40\!\cdots\!46\)\( \nu - \)\(22\!\cdots\!76\)\(\)\()/ \)\(36\!\cdots\!14\)\( \)
\(\beta_{17}\)\(=\)\((\)\(\)\(17\!\cdots\!37\)\( \nu^{19} - \)\(51\!\cdots\!91\)\( \nu^{18} + \)\(37\!\cdots\!36\)\( \nu^{17} - \)\(74\!\cdots\!09\)\( \nu^{16} + \)\(42\!\cdots\!47\)\( \nu^{15} - \)\(71\!\cdots\!30\)\( \nu^{14} + \)\(29\!\cdots\!75\)\( \nu^{13} - \)\(34\!\cdots\!35\)\( \nu^{12} + \)\(12\!\cdots\!05\)\( \nu^{11} - \)\(11\!\cdots\!48\)\( \nu^{10} + \)\(34\!\cdots\!35\)\( \nu^{9} - \)\(18\!\cdots\!59\)\( \nu^{8} + \)\(50\!\cdots\!20\)\( \nu^{7} - \)\(15\!\cdots\!33\)\( \nu^{6} + \)\(52\!\cdots\!35\)\( \nu^{5} - \)\(75\!\cdots\!18\)\( \nu^{4} + \)\(27\!\cdots\!78\)\( \nu^{3} - \)\(64\!\cdots\!90\)\( \nu^{2} + \)\(99\!\cdots\!48\)\( \nu - \)\(93\!\cdots\!88\)\(\)\()/ \)\(73\!\cdots\!28\)\( \)
\(\beta_{18}\)\(=\)\((\)\(-\)\(18\!\cdots\!67\)\( \nu^{19} + \)\(56\!\cdots\!39\)\( \nu^{18} - \)\(40\!\cdots\!14\)\( \nu^{17} + \)\(81\!\cdots\!51\)\( \nu^{16} - \)\(45\!\cdots\!23\)\( \nu^{15} + \)\(79\!\cdots\!40\)\( \nu^{14} - \)\(31\!\cdots\!25\)\( \nu^{13} + \)\(39\!\cdots\!15\)\( \nu^{12} - \)\(13\!\cdots\!05\)\( \nu^{11} + \)\(13\!\cdots\!38\)\( \nu^{10} - \)\(38\!\cdots\!65\)\( \nu^{9} + \)\(21\!\cdots\!11\)\( \nu^{8} - \)\(55\!\cdots\!66\)\( \nu^{7} + \)\(19\!\cdots\!07\)\( \nu^{6} - \)\(57\!\cdots\!55\)\( \nu^{5} + \)\(10\!\cdots\!88\)\( \nu^{4} - \)\(29\!\cdots\!42\)\( \nu^{3} + \)\(12\!\cdots\!90\)\( \nu^{2} - \)\(10\!\cdots\!32\)\( \nu + \)\(14\!\cdots\!12\)\(\)\()/ \)\(73\!\cdots\!28\)\( \)
\(\beta_{19}\)\(=\)\((\)\(\)\(18\!\cdots\!75\)\( \nu^{19} - \)\(55\!\cdots\!57\)\( \nu^{18} + \)\(40\!\cdots\!22\)\( \nu^{17} - \)\(79\!\cdots\!99\)\( \nu^{16} + \)\(45\!\cdots\!15\)\( \nu^{15} - \)\(76\!\cdots\!86\)\( \nu^{14} + \)\(31\!\cdots\!75\)\( \nu^{13} - \)\(37\!\cdots\!99\)\( \nu^{12} + \)\(13\!\cdots\!37\)\( \nu^{11} - \)\(12\!\cdots\!20\)\( \nu^{10} + \)\(37\!\cdots\!51\)\( \nu^{9} - \)\(20\!\cdots\!79\)\( \nu^{8} + \)\(54\!\cdots\!50\)\( \nu^{7} - \)\(17\!\cdots\!15\)\( \nu^{6} + \)\(56\!\cdots\!81\)\( \nu^{5} - \)\(92\!\cdots\!92\)\( \nu^{4} + \)\(29\!\cdots\!42\)\( \nu^{3} - \)\(13\!\cdots\!06\)\( \nu^{2} + \)\(10\!\cdots\!92\)\( \nu - \)\(11\!\cdots\!52\)\(\)\()/ \)\(73\!\cdots\!28\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{17} - 3 \beta_{9} + \beta_{8} - \beta_{2} + \beta_{1} - 3\)
\(\nu^{3}\)\(=\)\(-\beta_{14} - \beta_{12} - \beta_{11} - \beta_{10} + \beta_{7} - \beta_{5} + 7 \beta_{4} - \beta_{3}\)
\(\nu^{4}\)\(=\)\(\beta_{19} - 8 \beta_{17} - \beta_{16} - \beta_{14} - \beta_{10} + 17 \beta_{9} - 7 \beta_{8} + \beta_{7} + \beta_{6} + \beta_{4} - 7 \beta_{1}\)
\(\nu^{5}\)\(=\)\(-3 \beta_{19} - 10 \beta_{18} + 2 \beta_{17} - \beta_{16} - 2 \beta_{15} + 11 \beta_{14} + 10 \beta_{13} + 11 \beta_{12} + 11 \beta_{11} - 2 \beta_{10} - 12 \beta_{7} - 2 \beta_{6} - 41 \beta_{4} + 8 \beta_{3} - 29 \beta_{1}\)
\(\nu^{6}\)\(=\)\(13 \beta_{16} - 13 \beta_{15} + \beta_{14} + \beta_{12} + 12 \beta_{11} + 12 \beta_{10} - 14 \beta_{7} - 14 \beta_{6} + \beta_{5} - 15 \beta_{4} + 15 \beta_{3} + 46 \beta_{2} + 112\)
\(\nu^{7}\)\(=\)\(40 \beta_{19} + 83 \beta_{18} - 28 \beta_{17} + 28 \beta_{16} + 12 \beta_{15} - 26 \beta_{14} - 87 \beta_{13} - 30 \beta_{12} - 26 \beta_{11} + 99 \beta_{10} - 2 \beta_{8} + 14 \beta_{7} + 14 \beta_{6} + 87 \beta_{5} - 73 \beta_{4} + 179 \beta_{1}\)
\(\nu^{8}\)\(=\)\(-65 \beta_{19} + 444 \beta_{17} + 2 \beta_{16} + 131 \beta_{15} + 81 \beta_{14} - 20 \beta_{13} - 46 \beta_{12} - 141 \beta_{11} + 28 \beta_{10} - 779 \beta_{9} + 311 \beta_{8} + 48 \beta_{7} + 48 \beta_{6} + 40 \beta_{4} - 153 \beta_{3} - 311 \beta_{2} + 303 \beta_{1} - 779\)
\(\nu^{9}\)\(=\)\(-177 \beta_{16} + 177 \beta_{15} - 539 \beta_{14} - 539 \beta_{12} - 581 \beta_{11} - 581 \beta_{10} + 831 \beta_{7} + 179 \beta_{6} - 724 \beta_{5} + 2738 \beta_{4} - 474 \beta_{3} + 30 \beta_{2} + 4\)
\(\nu^{10}\)\(=\)\(433 \beta_{19} + 4 \beta_{18} - 3278 \beta_{17} - 1195 \beta_{16} - 44 \beta_{15} - 673 \beta_{14} + 261 \beta_{13} + 434 \beta_{12} + 284 \beta_{11} - 1262 \beta_{10} + 5582 \beta_{9} - 2176 \beta_{8} + 717 \beta_{7} + 717 \beta_{6} - 261 \beta_{5} + 978 \beta_{4} - 2026 \beta_{1}\)
\(\nu^{11}\)\(=\)\(-3452 \beta_{19} - 5001 \beta_{18} + 2464 \beta_{17} - 1026 \beta_{16} - 2722 \beta_{15} + 6345 \beta_{14} + 5889 \beta_{13} + 7015 \beta_{12} + 6741 \beta_{11} - 2152 \beta_{10} - 56 \beta_{9} + 302 \beta_{8} - 8041 \beta_{7} - 3040 \beta_{6} - 16038 \beta_{4} + 3727 \beta_{3} - 302 \beta_{2} - 7695 \beta_{1} - 56\)
\(\nu^{12}\)\(=\)\(9737 \beta_{16} - 9737 \beta_{15} + 332 \beta_{14} + 332 \beta_{12} + 8135 \beta_{11} + 8135 \beta_{10} - 10683 \beta_{7} - 10763 \beta_{6} + 2842 \beta_{5} - 11681 \beta_{4} + 11423 \beta_{3} + 15653 \beta_{2} + 40791\)
\(\nu^{13}\)\(=\)\(28749 \beta_{19} + 37910 \beta_{18} - 20356 \beta_{17} + 23956 \beta_{16} + 8829 \beta_{15} - 20000 \beta_{14} - 47244 \beta_{13} - 26298 \beta_{12} - 17578 \beta_{11} + 53651 \beta_{10} + 386 \beta_{9} - 2538 \beta_{8} + 11171 \beta_{7} + 11171 \beta_{6} + 47244 \beta_{5} - 36073 \beta_{4} + 52927 \beta_{1}\)
\(\nu^{14}\)\(=\)\(-16645 \beta_{19} - 1004 \beta_{18} + 179652 \beta_{17} + 6992 \beta_{16} + 86941 \beta_{15} + 30354 \beta_{14} - 28061 \beta_{13} - 42603 \beta_{12} - 87774 \beta_{11} + 21534 \beta_{10} - 302152 \beta_{9} + 114906 \beta_{8} + 49595 \beta_{7} + 50599 \beta_{6} + 30763 \beta_{4} - 92807 \beta_{3} - 114906 \beta_{2} + 96074 \beta_{1} - 302152\)
\(\nu^{15}\)\(=\)\(-129544 \beta_{16} + 129544 \beta_{15} - 218511 \beta_{14} - 218511 \beta_{12} - 282629 \beta_{11} - 282629 \beta_{10} + 422291 \beta_{7} + 136536 \beta_{6} - 375607 \beta_{5} + 1190219 \beta_{4} - 232425 \beta_{3} + 19022 \beta_{2} + 252\)
\(\nu^{16}\)\(=\)\(95613 \beta_{19} + 10008 \beta_{18} - 1336036 \beta_{17} - 715923 \beta_{16} - 71110 \beta_{15} - 200731 \beta_{14} + 260972 \beta_{13} + 372972 \beta_{12} + 176228 \beta_{11} - 709041 \beta_{10} + 2259523 \beta_{9} - 855587 \beta_{8} + 271841 \beta_{7} + 271841 \beta_{6} - 260972 \beta_{5} + 532813 \beta_{4} - 682063 \beta_{1}\)
\(\nu^{17}\)\(=\)\(-1851193 \beta_{19} - 2149200 \beta_{18} + 1263312 \beta_{17} - 613931 \beta_{16} - 1695944 \beta_{15} + 2978621 \beta_{14} + 2967752 \beta_{13} + 3446703 \beta_{12} + 3302323 \beta_{11} - 1092882 \beta_{10} + 36976 \beta_{9} + 129670 \beta_{8} - 4060634 \beta_{7} - 1911434 \beta_{6} - 6877705 \beta_{4} + 1834110 \beta_{3} - 129670 \beta_{2} - 2687401 \beta_{1} + 36976\)
\(\nu^{18}\)\(=\)\(5142647 \beta_{16} - 5142647 \beta_{15} - 145569 \beta_{14} - 145569 \beta_{12} + 4254666 \beta_{11} + 4254666 \beta_{10} - 5671250 \beta_{7} - 5756578 \beta_{6} + 2333197 \beta_{5} - 6504849 \beta_{4} + 5867259 \beta_{3} + 6433300 \beta_{2} + 17011682\)
\(\nu^{19}\)\(=\)\(14578856 \beta_{19} + 16161297 \beta_{18} - 9686600 \beta_{17} + 13906594 \beta_{16} + 5014166 \beta_{15} - 11112322 \beta_{14} - 23345353 \beta_{13} - 14990584 \beta_{12} - 8480700 \beta_{11} + 25727897 \beta_{10} - 706004 \beta_{9} - 796794 \beta_{8} + 6098156 \beta_{7} + 6098156 \beta_{6} + 23345353 \beta_{5} - 17247197 \beta_{4} + 19667087 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/483\mathbb{Z}\right)^\times\).

\(n\) \(323\) \(346\) \(442\)
\(\chi(n)\) \(1\) \(-1 - \beta_{9}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
277.1
1.39981 2.42454i
1.25760 2.17823i
1.13751 1.97023i
0.599601 1.03854i
0.432430 0.748990i
−0.0131282 + 0.0227388i
−0.384545 + 0.666052i
−0.526755 + 0.912367i
−1.03253 + 1.78839i
−1.37000 + 2.37290i
1.39981 + 2.42454i
1.25760 + 2.17823i
1.13751 + 1.97023i
0.599601 + 1.03854i
0.432430 + 0.748990i
−0.0131282 0.0227388i
−0.384545 0.666052i
−0.526755 0.912367i
−1.03253 1.78839i
−1.37000 2.37290i
−1.39981 2.42454i 0.500000 0.866025i −2.91893 + 5.05574i 0.973538 + 1.68622i −2.79962 −2.64333 + 0.113245i 10.7446 −0.500000 0.866025i 2.72553 4.72076i
277.2 −1.25760 2.17823i 0.500000 0.866025i −2.16313 + 3.74664i −2.15868 3.73894i −2.51520 1.42694 2.22797i 5.85100 −0.500000 0.866025i −5.42951 + 9.40419i
277.3 −1.13751 1.97023i 0.500000 0.866025i −1.58786 + 2.75026i 0.619347 + 1.07274i −2.27502 1.84753 + 1.89384i 2.67481 −0.500000 0.866025i 1.40903 2.44051i
277.4 −0.599601 1.03854i 0.500000 0.866025i 0.280958 0.486633i 0.286327 + 0.495932i −1.19920 −1.87398 1.86767i −3.07225 −0.500000 0.866025i 0.343363 0.594723i
277.5 −0.432430 0.748990i 0.500000 0.866025i 0.626009 1.08428i 1.24779 + 2.16123i −0.864859 −0.0271380 + 2.64561i −2.81254 −0.500000 0.866025i 1.07916 1.86916i
277.6 0.0131282 + 0.0227388i 0.500000 0.866025i 0.999655 1.73145i −1.38914 2.40606i 0.0262565 −2.58821 + 0.548771i 0.105008 −0.500000 0.866025i 0.0364740 0.0631748i
277.7 0.384545 + 0.666052i 0.500000 0.866025i 0.704250 1.21980i 2.01355 + 3.48756i 0.769091 2.63574 0.229917i 2.62145 −0.500000 0.866025i −1.54860 + 2.68225i
277.8 0.526755 + 0.912367i 0.500000 0.866025i 0.445058 0.770863i −1.08806 1.88457i 1.05351 1.36439 + 2.26681i 3.04477 −0.500000 0.866025i 1.14628 1.98542i
277.9 1.03253 + 1.78839i 0.500000 0.866025i −1.13223 + 1.96108i 0.304427 + 0.527282i 2.06506 2.05425 1.66735i −0.546135 −0.500000 0.866025i −0.628658 + 1.08887i
277.10 1.37000 + 2.37290i 0.500000 0.866025i −2.75378 + 4.76968i 1.69091 + 2.92874i 2.73999 −2.19620 1.47537i −9.61066 −0.500000 0.866025i −4.63307 + 8.02471i
415.1 −1.39981 + 2.42454i 0.500000 + 0.866025i −2.91893 5.05574i 0.973538 1.68622i −2.79962 −2.64333 0.113245i 10.7446 −0.500000 + 0.866025i 2.72553 + 4.72076i
415.2 −1.25760 + 2.17823i 0.500000 + 0.866025i −2.16313 3.74664i −2.15868 + 3.73894i −2.51520 1.42694 + 2.22797i 5.85100 −0.500000 + 0.866025i −5.42951 9.40419i
415.3 −1.13751 + 1.97023i 0.500000 + 0.866025i −1.58786 2.75026i 0.619347 1.07274i −2.27502 1.84753 1.89384i 2.67481 −0.500000 + 0.866025i 1.40903 + 2.44051i
415.4 −0.599601 + 1.03854i 0.500000 + 0.866025i 0.280958 + 0.486633i 0.286327 0.495932i −1.19920 −1.87398 + 1.86767i −3.07225 −0.500000 + 0.866025i 0.343363 + 0.594723i
415.5 −0.432430 + 0.748990i 0.500000 + 0.866025i 0.626009 + 1.08428i 1.24779 2.16123i −0.864859 −0.0271380 2.64561i −2.81254 −0.500000 + 0.866025i 1.07916 + 1.86916i
415.6 0.0131282 0.0227388i 0.500000 + 0.866025i 0.999655 + 1.73145i −1.38914 + 2.40606i 0.0262565 −2.58821 0.548771i 0.105008 −0.500000 + 0.866025i 0.0364740 + 0.0631748i
415.7 0.384545 0.666052i 0.500000 + 0.866025i 0.704250 + 1.21980i 2.01355 3.48756i 0.769091 2.63574 + 0.229917i 2.62145 −0.500000 + 0.866025i −1.54860 2.68225i
415.8 0.526755 0.912367i 0.500000 + 0.866025i 0.445058 + 0.770863i −1.08806 + 1.88457i 1.05351 1.36439 2.26681i 3.04477 −0.500000 + 0.866025i 1.14628 + 1.98542i
415.9 1.03253 1.78839i 0.500000 + 0.866025i −1.13223 1.96108i 0.304427 0.527282i 2.06506 2.05425 + 1.66735i −0.546135 −0.500000 + 0.866025i −0.628658 1.08887i
415.10 1.37000 2.37290i 0.500000 + 0.866025i −2.75378 4.76968i 1.69091 2.92874i 2.73999 −2.19620 + 1.47537i −9.61066 −0.500000 + 0.866025i −4.63307 8.02471i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 415.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 483.2.i.h 20
7.c even 3 1 inner 483.2.i.h 20
7.c even 3 1 3381.2.a.bi 10
7.d odd 6 1 3381.2.a.bj 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.i.h 20 1.a even 1 1 trivial
483.2.i.h 20 7.c even 3 1 inner
3381.2.a.bi 10 7.c even 3 1
3381.2.a.bj 10 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(483, [\chi])\):

\(T_{2}^{20} + \cdots\)
\(T_{5}^{20} - \cdots\)
\(T_{11}^{20} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 - 152 T + 5800 T^{2} + 84 T^{3} + 15786 T^{4} + 4026 T^{5} + 30661 T^{6} + 9009 T^{7} + 29358 T^{8} + 10565 T^{9} + 20315 T^{10} + 6640 T^{11} + 7135 T^{12} + 2021 T^{13} + 1707 T^{14} + 416 T^{15} + 245 T^{16} + 43 T^{17} + 22 T^{18} + 3 T^{19} + T^{20} \)
$3$ \( ( 1 - T + T^{2} )^{10} \)
$5$ \( 556516 - 2561764 T + 7945980 T^{2} - 14636660 T^{3} + 20588124 T^{4} - 19545514 T^{5} + 15668047 T^{6} - 9378557 T^{7} + 5625834 T^{8} - 2655713 T^{9} + 1332915 T^{10} - 485572 T^{11} + 197781 T^{12} - 56713 T^{13} + 20191 T^{14} - 4454 T^{15} + 1263 T^{16} - 187 T^{17} + 48 T^{18} - 5 T^{19} + T^{20} \)
$7$ \( 282475249 - 63412811 T^{2} + 31294634 T^{3} + 10706059 T^{4} - 7159782 T^{5} - 62426 T^{6} + 1051638 T^{7} - 298557 T^{8} - 87416 T^{9} + 50869 T^{10} - 12488 T^{11} - 6093 T^{12} + 3066 T^{13} - 26 T^{14} - 426 T^{15} + 91 T^{16} + 38 T^{17} - 11 T^{18} + T^{20} \)
$11$ \( 31719424 + 46137344 T + 101711872 T^{2} + 51494912 T^{3} + 107479040 T^{4} + 57151488 T^{5} + 67148032 T^{6} + 23564288 T^{7} + 20048768 T^{8} + 5391104 T^{9} + 4167744 T^{10} + 847040 T^{11} + 539456 T^{12} + 87344 T^{13} + 49792 T^{14} + 6744 T^{15} + 2504 T^{16} + 320 T^{17} + 88 T^{18} + 8 T^{19} + T^{20} \)
$13$ \( ( 30667 - 60372 T + 23181 T^{2} + 15500 T^{3} - 9422 T^{4} - 1256 T^{5} + 1150 T^{6} + 32 T^{7} - 57 T^{8} + T^{10} )^{2} \)
$17$ \( 53029799524 - 143498386044 T + 279747793416 T^{2} - 259814756552 T^{3} + 187551578044 T^{4} - 91754745020 T^{5} + 39228488265 T^{6} - 12828907041 T^{7} + 4214593952 T^{8} - 1100976623 T^{9} + 305557263 T^{10} - 62728712 T^{11} + 14627853 T^{12} - 2456119 T^{13} + 500805 T^{14} - 66378 T^{15} + 11129 T^{16} - 1123 T^{17} + 162 T^{18} - 11 T^{19} + T^{20} \)
$19$ \( 13176284944 + 44615799840 T + 98376185664 T^{2} + 128452491456 T^{3} + 123768262152 T^{4} + 82287882840 T^{5} + 42803649568 T^{6} + 16489024444 T^{7} + 5303109256 T^{8} + 1321992868 T^{9} + 316004204 T^{10} + 60380976 T^{11} + 12931600 T^{12} + 1860668 T^{13} + 358605 T^{14} + 34855 T^{15} + 7522 T^{16} + 423 T^{17} + 104 T^{18} + T^{19} + T^{20} \)
$23$ \( ( 1 + T + T^{2} )^{10} \)
$29$ \( ( 198784 + 29696 T - 366848 T^{2} - 47904 T^{3} + 91552 T^{4} - 2824 T^{5} - 6224 T^{6} + 714 T^{7} + 106 T^{8} - 22 T^{9} + T^{10} )^{2} \)
$31$ \( 3370484748544 + 2625613582080 T + 3839211133952 T^{2} - 412145371904 T^{3} + 1074502289728 T^{4} - 121968627584 T^{5} + 163260596736 T^{6} - 21263443056 T^{7} + 15904060352 T^{8} - 1434189056 T^{9} + 915041672 T^{10} - 58399464 T^{11} + 38144808 T^{12} - 1406882 T^{13} + 999963 T^{14} - 22761 T^{15} + 18644 T^{16} - 201 T^{17} + 168 T^{18} - 3 T^{19} + T^{20} \)
$37$ \( 345393601613824 - 342787571105792 T + 300929358411776 T^{2} - 109737410672640 T^{3} + 46717744947712 T^{4} - 11251348892160 T^{5} + 3887348812800 T^{6} - 738513244608 T^{7} + 206857180160 T^{8} - 27810809632 T^{9} + 6868431792 T^{10} - 690077408 T^{11} + 166439896 T^{12} - 11166112 T^{13} + 2654761 T^{14} - 121681 T^{15} + 31100 T^{16} - 815 T^{17} + 218 T^{18} - 3 T^{19} + T^{20} \)
$41$ \( ( -30634016 - 18552576 T + 2829792 T^{2} + 3978080 T^{3} + 736096 T^{4} - 63296 T^{5} - 33088 T^{6} - 2874 T^{7} + 102 T^{8} + 26 T^{9} + T^{10} )^{2} \)
$43$ \( ( 59884 + 185336 T - 362252 T^{2} - 163764 T^{3} + 130702 T^{4} + 17650 T^{5} - 14896 T^{6} + 1397 T^{7} + 151 T^{8} - 27 T^{9} + T^{10} )^{2} \)
$47$ \( 5161984 - 43404288 T + 240448128 T^{2} - 782782912 T^{3} + 1877618960 T^{4} - 2891325668 T^{5} + 3320375045 T^{6} - 2190817705 T^{7} + 1487747496 T^{8} - 763068975 T^{9} + 422283729 T^{10} - 157189626 T^{11} + 50844849 T^{12} - 9763163 T^{13} + 1629509 T^{14} - 106418 T^{15} + 16389 T^{16} - 151 T^{17} + 252 T^{18} + 11 T^{19} + T^{20} \)
$53$ \( 7340588259904 - 114378361666464 T + 1884159869988928 T^{2} + 1643932196475560 T^{3} + 986393504790896 T^{4} + 361275233649346 T^{5} + 105196589649495 T^{6} + 21999227829475 T^{7} + 4021702775176 T^{8} + 566157455801 T^{9} + 81564206285 T^{10} + 9057830272 T^{11} + 1149992739 T^{12} + 94756299 T^{13} + 10803703 T^{14} + 649986 T^{15} + 75675 T^{16} + 2763 T^{17} + 328 T^{18} + 5 T^{19} + T^{20} \)
$59$ \( 119638508544 - 39215397888 T + 252850581504 T^{2} - 45919472640 T^{3} + 418429294336 T^{4} - 100890694656 T^{5} + 190967660800 T^{6} - 78877837376 T^{7} + 74913648768 T^{8} - 23435582976 T^{9} + 7258705312 T^{10} - 1204088288 T^{11} + 228437936 T^{12} - 27881360 T^{13} + 4409140 T^{14} - 406244 T^{15} + 47056 T^{16} - 2592 T^{17} + 266 T^{18} - 10 T^{19} + T^{20} \)
$61$ \( 155728101376 + 157571784704 T + 207461457920 T^{2} + 154629163008 T^{3} + 147811002368 T^{4} + 96392692224 T^{5} + 59024289024 T^{6} + 25510784768 T^{7} + 10034363776 T^{8} + 3058932864 T^{9} + 921691072 T^{10} + 226045632 T^{11} + 54185632 T^{12} + 9979312 T^{13} + 1773456 T^{14} + 246456 T^{15} + 35552 T^{16} + 3840 T^{17} + 380 T^{18} + 22 T^{19} + T^{20} \)
$67$ \( 3442498027609 - 570419542886 T + 2076369706379 T^{2} + 250561243534 T^{3} + 833243777803 T^{4} + 92547780376 T^{5} + 148372492130 T^{6} + 18270430884 T^{7} + 18416173191 T^{8} + 1684261334 T^{9} + 1228839599 T^{10} + 62995052 T^{11} + 56934339 T^{12} + 1351914 T^{13} + 1517954 T^{14} - 32738 T^{15} + 26299 T^{16} - 314 T^{17} + 191 T^{18} - 2 T^{19} + T^{20} \)
$71$ \( ( 5367896 + 4154628 T - 44520154 T^{2} + 10898271 T^{3} + 1657557 T^{4} - 467797 T^{5} - 14987 T^{6} + 6221 T^{7} - 73 T^{8} - 27 T^{9} + T^{10} )^{2} \)
$73$ \( 60216094569409 - 5563819411388 T + 92472650807553 T^{2} - 93778837578140 T^{3} + 148384355838211 T^{4} - 81071032640840 T^{5} + 38896235892178 T^{6} - 10014665018100 T^{7} + 2661627851957 T^{8} - 428398968536 T^{9} + 100160180835 T^{10} - 12115086972 T^{11} + 2217306485 T^{12} - 109364700 T^{13} + 20679442 T^{14} - 812000 T^{15} + 132323 T^{16} - 2832 T^{17} + 449 T^{18} - 8 T^{19} + T^{20} \)
$79$ \( 2181141440046864 - 10033563009181824 T + 45450655065260928 T^{2} - 5388571515873024 T^{3} + 5165846567719848 T^{4} + 270282894010632 T^{5} + 385107193162128 T^{6} + 21167704620468 T^{7} + 12510827372128 T^{8} + 819471267476 T^{9} + 284208891644 T^{10} + 16895948424 T^{11} + 3818625748 T^{12} + 226272728 T^{13} + 36661733 T^{14} + 1823871 T^{15} + 197852 T^{16} + 8045 T^{17} + 706 T^{18} + 21 T^{19} + T^{20} \)
$83$ \( ( -947876608 + 972671872 T - 3040064 T^{2} - 63451744 T^{3} - 2806928 T^{4} + 1054760 T^{5} + 67052 T^{6} - 6282 T^{7} - 468 T^{8} + 12 T^{9} + T^{10} )^{2} \)
$89$ \( 175748762176 - 165680678592 T + 1194141034176 T^{2} + 1387079870272 T^{3} + 5716284516128 T^{4} + 1636417599072 T^{5} + 1532455731632 T^{6} - 93172121632 T^{7} + 223949829136 T^{8} - 12509759488 T^{9} + 14418013768 T^{10} - 1675420880 T^{11} + 594969952 T^{12} - 39266252 T^{13} + 7879448 T^{14} - 168204 T^{15} + 66164 T^{16} - 496 T^{17} + 336 T^{18} + 6 T^{19} + T^{20} \)
$97$ \( ( -21133952 + 161556160 T + 64563616 T^{2} - 23002288 T^{3} - 10632736 T^{4} - 273816 T^{5} + 150064 T^{6} + 2948 T^{7} - 684 T^{8} - 6 T^{9} + T^{10} )^{2} \)
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