Properties

Label 525.2.n.b
Level 525
Weight 2
Character orbit 525.n
Analytic conductor 4.192
Analytic rank 0
Dimension 20
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 525.n (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.19214610612\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 8 x^{19} + 31 x^{18} - 74 x^{17} + 109 x^{16} - 72 x^{15} - 51 x^{14} + 9 x^{13} + 866 x^{12} - 3240 x^{11} + 6385 x^{10} - 6775 x^{9} + 330 x^{8} + 11325 x^{7} - 18525 x^{6} + 12000 x^{5} + 5875 x^{4} - 21250 x^{3} + 22500 x^{2} - 12500 x + 3125\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-7496149849236434 \nu^{19} + 18187349602415911 \nu^{18} + 35298514756172379 \nu^{17} - 272373116958725125 \nu^{16} + 699057222419800998 \nu^{15} - 831453752127054346 \nu^{14} - 132218435730913209 \nu^{13} + 2302365829388940680 \nu^{12} - 2321607293890745758 \nu^{11} - 7461573878786628596 \nu^{10} + 32269007377540629260 \nu^{9} - 57653742754120458500 \nu^{8} + 34779391638584262730 \nu^{7} + 52163996031233910795 \nu^{6} - 131647813006646740875 \nu^{5} + 111289762393351421325 \nu^{4} + 14070216877555781000 \nu^{3} - 140084829910231601125 \nu^{2} + 149155237155614681875 \nu - 60128016980832161250\)\()/ 578956306290055625 \)
\(\beta_{2}\)\(=\)\((\)\(-8948581214751395 \nu^{19} - 1170118896593934 \nu^{18} + 182818748331086007 \nu^{17} - 741298990732606334 \nu^{16} + 1555890427130622926 \nu^{15} - 1554425621292432306 \nu^{14} - 586111784689534162 \nu^{13} + 3930825970896830214 \nu^{12} - 175648483779579816 \nu^{11} - 25617827564561798604 \nu^{10} + 78008454840040340810 \nu^{9} - 117854727899450871415 \nu^{8} + 49093609692883483325 \nu^{7} + 139156956826986132355 \nu^{6} - 284912274592766538300 \nu^{5} + 209808683597550057975 \nu^{4} + 68093578010226650500 \nu^{3} - 316484431229511796625 \nu^{2} + 313380196686340005000 \nu - 121470993205011475625\)\()/ 578956306290055625 \)
\(\beta_{3}\)\(=\)\((\)\(30594661097779031 \nu^{19} - 252380968252696836 \nu^{18} + 954565541101689750 \nu^{17} - 2147232010943421002 \nu^{16} + 2809293535610314076 \nu^{15} - 1044461212033380014 \nu^{14} - 2786283972348140180 \nu^{13} - 75935239443717053 \nu^{12} + 29594061956656362769 \nu^{11} - 100188439081754867358 \nu^{10} + 178040681206398405115 \nu^{9} - 150463637976148011980 \nu^{8} - 79908611372069691420 \nu^{7} + 389798727978811327635 \nu^{6} - 468282176719183324075 \nu^{5} + 152245654993671128825 \nu^{4} + 346447449214083481000 \nu^{3} - 609122953540557264000 \nu^{2} + 453440843027092372500 \nu - 141810590421272316875\)\()/ 578956306290055625 \)
\(\beta_{4}\)\(=\)\((\)\(-38056703563840256 \nu^{19} + 315722367041574732 \nu^{18} - 1198084500162315338 \nu^{17} + 2702715307576294593 \nu^{16} - 3550283407822405565 \nu^{15} + 1344780254010237538 \nu^{14} + 3494459325634686493 \nu^{13} + 4691666759628817 \nu^{12} - 37001863965856208370 \nu^{11} + 125881720274821433709 \nu^{10} - 224493117430571675355 \nu^{9} + 191007410812864952895 \nu^{8} + 98575884509150383070 \nu^{7} - 490537453012295272880 \nu^{6} + 592666201007179198425 \nu^{5} - 196252518084144004200 \nu^{4} - 433974739702589802000 \nu^{3} + 769035611391673859250 \nu^{2} - 574547145324886832500 \nu + 181305786683194126250\)\()/ 578956306290055625 \)
\(\beta_{5}\)\(=\)\((\)\(-38097937029278679 \nu^{19} + 221723873703511668 \nu^{18} - 612070387359621007 \nu^{17} + 954190866605994322 \nu^{16} - 495295329558946070 \nu^{15} - 1081359222747396908 \nu^{14} + 1745552547456165132 \nu^{13} + 5063868776863341868 \nu^{12} - 26477652889628181395 \nu^{11} + 56062825284570853406 \nu^{10} - 57897508073985384925 \nu^{9} - 18076971545862282840 \nu^{8} + 135249455797306238255 \nu^{7} - 167546321920121565670 \nu^{6} + 48874354804415707450 \nu^{5} + 136596709134275011725 \nu^{4} - 222101787533921132625 \nu^{3} + 133126929240788780250 \nu^{2} + 3873168849268583750 \nu - 32732967125916570625\)\()/ 578956306290055625 \)
\(\beta_{6}\)\(=\)\((\)\(-38141838390737218 \nu^{19} + 316215187202857788 \nu^{18} - 1199606050750009270 \nu^{17} + 2705550129578904531 \nu^{16} - 3553036508963672618 \nu^{15} + 1344474723728959837 \nu^{14} + 3497926831162465515 \nu^{13} + 12551860939584684 \nu^{12} - 37058578647224604067 \nu^{11} + 126029051155953777504 \nu^{10} - 224699811425932836880 \nu^{9} + 191106081651668733170 \nu^{8} + 98784803031503861135 \nu^{7} - 491031622751065884980 \nu^{6} + 593105948538016959950 \nu^{5} - 196246188892428021950 \nu^{4} - 434480787139032112625 \nu^{3} + 769685986274975508000 \nu^{2} - 574954660567099996875 \nu + 180483069624485484375\)\()/ 578956306290055625 \)
\(\beta_{7}\)\(=\)\((\)\(-49696489133711646 \nu^{19} + 410416243098932241 \nu^{18} - 1553564713209128140 \nu^{17} + 3498075828560026892 \nu^{16} - 4584292968340800326 \nu^{15} + 1721674654681180324 \nu^{14} + 4520231111703755330 \nu^{13} + 98158703589866788 \nu^{12} - 48099497805866511919 \nu^{11} + 163090642327590775653 \nu^{10} - 290236290641467278645 \nu^{9} + 246063616994527692790 \nu^{8} + 128618136434164176970 \nu^{7} - 634223169207332398710 \nu^{6} + 764384907323940345975 \nu^{5} - 251402150689214212525 \nu^{4} - 561767648269663086625 \nu^{3} + 992883926395422770125 \nu^{2} - 741162275616686191875 \nu + 233454836175605537500\)\()/ 578956306290055625 \)
\(\beta_{8}\)\(=\)\((\)\(-61609759914214484 \nu^{19} + 511052055433574644 \nu^{18} - 1939198378712729585 \nu^{17} + 4374357198267054553 \nu^{16} - 5745776759562930159 \nu^{15} + 2176013262253062106 \nu^{14} + 5655726470640855870 \nu^{13} + 10426458782486467 \nu^{12} - 59893671006946571996 \nu^{11} + 203744027247918181127 \nu^{10} - 363330813653705066115 \nu^{9} + 309111253480054798860 \nu^{8} + 159576940072626296230 \nu^{7} - 793923405524760404990 \nu^{6} + 959168700590145796225 \nu^{5} - 317569493991248050350 \nu^{4} - 702397734805328164875 \nu^{3} + 1244631137052870911500 \nu^{2} - 929844636965337158750 \nu + 292843864790051679375\)\()/ 578956306290055625 \)
\(\beta_{9}\)\(=\)\((\)\(67340933900117065 \nu^{19} - 435693982574621304 \nu^{18} + 1353769422427597412 \nu^{17} - 2493564866740195864 \nu^{16} + 2275099888895223026 \nu^{15} + 821157540741676264 \nu^{14} - 3928674427631158217 \nu^{13} - 6630772013253926706 \nu^{12} + 51788134254336462609 \nu^{11} - 131487093497346919069 \nu^{10} + 178989613863335971035 \nu^{9} - 63192931249734987690 \nu^{8} - 224650990489070663950 \nu^{7} + 446650939616994388455 \nu^{6} - 334298915288231629550 \nu^{5} - 94018894804930596775 \nu^{4} + 494015010288866266000 \nu^{3} - 526736802939415785750 \nu^{2} + 253017849077416814375 \nu - 35577348380697027500\)\()/ 578956306290055625 \)
\(\beta_{10}\)\(=\)\((\)\(68361049143743909 \nu^{19} - 516120783381980066 \nu^{18} + 1830505966511905946 \nu^{17} - 3887962653056874070 \nu^{16} + 4674733606379832987 \nu^{15} - 1037436084091379899 \nu^{14} - 5342330885213453501 \nu^{13} - 2844993832238376710 \nu^{12} + 60845240141129081448 \nu^{11} - 187719599188014956559 \nu^{10} + 310789905337078919385 \nu^{9} - 225786857511945841510 \nu^{8} - 200169299566702703655 \nu^{7} + 702966450092785026705 \nu^{6} - 760080017211615507450 \nu^{5} + 162277345215067700500 \nu^{4} + 665126575717762752125 \nu^{3} - 1026549923423361799875 \nu^{2} + 705619181786099604375 \nu - 202938880729760171250\)\()/ 578956306290055625 \)
\(\beta_{11}\)\(=\)\((\)\(-77638340774021523 \nu^{19} + 546415941644842565 \nu^{18} - 1835303039993145851 \nu^{17} + 3693920748641648798 \nu^{16} - 4055012821929969961 \nu^{15} + 194576300546097810 \nu^{14} + 5338092382984360496 \nu^{13} + 5284064919090048952 \nu^{12} - 64627052511574556904 \nu^{11} + 184261897782927904766 \nu^{10} - 284390435586059649150 \nu^{9} + 170979747541331899875 \nu^{8} + 241267297845167189710 \nu^{7} - 665551764203671078870 \nu^{6} + 640320054240683880950 \nu^{5} - 49319859618372664950 \nu^{4} - 666546433063326576375 \nu^{3} + 904935377260255383500 \nu^{2} - 565488834794789105000 \nu + 143537576149599546250\)\()/ 578956306290055625 \)
\(\beta_{12}\)\(=\)\((\)\(-80210143760530931 \nu^{19} + 543741149817217909 \nu^{18} - 1767519994124554469 \nu^{17} + 3435802244926096520 \nu^{16} - 3531508799191154063 \nu^{15} - 307843131993227339 \nu^{14} + 5112295656598369834 \nu^{13} + 6547899107047666725 \nu^{12} - 64390053533557709852 \nu^{11} + 175063008287809612211 \nu^{10} - 257716326936277737495 \nu^{9} + 132027275359787656810 \nu^{8} + 255668771356798552145 \nu^{7} - 616548226075085528070 \nu^{6} + 545171243083561232625 \nu^{5} + 17902460653381924225 \nu^{4} - 640345157785934755375 \nu^{3} + 797170694820233601500 \nu^{2} - 462532364382930348125 \nu + 105150546034648071875\)\()/ 578956306290055625 \)
\(\beta_{13}\)\(=\)\((\)\(-119431451199203498 \nu^{19} + 833067465542928897 \nu^{18} - 2776165043242942577 \nu^{17} + 5543980371684036095 \nu^{16} - 6001932201784979784 \nu^{15} + 123985982774260523 \nu^{14} + 8050308291131886307 \nu^{13} + 8523739068246955565 \nu^{12} - 98524615594161134086 \nu^{11} + 277837208919759934478 \nu^{10} - 424437283672547785770 \nu^{9} + 247233873731731738615 \nu^{8} + 372606823684327404235 \nu^{7} - 997280677615238422585 \nu^{6} + 943978754439002748750 \nu^{5} - 52689931442167264525 \nu^{4} - 1007120891253137788625 \nu^{3} + 1343167433846892595500 \nu^{2} - 827469878225313371250 \nu + 206644628443917063125\)\()/ 578956306290055625 \)
\(\beta_{14}\)\(=\)\((\)\(-6575345596843278 \nu^{19} + 43796072006877884 \nu^{18} - 140123130326421033 \nu^{17} + 267582239651426027 \nu^{16} - 265163599247354697 \nu^{15} - 44072018156723539 \nu^{14} + 404505714078291933 \nu^{13} + 576404641371480743 \nu^{12} - 5189780434553107878 \nu^{11} + 13785161474297947125 \nu^{10} - 19793640981177339180 \nu^{9} + 9172914209859934640 \nu^{8} + 21203232765151517885 \nu^{7} - 47944729624554219400 \nu^{6} + 40396290583422611900 \nu^{5} + 4056228991675321700 \nu^{4} - 50775342721979497750 \nu^{3} + 60372822317509633375 \nu^{2} - 33339420521565271875 \nu + 6919317791231319375\)\()/ 30471384541581875 \)
\(\beta_{15}\)\(=\)\((\)\(-147791714960405759 \nu^{19} + 1026654745452727610 \nu^{18} - 3411135769444114318 \nu^{17} + 6790345992839648859 \nu^{16} - 7307611478002251508 \nu^{15} + 63472130553556115 \nu^{14} + 9889686231202254418 \nu^{13} + 10737833995502311091 \nu^{12} - 121447001916870115762 \nu^{11} + 340978988560691838818 \nu^{10} - 518619386475600818430 \nu^{9} + 297891740299819768065 \nu^{8} + 462111882536519235030 \nu^{7} - 1221396403147585516960 \nu^{6} + 1146466511179183660150 \nu^{5} - 52120827956304170650 \nu^{4} - 1237845080087530323250 \nu^{3} + 1636283368731666981500 \nu^{2} - 1000034350928460114375 \nu + 246428841076479143750\)\()/ 578956306290055625 \)
\(\beta_{16}\)\(=\)\((\)\(-163857612058740414 \nu^{19} + 1106609603471731491 \nu^{18} - 3583892554698462036 \nu^{17} + 6934998576544117490 \nu^{16} - 7059941943483092152 \nu^{15} - 767670156260491406 \nu^{14} + 10387556848202746291 \nu^{13} + 13628203462061002440 \nu^{12} - 131134405844188808408 \nu^{11} + 354416596176207454374 \nu^{10} - 518325653524205301840 \nu^{9} + 258673556329431622335 \nu^{8} + 525953134439567224855 \nu^{7} - 1245107810641064857355 \nu^{6} + 1085483302507127775675 \nu^{5} + 56868510143459192625 \nu^{4} - 1301309736426862382875 \nu^{3} + 1598201130190251779500 \nu^{2} - 911623635245600988750 \nu + 199654771191282361875\)\()/ 578956306290055625 \)
\(\beta_{17}\)\(=\)\((\)\(318431387248873 \nu^{19} - 2112948400318823 \nu^{18} + 6731616959645245 \nu^{17} - 12784358399935751 \nu^{16} + 12515295621334803 \nu^{15} + 2437811930861708 \nu^{14} - 19478551828170840 \nu^{13} - 28473735940023989 \nu^{12} + 250593112482275332 \nu^{11} - 661007290511435809 \nu^{10} + 941424962669792265 \nu^{9} - 420574591594037470 \nu^{8} - 1035466569807101635 \nu^{7} + 2291301111889594355 \nu^{6} - 1895703658051444525 \nu^{5} - 239760301221578425 \nu^{4} + 2443957835195011625 \nu^{3} - 2855255565534627500 \nu^{2} + 1545102901420578125 \nu - 305044192027929375\)\()/ 982947888438125 \)
\(\beta_{18}\)\(=\)\((\)\(189226218835387071 \nu^{19} - 1249025161281691561 \nu^{18} + 3958909046318192030 \nu^{17} - 7472175246983029242 \nu^{16} + 7214164583292571306 \nu^{15} + 1637232533936390986 \nu^{14} - 11471425650121448585 \nu^{13} - 17282352960591483763 \nu^{12} + 148221744514515726414 \nu^{11} - 387902714167987098588 \nu^{10} + 547441748560436194865 \nu^{9} - 234312234885223513405 \nu^{8} - 619303676193905798645 \nu^{7} + 1339526286020501240660 \nu^{6} - 1086296463294557592200 \nu^{5} - 170073187058092349050 \nu^{4} + 1439979537760694127875 \nu^{3} - 1649793748075719243125 \nu^{2} + 872122189921419738750 \nu - 162766283559273554375\)\()/ 578956306290055625 \)
\(\beta_{19}\)\(=\)\((\)\(197301946117046295 \nu^{19} - 1389830825460437202 \nu^{18} + 4673808200443922396 \nu^{17} - 9422962982932261497 \nu^{16} + 10380766197926255788 \nu^{15} - 580336248249537423 \nu^{14} - 13562957547663939506 \nu^{13} - 13296822923757996038 \nu^{12} + 164283683720334618517 \nu^{11} - 469502229515924475892 \nu^{10} + 726460430032732806915 \nu^{9} - 440471841132698903590 \nu^{8} - 609511060070350716625 \nu^{7} + 1696886755484710020340 \nu^{6} - 1642057767825738448450 \nu^{5} + 137588830506818748075 \nu^{4} + 1696475148733317122500 \nu^{3} - 2315095108858845276125 \nu^{2} + 1454565048426592792500 \nu - 372860688146919923750\)\()/ 578956306290055625 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-3 \beta_{19} - 3 \beta_{18} + \beta_{15} - 4 \beta_{13} - 5 \beta_{12} - 4 \beta_{10} + 8 \beta_{9} - \beta_{8} - 2 \beta_{7} + \beta_{6} + 2 \beta_{4} + 4 \beta_{3} + 4 \beta_{2} - 4 \beta_{1} + 2\)\()/5\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{18} - 3 \beta_{17} - \beta_{16} + 2 \beta_{14} - 5 \beta_{13} - 4 \beta_{12} - \beta_{11} - 7 \beta_{10} + 3 \beta_{9} - 3 \beta_{8} - 2 \beta_{6} - 3 \beta_{5} + 6 \beta_{4} + 4 \beta_{2} - \beta_{1} - 4\)\()/5\)
\(\nu^{3}\)\(=\)\((\)\(-11 \beta_{19} + 7 \beta_{18} + 4 \beta_{17} + 8 \beta_{16} + 2 \beta_{15} + 4 \beta_{14} - 13 \beta_{13} - 3 \beta_{12} - 7 \beta_{11} - 17 \beta_{10} + 2 \beta_{9} - 3 \beta_{8} - 9 \beta_{7} - 2 \beta_{6} - 6 \beta_{5} + 6 \beta_{4} + 8 \beta_{3} + 11 \beta_{2} - 5 \beta_{1} - 9\)\()/5\)
\(\nu^{4}\)\(=\)\((\)\(-22 \beta_{19} + 12 \beta_{18} + 12 \beta_{17} + 9 \beta_{16} + 9 \beta_{15} + 2 \beta_{14} - 26 \beta_{13} - 4 \beta_{12} - \beta_{11} - 38 \beta_{10} + 10 \beta_{9} + 18 \beta_{8} - 23 \beta_{7} - 3 \beta_{6} + 2 \beta_{5} - 31 \beta_{4} + 6 \beta_{3} + 30 \beta_{2} - 17 \beta_{1} + 14\)\()/5\)
\(\nu^{5}\)\(=\)\((\)\(-6 \beta_{19} - 27 \beta_{18} + 12 \beta_{17} + 4 \beta_{16} - 13 \beta_{15} - 23 \beta_{14} + 27 \beta_{13} - 29 \beta_{12} + 4 \beta_{11} - 10 \beta_{10} + 34 \beta_{9} + 20 \beta_{8} + 6 \beta_{7} + 20 \beta_{6} + 7 \beta_{5} - 50 \beta_{4} + 13 \beta_{3} + 17 \beta_{2} + 11 \beta_{1} + 45\)\()/5\)
\(\nu^{6}\)\(=\)\((\)\(-36 \beta_{19} + 10 \beta_{18} - 32 \beta_{17} - 84 \beta_{16} + 22 \beta_{15} - 27 \beta_{14} + 37 \beta_{13} - 46 \beta_{12} + 81 \beta_{11} + 4 \beta_{10} + 43 \beta_{9} - 19 \beta_{8} - 44 \beta_{7} + 14 \beta_{6} + 33 \beta_{5} + 13 \beta_{4} + 8 \beta_{3} + 9 \beta_{2} + 8 \beta_{1} + 3\)\()/5\)
\(\nu^{7}\)\(=\)\((\)\(\beta_{19} + 61 \beta_{18} - 115 \beta_{17} - 145 \beta_{16} + 8 \beta_{15} + 18 \beta_{13} + 60 \beta_{12} + 55 \beta_{11} + 33 \beta_{10} - 86 \beta_{9} - 33 \beta_{8} - 11 \beta_{7} + 58 \beta_{6} - 35 \beta_{5} - 4 \beta_{4} - 3 \beta_{3} - 23 \beta_{2} - 7 \beta_{1} + 61\)\()/5\)
\(\nu^{8}\)\(=\)\((\)\(-240 \beta_{19} - 48 \beta_{18} - 29 \beta_{17} + 92 \beta_{16} - 65 \beta_{15} + 86 \beta_{14} - 195 \beta_{13} - 142 \beta_{12} - 188 \beta_{11} - 66 \beta_{10} + 254 \beta_{9} + 106 \beta_{8} - 80 \beta_{7} - 96 \beta_{6} - 134 \beta_{5} - 87 \beta_{4} + 190 \beta_{3} + 42 \beta_{2} - 168 \beta_{1} - 32\)\()/5\)
\(\nu^{9}\)\(=\)\((\)\(147 \beta_{19} - 204 \beta_{18} + 17 \beta_{17} + 279 \beta_{16} - 129 \beta_{15} + 277 \beta_{14} - 239 \beta_{13} - 169 \beta_{12} - 416 \beta_{11} - 121 \beta_{10} + 51 \beta_{9} + 476 \beta_{8} + 213 \beta_{7} - 116 \beta_{6} - 248 \beta_{5} - 652 \beta_{4} - 11 \beta_{3} - 227 \beta_{2} + 315 \beta_{1} + 303\)\()/5\)
\(\nu^{10}\)\(=\)\((\)\(-176 \beta_{19} - 154 \beta_{18} + 396 \beta_{17} + 567 \beta_{16} - 438 \beta_{15} + 431 \beta_{14} - 418 \beta_{13} - 537 \beta_{12} - 268 \beta_{11} - 439 \beta_{10} - 20 \beta_{9} - 186 \beta_{8} + 331 \beta_{7} + 611 \beta_{6} + 81 \beta_{5} - 313 \beta_{4} + 383 \beta_{3} - 300 \beta_{2} + 769 \beta_{1} + 812\)\()/5\)
\(\nu^{11}\)\(=\)\((\)\(-1708 \beta_{19} + 414 \beta_{18} + 211 \beta_{17} - 838 \beta_{16} + 246 \beta_{15} + 556 \beta_{14} - 2979 \beta_{13} - 662 \beta_{12} + 1717 \beta_{11} - 3115 \beta_{10} + 1287 \beta_{9} - 455 \beta_{8} - 1312 \beta_{7} + 960 \beta_{6} + 1191 \beta_{5} - 220 \beta_{4} + 899 \beta_{3} + 2066 \beta_{2} - 1287 \beta_{1} + 1095\)\()/5\)
\(\nu^{12}\)\(=\)\((\)\(-1258 \beta_{19} - 2635 \beta_{18} - 721 \beta_{17} - 1222 \beta_{16} - 1959 \beta_{15} - 881 \beta_{14} - 1564 \beta_{13} - 1798 \beta_{12} + 818 \beta_{11} - 3068 \beta_{10} + 3559 \beta_{9} - 642 \beta_{8} + 1183 \beta_{7} - 498 \beta_{6} + 49 \beta_{5} + 1524 \beta_{4} + 1989 \beta_{3} + 2982 \beta_{2} - 1431 \beta_{1} - 1461\)\()/5\)
\(\nu^{13}\)\(=\)\((\)\(-3827 \beta_{19} + 523 \beta_{18} - 1675 \beta_{17} - 2275 \beta_{16} - \beta_{15} - 615 \beta_{14} - 771 \beta_{13} - 3145 \beta_{12} + 960 \beta_{11} - 2101 \beta_{10} + 2972 \beta_{9} + 2401 \beta_{8} - 2378 \beta_{7} - 8366 \beta_{6} - 455 \beta_{5} + 4473 \beta_{4} + 2346 \beta_{3} + 1771 \beta_{2} - 241 \beta_{1} - 11147\)\()/5\)
\(\nu^{14}\)\(=\)\((\)\(5015 \beta_{19} + 10676 \beta_{18} - 3132 \beta_{17} - 3404 \beta_{16} + 1380 \beta_{15} + 1918 \beta_{14} + 4365 \beta_{13} + 9459 \beta_{12} + 411 \beta_{11} + 4767 \beta_{10} - 20928 \beta_{9} + 11328 \beta_{8} + 1715 \beta_{7} - 3653 \beta_{6} - 1682 \beta_{5} - 14671 \beta_{4} - 3955 \beta_{3} - 7469 \beta_{2} + 8736 \beta_{1} + 5379\)\()/5\)
\(\nu^{15}\)\(=\)\((\)\(-10119 \beta_{19} + 1103 \beta_{18} + 7151 \beta_{17} + 5087 \beta_{16} - 6917 \beta_{15} + 3826 \beta_{14} + 1828 \beta_{13} - 4752 \beta_{12} + 3342 \beta_{11} + 9152 \beta_{10} + 1423 \beta_{9} + 16668 \beta_{8} + 654 \beta_{7} + 10762 \beta_{6} + 6511 \beta_{5} - 37726 \beta_{4} + 10507 \beta_{3} - 8031 \beta_{2} + 15 \beta_{1} + 34064\)\()/5\)
\(\nu^{16}\)\(=\)\((\)\(23357 \beta_{19} - 39752 \beta_{18} - 11277 \beta_{17} - 25364 \beta_{16} - 18454 \beta_{15} + 11873 \beta_{14} + 9626 \beta_{13} - 16346 \beta_{12} + 22826 \beta_{11} + 23473 \beta_{10} + 22525 \beta_{9} + 17857 \beta_{8} + 32893 \beta_{7} + 19993 \beta_{6} + 7163 \beta_{5} - 48874 \beta_{4} - 3021 \beta_{3} - 28880 \beta_{2} + 18382 \beta_{1} + 50186\)\()/5\)
\(\nu^{17}\)\(=\)\((\)\(46866 \beta_{19} - 67818 \beta_{18} - 34272 \beta_{17} - 30209 \beta_{16} - 70412 \beta_{15} + 39763 \beta_{14} + 31373 \beta_{13} - 27126 \beta_{12} - 6529 \beta_{11} + 65470 \beta_{10} - 7534 \beta_{9} - 71615 \beta_{8} + 99389 \beta_{7} + 13510 \beta_{6} - 25197 \beta_{5} + 102345 \beta_{4} + 14532 \beta_{3} - 95772 \beta_{2} + 80414 \beta_{1} - 49695\)\()/5\)
\(\nu^{18}\)\(=\)\((\)\(-73009 \beta_{19} + 92310 \beta_{18} - 26653 \beta_{17} - 23706 \beta_{16} + 3818 \beta_{15} + 150447 \beta_{14} - 253507 \beta_{13} + 47641 \beta_{12} - 4746 \beta_{11} - 135989 \beta_{10} - 72553 \beta_{9} - 36306 \beta_{8} - 48941 \beta_{7} - 93529 \beta_{6} - 19398 \beta_{5} + 152312 \beta_{4} + 42772 \beta_{3} + 10011 \beta_{2} - 49633 \beta_{1} - 187618\)\()/5\)
\(\nu^{19}\)\(=\)\((\)\(11459 \beta_{19} - 57676 \beta_{18} + 117505 \beta_{17} + 266970 \beta_{16} - 171368 \beta_{15} + 150800 \beta_{14} - 317668 \beta_{13} + 74710 \beta_{12} - 241815 \beta_{11} - 254888 \beta_{10} - 89524 \beta_{9} + 205288 \beta_{8} + 178471 \beta_{7} - 128593 \beta_{6} - 76800 \beta_{5} - 203556 \beta_{4} + 98813 \beta_{3} + 73623 \beta_{2} - 24323 \beta_{1} - 2746\)\()/5\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(-\beta_{4}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
106.1
1.14677 + 0.258633i
−1.14371 0.271822i
0.891061 0.766796i
0.209942 1.15667i
0.895943 + 0.761087i
−1.67866 + 0.894495i
1.88955 0.218280i
0.670843 + 1.77989i
−0.238506 1.88710i
1.35678 + 1.33311i
−1.67866 0.894495i
1.88955 + 0.218280i
0.670843 1.77989i
−0.238506 + 1.88710i
1.35678 1.33311i
1.14677 0.258633i
−1.14371 + 0.271822i
0.891061 + 0.766796i
0.209942 + 1.15667i
0.895943 0.761087i
−0.724838 2.23082i 0.809017 0.587785i −2.83314 + 2.05840i 0.884129 + 2.05385i −1.89765 1.37872i −1.00000 2.85019 + 2.07078i 0.309017 0.951057i 3.94093 3.46105i
106.2 −0.401368 1.23528i 0.809017 0.587785i 0.253205 0.183964i −0.860419 2.06390i −1.05079 0.763447i −1.00000 −2.43047 1.76584i 0.309017 0.951057i −2.20416 + 1.89124i
106.3 −0.179936 0.553787i 0.809017 0.587785i 1.34373 0.976277i 2.17621 + 0.513897i −0.471080 0.342259i −1.00000 −1.72460 1.25299i 0.309017 0.951057i −0.106991 1.29763i
106.4 0.256297 + 0.788802i 0.809017 0.587785i 1.06151 0.771234i 2.01466 0.970131i 0.670995 + 0.487507i −1.00000 2.22241 + 1.61467i 0.309017 0.951057i 1.28159 + 1.34052i
106.5 0.549845 + 1.69225i 0.809017 0.587785i −0.943341 + 0.685378i −0.169498 + 2.22963i 1.43951 + 1.04587i −1.00000 1.20050 + 0.872217i 0.309017 0.951057i −3.86629 + 0.939121i
211.1 −2.02723 + 1.47287i −0.309017 + 0.951057i 1.32228 4.06957i 2.20175 0.390266i −0.774332 2.38315i −1.00000 1.76470 + 5.43119i −0.809017 0.587785i −3.88864 + 4.03404i
211.2 −0.870436 + 0.632409i −0.309017 + 0.951057i −0.260316 + 0.801170i −2.19187 0.442374i −0.332477 1.02326i −1.00000 −0.945033 2.90851i −0.809017 0.587785i 2.18765 1.00110i
211.3 0.140253 0.101900i −0.309017 + 0.951057i −0.608747 + 1.87353i −0.103443 2.23367i 0.0535718 + 0.164877i −1.00000 0.212677 + 0.654553i −0.809017 0.587785i −0.242119 0.302738i
211.4 0.269002 0.195442i −0.309017 + 0.951057i −0.583869 + 1.79696i −0.418871 + 2.19649i 0.102750 + 0.316231i −1.00000 0.399639 + 1.22996i −0.809017 0.587785i 0.316607 + 0.672725i
211.5 1.98841 1.44466i −0.309017 + 0.951057i 1.24868 3.84305i −1.03265 1.98334i 0.759505 + 2.33752i −1.00000 −1.55002 4.77046i −0.809017 0.587785i −4.91858 2.45187i
316.1 −2.02723 1.47287i −0.309017 0.951057i 1.32228 + 4.06957i 2.20175 + 0.390266i −0.774332 + 2.38315i −1.00000 1.76470 5.43119i −0.809017 + 0.587785i −3.88864 4.03404i
316.2 −0.870436 0.632409i −0.309017 0.951057i −0.260316 0.801170i −2.19187 + 0.442374i −0.332477 + 1.02326i −1.00000 −0.945033 + 2.90851i −0.809017 + 0.587785i 2.18765 + 1.00110i
316.3 0.140253 + 0.101900i −0.309017 0.951057i −0.608747 1.87353i −0.103443 + 2.23367i 0.0535718 0.164877i −1.00000 0.212677 0.654553i −0.809017 + 0.587785i −0.242119 + 0.302738i
316.4 0.269002 + 0.195442i −0.309017 0.951057i −0.583869 1.79696i −0.418871 2.19649i 0.102750 0.316231i −1.00000 0.399639 1.22996i −0.809017 + 0.587785i 0.316607 0.672725i
316.5 1.98841 + 1.44466i −0.309017 0.951057i 1.24868 + 3.84305i −1.03265 + 1.98334i 0.759505 2.33752i −1.00000 −1.55002 + 4.77046i −0.809017 + 0.587785i −4.91858 + 2.45187i
421.1 −0.724838 + 2.23082i 0.809017 + 0.587785i −2.83314 2.05840i 0.884129 2.05385i −1.89765 + 1.37872i −1.00000 2.85019 2.07078i 0.309017 + 0.951057i 3.94093 + 3.46105i
421.2 −0.401368 + 1.23528i 0.809017 + 0.587785i 0.253205 + 0.183964i −0.860419 + 2.06390i −1.05079 + 0.763447i −1.00000 −2.43047 + 1.76584i 0.309017 + 0.951057i −2.20416 1.89124i
421.3 −0.179936 + 0.553787i 0.809017 + 0.587785i 1.34373 + 0.976277i 2.17621 0.513897i −0.471080 + 0.342259i −1.00000 −1.72460 + 1.25299i 0.309017 + 0.951057i −0.106991 + 1.29763i
421.4 0.256297 0.788802i 0.809017 + 0.587785i 1.06151 + 0.771234i 2.01466 + 0.970131i 0.670995 0.487507i −1.00000 2.22241 1.61467i 0.309017 + 0.951057i 1.28159 1.34052i
421.5 0.549845 1.69225i 0.809017 + 0.587785i −0.943341 0.685378i −0.169498 2.22963i 1.43951 1.04587i −1.00000 1.20050 0.872217i 0.309017 + 0.951057i −3.86629 0.939121i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 421.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.n.b 20
25.d even 5 1 inner 525.2.n.b 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.n.b 20 1.a even 1 1 trivial
525.2.n.b 20 25.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{20} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(525, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T - 3 T^{2} - 8 T^{3} + 9 T^{4} + 24 T^{5} - 35 T^{6} - 76 T^{7} + 97 T^{8} + 174 T^{9} - 308 T^{10} - 375 T^{11} + 914 T^{12} + 753 T^{13} - 2368 T^{14} - 1381 T^{15} + 5578 T^{16} + 1957 T^{17} - 12263 T^{18} - 1170 T^{19} + 25773 T^{20} - 2340 T^{21} - 49052 T^{22} + 15656 T^{23} + 89248 T^{24} - 44192 T^{25} - 151552 T^{26} + 96384 T^{27} + 233984 T^{28} - 192000 T^{29} - 315392 T^{30} + 356352 T^{31} + 397312 T^{32} - 622592 T^{33} - 573440 T^{34} + 786432 T^{35} + 589824 T^{36} - 1048576 T^{37} - 786432 T^{38} + 1048576 T^{39} + 1048576 T^{40} \)
$3$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5} \)
$5$ \( 1 - 5 T + 20 T^{2} - 80 T^{3} + 205 T^{4} - 450 T^{5} + 850 T^{6} - 425 T^{7} - 1850 T^{8} + 9125 T^{9} - 28375 T^{10} + 45625 T^{11} - 46250 T^{12} - 53125 T^{13} + 531250 T^{14} - 1406250 T^{15} + 3203125 T^{16} - 6250000 T^{17} + 7812500 T^{18} - 9765625 T^{19} + 9765625 T^{20} \)
$7$ \( ( 1 + T )^{20} \)
$11$ \( 1 - 12 T + 5 T^{2} + 527 T^{3} - 2020 T^{4} - 7203 T^{5} + 56722 T^{6} - 4798 T^{7} - 711707 T^{8} + 1150473 T^{9} + 4220063 T^{10} - 12236295 T^{11} + 6043188 T^{12} - 37619910 T^{13} - 230334633 T^{14} + 2791361284 T^{15} - 3117216920 T^{16} - 35048914405 T^{17} + 113970259425 T^{18} + 155915532485 T^{19} - 1574551625415 T^{20} + 1715070857335 T^{21} + 13790401390425 T^{22} - 46650105073055 T^{23} - 45639172925720 T^{24} + 449551526149484 T^{25} - 408051852772113 T^{26} - 733105619174610 T^{27} + 1295411017352628 T^{28} - 28852543541644845 T^{29} + 109457565873969863 T^{30} + 328243373622849003 T^{31} - 2233641444710972747 T^{32} - 165639972866580938 T^{33} + 21540170060508596002 T^{34} - 30088718564300934153 T^{35} - 92818454324415765220 T^{36} + \)\(26\!\cdots\!17\)\( T^{37} + 27799586567461157405 T^{38} - \)\(73\!\cdots\!92\)\( T^{39} + \)\(67\!\cdots\!01\)\( T^{40} \)
$13$ \( 1 + 17 T + 118 T^{2} + 511 T^{3} + 2352 T^{4} + 11040 T^{5} + 35384 T^{6} + 109963 T^{7} + 467215 T^{8} + 971364 T^{9} - 724062 T^{10} - 1470107 T^{11} - 9009942 T^{12} - 143708236 T^{13} - 89788267 T^{14} + 1601061003 T^{15} + 3142834670 T^{16} + 22309519952 T^{17} + 208993081204 T^{18} + 830991825843 T^{19} + 2569268002331 T^{20} + 10802893735959 T^{21} + 35319830723476 T^{22} + 49014015334544 T^{23} + 89762501009870 T^{24} + 594462742986879 T^{25} - 433390815250003 T^{26} - 9017478689686012 T^{27} - 7349686483828182 T^{28} - 15589748759742911 T^{29} - 99818095325170638 T^{30} + 1740840088993356468 T^{31} + 10885214840499960415 T^{32} + 33305055346203916639 T^{33} + \)\(13\!\cdots\!76\)\( T^{34} + \)\(56\!\cdots\!80\)\( T^{35} + \)\(15\!\cdots\!32\)\( T^{36} + \)\(44\!\cdots\!63\)\( T^{37} + \)\(13\!\cdots\!22\)\( T^{38} + \)\(24\!\cdots\!09\)\( T^{39} + \)\(19\!\cdots\!01\)\( T^{40} \)
$17$ \( 1 + 9 T - 27 T^{2} - 620 T^{3} - 1290 T^{4} + 20557 T^{5} + 117068 T^{6} - 310247 T^{7} - 4476802 T^{8} - 4151366 T^{9} + 107746789 T^{10} + 390341611 T^{11} - 1579360503 T^{12} - 12891387238 T^{13} + 2870358378 T^{14} + 278972749333 T^{15} + 601233535747 T^{16} - 4051231959030 T^{17} - 20626010838649 T^{18} + 27189021791451 T^{19} + 423848770469272 T^{20} + 462213370454667 T^{21} - 5960917132369561 T^{22} - 19903702614714390 T^{23} + 50215626139125187 T^{24} + 396101410949705381 T^{25} + 69283473403703082 T^{26} - 5289834732370055174 T^{27} - 11017235780823752823 T^{28} + 46289782756908016667 T^{29} + \)\(21\!\cdots\!61\)\( T^{30} - \)\(14\!\cdots\!78\)\( T^{31} - \)\(26\!\cdots\!22\)\( T^{32} - \)\(30\!\cdots\!39\)\( T^{33} + \)\(19\!\cdots\!72\)\( T^{34} + \)\(58\!\cdots\!01\)\( T^{35} - \)\(62\!\cdots\!90\)\( T^{36} - \)\(51\!\cdots\!40\)\( T^{37} - \)\(37\!\cdots\!43\)\( T^{38} + \)\(21\!\cdots\!77\)\( T^{39} + \)\(40\!\cdots\!01\)\( T^{40} \)
$19$ \( 1 + 9 T + 9 T^{2} + 10 T^{3} + 1521 T^{4} + 5182 T^{5} - 28341 T^{6} - 73578 T^{7} + 621886 T^{8} - 1041450 T^{9} - 21412968 T^{10} + 22324377 T^{11} + 309389811 T^{12} - 612021092 T^{13} + 809994731 T^{14} + 38047928694 T^{15} + 10122189378 T^{16} - 287816743238 T^{17} + 1931412339679 T^{18} + 1204750584038 T^{19} - 56973551738703 T^{20} + 22890261096722 T^{21} + 697239854624119 T^{22} - 1974135041869442 T^{23} + 1319133841930338 T^{24} + 94210438191284706 T^{25} + 38106915725253011 T^{26} - 547068357810718988 T^{27} + 5254541359361575251 T^{28} + 7203801818480458683 T^{29} - \)\(13\!\cdots\!68\)\( T^{30} - \)\(12\!\cdots\!50\)\( T^{31} + \)\(13\!\cdots\!46\)\( T^{32} - \)\(30\!\cdots\!02\)\( T^{33} - \)\(22\!\cdots\!61\)\( T^{34} + \)\(78\!\cdots\!18\)\( T^{35} + \)\(43\!\cdots\!01\)\( T^{36} + \)\(54\!\cdots\!90\)\( T^{37} + \)\(93\!\cdots\!69\)\( T^{38} + \)\(17\!\cdots\!11\)\( T^{39} + \)\(37\!\cdots\!01\)\( T^{40} \)
$23$ \( 1 - 7 T + 5 T^{2} + 232 T^{3} - 1052 T^{4} - 3365 T^{5} + 37947 T^{6} - 130501 T^{7} - 60947 T^{8} + 2537516 T^{9} - 10158836 T^{10} - 41777788 T^{11} + 620616929 T^{12} - 1197621959 T^{13} - 12815167944 T^{14} + 50019431341 T^{15} + 259769120270 T^{16} - 2481947544649 T^{17} + 5320406087173 T^{18} + 22264384484774 T^{19} - 176619524520221 T^{20} + 512080843149802 T^{21} + 2814494820114517 T^{22} - 30197855775744383 T^{23} + 72694050385477070 T^{24} + 321942216775625963 T^{25} - 1897104779274342216 T^{26} - 4077693721889190673 T^{27} + 48601123192058422049 T^{28} - 75248174046236983844 T^{29} - \)\(42\!\cdots\!64\)\( T^{30} + \)\(24\!\cdots\!32\)\( T^{31} - \)\(13\!\cdots\!87\)\( T^{32} - \)\(65\!\cdots\!83\)\( T^{33} + \)\(43\!\cdots\!23\)\( T^{34} - \)\(89\!\cdots\!55\)\( T^{35} - \)\(64\!\cdots\!72\)\( T^{36} + \)\(32\!\cdots\!96\)\( T^{37} + \)\(16\!\cdots\!45\)\( T^{38} - \)\(52\!\cdots\!09\)\( T^{39} + \)\(17\!\cdots\!01\)\( T^{40} \)
$29$ \( 1 - 28 T + 292 T^{2} - 1111 T^{3} - 2637 T^{4} + 48056 T^{5} - 354977 T^{6} + 1751017 T^{7} + 1171058 T^{8} - 59724245 T^{9} + 265622419 T^{10} - 1367742638 T^{11} + 9549291711 T^{12} - 12436162284 T^{13} - 128894774032 T^{14} + 796946984854 T^{15} - 10162058212054 T^{16} + 58858081368164 T^{17} + 81427480890353 T^{18} - 1834790782280269 T^{19} + 9689391931459613 T^{20} - 53208932686127801 T^{21} + 68480511428786873 T^{22} + 1435489746488151796 T^{23} - 7187430694279765174 T^{24} + 16346298351441137246 T^{25} - 76669617549258800272 T^{26} - \)\(21\!\cdots\!56\)\( T^{27} + \)\(47\!\cdots\!71\)\( T^{28} - \)\(19\!\cdots\!22\)\( T^{29} + \)\(11\!\cdots\!19\)\( T^{30} - \)\(72\!\cdots\!05\)\( T^{31} + \)\(41\!\cdots\!78\)\( T^{32} + \)\(17\!\cdots\!13\)\( T^{33} - \)\(10\!\cdots\!37\)\( T^{34} + \)\(41\!\cdots\!44\)\( T^{35} - \)\(65\!\cdots\!77\)\( T^{36} - \)\(80\!\cdots\!99\)\( T^{37} + \)\(61\!\cdots\!12\)\( T^{38} - \)\(17\!\cdots\!32\)\( T^{39} + \)\(17\!\cdots\!01\)\( T^{40} \)
$31$ \( 1 - 6 T - 140 T^{2} + 841 T^{3} + 10547 T^{4} - 68492 T^{5} - 515260 T^{6} + 4069562 T^{7} + 17326096 T^{8} - 190777850 T^{9} - 393262670 T^{10} + 7140215105 T^{11} + 5355496331 T^{12} - 210301021603 T^{13} - 32554607310 T^{14} + 4743770235228 T^{15} + 963101506732 T^{16} - 77868139958804 T^{17} - 93076859779940 T^{18} + 691653607625279 T^{19} + 3973675961861677 T^{20} + 21441261836383649 T^{21} - 89446862248522340 T^{22} - 2319769757512729964 T^{23} + 889444466598643372 T^{24} + \)\(13\!\cdots\!28\)\( T^{25} - 28892333821134508110 T^{26} - \)\(57\!\cdots\!33\)\( T^{27} + \)\(45\!\cdots\!71\)\( T^{28} + \)\(18\!\cdots\!55\)\( T^{29} - \)\(32\!\cdots\!70\)\( T^{30} - \)\(48\!\cdots\!50\)\( T^{31} + \)\(13\!\cdots\!56\)\( T^{32} + \)\(99\!\cdots\!42\)\( T^{33} - \)\(39\!\cdots\!60\)\( T^{34} - \)\(16\!\cdots\!92\)\( T^{35} + \)\(76\!\cdots\!07\)\( T^{36} + \)\(18\!\cdots\!51\)\( T^{37} - \)\(97\!\cdots\!40\)\( T^{38} - \)\(13\!\cdots\!26\)\( T^{39} + \)\(67\!\cdots\!01\)\( T^{40} \)
$37$ \( 1 + 5 T - 81 T^{2} - 679 T^{3} + 3744 T^{4} + 60925 T^{5} + 60905 T^{6} - 3234461 T^{7} - 14856815 T^{8} + 110399361 T^{9} + 1143959304 T^{10} - 675160546 T^{11} - 47596373540 T^{12} - 138591309660 T^{13} + 1351597549496 T^{14} + 10023063834413 T^{15} - 4790109168240 T^{16} - 343820155009758 T^{17} - 1189329408355585 T^{18} + 5609372386700900 T^{19} + 73074032491390783 T^{20} + 207546778307933300 T^{21} - 1628191960038795865 T^{22} - 17415522311709271974 T^{23} - 8977435788857846640 T^{24} + \)\(69\!\cdots\!41\)\( T^{25} + \)\(34\!\cdots\!64\)\( T^{26} - \)\(13\!\cdots\!80\)\( T^{27} - \)\(16\!\cdots\!40\)\( T^{28} - \)\(87\!\cdots\!42\)\( T^{29} + \)\(55\!\cdots\!96\)\( T^{30} + \)\(19\!\cdots\!93\)\( T^{31} - \)\(97\!\cdots\!15\)\( T^{32} - \)\(78\!\cdots\!17\)\( T^{33} + \)\(54\!\cdots\!45\)\( T^{34} + \)\(20\!\cdots\!25\)\( T^{35} + \)\(46\!\cdots\!04\)\( T^{36} - \)\(30\!\cdots\!43\)\( T^{37} - \)\(13\!\cdots\!49\)\( T^{38} + \)\(31\!\cdots\!65\)\( T^{39} + \)\(23\!\cdots\!01\)\( T^{40} \)
$41$ \( 1 - 11 T - 74 T^{2} + 1250 T^{3} + 74 T^{4} - 40033 T^{5} - 23006 T^{6} + 264250 T^{7} + 14104671 T^{8} - 64818221 T^{9} - 407915891 T^{10} + 4491189267 T^{11} - 20807200322 T^{12} + 22786129000 T^{13} + 615571236857 T^{14} - 4246961124649 T^{15} + 23630290658722 T^{16} - 244694380689250 T^{17} + 940209666208138 T^{18} + 10616650231934837 T^{19} - 133412201560405859 T^{20} + 435282659509328317 T^{21} + 1580492448895879978 T^{22} - 16864581411483799250 T^{23} + 66773553762080937442 T^{24} - \)\(49\!\cdots\!49\)\( T^{25} + \)\(29\!\cdots\!37\)\( T^{26} + \)\(44\!\cdots\!00\)\( T^{27} - \)\(16\!\cdots\!62\)\( T^{28} + \)\(14\!\cdots\!87\)\( T^{29} - \)\(54\!\cdots\!91\)\( T^{30} - \)\(35\!\cdots\!61\)\( T^{31} + \)\(31\!\cdots\!51\)\( T^{32} + \)\(24\!\cdots\!50\)\( T^{33} - \)\(87\!\cdots\!66\)\( T^{34} - \)\(62\!\cdots\!33\)\( T^{35} + \)\(47\!\cdots\!34\)\( T^{36} + \)\(32\!\cdots\!50\)\( T^{37} - \)\(79\!\cdots\!54\)\( T^{38} - \)\(48\!\cdots\!71\)\( T^{39} + \)\(18\!\cdots\!01\)\( T^{40} \)
$43$ \( ( 1 - 14 T + 271 T^{2} - 2568 T^{3} + 26841 T^{4} - 162823 T^{5} + 953702 T^{6} - 904287 T^{7} - 22940467 T^{8} + 388741980 T^{9} - 2790795889 T^{10} + 16715905140 T^{11} - 42416923483 T^{12} - 71897146509 T^{13} + 3260517351302 T^{14} - 23936355714589 T^{15} + 169671705598209 T^{16} - 698030193322776 T^{17} + 3167502275229871 T^{18} - 7036296567115802 T^{19} + 21611482313284249 T^{20} )^{2} \)
$47$ \( 1 + 24 T + 214 T^{2} + 1028 T^{3} + 12897 T^{4} + 148620 T^{5} + 168624 T^{6} - 6740131 T^{7} - 13459903 T^{8} + 17951408 T^{9} - 4846961466 T^{10} - 43154146793 T^{11} - 4688333873 T^{12} + 105775993364 T^{13} - 13620645345307 T^{14} - 46895778039065 T^{15} + 646122293050745 T^{16} + 2673706913658631 T^{17} - 13814120092137086 T^{18} + 122417875161538166 T^{19} + 2510353529002371099 T^{20} + 5753640132592293802 T^{21} - 30515391283530822974 T^{22} + \)\(27\!\cdots\!13\)\( T^{23} + \)\(31\!\cdots\!45\)\( T^{24} - \)\(10\!\cdots\!55\)\( T^{25} - \)\(14\!\cdots\!03\)\( T^{26} + \)\(53\!\cdots\!32\)\( T^{27} - \)\(11\!\cdots\!53\)\( T^{28} - \)\(48\!\cdots\!31\)\( T^{29} - \)\(25\!\cdots\!34\)\( T^{30} + \)\(44\!\cdots\!24\)\( T^{31} - \)\(15\!\cdots\!23\)\( T^{32} - \)\(36\!\cdots\!37\)\( T^{33} + \)\(43\!\cdots\!56\)\( T^{34} + \)\(17\!\cdots\!60\)\( T^{35} + \)\(73\!\cdots\!37\)\( T^{36} + \)\(27\!\cdots\!36\)\( T^{37} + \)\(26\!\cdots\!46\)\( T^{38} + \)\(14\!\cdots\!92\)\( T^{39} + \)\(27\!\cdots\!01\)\( T^{40} \)
$53$ \( 1 + 26 T + 189 T^{2} - 687 T^{3} - 12003 T^{4} + 33403 T^{5} + 552417 T^{6} - 5988920 T^{7} - 70924623 T^{8} + 151543065 T^{9} + 3746438244 T^{10} - 1341982202 T^{11} - 133505239607 T^{12} + 94806090571 T^{13} + 3402269873522 T^{14} - 28374866549812 T^{15} - 304279860832395 T^{16} - 79730970585402 T^{17} + 9927808060918525 T^{18} + 92743126415480970 T^{19} + 733823132261482620 T^{20} + 4915385700020491410 T^{21} + 27887212843120136725 T^{22} - 11870107707842893554 T^{23} - \)\(24\!\cdots\!95\)\( T^{24} - \)\(11\!\cdots\!16\)\( T^{25} + \)\(75\!\cdots\!38\)\( T^{26} + \)\(11\!\cdots\!27\)\( T^{27} - \)\(83\!\cdots\!27\)\( T^{28} - \)\(44\!\cdots\!66\)\( T^{29} + \)\(65\!\cdots\!56\)\( T^{30} + \)\(14\!\cdots\!05\)\( T^{31} - \)\(34\!\cdots\!43\)\( T^{32} - \)\(15\!\cdots\!60\)\( T^{33} + \)\(76\!\cdots\!73\)\( T^{34} + \)\(24\!\cdots\!71\)\( T^{35} - \)\(46\!\cdots\!63\)\( T^{36} - \)\(14\!\cdots\!31\)\( T^{37} + \)\(20\!\cdots\!21\)\( T^{38} + \)\(15\!\cdots\!42\)\( T^{39} + \)\(30\!\cdots\!01\)\( T^{40} \)
$59$ \( 1 - 64 T + 1835 T^{2} - 30211 T^{3} + 294923 T^{4} - 1319681 T^{5} - 6472851 T^{6} + 158363147 T^{7} - 1301770421 T^{8} + 5307853640 T^{9} + 6106474336 T^{10} - 292966186104 T^{11} + 2789886458001 T^{12} - 14757440708664 T^{13} - 27120694739231 T^{14} + 1672303716112849 T^{15} - 19763683840572248 T^{16} + 114997595957857866 T^{17} + 62338242339194563 T^{18} - 8477577091481604596 T^{19} + 91524505213402944705 T^{20} - \)\(50\!\cdots\!64\)\( T^{21} + \)\(21\!\cdots\!03\)\( T^{22} + \)\(23\!\cdots\!14\)\( T^{23} - \)\(23\!\cdots\!28\)\( T^{24} + \)\(11\!\cdots\!51\)\( T^{25} - \)\(11\!\cdots\!71\)\( T^{26} - \)\(36\!\cdots\!16\)\( T^{27} + \)\(40\!\cdots\!21\)\( T^{28} - \)\(25\!\cdots\!56\)\( T^{29} + \)\(31\!\cdots\!36\)\( T^{30} + \)\(16\!\cdots\!60\)\( T^{31} - \)\(23\!\cdots\!01\)\( T^{32} + \)\(16\!\cdots\!13\)\( T^{33} - \)\(40\!\cdots\!11\)\( T^{34} - \)\(48\!\cdots\!19\)\( T^{35} + \)\(63\!\cdots\!43\)\( T^{36} - \)\(38\!\cdots\!09\)\( T^{37} + \)\(13\!\cdots\!35\)\( T^{38} - \)\(28\!\cdots\!96\)\( T^{39} + \)\(26\!\cdots\!01\)\( T^{40} \)
$61$ \( 1 - 8 T - 250 T^{2} + 2547 T^{3} + 23448 T^{4} - 409872 T^{5} + 30251 T^{6} + 40503616 T^{7} - 251606971 T^{8} - 2220692740 T^{9} + 32366729049 T^{10} - 15508614507 T^{11} - 2274901784809 T^{12} + 15563607606792 T^{13} + 79216530591596 T^{14} - 1576254516302698 T^{15} + 2970947510282232 T^{16} + 87783662074531477 T^{17} - 687319481699616148 T^{18} - 2134990650502795753 T^{19} + 55437519869949000215 T^{20} - \)\(13\!\cdots\!33\)\( T^{21} - \)\(25\!\cdots\!08\)\( T^{22} + \)\(19\!\cdots\!37\)\( T^{23} + \)\(41\!\cdots\!12\)\( T^{24} - \)\(13\!\cdots\!98\)\( T^{25} + \)\(40\!\cdots\!56\)\( T^{26} + \)\(48\!\cdots\!32\)\( T^{27} - \)\(43\!\cdots\!29\)\( T^{28} - \)\(18\!\cdots\!87\)\( T^{29} + \)\(23\!\cdots\!49\)\( T^{30} - \)\(96\!\cdots\!40\)\( T^{31} - \)\(66\!\cdots\!91\)\( T^{32} + \)\(65\!\cdots\!96\)\( T^{33} + \)\(29\!\cdots\!91\)\( T^{34} - \)\(24\!\cdots\!72\)\( T^{35} + \)\(86\!\cdots\!28\)\( T^{36} + \)\(57\!\cdots\!87\)\( T^{37} - \)\(34\!\cdots\!50\)\( T^{38} - \)\(66\!\cdots\!28\)\( T^{39} + \)\(50\!\cdots\!01\)\( T^{40} \)
$67$ \( 1 + 3 T - 311 T^{2} - 1039 T^{3} + 45007 T^{4} + 100608 T^{5} - 4203051 T^{6} + 2712957 T^{7} + 286251229 T^{8} - 1634937209 T^{9} - 12746581716 T^{10} + 220418749894 T^{11} + 98093010794 T^{12} - 19071644840712 T^{13} + 44213444347269 T^{14} + 1156522239768568 T^{15} - 5805695400666044 T^{16} - 49425850295776060 T^{17} + 521799371255402099 T^{18} + 1103857348682193140 T^{19} - 37995128081047056923 T^{20} + 73958442361706940380 T^{21} + \)\(23\!\cdots\!11\)\( T^{22} - \)\(14\!\cdots\!80\)\( T^{23} - \)\(11\!\cdots\!24\)\( T^{24} + \)\(15\!\cdots\!76\)\( T^{25} + \)\(39\!\cdots\!61\)\( T^{26} - \)\(11\!\cdots\!76\)\( T^{27} + \)\(39\!\cdots\!54\)\( T^{28} + \)\(59\!\cdots\!18\)\( T^{29} - \)\(23\!\cdots\!84\)\( T^{30} - \)\(19\!\cdots\!47\)\( T^{31} + \)\(23\!\cdots\!69\)\( T^{32} + \)\(14\!\cdots\!59\)\( T^{33} - \)\(15\!\cdots\!79\)\( T^{34} + \)\(24\!\cdots\!44\)\( T^{35} + \)\(74\!\cdots\!67\)\( T^{36} - \)\(11\!\cdots\!53\)\( T^{37} - \)\(23\!\cdots\!99\)\( T^{38} + \)\(14\!\cdots\!09\)\( T^{39} + \)\(33\!\cdots\!01\)\( T^{40} \)
$71$ \( 1 - 19 T + 23 T^{2} + 4108 T^{3} - 53279 T^{4} + 41926 T^{5} + 6067035 T^{6} - 66073585 T^{7} + 97395390 T^{8} + 4538324065 T^{9} - 50695622941 T^{10} + 177600768129 T^{11} + 1606952963487 T^{12} - 30747394363113 T^{13} + 239696534777059 T^{14} - 283347904916686 T^{15} - 18611857528547495 T^{16} + 235177810223507470 T^{17} - 873303489026431455 T^{18} - 10620340083202397445 T^{19} + \)\(16\!\cdots\!75\)\( T^{20} - \)\(75\!\cdots\!95\)\( T^{21} - \)\(44\!\cdots\!55\)\( T^{22} + \)\(84\!\cdots\!70\)\( T^{23} - \)\(47\!\cdots\!95\)\( T^{24} - \)\(51\!\cdots\!86\)\( T^{25} + \)\(30\!\cdots\!39\)\( T^{26} - \)\(27\!\cdots\!83\)\( T^{27} + \)\(10\!\cdots\!07\)\( T^{28} + \)\(81\!\cdots\!99\)\( T^{29} - \)\(16\!\cdots\!41\)\( T^{30} + \)\(10\!\cdots\!15\)\( T^{31} + \)\(15\!\cdots\!90\)\( T^{32} - \)\(76\!\cdots\!35\)\( T^{33} + \)\(50\!\cdots\!35\)\( T^{34} + \)\(24\!\cdots\!26\)\( T^{35} - \)\(22\!\cdots\!59\)\( T^{36} + \)\(12\!\cdots\!28\)\( T^{37} + \)\(48\!\cdots\!03\)\( T^{38} - \)\(28\!\cdots\!89\)\( T^{39} + \)\(10\!\cdots\!01\)\( T^{40} \)
$73$ \( 1 - 31 T + 534 T^{2} - 7966 T^{3} + 111434 T^{4} - 1603743 T^{5} + 22351367 T^{6} - 281646142 T^{7} + 3353687638 T^{8} - 38445124926 T^{9} + 434550976103 T^{10} - 4822841787689 T^{11} + 51216166388444 T^{12} - 524844739003705 T^{13} + 5218228291241591 T^{14} - 50848328799375207 T^{15} + 487847407358487847 T^{16} - 4547910484654680503 T^{17} + 41187613457894448137 T^{18} - \)\(36\!\cdots\!40\)\( T^{19} + \)\(31\!\cdots\!33\)\( T^{20} - \)\(26\!\cdots\!20\)\( T^{21} + \)\(21\!\cdots\!73\)\( T^{22} - \)\(17\!\cdots\!51\)\( T^{23} + \)\(13\!\cdots\!27\)\( T^{24} - \)\(10\!\cdots\!51\)\( T^{25} + \)\(78\!\cdots\!99\)\( T^{26} - \)\(57\!\cdots\!85\)\( T^{27} + \)\(41\!\cdots\!64\)\( T^{28} - \)\(28\!\cdots\!57\)\( T^{29} + \)\(18\!\cdots\!47\)\( T^{30} - \)\(12\!\cdots\!02\)\( T^{31} + \)\(76\!\cdots\!98\)\( T^{32} - \)\(47\!\cdots\!86\)\( T^{33} + \)\(27\!\cdots\!03\)\( T^{34} - \)\(14\!\cdots\!51\)\( T^{35} + \)\(72\!\cdots\!74\)\( T^{36} - \)\(37\!\cdots\!98\)\( T^{37} + \)\(18\!\cdots\!46\)\( T^{38} - \)\(78\!\cdots\!47\)\( T^{39} + \)\(18\!\cdots\!01\)\( T^{40} \)
$79$ \( 1 - 43 T + 1010 T^{2} - 15819 T^{3} + 185753 T^{4} - 1522733 T^{5} + 5028910 T^{6} + 100355235 T^{7} - 2341403015 T^{8} + 30188080151 T^{9} - 275046275717 T^{10} + 1696906291579 T^{11} - 1741732498173 T^{12} - 115433993429912 T^{13} + 1897867614882981 T^{14} - 18867870447247377 T^{15} + 133342713160758839 T^{16} - 441348550385955580 T^{17} - 3733471051129577399 T^{18} + 87409733638924456455 T^{19} - \)\(93\!\cdots\!19\)\( T^{20} + \)\(69\!\cdots\!45\)\( T^{21} - \)\(23\!\cdots\!59\)\( T^{22} - \)\(21\!\cdots\!20\)\( T^{23} + \)\(51\!\cdots\!59\)\( T^{24} - \)\(58\!\cdots\!23\)\( T^{25} + \)\(46\!\cdots\!01\)\( T^{26} - \)\(22\!\cdots\!08\)\( T^{27} - \)\(26\!\cdots\!53\)\( T^{28} + \)\(20\!\cdots\!01\)\( T^{29} - \)\(26\!\cdots\!17\)\( T^{30} + \)\(22\!\cdots\!29\)\( T^{31} - \)\(13\!\cdots\!15\)\( T^{32} + \)\(46\!\cdots\!65\)\( T^{33} + \)\(18\!\cdots\!10\)\( T^{34} - \)\(44\!\cdots\!67\)\( T^{35} + \)\(42\!\cdots\!13\)\( T^{36} - \)\(28\!\cdots\!21\)\( T^{37} + \)\(14\!\cdots\!10\)\( T^{38} - \)\(48\!\cdots\!17\)\( T^{39} + \)\(89\!\cdots\!01\)\( T^{40} \)
$83$ \( 1 - 32 T + 406 T^{2} - 3496 T^{3} + 33502 T^{4} - 271710 T^{5} + 2069101 T^{6} - 15131563 T^{7} - 71284893 T^{8} + 1838127961 T^{9} - 6132967089 T^{10} + 19871695526 T^{11} - 29200068963 T^{12} - 5255246673042 T^{13} + 40994603541527 T^{14} - 656705281023150 T^{15} + 4381223353664725 T^{16} + 91171220352701247 T^{17} - 1466354427687287544 T^{18} + 14288651913681658093 T^{19} - \)\(14\!\cdots\!41\)\( T^{20} + \)\(11\!\cdots\!19\)\( T^{21} - \)\(10\!\cdots\!16\)\( T^{22} + \)\(52\!\cdots\!89\)\( T^{23} + \)\(20\!\cdots\!25\)\( T^{24} - \)\(25\!\cdots\!50\)\( T^{25} + \)\(13\!\cdots\!63\)\( T^{26} - \)\(14\!\cdots\!34\)\( T^{27} - \)\(65\!\cdots\!83\)\( T^{28} + \)\(37\!\cdots\!78\)\( T^{29} - \)\(95\!\cdots\!61\)\( T^{30} + \)\(23\!\cdots\!87\)\( T^{31} - \)\(76\!\cdots\!73\)\( T^{32} - \)\(13\!\cdots\!69\)\( T^{33} + \)\(15\!\cdots\!29\)\( T^{34} - \)\(16\!\cdots\!70\)\( T^{35} + \)\(16\!\cdots\!62\)\( T^{36} - \)\(14\!\cdots\!08\)\( T^{37} + \)\(14\!\cdots\!54\)\( T^{38} - \)\(92\!\cdots\!04\)\( T^{39} + \)\(24\!\cdots\!01\)\( T^{40} \)
$89$ \( 1 + 47 T + 1272 T^{2} + 26378 T^{3} + 462421 T^{4} + 7095712 T^{5} + 97695631 T^{6} + 1229767604 T^{7} + 14280412620 T^{8} + 153298352320 T^{9} + 1534900173114 T^{10} + 14514559954090 T^{11} + 131455544472509 T^{12} + 1167821070144865 T^{13} + 10554337797718648 T^{14} + 99660514974136256 T^{15} + 988915111864240255 T^{16} + 10175025404199544427 T^{17} + \)\(10\!\cdots\!12\)\( T^{18} + \)\(10\!\cdots\!71\)\( T^{19} + \)\(10\!\cdots\!15\)\( T^{20} + \)\(94\!\cdots\!19\)\( T^{21} + \)\(83\!\cdots\!52\)\( T^{22} + \)\(71\!\cdots\!63\)\( T^{23} + \)\(62\!\cdots\!55\)\( T^{24} + \)\(55\!\cdots\!44\)\( T^{25} + \)\(52\!\cdots\!28\)\( T^{26} + \)\(51\!\cdots\!85\)\( T^{27} + \)\(51\!\cdots\!29\)\( T^{28} + \)\(50\!\cdots\!10\)\( T^{29} + \)\(47\!\cdots\!14\)\( T^{30} + \)\(42\!\cdots\!80\)\( T^{31} + \)\(35\!\cdots\!20\)\( T^{32} + \)\(27\!\cdots\!76\)\( T^{33} + \)\(19\!\cdots\!71\)\( T^{34} + \)\(12\!\cdots\!88\)\( T^{35} + \)\(71\!\cdots\!81\)\( T^{36} + \)\(36\!\cdots\!62\)\( T^{37} + \)\(15\!\cdots\!32\)\( T^{38} + \)\(51\!\cdots\!23\)\( T^{39} + \)\(97\!\cdots\!01\)\( T^{40} \)
$97$ \( 1 + 45 T + 424 T^{2} - 9536 T^{3} - 203356 T^{4} + 44605 T^{5} + 35372400 T^{6} + 374377366 T^{7} - 1208346180 T^{8} - 73904977716 T^{9} - 534049014451 T^{10} + 5845040218546 T^{11} + 113608841594705 T^{12} + 304677990329790 T^{13} - 9760410996893029 T^{14} - 128654933544812038 T^{15} + 42375802710219630 T^{16} + 15203142043841323748 T^{17} + \)\(10\!\cdots\!85\)\( T^{18} - \)\(65\!\cdots\!30\)\( T^{19} - \)\(14\!\cdots\!02\)\( T^{20} - \)\(63\!\cdots\!10\)\( T^{21} + \)\(99\!\cdots\!65\)\( T^{22} + \)\(13\!\cdots\!04\)\( T^{23} + \)\(37\!\cdots\!30\)\( T^{24} - \)\(11\!\cdots\!66\)\( T^{25} - \)\(81\!\cdots\!41\)\( T^{26} + \)\(24\!\cdots\!70\)\( T^{27} + \)\(89\!\cdots\!05\)\( T^{28} + \)\(44\!\cdots\!82\)\( T^{29} - \)\(39\!\cdots\!99\)\( T^{30} - \)\(52\!\cdots\!48\)\( T^{31} - \)\(83\!\cdots\!80\)\( T^{32} + \)\(25\!\cdots\!82\)\( T^{33} + \)\(23\!\cdots\!00\)\( T^{34} + \)\(28\!\cdots\!65\)\( T^{35} - \)\(12\!\cdots\!76\)\( T^{36} - \)\(56\!\cdots\!32\)\( T^{37} + \)\(24\!\cdots\!36\)\( T^{38} + \)\(25\!\cdots\!85\)\( T^{39} + \)\(54\!\cdots\!01\)\( T^{40} \)
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