Properties

Label 35.3.g.a
Level $35$
Weight $3$
Character orbit 35.g
Analytic conductor $0.954$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 35.g (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.953680925261\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 4 x^{11} + 8 x^{10} + 8 x^{9} + 70 x^{8} - 248 x^{7} + 464 x^{6} + 432 x^{5} + 1129 x^{4} - 2708 x^{3} + 3528 x^{2} + 840 x + 100\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + \beta_{7} q^{3} + ( \beta_{1} + \beta_{3} + \beta_{4} + \beta_{10} ) q^{4} + ( -\beta_{1} + \beta_{2} + \beta_{3} - \beta_{11} ) q^{5} + ( -4 + \beta_{1} - \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} ) q^{6} + \beta_{5} q^{7} + ( 2 - \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{8} - \beta_{10} ) q^{8} + ( -2 \beta_{1} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - \beta_{10} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} + \beta_{7} q^{3} + ( \beta_{1} + \beta_{3} + \beta_{4} + \beta_{10} ) q^{4} + ( -\beta_{1} + \beta_{2} + \beta_{3} - \beta_{11} ) q^{5} + ( -4 + \beta_{1} - \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} ) q^{6} + \beta_{5} q^{7} + ( 2 - \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{8} - \beta_{10} ) q^{8} + ( -2 \beta_{1} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - \beta_{10} ) q^{9} + ( 4 \beta_{1} - \beta_{2} + 3 \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{11} ) q^{10} + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{11} ) q^{11} + ( 3 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{9} + \beta_{10} ) q^{12} + ( -2 + \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + \beta_{5} - \beta_{6} - \beta_{8} - 2 \beta_{10} + 2 \beta_{11} ) q^{13} + ( -\beta_{4} - \beta_{6} + \beta_{7} - \beta_{10} ) q^{14} + ( -6 - 3 \beta_{1} + \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + 4 \beta_{8} - \beta_{10} + 2 \beta_{11} ) q^{15} + ( 3 - \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} ) q^{16} + ( -3 + 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{9} - \beta_{10} ) q^{17} + ( -1 - 2 \beta_{1} + 4 \beta_{2} + 9 \beta_{3} + 5 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - 4 \beta_{8} + 4 \beta_{10} - 4 \beta_{11} ) q^{18} + ( 3 \beta_{1} + 3 \beta_{3} + 4 \beta_{4} - 3 \beta_{5} - \beta_{6} + \beta_{7} - 3 \beta_{8} + \beta_{9} + 4 \beta_{10} - \beta_{11} ) q^{19} + ( 6 - 3 \beta_{1} - \beta_{2} - 3 \beta_{3} - 10 \beta_{4} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + \beta_{9} - 2 \beta_{10} ) q^{20} + ( 3 \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{9} + \beta_{11} ) q^{21} + ( -10 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - 6 \beta_{4} - 2 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + 4 \beta_{9} + \beta_{10} ) q^{22} + ( -1 + 3 \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{7} + 3 \beta_{10} ) q^{23} + ( -7 \beta_{1} - 7 \beta_{3} - 4 \beta_{4} + \beta_{5} - 4 \beta_{6} + 4 \beta_{7} + \beta_{8} - \beta_{9} - 4 \beta_{10} + \beta_{11} ) q^{24} + ( 8 - 2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + 4 \beta_{10} - 2 \beta_{11} ) q^{25} + ( -6 \beta_{1} + 4 \beta_{2} + 6 \beta_{3} - 4 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 4 \beta_{8} - \beta_{9} - \beta_{11} ) q^{26} + ( 13 - 3 \beta_{1} + \beta_{2} + 13 \beta_{4} + 5 \beta_{5} - 4 \beta_{6} - \beta_{10} ) q^{27} + ( -1 + \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{11} ) q^{28} + ( 5 \beta_{1} + 5 \beta_{3} + 6 \beta_{4} + 2 \beta_{5} + 2 \beta_{8} - 2 \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{29} + ( -2 + 9 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + 11 \beta_{4} + 6 \beta_{6} + 8 \beta_{7} + 3 \beta_{10} - 2 \beta_{11} ) q^{30} + ( -2 - 7 \beta_{1} - 4 \beta_{2} + 7 \beta_{3} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - \beta_{8} + 2 \beta_{9} + 2 \beta_{11} ) q^{31} + ( 10 - 9 \beta_{1} + \beta_{2} + 2 \beta_{3} + 6 \beta_{4} - 4 \beta_{5} + 6 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 4 \beta_{9} - \beta_{10} ) q^{32} + ( 7 - \beta_{1} - 3 \beta_{2} - 12 \beta_{3} - 5 \beta_{4} - \beta_{5} + \beta_{6} - 5 \beta_{7} - 3 \beta_{10} - 2 \beta_{11} ) q^{33} + ( 3 \beta_{1} + 3 \beta_{3} - 16 \beta_{4} + \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + \beta_{8} - 2 \beta_{10} ) q^{34} + ( -2 - \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} ) q^{35} + ( 13 + 10 \beta_{1} - \beta_{2} - 10 \beta_{3} + 2 \beta_{5} - 2 \beta_{8} ) q^{36} + ( -12 + 6 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 16 \beta_{4} + 4 \beta_{5} - 10 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 4 \beta_{9} - 2 \beta_{10} ) q^{37} + ( -5 + 2 \beta_{1} - \beta_{2} - 19 \beta_{3} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 9 \beta_{7} - 3 \beta_{8} - \beta_{10} + 4 \beta_{11} ) q^{38} + ( -8 \beta_{4} - 6 \beta_{5} - 6 \beta_{8} + 2 \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{39} + ( -11 - 5 \beta_{1} + 4 \beta_{2} + 6 \beta_{3} + 5 \beta_{4} - 2 \beta_{5} - \beta_{6} - 7 \beta_{7} - 3 \beta_{8} - \beta_{9} - 3 \beta_{10} + 2 \beta_{11} ) q^{40} + ( -24 + 9 \beta_{1} - 2 \beta_{2} - 9 \beta_{3} + 5 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 5 \beta_{8} ) q^{41} + ( -9 - 7 \beta_{1} + 2 \beta_{2} - 9 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} - 2 \beta_{10} ) q^{42} + ( 12 - \beta_{2} + 10 \beta_{3} - 12 \beta_{4} + 6 \beta_{7} + 6 \beta_{8} - \beta_{10} ) q^{43} + ( 2 \beta_{1} + 2 \beta_{3} + 18 \beta_{4} + 8 \beta_{5} + 6 \beta_{6} - 6 \beta_{7} + 8 \beta_{8} + 2 \beta_{9} + 4 \beta_{10} - 2 \beta_{11} ) q^{44} + ( 5 + 13 \beta_{1} + 10 \beta_{3} - 9 \beta_{4} - 12 \beta_{5} + 2 \beta_{6} - 6 \beta_{7} - \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{11} ) q^{45} + ( -16 + 11 \beta_{1} - 2 \beta_{2} - 11 \beta_{3} + 4 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 4 \beta_{8} + 2 \beta_{9} + 2 \beta_{11} ) q^{46} + ( -17 + 4 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 11 \beta_{4} - 8 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} - 3 \beta_{8} + 6 \beta_{9} + 3 \beta_{10} ) q^{47} + ( -25 - \beta_{1} + 3 \beta_{2} + 14 \beta_{3} + 27 \beta_{4} - \beta_{5} + \beta_{6} + 8 \beta_{7} + 10 \beta_{8} + 3 \beta_{10} - 2 \beta_{11} ) q^{48} + 7 \beta_{4} q^{49} + ( -11 \beta_{1} - \beta_{2} - 5 \beta_{3} + 15 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 4 \beta_{7} - 14 \beta_{8} - 4 \beta_{11} ) q^{50} + ( 24 - 11 \beta_{1} + 5 \beta_{2} + 11 \beta_{3} + 8 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 8 \beta_{8} - 2 \beta_{9} - 2 \beta_{11} ) q^{51} + ( 19 + 9 \beta_{1} - \beta_{2} + 2 \beta_{3} + 15 \beta_{4} - \beta_{5} + 3 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 4 \beta_{9} + \beta_{10} ) q^{52} + ( -18 + 4 \beta_{1} - 3 \beta_{2} + 8 \beta_{3} + 10 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} + 16 \beta_{8} - 3 \beta_{10} + 8 \beta_{11} ) q^{53} + ( -8 \beta_{1} - 8 \beta_{3} + 12 \beta_{4} + 10 \beta_{5} - 5 \beta_{6} + 5 \beta_{7} + 10 \beta_{8} - 4 \beta_{9} - 2 \beta_{10} + 4 \beta_{11} ) q^{54} + ( 9 - 3 \beta_{1} + 7 \beta_{2} + 2 \beta_{3} - 33 \beta_{4} - 7 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} + 8 \beta_{8} - 2 \beta_{9} + 5 \beta_{10} + 2 \beta_{11} ) q^{55} + ( 15 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} - 2 \beta_{11} ) q^{56} + ( -3 - 18 \beta_{1} + 2 \beta_{2} + 6 \beta_{3} - 15 \beta_{4} + 6 \beta_{5} + 10 \beta_{6} + 6 \beta_{7} + 6 \beta_{8} - 12 \beta_{9} - 2 \beta_{10} ) q^{57} + ( 32 - 2 \beta_{1} - 5 \beta_{2} - 14 \beta_{3} - 28 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 8 \beta_{7} - 2 \beta_{8} - 5 \beta_{10} - 4 \beta_{11} ) q^{58} + ( -8 \beta_{1} - 8 \beta_{3} - 6 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} + 3 \beta_{9} + 8 \beta_{10} - 3 \beta_{11} ) q^{59} + ( 1 - 7 \beta_{1} - 18 \beta_{3} - 30 \beta_{4} - 6 \beta_{5} - 12 \beta_{6} - 2 \beta_{7} - 8 \beta_{8} + 6 \beta_{9} - 7 \beta_{10} + 4 \beta_{11} ) q^{60} + ( 28 - 2 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} + 14 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} - 14 \beta_{8} + \beta_{9} + \beta_{11} ) q^{61} + ( 44 + 12 \beta_{1} - 8 \beta_{2} - 2 \beta_{3} + 48 \beta_{4} - 16 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + 4 \beta_{9} + 8 \beta_{10} ) q^{62} + ( 16 - 3 \beta_{1} + \beta_{2} + 8 \beta_{3} - 10 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} + \beta_{7} - 3 \beta_{8} + \beta_{10} - 6 \beta_{11} ) q^{63} + ( 5 \beta_{1} + 5 \beta_{3} + 15 \beta_{4} - 4 \beta_{5} + 8 \beta_{6} - 8 \beta_{7} - 4 \beta_{8} + 8 \beta_{9} + 9 \beta_{10} - 8 \beta_{11} ) q^{64} + ( -19 + 2 \beta_{2} + \beta_{3} + 26 \beta_{4} - 2 \beta_{5} - 4 \beta_{6} - 2 \beta_{8} - 2 \beta_{9} - 8 \beta_{10} + 6 \beta_{11} ) q^{65} + ( -26 - 29 \beta_{1} + 12 \beta_{2} + 29 \beta_{3} - \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + \beta_{8} - 4 \beta_{9} - 4 \beta_{11} ) q^{66} + ( 16 + 4 \beta_{1} - \beta_{2} - 6 \beta_{3} + 28 \beta_{4} + 8 \beta_{5} - 6 \beta_{6} - 6 \beta_{7} - 6 \beta_{8} + 12 \beta_{9} + \beta_{10} ) q^{67} + ( 18 + 2 \beta_{1} + 8 \beta_{3} - 22 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 8 \beta_{7} - 2 \beta_{8} + 4 \beta_{11} ) q^{68} + ( 5 \beta_{1} + 5 \beta_{3} - 22 \beta_{4} - 3 \beta_{5} + 5 \beta_{6} - 5 \beta_{7} - 3 \beta_{8} - 2 \beta_{9} + 4 \beta_{10} + 2 \beta_{11} ) q^{69} + ( -6 - 6 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} + 16 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 3 \beta_{8} - 4 \beta_{9} + \beta_{10} + \beta_{11} ) q^{70} + ( \beta_{1} + 10 \beta_{2} - \beta_{3} - 4 \beta_{5} - 10 \beta_{6} - 10 \beta_{7} + 4 \beta_{8} + 2 \beta_{9} + 2 \beta_{11} ) q^{71} + ( -45 + \beta_{1} - 4 \beta_{2} - 8 \beta_{3} - 29 \beta_{4} - 18 \beta_{5} + 4 \beta_{6} - 8 \beta_{7} - 8 \beta_{8} + 16 \beta_{9} + 4 \beta_{10} ) q^{72} + ( -12 - 6 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} + 24 \beta_{4} - 6 \beta_{5} + 6 \beta_{6} + 10 \beta_{7} + 14 \beta_{8} + 4 \beta_{10} - 12 \beta_{11} ) q^{73} + ( 10 \beta_{1} + 10 \beta_{3} - 30 \beta_{4} + 22 \beta_{5} - 8 \beta_{6} + 8 \beta_{7} + 22 \beta_{8} - 8 \beta_{9} - 10 \beta_{10} + 8 \beta_{11} ) q^{74} + ( -3 + 26 \beta_{1} - 11 \beta_{2} - 17 \beta_{3} + 33 \beta_{4} + 2 \beta_{5} - 8 \beta_{6} + 8 \beta_{7} + 11 \beta_{8} + 3 \beta_{9} + 3 \beta_{10} + \beta_{11} ) q^{75} + ( -40 - 11 \beta_{1} + 6 \beta_{2} + 11 \beta_{3} - \beta_{5} + 4 \beta_{6} + 4 \beta_{7} + \beta_{8} - 7 \beta_{9} - 7 \beta_{11} ) q^{76} + ( -9 + 12 \beta_{1} - 6 \beta_{2} - \beta_{3} - 7 \beta_{4} - 3 \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{9} + 6 \beta_{10} ) q^{77} + ( -5 + 2 \beta_{1} + 4 \beta_{2} + 10 \beta_{3} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 16 \beta_{7} + 6 \beta_{8} + 4 \beta_{10} + 4 \beta_{11} ) q^{78} + ( -22 \beta_{1} - 22 \beta_{3} + 8 \beta_{4} - 4 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 4 \beta_{8} - 10 \beta_{9} - 11 \beta_{10} + 10 \beta_{11} ) q^{79} + ( 11 + 14 \beta_{1} + 6 \beta_{2} + 19 \beta_{3} - 17 \beta_{4} + 11 \beta_{5} - 3 \beta_{6} + 9 \beta_{7} - 10 \beta_{8} - 8 \beta_{9} + 5 \beta_{10} - 5 \beta_{11} ) q^{80} + ( -41 + 25 \beta_{1} - 8 \beta_{2} - 25 \beta_{3} + 8 \beta_{5} - 8 \beta_{8} + 8 \beta_{9} + 8 \beta_{11} ) q^{81} + ( -36 + 14 \beta_{2} + 2 \beta_{3} - 40 \beta_{4} + 2 \beta_{5} - 8 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 4 \beta_{9} - 14 \beta_{10} ) q^{82} + ( -15 - 6 \beta_{1} + 3 \beta_{2} + 5 \beta_{3} + 27 \beta_{4} - 6 \beta_{5} + 6 \beta_{6} - 15 \beta_{7} - 23 \beta_{8} + 3 \beta_{10} - 12 \beta_{11} ) q^{83} + ( 12 \beta_{1} + 12 \beta_{3} + 12 \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} + 6 \beta_{10} + \beta_{11} ) q^{84} + ( -20 - 5 \beta_{1} - 3 \beta_{2} - 5 \beta_{3} + 26 \beta_{4} - 18 \beta_{5} + 16 \beta_{6} - 4 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + 4 \beta_{10} - 8 \beta_{11} ) q^{85} + ( 38 + 2 \beta_{1} - 16 \beta_{2} - 2 \beta_{3} - 8 \beta_{5} + 8 \beta_{8} + 6 \beta_{9} + 6 \beta_{11} ) q^{86} + ( 23 + 18 \beta_{1} - 5 \beta_{2} + 3 \beta_{3} + 17 \beta_{4} + 10 \beta_{5} + 9 \beta_{6} + 3 \beta_{7} + 3 \beta_{8} - 6 \beta_{9} + 5 \beta_{10} ) q^{87} + ( -12 + 4 \beta_{1} - 16 \beta_{2} - 32 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} - 14 \beta_{8} - 16 \beta_{10} + 8 \beta_{11} ) q^{88} + ( 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} - 4 \beta_{7} - 2 \beta_{8} + 6 \beta_{9} - 14 \beta_{10} - 6 \beta_{11} ) q^{89} + ( 76 - 12 \beta_{1} - 11 \beta_{2} - 6 \beta_{3} - 60 \beta_{4} + 3 \beta_{5} + 17 \beta_{6} - 10 \beta_{7} - 5 \beta_{8} - 4 \beta_{9} - 2 \beta_{10} - 5 \beta_{11} ) q^{90} + ( 12 - 2 \beta_{1} + 5 \beta_{2} + 2 \beta_{3} - 3 \beta_{5} + 5 \beta_{6} + 5 \beta_{7} + 3 \beta_{8} - 3 \beta_{9} - 3 \beta_{11} ) q^{91} + ( -35 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - 39 \beta_{4} - 8 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 4 \beta_{9} - \beta_{10} ) q^{92} + ( -2 + 8 \beta_{1} - 3 \beta_{2} - 16 \beta_{3} - 14 \beta_{4} + 8 \beta_{5} - 8 \beta_{6} - 20 \beta_{7} + 10 \beta_{8} - 3 \beta_{10} + 16 \beta_{11} ) q^{93} + ( 7 \beta_{1} + 7 \beta_{3} + 6 \beta_{4} - 9 \beta_{5} + 14 \beta_{6} - 14 \beta_{7} - 9 \beta_{8} + 4 \beta_{10} ) q^{94} + ( 20 - 21 \beta_{1} + 2 \beta_{2} + 10 \beta_{3} - 41 \beta_{4} + 16 \beta_{5} + 12 \beta_{6} + 8 \beta_{7} + 4 \beta_{8} - 2 \beta_{9} - 9 \beta_{10} + 4 \beta_{11} ) q^{95} + ( 18 + 22 \beta_{1} - 8 \beta_{2} - 22 \beta_{3} - 2 \beta_{5} - 13 \beta_{6} - 13 \beta_{7} + 2 \beta_{8} + 5 \beta_{9} + 5 \beta_{11} ) q^{96} + ( 69 - 52 \beta_{1} + 19 \beta_{2} + 7 \beta_{3} + 55 \beta_{4} + 2 \beta_{5} + 5 \beta_{6} + 7 \beta_{7} + 7 \beta_{8} - 14 \beta_{9} - 19 \beta_{10} ) q^{97} -7 \beta_{3} q^{98} + ( 2 \beta_{1} + 2 \beta_{3} + 18 \beta_{4} - 8 \beta_{5} - 16 \beta_{6} + 16 \beta_{7} - 8 \beta_{8} - 6 \beta_{9} - 14 \beta_{10} + 6 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 4q^{2} - 4q^{3} - 8q^{5} - 24q^{6} + 24q^{8} + O(q^{10}) \) \( 12q - 4q^{2} - 4q^{3} - 8q^{5} - 24q^{6} + 24q^{8} + 28q^{10} - 12q^{11} + 16q^{12} - 4q^{13} - 64q^{15} + 40q^{16} - 12q^{17} - 56q^{18} + 60q^{20} + 28q^{21} - 68q^{22} - 16q^{23} + 64q^{25} - 56q^{26} + 164q^{27} - 76q^{30} - 96q^{31} + 32q^{32} + 124q^{33} + 232q^{36} - 104q^{37} + 80q^{38} - 124q^{40} - 208q^{41} - 140q^{42} + 76q^{43} + 92q^{45} - 80q^{46} - 164q^{47} - 392q^{48} - 52q^{50} + 220q^{51} + 216q^{52} - 204q^{53} + 116q^{55} + 168q^{56} - 236q^{57} + 356q^{58} + 152q^{60} + 280q^{61} + 568q^{62} + 112q^{63} - 192q^{65} - 544q^{66} + 324q^{67} + 184q^{68} - 112q^{70} + 144q^{71} - 440q^{72} - 248q^{73} + 108q^{75} - 632q^{76} - 56q^{77} + 12q^{78} + 60q^{80} - 260q^{81} - 376q^{82} - 224q^{83} - 324q^{85} + 456q^{86} + 244q^{87} - 24q^{88} + 780q^{90} + 84q^{91} - 424q^{92} + 236q^{93} + 52q^{95} + 504q^{96} + 564q^{97} + 28q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 4 x^{11} + 8 x^{10} + 8 x^{9} + 70 x^{8} - 248 x^{7} + 464 x^{6} + 432 x^{5} + 1129 x^{4} - 2708 x^{3} + 3528 x^{2} + 840 x + 100\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-216885938 \nu^{11} + 1060104974 \nu^{10} - 3418831332 \nu^{9} + 4776047991 \nu^{8} - 23945884771 \nu^{7} + 63915737188 \nu^{6} - 204203079084 \nu^{5} + 506142337977 \nu^{4} - 961485749639 \nu^{3} + 623382228558 \nu^{2} + 159903996190 \nu + 9734414919250\)\()/ 2042321889134 \)
\(\beta_{3}\)\(=\)\((\)\(650167512 \nu^{11} - 2167388474 \nu^{10} + 4094056596 \nu^{9} + 5876565360 \nu^{8} + 56164339191 \nu^{7} - 129023088556 \nu^{6} + 236570374200 \nu^{5} + 313239660300 \nu^{4} + 1553421119325 \nu^{3} - 1168718252810 \nu^{2} + 1748454958084 \nu + 412177775220\)\()/ 2042321889134 \)
\(\beta_{4}\)\(=\)\((\)\(-20608888761 \nu^{11} + 85686392604 \nu^{10} - 175708052458 \nu^{9} - 144400827108 \nu^{8} - 1413239386470 \nu^{7} + 5391826108683 \nu^{6} - 10207639827884 \nu^{5} - 7720188073752 \nu^{4} - 21701237109669 \nu^{3} + 63575976361413 \nu^{2} - 78551750812858 \nu - 8569191768820\)\()/ 10211609445670 \)
\(\beta_{5}\)\(=\)\((\)\(-62789472393 \nu^{11} + 265477896952 \nu^{10} - 559679834849 \nu^{9} - 389383135769 \nu^{8} - 4290358282230 \nu^{7} + 16693318345309 \nu^{6} - 33622104799617 \nu^{5} - 19577456593071 \nu^{4} - 65283986871277 \nu^{3} + 186102541156969 \nu^{2} - 305661466865764 \nu + 4818810061550\)\()/ 20423218891340 \)
\(\beta_{6}\)\(=\)\((\)\(-137676730578 \nu^{11} + 596855018097 \nu^{10} - 1287825684554 \nu^{9} - 649638989989 \nu^{8} - 9365288143960 \nu^{7} + 36808958000879 \nu^{6} - 73044528683602 \nu^{5} - 33680389194101 \nu^{4} - 140853910278262 \nu^{3} + 388722134797854 \nu^{2} - 533770796318264 \nu + 10093227434800\)\()/ 40846437782680 \)
\(\beta_{7}\)\(=\)\((\)\(150422124598 \nu^{11} - 616051382677 \nu^{10} + 1215756736104 \nu^{9} + 1333433365979 \nu^{8} + 9880196872050 \nu^{7} - 37904218748399 \nu^{6} + 70451110117132 \nu^{5} + 72367171859091 \nu^{4} + 143160161256492 \nu^{3} - 395992267619354 \nu^{2} + 532117745999764 \nu + 126398845752920\)\()/ 40846437782680 \)
\(\beta_{8}\)\(=\)\((\)\(-86748937073 \nu^{11} + 338454191902 \nu^{10} - 673792264554 \nu^{9} - 692145335679 \nu^{8} - 6366562303195 \nu^{7} + 21016823182349 \nu^{6} - 39064719056232 \nu^{5} - 38220862323141 \nu^{4} - 112221517908142 \nu^{3} + 224148138608179 \nu^{2} - 296181361297214 \nu - 70443542020170\)\()/ 20423218891340 \)
\(\beta_{9}\)\(=\)\((\)\(-360571142916 \nu^{11} + 1533377436449 \nu^{10} - 3341673499113 \nu^{9} - 1848600783168 \nu^{8} - 25104295424530 \nu^{7} + 95079862089563 \nu^{6} - 196282984532029 \nu^{5} - 99316547026692 \nu^{4} - 391813287370614 \nu^{3} + 1044746818796418 \nu^{2} - 1594030930721148 \nu + 64434080701620\)\()/ 40846437782680 \)
\(\beta_{10}\)\(=\)\((\)\(19958721249 \nu^{11} - 83519004130 \nu^{10} + 171613995862 \nu^{9} + 138524261748 \nu^{8} + 1357075047279 \nu^{7} - 5262803020127 \nu^{6} + 9971069453684 \nu^{5} + 7406948413452 \nu^{4} + 20147815990344 \nu^{3} - 60364936219469 \nu^{2} + 74760973965640 \nu + 8157013993600\)\()/ 2042321889134 \)
\(\beta_{11}\)\(=\)\((\)\(448601700821 \nu^{11} - 1754253899729 \nu^{10} + 3449654172798 \nu^{9} + 3904802715988 \nu^{8} + 31494430826465 \nu^{7} - 108057838585943 \nu^{6} + 201921387930034 \nu^{5} + 208221351772472 \nu^{4} + 507712177017044 \nu^{3} - 1152911001994558 \nu^{2} + 1567496069553948 \nu + 409254215744460\)\()/ 40846437782680 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{10} + 5 \beta_{4} + \beta_{3} + \beta_{1}\)
\(\nu^{3}\)\(=\)\(\beta_{10} + 2 \beta_{8} + 2 \beta_{4} + 9 \beta_{3} + \beta_{2} - 2\)
\(\nu^{4}\)\(=\)\(2 \beta_{8} - 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + 13 \beta_{3} + 11 \beta_{2} - 13 \beta_{1} - 41\)
\(\nu^{5}\)\(=\)\(-15 \beta_{10} + 4 \beta_{9} - 2 \beta_{8} - 2 \beta_{7} - 6 \beta_{6} - 28 \beta_{5} - 38 \beta_{4} - 2 \beta_{3} + 15 \beta_{2} - 87 \beta_{1} - 42\)
\(\nu^{6}\)\(=\)\(-8 \beta_{11} - 115 \beta_{10} + 8 \beta_{9} - 44 \beta_{8} + 32 \beta_{7} - 32 \beta_{6} - 44 \beta_{5} - 389 \beta_{4} - 159 \beta_{3} - 159 \beta_{1}\)
\(\nu^{7}\)\(=\)\(-80 \beta_{11} - 195 \beta_{10} - 342 \beta_{8} + 136 \beta_{7} + 40 \beta_{6} - 40 \beta_{5} - 550 \beta_{4} - 905 \beta_{3} - 195 \beta_{2} - 40 \beta_{1} + 630\)
\(\nu^{8}\)\(=\)\(-176 \beta_{11} - 176 \beta_{9} - 526 \beta_{8} + 598 \beta_{7} + 598 \beta_{6} + 526 \beta_{5} - 1749 \beta_{3} - 1247 \beta_{2} + 1749 \beta_{1} + 4401\)
\(\nu^{9}\)\(=\)\(2451 \beta_{10} - 1196 \beta_{9} + 598 \beta_{8} + 598 \beta_{7} + 2178 \beta_{6} + 4112 \beta_{5} + 7254 \beta_{4} + 598 \beta_{3} - 2451 \beta_{2} + 9971 \beta_{1} + 8450\)
\(\nu^{10}\)\(=\)\(2776 \beta_{11} + 14083 \beta_{10} - 2776 \beta_{9} + 9856 \beta_{8} - 5308 \beta_{7} + 5308 \beta_{6} + 9856 \beta_{5} + 44813 \beta_{4} + 23323 \beta_{3} + 23323 \beta_{1}\)
\(\nu^{11}\)\(=\)\(16168 \beta_{11} + 30403 \beta_{10} + 49642 \beta_{8} - 30572 \beta_{7} - 8084 \beta_{6} + 8084 \beta_{5} + 92062 \beta_{4} + 114317 \beta_{3} + 30403 \beta_{2} + 8084 \beta_{1} - 108230\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/35\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\)
\(\chi(n)\) \(\beta_{4}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1
2.47480 2.47480i
1.90845 1.90845i
0.950261 0.950261i
−0.109772 + 0.109772i
−1.36503 + 1.36503i
−1.85871 + 1.85871i
2.47480 + 2.47480i
1.90845 + 1.90845i
0.950261 + 0.950261i
−0.109772 0.109772i
−1.36503 1.36503i
−1.85871 1.85871i
−2.47480 + 2.47480i 2.98009 + 2.98009i 8.24929i −4.25027 + 2.63347i −14.7503 1.87083 1.87083i 10.5161 + 10.5161i 8.76185i 4.00127 17.0359i
8.2 −1.90845 + 1.90845i −3.30412 3.30412i 3.28438i −3.50673 3.56410i 12.6115 −1.87083 + 1.87083i −1.36573 1.36573i 12.8345i 13.4943 + 0.109487i
8.3 −0.950261 + 0.950261i 0.269488 + 0.269488i 2.19401i 4.00169 + 2.99774i −0.512169 −1.87083 + 1.87083i −5.88593 5.88593i 8.85475i −6.65129 + 0.954016i
8.4 0.109772 0.109772i 1.67281 + 1.67281i 3.97590i −0.861142 4.92529i 0.367256 1.87083 1.87083i 0.875529 + 0.875529i 3.40339i −0.635186 0.446128i
8.5 1.36503 1.36503i −3.78207 3.78207i 0.273387i 4.98225 + 0.420986i −10.3253 1.87083 1.87083i 5.83330 + 5.83330i 19.6082i 7.37557 6.22626i
8.6 1.85871 1.85871i 0.163806 + 0.163806i 2.90963i −4.36579 + 2.43719i 0.608937 −1.87083 + 1.87083i 2.02669 + 2.02669i 8.94634i −3.58471 + 12.6448i
22.1 −2.47480 2.47480i 2.98009 2.98009i 8.24929i −4.25027 2.63347i −14.7503 1.87083 + 1.87083i 10.5161 10.5161i 8.76185i 4.00127 + 17.0359i
22.2 −1.90845 1.90845i −3.30412 + 3.30412i 3.28438i −3.50673 + 3.56410i 12.6115 −1.87083 1.87083i −1.36573 + 1.36573i 12.8345i 13.4943 0.109487i
22.3 −0.950261 0.950261i 0.269488 0.269488i 2.19401i 4.00169 2.99774i −0.512169 −1.87083 1.87083i −5.88593 + 5.88593i 8.85475i −6.65129 0.954016i
22.4 0.109772 + 0.109772i 1.67281 1.67281i 3.97590i −0.861142 + 4.92529i 0.367256 1.87083 + 1.87083i 0.875529 0.875529i 3.40339i −0.635186 + 0.446128i
22.5 1.36503 + 1.36503i −3.78207 + 3.78207i 0.273387i 4.98225 0.420986i −10.3253 1.87083 + 1.87083i 5.83330 5.83330i 19.6082i 7.37557 + 6.22626i
22.6 1.85871 + 1.85871i 0.163806 0.163806i 2.90963i −4.36579 2.43719i 0.608937 −1.87083 1.87083i 2.02669 2.02669i 8.94634i −3.58471 12.6448i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 22.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 35.3.g.a 12
3.b odd 2 1 315.3.o.a 12
4.b odd 2 1 560.3.bh.e 12
5.b even 2 1 175.3.g.b 12
5.c odd 4 1 inner 35.3.g.a 12
5.c odd 4 1 175.3.g.b 12
7.b odd 2 1 245.3.g.a 12
7.c even 3 2 245.3.m.d 24
7.d odd 6 2 245.3.m.c 24
15.e even 4 1 315.3.o.a 12
20.e even 4 1 560.3.bh.e 12
35.f even 4 1 245.3.g.a 12
35.k even 12 2 245.3.m.c 24
35.l odd 12 2 245.3.m.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.3.g.a 12 1.a even 1 1 trivial
35.3.g.a 12 5.c odd 4 1 inner
175.3.g.b 12 5.b even 2 1
175.3.g.b 12 5.c odd 4 1
245.3.g.a 12 7.b odd 2 1
245.3.g.a 12 35.f even 4 1
245.3.m.c 24 7.d odd 6 2
245.3.m.c 24 35.k even 12 2
245.3.m.d 24 7.c even 3 2
245.3.m.d 24 35.l odd 12 2
315.3.o.a 12 3.b odd 2 1
315.3.o.a 12 15.e even 4 1
560.3.bh.e 12 4.b odd 2 1
560.3.bh.e 12 20.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(35, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 4 T + 8 T^{2} + 8 T^{3} - 26 T^{4} - 104 T^{5} - 176 T^{6} - 208 T^{7} + 9 T^{8} + 468 T^{9} + 968 T^{10} + 2568 T^{11} + 5732 T^{12} + 10272 T^{13} + 15488 T^{14} + 29952 T^{15} + 2304 T^{16} - 212992 T^{17} - 720896 T^{18} - 1703936 T^{19} - 1703936 T^{20} + 2097152 T^{21} + 8388608 T^{22} + 16777216 T^{23} + 16777216 T^{24} \)
$3$ \( 1 + 4 T + 8 T^{2} - 32 T^{3} - 185 T^{4} - 104 T^{5} + 1576 T^{6} + 6004 T^{7} - 7229 T^{8} - 84904 T^{9} - 176944 T^{10} + 302584 T^{11} + 2631886 T^{12} + 2723256 T^{13} - 14332464 T^{14} - 61895016 T^{15} - 47429469 T^{16} + 354530196 T^{17} + 837551016 T^{18} - 497428776 T^{19} - 7963643385 T^{20} - 12397455648 T^{21} + 27894275208 T^{22} + 125524238436 T^{23} + 282429536481 T^{24} \)
$5$ \( 1 + 8 T - 328 T^{3} - 1231 T^{4} + 2600 T^{5} + 41200 T^{6} + 65000 T^{7} - 769375 T^{8} - 5125000 T^{9} + 78125000 T^{11} + 244140625 T^{12} \)
$7$ \( ( 1 + 49 T^{4} )^{3} \)
$11$ \( ( 1 + 6 T + 319 T^{2} + 1166 T^{3} + 55579 T^{4} + 201340 T^{5} + 7907818 T^{6} + 24362140 T^{7} + 813732139 T^{8} + 2065640126 T^{9} + 68380483039 T^{10} + 155624547606 T^{11} + 3138428376721 T^{12} )^{2} \)
$13$ \( 1 + 4 T + 8 T^{2} - 2424 T^{3} + 42679 T^{4} + 821112 T^{5} + 5880904 T^{6} - 59122612 T^{7} - 455341821 T^{8} + 28627711344 T^{9} + 347588147888 T^{10} + 1969414999296 T^{11} - 34074574484818 T^{12} + 332831134881024 T^{13} + 9927465091829168 T^{14} + 138180494764621296 T^{15} - 371436311945782941 T^{16} - 8150554124493589588 T^{17} + \)\(13\!\cdots\!24\)\( T^{18} + \)\(32\!\cdots\!68\)\( T^{19} + \)\(28\!\cdots\!39\)\( T^{20} - \)\(27\!\cdots\!96\)\( T^{21} + \)\(15\!\cdots\!08\)\( T^{22} + \)\(12\!\cdots\!76\)\( T^{23} + \)\(54\!\cdots\!61\)\( T^{24} \)
$17$ \( 1 + 12 T + 72 T^{2} + 2784 T^{3} + 76619 T^{4} + 611296 T^{5} + 5694312 T^{6} + 329614108 T^{7} - 1813984861 T^{8} - 125752615304 T^{9} - 710229888304 T^{10} - 21298315351968 T^{11} - 439455529642318 T^{12} - 6155213136718752 T^{13} - 59319110501038384 T^{14} - 3035362428830755976 T^{15} - 12653918391982100701 T^{16} + \)\(66\!\cdots\!92\)\( T^{17} + \)\(33\!\cdots\!32\)\( T^{18} + \)\(10\!\cdots\!84\)\( T^{19} + \)\(37\!\cdots\!39\)\( T^{20} + \)\(39\!\cdots\!56\)\( T^{21} + \)\(29\!\cdots\!72\)\( T^{22} + \)\(14\!\cdots\!68\)\( T^{23} + \)\(33\!\cdots\!21\)\( T^{24} \)
$19$ \( 1 - 2208 T^{2} + 2306338 T^{4} - 1539873472 T^{6} + 753579200867 T^{8} - 300967197557024 T^{10} + 110083881158431556 T^{12} - 39222346152828924704 T^{14} + \)\(12\!\cdots\!47\)\( T^{16} - \)\(34\!\cdots\!92\)\( T^{18} + \)\(66\!\cdots\!78\)\( T^{20} - \)\(82\!\cdots\!08\)\( T^{22} + \)\(48\!\cdots\!21\)\( T^{24} \)
$23$ \( 1 + 16 T + 128 T^{2} + 12576 T^{3} + 414846 T^{4} - 3136832 T^{5} - 24211712 T^{6} - 1482627024 T^{7} + 34808010943 T^{8} + 851590367168 T^{9} + 5488492394624 T^{10} + 645053822277920 T^{11} + 70544571658994180 T^{12} + 341233471985019680 T^{13} + 1535905200203974784 T^{14} + \)\(12\!\cdots\!52\)\( T^{15} + \)\(27\!\cdots\!83\)\( T^{16} - \)\(61\!\cdots\!76\)\( T^{17} - \)\(53\!\cdots\!52\)\( T^{18} - \)\(36\!\cdots\!88\)\( T^{19} + \)\(25\!\cdots\!06\)\( T^{20} + \)\(40\!\cdots\!44\)\( T^{21} + \)\(21\!\cdots\!28\)\( T^{22} + \)\(14\!\cdots\!64\)\( T^{23} + \)\(48\!\cdots\!41\)\( T^{24} \)
$29$ \( 1 - 7018 T^{2} + 22990943 T^{4} - 47164435002 T^{6} + 68979050432827 T^{8} - 78004905252931004 T^{10} + 71933262446730943946 T^{12} - \)\(55\!\cdots\!24\)\( T^{14} + \)\(34\!\cdots\!47\)\( T^{16} - \)\(16\!\cdots\!82\)\( T^{18} + \)\(57\!\cdots\!03\)\( T^{20} - \)\(12\!\cdots\!18\)\( T^{22} + \)\(12\!\cdots\!81\)\( T^{24} \)
$31$ \( ( 1 + 48 T + 3144 T^{2} + 79440 T^{3} + 3260215 T^{4} + 42545920 T^{5} + 2350062944 T^{6} + 40886629120 T^{7} + 3010877017015 T^{8} + 70503292418640 T^{9} + 2681489421714504 T^{10} + 39342157775078448 T^{11} + 787662783788549761 T^{12} )^{2} \)
$37$ \( 1 + 104 T + 5408 T^{2} + 174344 T^{3} + 2697254 T^{4} + 10557800 T^{5} + 1709176736 T^{6} + 138455876040 T^{7} + 4386451218735 T^{8} + 70184122125200 T^{9} + 3161821544956736 T^{10} + 334287451090079440 T^{11} + 17427941461580290900 T^{12} + \)\(45\!\cdots\!60\)\( T^{13} + \)\(59\!\cdots\!96\)\( T^{14} + \)\(18\!\cdots\!00\)\( T^{15} + \)\(15\!\cdots\!35\)\( T^{16} + \)\(66\!\cdots\!60\)\( T^{17} + \)\(11\!\cdots\!16\)\( T^{18} + \)\(95\!\cdots\!00\)\( T^{19} + \)\(33\!\cdots\!14\)\( T^{20} + \)\(29\!\cdots\!76\)\( T^{21} + \)\(12\!\cdots\!08\)\( T^{22} + \)\(32\!\cdots\!76\)\( T^{23} + \)\(43\!\cdots\!61\)\( T^{24} \)
$41$ \( ( 1 + 104 T + 11808 T^{2} + 782920 T^{3} + 51161335 T^{4} + 2462767696 T^{5} + 114809897552 T^{6} + 4139912496976 T^{7} + 144569705150935 T^{8} + 3718951612363720 T^{9} + 94285997105460768 T^{10} + 1395956568255849704 T^{11} + 22563490300366186081 T^{12} )^{2} \)
$43$ \( 1 - 76 T + 2888 T^{2} - 67684 T^{3} + 11551602 T^{4} - 935583796 T^{5} + 40033903848 T^{6} - 1100447483004 T^{7} + 64203765803999 T^{8} - 5036169803944728 T^{9} + 247013912808013264 T^{10} - 7962007386466391112 T^{11} + \)\(24\!\cdots\!96\)\( T^{12} - \)\(14\!\cdots\!88\)\( T^{13} + \)\(84\!\cdots\!64\)\( T^{14} - \)\(31\!\cdots\!72\)\( T^{15} + \)\(75\!\cdots\!99\)\( T^{16} - \)\(23\!\cdots\!96\)\( T^{17} + \)\(15\!\cdots\!48\)\( T^{18} - \)\(69\!\cdots\!04\)\( T^{19} + \)\(15\!\cdots\!02\)\( T^{20} - \)\(17\!\cdots\!16\)\( T^{21} + \)\(13\!\cdots\!88\)\( T^{22} - \)\(65\!\cdots\!24\)\( T^{23} + \)\(15\!\cdots\!01\)\( T^{24} \)
$47$ \( 1 + 164 T + 13448 T^{2} + 868672 T^{3} + 63690059 T^{4} + 4678476896 T^{5} + 288061819304 T^{6} + 15844760579700 T^{7} + 894357153623779 T^{8} + 50754877518397512 T^{9} + 2657783539457568848 T^{10} + \)\(12\!\cdots\!80\)\( T^{11} + \)\(60\!\cdots\!22\)\( T^{12} + \)\(28\!\cdots\!20\)\( T^{13} + \)\(12\!\cdots\!88\)\( T^{14} + \)\(54\!\cdots\!48\)\( T^{15} + \)\(21\!\cdots\!19\)\( T^{16} + \)\(83\!\cdots\!00\)\( T^{17} + \)\(33\!\cdots\!64\)\( T^{18} + \)\(12\!\cdots\!24\)\( T^{19} + \)\(36\!\cdots\!39\)\( T^{20} + \)\(10\!\cdots\!08\)\( T^{21} + \)\(37\!\cdots\!48\)\( T^{22} + \)\(10\!\cdots\!76\)\( T^{23} + \)\(13\!\cdots\!81\)\( T^{24} \)
$53$ \( 1 + 204 T + 20808 T^{2} + 1399092 T^{3} + 75768866 T^{4} + 3705029700 T^{5} + 157956707304 T^{6} + 4490093784828 T^{7} - 72880136612449 T^{8} - 22948283226553704 T^{9} - 1987483652559492912 T^{10} - \)\(11\!\cdots\!40\)\( T^{11} - \)\(61\!\cdots\!96\)\( T^{12} - \)\(33\!\cdots\!60\)\( T^{13} - \)\(15\!\cdots\!72\)\( T^{14} - \)\(50\!\cdots\!16\)\( T^{15} - \)\(45\!\cdots\!89\)\( T^{16} + \)\(78\!\cdots\!72\)\( T^{17} + \)\(77\!\cdots\!64\)\( T^{18} + \)\(51\!\cdots\!00\)\( T^{19} + \)\(29\!\cdots\!86\)\( T^{20} + \)\(15\!\cdots\!88\)\( T^{21} + \)\(63\!\cdots\!08\)\( T^{22} + \)\(17\!\cdots\!36\)\( T^{23} + \)\(24\!\cdots\!81\)\( T^{24} \)
$59$ \( 1 - 27144 T^{2} + 365962322 T^{4} - 3231050981288 T^{6} + 20796912147061923 T^{8} - \)\(10\!\cdots\!08\)\( T^{10} + \)\(40\!\cdots\!08\)\( T^{12} - \)\(12\!\cdots\!88\)\( T^{14} + \)\(30\!\cdots\!83\)\( T^{16} - \)\(57\!\cdots\!28\)\( T^{18} + \)\(78\!\cdots\!02\)\( T^{20} - \)\(70\!\cdots\!44\)\( T^{22} + \)\(31\!\cdots\!61\)\( T^{24} \)
$61$ \( ( 1 - 140 T + 18986 T^{2} - 1612508 T^{3} + 146964673 T^{4} - 10078896872 T^{5} + 706860431120 T^{6} - 37503575260712 T^{7} + 2034849494974993 T^{8} - 83077015820107388 T^{9} + 3639755044566377066 T^{10} - 99868007632803564140 T^{11} + \)\(26\!\cdots\!21\)\( T^{12} )^{2} \)
$67$ \( 1 - 324 T + 52488 T^{2} - 5922460 T^{3} + 578352418 T^{4} - 55559962188 T^{5} + 5182632258728 T^{6} - 447217160038868 T^{7} + 35870583162738015 T^{8} - 2754817067496129224 T^{9} + \)\(20\!\cdots\!04\)\( T^{10} - \)\(14\!\cdots\!28\)\( T^{11} + \)\(99\!\cdots\!72\)\( T^{12} - \)\(65\!\cdots\!92\)\( T^{13} + \)\(41\!\cdots\!84\)\( T^{14} - \)\(24\!\cdots\!56\)\( T^{15} + \)\(14\!\cdots\!15\)\( T^{16} - \)\(81\!\cdots\!32\)\( T^{17} + \)\(42\!\cdots\!08\)\( T^{18} - \)\(20\!\cdots\!52\)\( T^{19} + \)\(95\!\cdots\!58\)\( T^{20} - \)\(43\!\cdots\!40\)\( T^{21} + \)\(17\!\cdots\!88\)\( T^{22} - \)\(48\!\cdots\!36\)\( T^{23} + \)\(66\!\cdots\!21\)\( T^{24} \)
$71$ \( ( 1 - 72 T + 18698 T^{2} - 925624 T^{3} + 153299139 T^{4} - 5976553984 T^{5} + 867379084884 T^{6} - 30127808633344 T^{7} + 3895588817842659 T^{8} - 118572697204091704 T^{9} + 12074299527233239178 T^{10} - \)\(23\!\cdots\!72\)\( T^{11} + \)\(16\!\cdots\!41\)\( T^{12} )^{2} \)
$73$ \( 1 + 248 T + 30752 T^{2} + 3080952 T^{3} + 254433510 T^{4} + 15775678040 T^{5} + 834161467552 T^{6} + 36580397057880 T^{7} + 633277440059215 T^{8} - 33802054838880848 T^{9} - 2957631245611274944 T^{10} - \)\(15\!\cdots\!12\)\( T^{11} - \)\(73\!\cdots\!92\)\( T^{12} - \)\(81\!\cdots\!48\)\( T^{13} - \)\(83\!\cdots\!04\)\( T^{14} - \)\(51\!\cdots\!72\)\( T^{15} + \)\(51\!\cdots\!15\)\( T^{16} + \)\(15\!\cdots\!20\)\( T^{17} + \)\(19\!\cdots\!92\)\( T^{18} + \)\(19\!\cdots\!60\)\( T^{19} + \)\(16\!\cdots\!10\)\( T^{20} + \)\(10\!\cdots\!88\)\( T^{21} + \)\(56\!\cdots\!52\)\( T^{22} + \)\(24\!\cdots\!92\)\( T^{23} + \)\(52\!\cdots\!41\)\( T^{24} \)
$79$ \( 1 - 30162 T^{2} + 585976951 T^{4} - 7906141610906 T^{6} + 83787919669756051 T^{8} - \)\(70\!\cdots\!12\)\( T^{10} + \)\(48\!\cdots\!94\)\( T^{12} - \)\(27\!\cdots\!72\)\( T^{14} + \)\(12\!\cdots\!11\)\( T^{16} - \)\(46\!\cdots\!46\)\( T^{18} + \)\(13\!\cdots\!71\)\( T^{20} - \)\(27\!\cdots\!62\)\( T^{22} + \)\(34\!\cdots\!81\)\( T^{24} \)
$83$ \( 1 + 224 T + 25088 T^{2} + 4058120 T^{3} + 584315970 T^{4} + 50039102632 T^{5} + 4783608901408 T^{6} + 554051313045392 T^{7} + 38707506803414307 T^{8} + 2439696738179139952 T^{9} + \)\(28\!\cdots\!24\)\( T^{10} + \)\(19\!\cdots\!36\)\( T^{11} + \)\(97\!\cdots\!44\)\( T^{12} + \)\(13\!\cdots\!04\)\( T^{13} + \)\(13\!\cdots\!04\)\( T^{14} + \)\(79\!\cdots\!88\)\( T^{15} + \)\(87\!\cdots\!87\)\( T^{16} + \)\(85\!\cdots\!08\)\( T^{17} + \)\(51\!\cdots\!88\)\( T^{18} + \)\(36\!\cdots\!28\)\( T^{19} + \)\(29\!\cdots\!70\)\( T^{20} + \)\(14\!\cdots\!80\)\( T^{21} + \)\(60\!\cdots\!88\)\( T^{22} + \)\(37\!\cdots\!36\)\( T^{23} + \)\(11\!\cdots\!21\)\( T^{24} \)
$89$ \( 1 - 55228 T^{2} + 1551237378 T^{4} - 29097366482252 T^{6} + 406334885845730607 T^{8} - \)\(44\!\cdots\!64\)\( T^{10} + \)\(39\!\cdots\!36\)\( T^{12} - \)\(27\!\cdots\!24\)\( T^{14} + \)\(15\!\cdots\!67\)\( T^{16} - \)\(71\!\cdots\!92\)\( T^{18} + \)\(24\!\cdots\!58\)\( T^{20} - \)\(53\!\cdots\!28\)\( T^{22} + \)\(61\!\cdots\!41\)\( T^{24} \)
$97$ \( 1 - 564 T + 159048 T^{2} - 28992320 T^{3} + 3611144475 T^{4} - 287383360192 T^{5} + 8016218179688 T^{6} + 1344437255154268 T^{7} - 202827254806258013 T^{8} + 6263140923061522168 T^{9} + \)\(20\!\cdots\!44\)\( T^{10} - \)\(45\!\cdots\!96\)\( T^{11} + \)\(54\!\cdots\!74\)\( T^{12} - \)\(43\!\cdots\!64\)\( T^{13} + \)\(18\!\cdots\!64\)\( T^{14} + \)\(52\!\cdots\!72\)\( T^{15} - \)\(15\!\cdots\!93\)\( T^{16} + \)\(99\!\cdots\!32\)\( T^{17} + \)\(55\!\cdots\!08\)\( T^{18} - \)\(18\!\cdots\!48\)\( T^{19} + \)\(22\!\cdots\!75\)\( T^{20} - \)\(16\!\cdots\!80\)\( T^{21} + \)\(86\!\cdots\!48\)\( T^{22} - \)\(28\!\cdots\!76\)\( T^{23} + \)\(48\!\cdots\!81\)\( T^{24} \)
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