Properties

Label 105.3.n.b
Level 105
Weight 3
Character orbit 105.n
Analytic conductor 2.861
Analytic rank 0
Dimension 12
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 105.n (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.86104277578\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2}\cdot 7 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{2} - \beta_{10} ) q^{2} + ( 2 + \beta_{3} ) q^{3} + ( 4 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{11} ) q^{4} + \beta_{4} q^{5} + ( -\beta_{2} - 2 \beta_{10} ) q^{6} + ( 2 + \beta_{2} + \beta_{5} - \beta_{7} - \beta_{9} ) q^{7} + ( 4 - \beta_{1} + 3 \beta_{2} - \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{8} + ( 3 + 3 \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{2} - \beta_{10} ) q^{2} + ( 2 + \beta_{3} ) q^{3} + ( 4 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{11} ) q^{4} + \beta_{4} q^{5} + ( -\beta_{2} - 2 \beta_{10} ) q^{6} + ( 2 + \beta_{2} + \beta_{5} - \beta_{7} - \beta_{9} ) q^{7} + ( 4 - \beta_{1} + 3 \beta_{2} - \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{8} + ( 3 + 3 \beta_{3} ) q^{9} + ( -\beta_{6} + \beta_{8} ) q^{10} + ( -4 \beta_{3} + 2 \beta_{4} + \beta_{6} - 2 \beta_{7} + \beta_{8} - 2 \beta_{9} + 2 \beta_{11} ) q^{11} + ( -4 - \beta_{1} + 4 \beta_{3} + 2 \beta_{4} - \beta_{11} ) q^{12} + ( 3 + \beta_{2} + 5 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{13} + ( -2 - 9 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{14} + ( \beta_{4} - \beta_{5} ) q^{15} + ( -15 - \beta_{1} - 4 \beta_{2} - 14 \beta_{3} - 6 \beta_{4} - 13 \beta_{5} - 2 \beta_{6} + \beta_{7} - 3 \beta_{8} - 4 \beta_{10} + 2 \beta_{11} ) q^{16} + ( -10 - 6 \beta_{3} + \beta_{4} + 3 \beta_{5} + 3 \beta_{6} - 2 \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{11} ) q^{17} -3 \beta_{10} q^{18} + ( -2 \beta_{2} + \beta_{3} - \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + \beta_{8} + 2 \beta_{10} + \beta_{11} ) q^{19} + ( -4 - \beta_{2} - 7 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{10} + \beta_{11} ) q^{20} + ( 4 + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} ) q^{21} + ( 2 + 4 \beta_{1} - 7 \beta_{2} - \beta_{3} - 6 \beta_{4} + 8 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} - \beta_{8} + 2 \beta_{9} - \beta_{11} ) q^{22} + ( 2 - 2 \beta_{1} + 7 \beta_{2} + 2 \beta_{3} - 4 \beta_{5} + 7 \beta_{10} + 2 \beta_{11} ) q^{23} + ( 8 - 2 \beta_{1} + 6 \beta_{2} + 4 \beta_{3} + \beta_{4} + 3 \beta_{5} - 3 \beta_{6} + 3 \beta_{8} + 3 \beta_{10} + \beta_{11} ) q^{24} -5 \beta_{3} q^{25} + ( 4 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} - 7 \beta_{6} + 9 \beta_{7} + 2 \beta_{8} + 3 \beta_{9} - 2 \beta_{10} ) q^{26} + ( 3 + 6 \beta_{3} ) q^{27} + ( -12 + \beta_{1} - 8 \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} - 5 \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{28} + ( -6 + 2 \beta_{2} + \beta_{3} - 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - 4 \beta_{9} + \beta_{11} ) q^{29} + ( -\beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{30} + ( 15 + 7 \beta_{1} + 8 \beta_{3} - 4 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - 3 \beta_{11} ) q^{31} + ( 22 \beta_{3} + 13 \beta_{4} + 9 \beta_{5} + 5 \beta_{6} - 10 \beta_{7} + 5 \beta_{8} - 10 \beta_{9} + 7 \beta_{10} - 5 \beta_{11} ) q^{32} + ( 4 + 2 \beta_{1} - 4 \beta_{3} + 2 \beta_{4} + 3 \beta_{6} - 3 \beta_{7} - 3 \beta_{9} + 2 \beta_{11} ) q^{33} + ( 4 + \beta_{1} + 6 \beta_{2} + 7 \beta_{3} - 9 \beta_{4} - 7 \beta_{5} + \beta_{6} + 8 \beta_{7} - 6 \beta_{8} + 7 \beta_{9} + 12 \beta_{10} - 3 \beta_{11} ) q^{34} + ( -5 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - 5 \beta_{10} - \beta_{11} ) q^{35} + ( -12 - 3 \beta_{1} + 3 \beta_{4} - 3 \beta_{5} ) q^{36} + ( -3 + 5 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} - 6 \beta_{4} - 2 \beta_{5} + 3 \beta_{10} - 5 \beta_{11} ) q^{37} + ( 14 + 2 \beta_{1} - 6 \beta_{2} + 8 \beta_{3} - 2 \beta_{4} - 6 \beta_{5} - 6 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - 3 \beta_{10} ) q^{38} + ( 1 - \beta_{1} + 8 \beta_{3} + 4 \beta_{4} + \beta_{5} - \beta_{6} + 5 \beta_{7} + \beta_{8} + \beta_{9} + 3 \beta_{10} - \beta_{11} ) q^{39} + ( -7 + \beta_{1} - 4 \beta_{2} + 6 \beta_{3} + 4 \beta_{4} + \beta_{5} + 4 \beta_{6} - 5 \beta_{7} - \beta_{8} - 2 \beta_{9} + 4 \beta_{10} ) q^{40} + ( 8 - 2 \beta_{1} + 6 \beta_{2} + 16 \beta_{3} - 8 \beta_{4} - 12 \beta_{5} + 3 \beta_{7} - 3 \beta_{8} + 3 \beta_{9} + 12 \beta_{10} + 4 \beta_{11} ) q^{41} + ( 5 + \beta_{1} + 2 \beta_{2} - 11 \beta_{3} + 2 \beta_{4} + 11 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} - \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{42} + ( -11 - 2 \beta_{2} - 10 \beta_{4} + 10 \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{43} + ( 37 + 5 \beta_{1} - 10 \beta_{2} + 36 \beta_{3} + 9 \beta_{5} + 8 \beta_{6} + 5 \beta_{7} - 9 \beta_{8} + 6 \beta_{9} - 10 \beta_{10} - 6 \beta_{11} ) q^{44} -3 \beta_{5} q^{45} + ( -48 \beta_{3} - \beta_{4} - 5 \beta_{5} + 2 \beta_{6} - 4 \beta_{7} + 2 \beta_{8} - 4 \beta_{9} + 4 \beta_{10} + 9 \beta_{11} ) q^{46} + ( 2 + 3 \beta_{1} - \beta_{2} - 6 \beta_{3} + 4 \beta_{5} - \beta_{6} - 3 \beta_{7} - 4 \beta_{8} + 9 \beta_{9} + \beta_{10} - \beta_{11} ) q^{47} + ( -16 - 4 \beta_{2} - 29 \beta_{3} - 18 \beta_{4} - 18 \beta_{5} - 3 \beta_{6} - 6 \beta_{8} + 3 \beta_{9} - 8 \beta_{10} + 3 \beta_{11} ) q^{48} + ( -4 - 6 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + 13 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} + 4 \beta_{7} + 4 \beta_{8} - 7 \beta_{9} + 8 \beta_{10} + 8 \beta_{11} ) q^{49} -5 \beta_{2} q^{50} + ( -14 - \beta_{1} - 16 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} + 5 \beta_{6} - \beta_{7} + 4 \beta_{8} + \beta_{9} - \beta_{11} ) q^{51} + ( -53 - 11 \beta_{1} - 6 \beta_{2} - 25 \beta_{3} + 4 \beta_{4} - 39 \beta_{5} - 12 \beta_{6} + 3 \beta_{7} + 6 \beta_{8} - 3 \beta_{9} - 3 \beta_{10} + 7 \beta_{11} ) q^{52} + ( 10 \beta_{3} - 9 \beta_{4} - 5 \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} - 2 \beta_{9} - 7 \beta_{10} + \beta_{11} ) q^{53} + ( 3 \beta_{2} - 3 \beta_{10} ) q^{54} + ( -2 - 3 \beta_{2} - 6 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 6 \beta_{10} - 2 \beta_{11} ) q^{55} + ( 44 + 8 \beta_{2} + 67 \beta_{3} + 20 \beta_{4} + 36 \beta_{5} - 7 \beta_{6} + 6 \beta_{7} + 6 \beta_{8} + \beta_{9} + 10 \beta_{10} - 9 \beta_{11} ) q^{56} + ( -1 + \beta_{1} - 6 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - 5 \beta_{6} + 4 \beta_{7} + 2 \beta_{8} - \beta_{9} + \beta_{11} ) q^{57} + ( -7 - 2 \beta_{1} + 8 \beta_{2} - 8 \beta_{3} + 16 \beta_{4} + 27 \beta_{5} - 3 \beta_{7} + 7 \beta_{8} - 2 \beta_{9} + 8 \beta_{10} + \beta_{11} ) q^{58} + ( -31 - 9 \beta_{1} + 4 \beta_{2} - 15 \beta_{3} + 4 \beta_{4} - 15 \beta_{5} + \beta_{7} - 2 \beta_{8} - \beta_{9} + 2 \beta_{10} + 5 \beta_{11} ) q^{59} + ( -1 + \beta_{1} - 11 \beta_{3} - 8 \beta_{4} - 3 \beta_{5} - 3 \beta_{7} - 2 \beta_{8} + \beta_{9} - 3 \beta_{10} + \beta_{11} ) q^{60} + ( -5 + 10 \beta_{2} + 5 \beta_{5} + 4 \beta_{6} - 9 \beta_{7} - 5 \beta_{8} + 6 \beta_{9} - 10 \beta_{10} - 5 \beta_{11} ) q^{61} + ( -22 + 3 \beta_{1} - 20 \beta_{2} - 42 \beta_{3} - 25 \beta_{4} - 19 \beta_{5} - 2 \beta_{6} - 7 \beta_{7} + 3 \beta_{8} - 5 \beta_{9} - 40 \beta_{10} - 4 \beta_{11} ) q^{62} + ( 6 + 3 \beta_{2} + 6 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} - 3 \beta_{8} + 3 \beta_{10} ) q^{63} + ( 70 + 4 \beta_{1} + 36 \beta_{2} - \beta_{3} - 28 \beta_{4} + 30 \beta_{5} - 7 \beta_{6} + 2 \beta_{7} + 4 \beta_{8} + 7 \beta_{9} - \beta_{11} ) q^{64} + ( -10 - 5 \beta_{1} + 5 \beta_{2} - 10 \beta_{3} + \beta_{4} - 8 \beta_{5} + 5 \beta_{10} + 5 \beta_{11} ) q^{65} + ( 5 + 7 \beta_{1} - 14 \beta_{2} + \beta_{3} - 2 \beta_{4} + 21 \beta_{5} + 6 \beta_{6} - 3 \beta_{7} + 3 \beta_{9} - 7 \beta_{10} - 5 \beta_{11} ) q^{66} + ( -3 + 3 \beta_{1} + 4 \beta_{3} + 11 \beta_{4} + 12 \beta_{5} - 2 \beta_{6} - 5 \beta_{7} - 8 \beta_{8} + 7 \beta_{9} - 4 \beta_{10} - 4 \beta_{11} ) q^{67} + ( 6 - 6 \beta_{3} + 30 \beta_{4} ) q^{68} + ( 2 - 2 \beta_{1} + 7 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - 6 \beta_{5} + 14 \beta_{10} + 4 \beta_{11} ) q^{69} + ( -18 - \beta_{1} + 4 \beta_{2} + 14 \beta_{3} + 6 \beta_{4} + 10 \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{10} - 5 \beta_{11} ) q^{70} + ( -56 - 10 \beta_{1} - 6 \beta_{2} + \beta_{3} - 6 \beta_{4} + 4 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} + \beta_{8} - 2 \beta_{9} + \beta_{11} ) q^{71} + ( 12 - 3 \beta_{1} + 9 \beta_{2} + 12 \beta_{3} + 6 \beta_{4} + 6 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} + 6 \beta_{8} - 3 \beta_{9} + 9 \beta_{10} + 3 \beta_{11} ) q^{72} + ( 22 - 8 \beta_{1} + 4 \beta_{2} + 9 \beta_{3} + 6 \beta_{4} + 9 \beta_{6} - 4 \beta_{7} - \beta_{8} + 4 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{73} + ( -44 \beta_{3} - 18 \beta_{4} - 8 \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} - 2 \beta_{9} - 12 \beta_{10} - 2 \beta_{11} ) q^{74} + ( 5 - 5 \beta_{3} ) q^{75} + ( 8 - \beta_{1} - 6 \beta_{2} + 16 \beta_{3} - 13 \beta_{4} - 15 \beta_{5} - 6 \beta_{7} + 6 \beta_{8} - 6 \beta_{9} - 12 \beta_{10} + 2 \beta_{11} ) q^{76} + ( -34 - 7 \beta_{1} + 4 \beta_{2} - 38 \beta_{3} + 22 \beta_{4} - 10 \beta_{5} - \beta_{6} - 11 \beta_{7} + 10 \beta_{8} - 9 \beta_{9} - 7 \beta_{10} + 11 \beta_{11} ) q^{77} + ( 10 - 4 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} - 8 \beta_{5} - 16 \beta_{6} + 11 \beta_{7} + 7 \beta_{8} + \beta_{9} + 2 \beta_{11} ) q^{78} + ( 7 - 9 \beta_{1} - 26 \beta_{2} + 9 \beta_{3} - 16 \beta_{5} - 10 \beta_{6} - 4 \beta_{7} + 6 \beta_{8} - 6 \beta_{9} - 26 \beta_{10} + 11 \beta_{11} ) q^{79} + ( 60 + 10 \beta_{1} + 10 \beta_{2} + 30 \beta_{3} - 5 \beta_{4} + 19 \beta_{5} - \beta_{6} + \beta_{8} + 5 \beta_{10} - 5 \beta_{11} ) q^{80} + 9 \beta_{3} q^{81} + ( 31 + 5 \beta_{1} - \beta_{2} - 28 \beta_{3} + 20 \beta_{4} - 3 \beta_{5} + 9 \beta_{6} - 6 \beta_{7} + 3 \beta_{8} - 15 \beta_{9} + \beta_{10} + 8 \beta_{11} ) q^{82} + ( -2 + 3 \beta_{1} + 2 \beta_{2} - 6 \beta_{3} - 29 \beta_{4} - 23 \beta_{5} + 2 \beta_{6} + 7 \beta_{7} - 3 \beta_{8} + 5 \beta_{9} + 4 \beta_{10} - 8 \beta_{11} ) q^{83} + ( -25 - 17 \beta_{2} - 11 \beta_{3} + 3 \beta_{4} + 12 \beta_{7} - 9 \beta_{8} + 6 \beta_{9} - 7 \beta_{10} - 3 \beta_{11} ) q^{84} + ( -10 - 8 \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + 4 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + 4 \beta_{9} - \beta_{11} ) q^{85} + ( 27 + 3 \beta_{1} + 14 \beta_{2} + 26 \beta_{3} + 8 \beta_{4} + 21 \beta_{5} + 13 \beta_{6} + 10 \beta_{7} - 19 \beta_{8} + 11 \beta_{9} + 14 \beta_{10} - 4 \beta_{11} ) q^{86} + ( -13 + \beta_{1} + 4 \beta_{2} - 5 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} + 3 \beta_{7} - 6 \beta_{8} - 3 \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{87} + ( 2 - 2 \beta_{1} + 112 \beta_{3} + 33 \beta_{4} + 19 \beta_{5} - 7 \beta_{6} + 20 \beta_{7} - 3 \beta_{8} + 12 \beta_{9} - 7 \beta_{10} - 11 \beta_{11} ) q^{88} + ( -9 + \beta_{1} + 10 \beta_{3} + 16 \beta_{4} - \beta_{5} - 11 \beta_{6} + 12 \beta_{7} + \beta_{8} + 9 \beta_{9} + 2 \beta_{11} ) q^{89} + ( -3 \beta_{7} + 3 \beta_{8} - 3 \beta_{9} ) q^{90} + ( 41 + 8 \beta_{1} - 22 \beta_{2} + 21 \beta_{3} - 7 \beta_{4} + \beta_{5} + 8 \beta_{6} - 6 \beta_{7} - 4 \beta_{8} - 8 \beta_{10} - 11 \beta_{11} ) q^{91} + ( 24 + 9 \beta_{1} - 41 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + 7 \beta_{5} + 16 \beta_{6} - 11 \beta_{7} - 7 \beta_{8} - \beta_{9} - 2 \beta_{11} ) q^{92} + ( 22 + 11 \beta_{1} + 23 \beta_{3} - 14 \beta_{4} - 5 \beta_{5} - 4 \beta_{6} - \beta_{7} + \beta_{8} - 2 \beta_{9} - 10 \beta_{11} ) q^{93} + ( 1 + 7 \beta_{1} - 16 \beta_{2} + \beta_{3} - 4 \beta_{4} - 39 \beta_{5} + 9 \beta_{6} + \beta_{7} - 11 \beta_{8} - \beta_{9} - 8 \beta_{10} - 3 \beta_{11} ) q^{94} + ( 1 - \beta_{1} + \beta_{3} + \beta_{4} - 3 \beta_{5} - \beta_{6} + 5 \beta_{7} + \beta_{8} + \beta_{9} + 8 \beta_{10} + 4 \beta_{11} ) q^{95} + ( -22 - 5 \beta_{1} - 7 \beta_{2} + 22 \beta_{3} + 22 \beta_{4} + 15 \beta_{6} - 15 \beta_{7} - 15 \beta_{9} + 7 \beta_{10} - 5 \beta_{11} ) q^{96} + ( 38 - \beta_{1} - 10 \beta_{2} + 77 \beta_{3} + 3 \beta_{4} + \beta_{5} - \beta_{6} - 8 \beta_{7} + 6 \beta_{8} - 7 \beta_{9} - 20 \beta_{10} + 3 \beta_{11} ) q^{97} + ( 18 + 2 \beta_{1} - 5 \beta_{2} - 28 \beta_{3} + \beta_{4} + 17 \beta_{5} - 17 \beta_{6} - 4 \beta_{7} + 15 \beta_{8} - 2 \beta_{9} + 15 \beta_{10} + 13 \beta_{11} ) q^{98} + ( 12 + 6 \beta_{1} + 6 \beta_{6} - 3 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 2q^{2} + 18q^{3} - 22q^{4} + 22q^{7} + 40q^{8} + 18q^{9} + O(q^{10}) \) \( 12q + 2q^{2} + 18q^{3} - 22q^{4} + 22q^{7} + 40q^{8} + 18q^{9} + 20q^{11} - 66q^{12} + 32q^{14} - 82q^{16} - 78q^{17} - 6q^{18} - 6q^{19} + 36q^{21} + 56q^{22} + 2q^{23} + 60q^{24} + 30q^{25} + 36q^{26} - 128q^{28} - 100q^{29} + 108q^{31} - 108q^{32} + 60q^{33} - 60q^{35} - 132q^{36} - 34q^{37} + 126q^{38} - 42q^{39} - 90q^{40} + 114q^{42} - 124q^{43} + 234q^{44} + 278q^{46} + 96q^{47} - 60q^{49} + 20q^{50} - 78q^{51} - 444q^{52} - 76q^{53} - 18q^{54} + 112q^{56} - 12q^{57} - 52q^{58} - 270q^{59} + 60q^{60} - 60q^{61} + 42q^{63} + 700q^{64} - 60q^{65} + 84q^{66} - 18q^{67} + 108q^{68} - 300q^{70} - 628q^{71} + 60q^{72} + 234q^{73} + 244q^{74} + 90q^{75} - 196q^{77} + 72q^{78} + 108q^{79} + 480q^{80} - 54q^{81} + 480q^{82} - 192q^{84} - 60q^{85} + 130q^{86} - 150q^{87} - 668q^{88} - 186q^{89} + 444q^{91} + 456q^{92} + 108q^{93} + 30q^{94} - 324q^{96} + 416q^{98} + 120q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 6 x^{11} + 39 x^{10} - 140 x^{9} + 456 x^{8} - 1050 x^{7} + 1999 x^{6} - 2844 x^{5} + 2949 x^{4} - 2136 x^{3} + 1020 x^{2} - 288 x + 36\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{8} - 4 \nu^{7} + 24 \nu^{6} - 58 \nu^{5} + 154 \nu^{4} - 216 \nu^{3} + 267 \nu^{2} - 168 \nu + 60 \)\()/6\)
\(\beta_{2}\)\(=\)\((\)\( 3 \nu^{10} - 15 \nu^{9} + 101 \nu^{8} - 314 \nu^{7} + 1020 \nu^{6} - 2024 \nu^{5} + 3629 \nu^{4} - 4221 \nu^{3} + 3417 \nu^{2} - 1596 \nu + 342 \)\()/12\)
\(\beta_{3}\)\(=\)\((\)\( 206 \nu^{11} - 1133 \nu^{10} + 7471 \nu^{9} - 25122 \nu^{8} + 81494 \nu^{7} - 175924 \nu^{6} + 325036 \nu^{5} - 425733 \nu^{4} + 398817 \nu^{3} - 245406 \nu^{2} + 90966 \nu - 15420 \)\()/168\)
\(\beta_{4}\)\(=\)\((\)\( 272 \nu^{11} - 1489 \nu^{10} + 9825 \nu^{9} - 32912 \nu^{8} + 106722 \nu^{7} - 229488 \nu^{6} + 423174 \nu^{5} - 551705 \nu^{4} + 513735 \nu^{3} - 314580 \nu^{2} + 116286 \nu - 19920 \)\()/168\)
\(\beta_{5}\)\(=\)\((\)\( 272 \nu^{11} - 1503 \nu^{10} + 9895 \nu^{9} - 33388 \nu^{8} + 108206 \nu^{7} - 234304 \nu^{6} + 432722 \nu^{5} - 568295 \nu^{4} + 532593 \nu^{3} - 327096 \nu^{2} + 120738 \nu - 19920 \)\()/168\)
\(\beta_{6}\)\(=\)\((\)\( -87 \nu^{11} + 468 \nu^{10} - 3096 \nu^{9} + 10226 \nu^{8} - 33110 \nu^{7} + 70140 \nu^{6} - 128273 \nu^{5} + 163730 \nu^{4} - 147846 \nu^{3} + 86226 \nu^{2} - 29604 \nu + 4542 \)\()/42\)
\(\beta_{7}\)\(=\)\((\)\( -708 \nu^{11} + 3887 \nu^{10} - 25609 \nu^{9} + 85962 \nu^{8} - 278334 \nu^{7} + 599452 \nu^{6} - 1103022 \nu^{5} + 1438675 \nu^{4} - 1332291 \nu^{3} + 809130 \nu^{2} - 292458 \nu + 47532 \)\()/168\)
\(\beta_{8}\)\(=\)\((\)\( 432 \nu^{11} - 2355 \nu^{10} + 15562 \nu^{9} - 51971 \nu^{8} + 168658 \nu^{7} - 361606 \nu^{6} + 666770 \nu^{5} - 865921 \nu^{4} + 803904 \nu^{3} - 487725 \nu^{2} + 177174 \nu - 28878 \)\()/84\)
\(\beta_{9}\)\(=\)\((\)\( -852 \nu^{11} + 4735 \nu^{10} - 31123 \nu^{9} + 105458 \nu^{8} - 341502 \nu^{7} + 742336 \nu^{6} - 1370858 \nu^{5} + 1808051 \nu^{4} - 1697013 \nu^{3} + 1048110 \nu^{2} - 387678 \nu + 65460 \)\()/168\)
\(\beta_{10}\)\(=\)\((\)\( 471 \nu^{11} - 2601 \nu^{10} + 17121 \nu^{9} - 57733 \nu^{8} + 187012 \nu^{7} - 404628 \nu^{6} + 746383 \nu^{5} - 979237 \nu^{4} + 914673 \nu^{3} - 561435 \nu^{2} + 206160 \nu - 34290 \)\()/84\)
\(\beta_{11}\)\(=\)\((\)\( -575 \nu^{11} + 3159 \nu^{10} - 20828 \nu^{9} + 69974 \nu^{8} - 226856 \nu^{7} + 489230 \nu^{6} - 902535 \nu^{5} + 1180277 \nu^{4} - 1100976 \nu^{3} + 674814 \nu^{2} - 248820 \nu + 41988 \)\()/84\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(5 \beta_{11} - 28 \beta_{10} - 9 \beta_{9} + 6 \beta_{8} - 12 \beta_{7} - 3 \beta_{6} + 15 \beta_{5} + 17 \beta_{4} + 5 \beta_{3} - 14 \beta_{2} - \beta_{1} + 22\)\()/42\)
\(\nu^{2}\)\(=\)\((\)\(-3 \beta_{11} - 28 \beta_{10} + 11 \beta_{9} + 2 \beta_{8} - 32 \beta_{7} + 13 \beta_{6} + 47 \beta_{5} + \beta_{4} - 3 \beta_{3} - 14 \beta_{2} + 9 \beta_{1} - 156\)\()/42\)
\(\nu^{3}\)\(=\)\((\)\(-23 \beta_{11} + 56 \beta_{10} + 33 \beta_{9} - 22 \beta_{8} + 16 \beta_{7} + 11 \beta_{6} - 13 \beta_{5} - 53 \beta_{4} + 5 \beta_{3} + 28 \beta_{2} + 13 \beta_{1} - 76\)\()/14\)
\(\nu^{4}\)\(=\)\((\)\(-69 \beta_{11} + 364 \beta_{10} + \beta_{9} - 122 \beta_{8} + 272 \beta_{7} - 37 \beta_{6} - 389 \beta_{5} - 187 \beta_{4} + 99 \beta_{3} + 266 \beta_{2} - 3 \beta_{1} + 1032\)\()/42\)
\(\nu^{5}\)\(=\)\((\)\(501 \beta_{11} - 952 \beta_{10} - 829 \beta_{9} + 422 \beta_{8} - 116 \beta_{7} - 197 \beta_{6} + 47 \beta_{5} + 1261 \beta_{4} - 507 \beta_{3} - 266 \beta_{2} - 327 \beta_{1} + 2574\)\()/42\)
\(\nu^{6}\)\(=\)\((\)\(375 \beta_{11} - 1260 \beta_{10} - 255 \beta_{9} + 548 \beta_{8} - 690 \beta_{7} - 29 \beta_{6} + 1153 \beta_{5} + 1009 \beta_{4} - 773 \beta_{3} - 938 \beta_{2} - 145 \beta_{1} - 2102\)\()/14\)
\(\nu^{7}\)\(=\)\((\)\(-423 \beta_{11} + 574 \beta_{10} + 907 \beta_{9} - 248 \beta_{8} - 100 \beta_{7} + 71 \beta_{6} + 373 \beta_{5} - 1183 \beta_{4} + 651 \beta_{3} - 280 \beta_{2} + 309 \beta_{1} - 3696\)\()/6\)
\(\nu^{8}\)\(=\)\((\)\(-12423 \beta_{11} + 34580 \beta_{10} + 12455 \beta_{9} - 16918 \beta_{8} + 15160 \beta_{7} + 1501 \beta_{6} - 28393 \beta_{5} - 35591 \beta_{4} + 34113 \beta_{3} + 22834 \beta_{2} + 6693 \beta_{1} + 31800\)\()/42\)
\(\nu^{9}\)\(=\)\((\)\(3907 \beta_{11} - 112 \beta_{10} - 14405 \beta_{9} - 1382 \beta_{8} + 5508 \beta_{7} + 1685 \beta_{6} - 16615 \beta_{5} + 12749 \beta_{4} - 4577 \beta_{3} + 16492 \beta_{2} - 3069 \beta_{1} + 80608\)\()/14\)
\(\nu^{10}\)\(=\)\((\)\(114897 \beta_{11} - 289352 \beta_{10} - 149297 \beta_{9} + 148522 \beta_{8} - 106924 \beta_{7} - 1783 \beta_{6} + 205741 \beta_{5} + 354695 \beta_{4} - 386331 \beta_{3} - 144130 \beta_{2} - 69969 \beta_{1} - 73356\)\()/42\)
\(\nu^{11}\)\(=\)\((\)\(18675 \beta_{11} - 271684 \beta_{10} + 235577 \beta_{9} + 204326 \beta_{8} - 210020 \beta_{7} - 93707 \beta_{6} + 644585 \beta_{5} + 9763 \beta_{4} - 280113 \beta_{3} - 633878 \beta_{2} - 16461 \beta_{1} - 2125518\)\()/42\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\) \(71\)
\(\chi(n)\) \(1\) \(1 + \beta_{3}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
0.500000 + 0.0182799i
0.500000 2.96550i
0.500000 0.792243i
0.500000 + 2.68684i
0.500000 + 2.38770i
0.500000 + 0.396977i
0.500000 0.0182799i
0.500000 + 2.96550i
0.500000 + 0.792243i
0.500000 2.68684i
0.500000 2.38770i
0.500000 0.396977i
−1.99068 3.44796i 1.50000 + 0.866025i −5.92561 + 10.2635i −1.93649 + 1.11803i 6.89592i 2.28451 + 6.61672i 31.2586 1.50000 + 2.59808i 7.70987 + 4.45130i
31.2 −1.25588 2.17524i 1.50000 + 0.866025i −1.15446 + 1.99958i 1.93649 1.11803i 4.35049i 6.75110 1.85004i −4.24760 1.50000 + 2.59808i −4.86399 2.80823i
31.3 0.288563 + 0.499806i 1.50000 + 0.866025i 1.83346 3.17565i 1.93649 1.11803i 0.999611i −1.64406 6.80420i 4.42478 1.50000 + 2.59808i 1.11760 + 0.645246i
31.4 0.687692 + 1.19112i 1.50000 + 0.866025i 1.05416 1.82586i −1.93649 + 1.11803i 2.38224i 6.56639 + 2.42539i 8.40129 1.50000 + 2.59808i −2.66342 1.53773i
31.5 1.46731 + 2.54146i 1.50000 + 0.866025i −2.30602 + 3.99415i 1.93649 1.11803i 5.08293i −5.41652 + 4.43411i −1.79613 1.50000 + 2.59808i 5.68288 + 3.28101i
31.6 1.80299 + 3.12287i 1.50000 + 0.866025i −4.50153 + 7.79688i −1.93649 + 1.11803i 6.24573i 2.45857 6.55404i −18.0409 1.50000 + 2.59808i −6.98294 4.03160i
61.1 −1.99068 + 3.44796i 1.50000 0.866025i −5.92561 10.2635i −1.93649 1.11803i 6.89592i 2.28451 6.61672i 31.2586 1.50000 2.59808i 7.70987 4.45130i
61.2 −1.25588 + 2.17524i 1.50000 0.866025i −1.15446 1.99958i 1.93649 + 1.11803i 4.35049i 6.75110 + 1.85004i −4.24760 1.50000 2.59808i −4.86399 + 2.80823i
61.3 0.288563 0.499806i 1.50000 0.866025i 1.83346 + 3.17565i 1.93649 + 1.11803i 0.999611i −1.64406 + 6.80420i 4.42478 1.50000 2.59808i 1.11760 0.645246i
61.4 0.687692 1.19112i 1.50000 0.866025i 1.05416 + 1.82586i −1.93649 1.11803i 2.38224i 6.56639 2.42539i 8.40129 1.50000 2.59808i −2.66342 + 1.53773i
61.5 1.46731 2.54146i 1.50000 0.866025i −2.30602 3.99415i 1.93649 + 1.11803i 5.08293i −5.41652 4.43411i −1.79613 1.50000 2.59808i 5.68288 3.28101i
61.6 1.80299 3.12287i 1.50000 0.866025i −4.50153 7.79688i −1.93649 1.11803i 6.24573i 2.45857 + 6.55404i −18.0409 1.50000 2.59808i −6.98294 + 4.03160i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 61.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 105.3.n.b 12
3.b odd 2 1 315.3.w.b 12
5.b even 2 1 525.3.o.m 12
5.c odd 4 2 525.3.s.j 24
7.c even 3 1 735.3.h.b 12
7.d odd 6 1 inner 105.3.n.b 12
7.d odd 6 1 735.3.h.b 12
21.g even 6 1 315.3.w.b 12
35.i odd 6 1 525.3.o.m 12
35.k even 12 2 525.3.s.j 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.3.n.b 12 1.a even 1 1 trivial
105.3.n.b 12 7.d odd 6 1 inner
315.3.w.b 12 3.b odd 2 1
315.3.w.b 12 21.g even 6 1
525.3.o.m 12 5.b even 2 1
525.3.o.m 12 35.i odd 6 1
525.3.s.j 24 5.c odd 4 2
525.3.s.j 24 35.k even 12 2
735.3.h.b 12 7.c even 3 1
735.3.h.b 12 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{12} - \cdots\) acting on \(S_{3}^{\mathrm{new}}(105, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T + T^{2} - 18 T^{3} + 35 T^{4} - 10 T^{5} + 142 T^{6} - 148 T^{7} - 208 T^{8} - 424 T^{9} - 1176 T^{10} + 4080 T^{11} + 144 T^{12} + 16320 T^{13} - 18816 T^{14} - 27136 T^{15} - 53248 T^{16} - 151552 T^{17} + 581632 T^{18} - 163840 T^{19} + 2293760 T^{20} - 4718592 T^{21} + 1048576 T^{22} - 8388608 T^{23} + 16777216 T^{24} \)
$3$ \( ( 1 - 3 T + 3 T^{2} )^{6} \)
$5$ \( ( 1 - 5 T^{2} + 25 T^{4} )^{3} \)
$7$ \( 1 - 22 T + 272 T^{2} - 2568 T^{3} + 19600 T^{4} - 138362 T^{5} + 991858 T^{6} - 6779738 T^{7} + 47059600 T^{8} - 302122632 T^{9} + 1568025872 T^{10} - 6214455478 T^{11} + 13841287201 T^{12} \)
$11$ \( 1 - 20 T - 77 T^{2} + 7872 T^{3} - 76264 T^{4} - 640168 T^{5} + 22115011 T^{6} - 150830740 T^{7} - 1492575946 T^{8} + 41825299028 T^{9} - 261075878769 T^{10} - 2649707325072 T^{11} + 64192943543664 T^{12} - 320614586333712 T^{13} - 3822411941056929 T^{14} + 74096068571342708 T^{15} - 319946909592076426 T^{16} - 3912160946263034740 T^{17} + 69406378073897058931 T^{18} - \)\(24\!\cdots\!88\)\( T^{19} - \)\(35\!\cdots\!04\)\( T^{20} + \)\(43\!\cdots\!32\)\( T^{21} - \)\(51\!\cdots\!77\)\( T^{22} - \)\(16\!\cdots\!20\)\( T^{23} + \)\(98\!\cdots\!41\)\( T^{24} \)
$13$ \( 1 - 1050 T^{2} + 558861 T^{4} - 197481550 T^{6} + 52237691370 T^{8} - 11158031023650 T^{10} + 2025783626034165 T^{12} - 318684524066467650 T^{14} + 42611889644625577770 T^{16} - \)\(46\!\cdots\!50\)\( T^{18} + \)\(37\!\cdots\!01\)\( T^{20} - \)\(19\!\cdots\!50\)\( T^{22} + \)\(54\!\cdots\!61\)\( T^{24} \)
$17$ \( 1 + 78 T + 3456 T^{2} + 111384 T^{3} + 2892825 T^{4} + 64908048 T^{5} + 1326815384 T^{6} + 25384395654 T^{7} + 459528789294 T^{8} + 7866410803470 T^{9} + 127737192900600 T^{10} + 2033291698334568 T^{11} + 33507834279238509 T^{12} + 587621300818690152 T^{13} + 10668738088251012600 T^{14} + \)\(18\!\cdots\!30\)\( T^{15} + \)\(32\!\cdots\!54\)\( T^{16} + \)\(51\!\cdots\!46\)\( T^{17} + \)\(77\!\cdots\!24\)\( T^{18} + \)\(10\!\cdots\!92\)\( T^{19} + \)\(14\!\cdots\!25\)\( T^{20} + \)\(15\!\cdots\!56\)\( T^{21} + \)\(14\!\cdots\!56\)\( T^{22} + \)\(91\!\cdots\!42\)\( T^{23} + \)\(33\!\cdots\!21\)\( T^{24} \)
$19$ \( 1 + 6 T + 1695 T^{2} + 10098 T^{3} + 1583475 T^{4} + 6257352 T^{5} + 993857576 T^{6} + 1495351404 T^{7} + 468684238005 T^{8} - 646226336898 T^{9} + 184440578012361 T^{10} - 649631264996682 T^{11} + 67037012122357518 T^{12} - 234516886663802202 T^{13} + 24036480567148897881 T^{14} - 30402287344769217138 T^{15} + \)\(79\!\cdots\!05\)\( T^{16} + \)\(91\!\cdots\!04\)\( T^{17} + \)\(21\!\cdots\!36\)\( T^{18} + \)\(49\!\cdots\!92\)\( T^{19} + \)\(45\!\cdots\!75\)\( T^{20} + \)\(10\!\cdots\!18\)\( T^{21} + \)\(63\!\cdots\!95\)\( T^{22} + \)\(81\!\cdots\!66\)\( T^{23} + \)\(48\!\cdots\!21\)\( T^{24} \)
$23$ \( 1 - 2 T - 1961 T^{2} - 16242 T^{3} + 1952552 T^{4} + 28088402 T^{5} - 1236738581 T^{6} - 18913143826 T^{7} + 643866095426 T^{8} + 6409263044018 T^{9} - 382808809220061 T^{10} - 936062983320174 T^{11} + 225254373554835492 T^{12} - 495177318176372046 T^{13} - \)\(10\!\cdots\!01\)\( T^{14} + \)\(94\!\cdots\!02\)\( T^{15} + \)\(50\!\cdots\!06\)\( T^{16} - \)\(78\!\cdots\!74\)\( T^{17} - \)\(27\!\cdots\!01\)\( T^{18} + \)\(32\!\cdots\!18\)\( T^{19} + \)\(11\!\cdots\!72\)\( T^{20} - \)\(52\!\cdots\!98\)\( T^{21} - \)\(33\!\cdots\!61\)\( T^{22} - \)\(18\!\cdots\!58\)\( T^{23} + \)\(48\!\cdots\!41\)\( T^{24} \)
$29$ \( ( 1 + 50 T + 4152 T^{2} + 144894 T^{3} + 7271923 T^{4} + 198551984 T^{5} + 7580628328 T^{6} + 166982218544 T^{7} + 5143292971363 T^{8} + 86186330272974 T^{9} + 2077023106614072 T^{10} + 21035361665010050 T^{11} + 353814783205469041 T^{12} )^{2} \)
$31$ \( 1 - 108 T + 7296 T^{2} - 368064 T^{3} + 14679372 T^{4} - 497355660 T^{5} + 14549964692 T^{6} - 358253632404 T^{7} + 6912430087356 T^{8} - 59089176847584 T^{9} - 2401476398747496 T^{10} + 161517767136947820 T^{11} - 6040783753357557738 T^{12} + \)\(15\!\cdots\!20\)\( T^{13} - \)\(22\!\cdots\!16\)\( T^{14} - \)\(52\!\cdots\!04\)\( T^{15} + \)\(58\!\cdots\!96\)\( T^{16} - \)\(29\!\cdots\!04\)\( T^{17} + \)\(11\!\cdots\!12\)\( T^{18} - \)\(37\!\cdots\!60\)\( T^{19} + \)\(10\!\cdots\!32\)\( T^{20} - \)\(25\!\cdots\!24\)\( T^{21} + \)\(49\!\cdots\!96\)\( T^{22} - \)\(69\!\cdots\!88\)\( T^{23} + \)\(62\!\cdots\!21\)\( T^{24} \)
$37$ \( 1 + 34 T - 4909 T^{2} - 226690 T^{3} + 12133431 T^{4} + 708409644 T^{5} - 18171294956 T^{6} - 1398986295020 T^{7} + 16325526424229 T^{8} + 1786096223883250 T^{9} - 2380658436212215 T^{10} - 1019433693058130642 T^{11} - 10072912633170313306 T^{12} - \)\(13\!\cdots\!98\)\( T^{13} - \)\(44\!\cdots\!15\)\( T^{14} + \)\(45\!\cdots\!50\)\( T^{15} + \)\(57\!\cdots\!09\)\( T^{16} - \)\(67\!\cdots\!80\)\( T^{17} - \)\(11\!\cdots\!36\)\( T^{18} + \)\(63\!\cdots\!16\)\( T^{19} + \)\(14\!\cdots\!71\)\( T^{20} - \)\(38\!\cdots\!10\)\( T^{21} - \)\(11\!\cdots\!09\)\( T^{22} + \)\(10\!\cdots\!46\)\( T^{23} + \)\(43\!\cdots\!61\)\( T^{24} \)
$41$ \( 1 - 10494 T^{2} + 51364035 T^{4} - 152997660886 T^{6} + 310636210241223 T^{8} - 486286748354299068 T^{10} + \)\(74\!\cdots\!22\)\( T^{12} - \)\(13\!\cdots\!48\)\( T^{14} + \)\(24\!\cdots\!83\)\( T^{16} - \)\(34\!\cdots\!66\)\( T^{18} + \)\(32\!\cdots\!35\)\( T^{20} - \)\(18\!\cdots\!94\)\( T^{22} + \)\(50\!\cdots\!61\)\( T^{24} \)
$43$ \( ( 1 + 62 T + 7340 T^{2} + 336972 T^{3} + 27181828 T^{4} + 1019437474 T^{5} + 62011212778 T^{6} + 1884939889426 T^{7} + 92929260748228 T^{8} + 2130122349347628 T^{9} + 85791390037591340 T^{10} + 1339911903423623438 T^{11} + 39959630797262576401 T^{12} )^{2} \)
$47$ \( 1 - 96 T + 10053 T^{2} - 670176 T^{3} + 37816182 T^{4} - 1580720664 T^{5} + 53466325187 T^{6} - 1308296276496 T^{7} + 51809528884176 T^{8} - 4886895580883496 T^{9} + 455990427411620625 T^{10} - 31230646812310303944 T^{11} + \)\(16\!\cdots\!56\)\( T^{12} - \)\(68\!\cdots\!96\)\( T^{13} + \)\(22\!\cdots\!25\)\( T^{14} - \)\(52\!\cdots\!84\)\( T^{15} + \)\(12\!\cdots\!36\)\( T^{16} - \)\(68\!\cdots\!04\)\( T^{17} + \)\(62\!\cdots\!67\)\( T^{18} - \)\(40\!\cdots\!16\)\( T^{19} + \)\(21\!\cdots\!22\)\( T^{20} - \)\(83\!\cdots\!64\)\( T^{21} + \)\(27\!\cdots\!53\)\( T^{22} - \)\(58\!\cdots\!64\)\( T^{23} + \)\(13\!\cdots\!81\)\( T^{24} \)
$53$ \( 1 + 76 T - 11231 T^{2} - 724524 T^{3} + 92836166 T^{4} + 4449767804 T^{5} - 540109351793 T^{6} - 16759328047612 T^{7} + 2565935011756424 T^{8} + 43462405312053452 T^{9} - 9681885823291571571 T^{10} - 46904208416898252492 T^{11} + \)\(30\!\cdots\!44\)\( T^{12} - \)\(13\!\cdots\!28\)\( T^{13} - \)\(76\!\cdots\!51\)\( T^{14} + \)\(96\!\cdots\!08\)\( T^{15} + \)\(15\!\cdots\!64\)\( T^{16} - \)\(29\!\cdots\!88\)\( T^{17} - \)\(26\!\cdots\!13\)\( T^{18} + \)\(61\!\cdots\!76\)\( T^{19} + \)\(35\!\cdots\!86\)\( T^{20} - \)\(78\!\cdots\!36\)\( T^{21} - \)\(34\!\cdots\!31\)\( T^{22} + \)\(65\!\cdots\!84\)\( T^{23} + \)\(24\!\cdots\!81\)\( T^{24} \)
$59$ \( 1 + 270 T + 48372 T^{2} + 6499440 T^{3} + 729674841 T^{4} + 72198474768 T^{5} + 6488082937652 T^{6} + 539451035302806 T^{7} + 41805568251303822 T^{8} + 3030722232785131878 T^{9} + \)\(20\!\cdots\!28\)\( T^{10} + \)\(13\!\cdots\!68\)\( T^{11} + \)\(80\!\cdots\!17\)\( T^{12} + \)\(46\!\cdots\!08\)\( T^{13} + \)\(25\!\cdots\!08\)\( T^{14} + \)\(12\!\cdots\!98\)\( T^{15} + \)\(61\!\cdots\!62\)\( T^{16} + \)\(27\!\cdots\!06\)\( T^{17} + \)\(11\!\cdots\!12\)\( T^{18} + \)\(44\!\cdots\!48\)\( T^{19} + \)\(15\!\cdots\!81\)\( T^{20} + \)\(48\!\cdots\!40\)\( T^{21} + \)\(12\!\cdots\!72\)\( T^{22} + \)\(24\!\cdots\!70\)\( T^{23} + \)\(31\!\cdots\!61\)\( T^{24} \)
$61$ \( 1 + 60 T + 13086 T^{2} + 713160 T^{3} + 79444341 T^{4} + 3022528800 T^{5} + 251078089370 T^{6} + 4114433299500 T^{7} + 360604571968890 T^{8} - 6628548155521500 T^{9} + 198447695188114806 T^{10} - 25628280588886519440 T^{11} + \)\(58\!\cdots\!41\)\( T^{12} - \)\(95\!\cdots\!40\)\( T^{13} + \)\(27\!\cdots\!46\)\( T^{14} - \)\(34\!\cdots\!00\)\( T^{15} + \)\(69\!\cdots\!90\)\( T^{16} + \)\(29\!\cdots\!00\)\( T^{17} + \)\(66\!\cdots\!70\)\( T^{18} + \)\(29\!\cdots\!00\)\( T^{19} + \)\(29\!\cdots\!01\)\( T^{20} + \)\(97\!\cdots\!60\)\( T^{21} + \)\(66\!\cdots\!86\)\( T^{22} + \)\(11\!\cdots\!60\)\( T^{23} + \)\(70\!\cdots\!41\)\( T^{24} \)
$67$ \( 1 + 18 T - 15576 T^{2} + 716224 T^{3} + 161647428 T^{4} - 11551101282 T^{5} - 593219478248 T^{6} + 115316141768802 T^{7} - 912928968126828 T^{8} - 541451373822575840 T^{9} + 35055175141983029184 T^{10} + \)\(12\!\cdots\!10\)\( T^{11} - \)\(20\!\cdots\!66\)\( T^{12} + \)\(55\!\cdots\!90\)\( T^{13} + \)\(70\!\cdots\!64\)\( T^{14} - \)\(48\!\cdots\!60\)\( T^{15} - \)\(37\!\cdots\!48\)\( T^{16} + \)\(21\!\cdots\!98\)\( T^{17} - \)\(48\!\cdots\!28\)\( T^{18} - \)\(42\!\cdots\!78\)\( T^{19} + \)\(26\!\cdots\!68\)\( T^{20} + \)\(53\!\cdots\!16\)\( T^{21} - \)\(51\!\cdots\!76\)\( T^{22} + \)\(26\!\cdots\!02\)\( T^{23} + \)\(66\!\cdots\!21\)\( T^{24} \)
$71$ \( ( 1 + 314 T + 58704 T^{2} + 7369374 T^{3} + 727286203 T^{4} + 59185452824 T^{5} + 4395000293464 T^{6} + 298353867685784 T^{7} + 18481564986337243 T^{8} + 944018901720035454 T^{9} + 37908315298251153744 T^{10} + \)\(10\!\cdots\!14\)\( T^{11} + \)\(16\!\cdots\!41\)\( T^{12} )^{2} \)
$73$ \( 1 - 234 T + 50280 T^{2} - 7494552 T^{3} + 1045125336 T^{4} - 124045277670 T^{5} + 13798749511760 T^{6} - 1393240344068466 T^{7} + 132320540590792632 T^{8} - 11749990134656584872 T^{9} + \)\(98\!\cdots\!48\)\( T^{10} - \)\(77\!\cdots\!58\)\( T^{11} + \)\(58\!\cdots\!42\)\( T^{12} - \)\(41\!\cdots\!82\)\( T^{13} + \)\(27\!\cdots\!68\)\( T^{14} - \)\(17\!\cdots\!08\)\( T^{15} + \)\(10\!\cdots\!92\)\( T^{16} - \)\(59\!\cdots\!34\)\( T^{17} + \)\(31\!\cdots\!60\)\( T^{18} - \)\(15\!\cdots\!30\)\( T^{19} + \)\(67\!\cdots\!96\)\( T^{20} - \)\(25\!\cdots\!88\)\( T^{21} + \)\(92\!\cdots\!80\)\( T^{22} - \)\(23\!\cdots\!86\)\( T^{23} + \)\(52\!\cdots\!41\)\( T^{24} \)
$79$ \( 1 - 108 T - 6108 T^{2} + 1752736 T^{3} - 90950172 T^{4} - 3971475972 T^{5} + 1036682005132 T^{6} - 72467068572108 T^{7} - 540849229955460 T^{8} + 549266052938802784 T^{9} - 33282724854561757956 T^{10} - \)\(11\!\cdots\!32\)\( T^{11} + \)\(26\!\cdots\!02\)\( T^{12} - \)\(70\!\cdots\!12\)\( T^{13} - \)\(12\!\cdots\!36\)\( T^{14} + \)\(13\!\cdots\!64\)\( T^{15} - \)\(82\!\cdots\!60\)\( T^{16} - \)\(68\!\cdots\!08\)\( T^{17} + \)\(61\!\cdots\!12\)\( T^{18} - \)\(14\!\cdots\!32\)\( T^{19} - \)\(20\!\cdots\!12\)\( T^{20} + \)\(25\!\cdots\!96\)\( T^{21} - \)\(54\!\cdots\!08\)\( T^{22} - \)\(60\!\cdots\!28\)\( T^{23} + \)\(34\!\cdots\!81\)\( T^{24} \)
$83$ \( 1 - 52956 T^{2} + 1343643534 T^{4} - 21859222482508 T^{6} + 258399723525197679 T^{8} - \)\(23\!\cdots\!76\)\( T^{10} + \)\(18\!\cdots\!16\)\( T^{12} - \)\(11\!\cdots\!96\)\( T^{14} + \)\(58\!\cdots\!39\)\( T^{16} - \)\(23\!\cdots\!88\)\( T^{18} + \)\(68\!\cdots\!54\)\( T^{20} - \)\(12\!\cdots\!56\)\( T^{22} + \)\(11\!\cdots\!21\)\( T^{24} \)
$89$ \( 1 + 186 T + 44532 T^{2} + 6138000 T^{3} + 889877673 T^{4} + 104389916664 T^{5} + 11976622429892 T^{6} + 1266314492339586 T^{7} + 128180246542580718 T^{8} + 12521360973919319370 T^{9} + \)\(11\!\cdots\!68\)\( T^{10} + \)\(10\!\cdots\!44\)\( T^{11} + \)\(99\!\cdots\!61\)\( T^{12} + \)\(86\!\cdots\!24\)\( T^{13} + \)\(74\!\cdots\!88\)\( T^{14} + \)\(62\!\cdots\!70\)\( T^{15} + \)\(50\!\cdots\!58\)\( T^{16} + \)\(39\!\cdots\!86\)\( T^{17} + \)\(29\!\cdots\!32\)\( T^{18} + \)\(20\!\cdots\!24\)\( T^{19} + \)\(13\!\cdots\!53\)\( T^{20} + \)\(75\!\cdots\!00\)\( T^{21} + \)\(43\!\cdots\!32\)\( T^{22} + \)\(14\!\cdots\!06\)\( T^{23} + \)\(61\!\cdots\!41\)\( T^{24} \)
$97$ \( 1 - 64740 T^{2} + 2084566242 T^{4} - 44252577868756 T^{6} + 697286703828717423 T^{8} - \)\(87\!\cdots\!56\)\( T^{10} + \)\(89\!\cdots\!48\)\( T^{12} - \)\(77\!\cdots\!36\)\( T^{14} + \)\(54\!\cdots\!03\)\( T^{16} - \)\(30\!\cdots\!96\)\( T^{18} + \)\(12\!\cdots\!82\)\( T^{20} - \)\(35\!\cdots\!40\)\( T^{22} + \)\(48\!\cdots\!81\)\( T^{24} \)
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