# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$35 = 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 35.h (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.953680925261$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 2 x^{11} + 19 x^{10} - 26 x^{9} + 244 x^{8} - 338 x^{7} + 1249 x^{6} - 986 x^{5} + 3532 x^{4} - 3618 x^{3} + 3579 x^{2} - 1386 x + 441$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} + \beta_{2} ) q^{2} + ( -1 + \beta_{7} ) q^{3} + ( -2 + 2 \beta_{3} - \beta_{6} + \beta_{8} + 2 \beta_{10} ) q^{4} -\beta_{10} q^{5} + ( 1 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{7} - \beta_{9} + \beta_{11} ) q^{6} + ( 1 - 2 \beta_{3} - \beta_{5} + \beta_{6} - \beta_{9} - \beta_{11} ) q^{7} + ( -\beta_{2} - \beta_{4} - \beta_{8} + \beta_{9} ) q^{8} + ( 1 - \beta_{1} - 2 \beta_{2} + 2 \beta_{4} + 4 \beta_{5} - 2 \beta_{7} + \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{1} + \beta_{2} ) q^{2} + ( -1 + \beta_{7} ) q^{3} + ( -2 + 2 \beta_{3} - \beta_{6} + \beta_{8} + 2 \beta_{10} ) q^{4} -\beta_{10} q^{5} + ( 1 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{7} - \beta_{9} + \beta_{11} ) q^{6} + ( 1 - 2 \beta_{3} - \beta_{5} + \beta_{6} - \beta_{9} - \beta_{11} ) q^{7} + ( -\beta_{2} - \beta_{4} - \beta_{8} + \beta_{9} ) q^{8} + ( 1 - \beta_{1} - 2 \beta_{2} + 2 \beta_{4} + 4 \beta_{5} - 2 \beta_{7} + \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{9} + ( \beta_{1} + \beta_{2} + \beta_{4} ) q^{10} + ( -2 + 3 \beta_{1} + \beta_{2} + 3 \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{11} + ( 3 + \beta_{2} - 2 \beta_{3} - \beta_{5} + \beta_{6} - 2 \beta_{8} + \beta_{9} - 6 \beta_{10} + \beta_{11} ) q^{12} + ( -2 \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{11} ) q^{13} + ( -1 - 3 \beta_{1} + \beta_{2} + 4 \beta_{3} + \beta_{4} - 3 \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{10} ) q^{14} + ( -1 + \beta_{5} - \beta_{7} + \beta_{8} + \beta_{10} - \beta_{11} ) q^{15} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + 4 \beta_{5} + 4 \beta_{7} - 2 \beta_{10} - 2 \beta_{11} ) q^{16} + ( 4 - 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} ) q^{17} + ( 11 - 3 \beta_{1} - \beta_{2} - 12 \beta_{3} - \beta_{4} - 3 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} + 4 \beta_{10} + 2 \beta_{11} ) q^{18} + ( -7 + 3 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} + 4 \beta_{8} - \beta_{9} - \beta_{11} ) q^{19} + ( -3 + 6 \beta_{3} - 2 \beta_{6} - 2 \beta_{7} + \beta_{8} + 2 \beta_{10} + 2 \beta_{11} ) q^{20} + ( -8 - \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - 5 \beta_{5} - \beta_{8} - \beta_{9} + \beta_{10} + 5 \beta_{11} ) q^{21} + ( -12 - \beta_{2} + 6 \beta_{5} + 3 \beta_{7} + 4 \beta_{8} + 6 \beta_{10} + 3 \beta_{11} ) q^{22} + ( 5 \beta_{1} + 6 \beta_{2} - 2 \beta_{4} + 7 \beta_{5} + \beta_{6} - \beta_{9} - 4 \beta_{10} ) q^{23} + ( 1 + 5 \beta_{1} + 9 \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} - 3 \beta_{7} + \beta_{8} ) q^{24} + ( 5 - 5 \beta_{3} ) q^{25} + ( 9 - 3 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - \beta_{9} - 6 \beta_{10} - \beta_{11} ) q^{26} + ( -4 + 8 \beta_{3} + \beta_{4} - 6 \beta_{5} - 6 \beta_{6} + 7 \beta_{7} + 3 \beta_{8} + \beta_{9} + 12 \beta_{10} - 7 \beta_{11} ) q^{27} + ( 12 + 11 \beta_{1} + \beta_{2} + 8 \beta_{3} + \beta_{4} - 9 \beta_{5} - \beta_{6} + 3 \beta_{7} - \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - 3 \beta_{11} ) q^{28} + ( 7 + 5 \beta_{2} + \beta_{4} - 4 \beta_{5} + \beta_{7} - 4 \beta_{8} - \beta_{9} - 4 \beta_{10} + \beta_{11} ) q^{29} + ( -1 - 5 \beta_{1} - 5 \beta_{2} + 3 \beta_{3} + \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{11} ) q^{30} + ( 5 - 4 \beta_{1} - 6 \beta_{2} + 7 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{31} + ( -8 - \beta_{1} + 6 \beta_{3} - 4 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + 4 \beta_{10} + 4 \beta_{11} ) q^{32} + ( -16 - 6 \beta_{1} + 8 \beta_{2} + 8 \beta_{3} + 2 \beta_{9} - 6 \beta_{10} ) q^{33} + ( 7 + 12 \beta_{1} + 6 \beta_{2} - 14 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + 6 \beta_{6} + 4 \beta_{7} - 3 \beta_{8} + 2 \beta_{9} - 9 \beta_{10} - 4 \beta_{11} ) q^{34} + ( 2 - 3 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{35} + ( 20 - \beta_{2} + \beta_{4} + 2 \beta_{5} - 7 \beta_{7} - 2 \beta_{8} - \beta_{9} + 2 \beta_{10} - 7 \beta_{11} ) q^{36} + ( 5 - 3 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + 10 \beta_{5} - 2 \beta_{6} - 10 \beta_{7} - 4 \beta_{10} + 5 \beta_{11} ) q^{37} + ( -19 - 7 \beta_{1} - 15 \beta_{2} - 20 \beta_{3} + \beta_{4} + 5 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} ) q^{38} + ( -5 + 4 \beta_{1} + 3 \beta_{3} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} + 2 \beta_{10} + 4 \beta_{11} ) q^{39} + ( -2 + 5 \beta_{1} - 5 \beta_{2} + 2 \beta_{3} - \beta_{5} + \beta_{6} - 2 \beta_{8} - 2 \beta_{11} ) q^{40} + ( 12 + 8 \beta_{1} + 4 \beta_{2} - 24 \beta_{3} + 6 \beta_{4} - 5 \beta_{5} + 2 \beta_{6} - 12 \beta_{7} - \beta_{8} + 6 \beta_{9} + 3 \beta_{10} + 12 \beta_{11} ) q^{41} + ( -1 - 2 \beta_{2} - 14 \beta_{3} - \beta_{4} + 4 \beta_{5} + 4 \beta_{6} - 3 \beta_{7} - \beta_{8} + 5 \beta_{9} - 10 \beta_{10} - 4 \beta_{11} ) q^{42} + ( -4 - 4 \beta_{2} + 4 \beta_{4} - 2 \beta_{5} + \beta_{7} - 4 \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{43} + ( -1 - 29 \beta_{1} - 24 \beta_{2} - 3 \beta_{3} - 10 \beta_{4} - 7 \beta_{5} + \beta_{6} + 2 \beta_{7} - 5 \beta_{9} + 3 \beta_{10} - \beta_{11} ) q^{44} + ( -10 + \beta_{1} + 6 \beta_{2} - 7 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} - \beta_{6} + 3 \beta_{7} - \beta_{8} ) q^{45} + ( -37 - 14 \beta_{1} - 4 \beta_{2} + 39 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} - 6 \beta_{6} + 2 \beta_{7} + 6 \beta_{8} - 8 \beta_{9} + 4 \beta_{10} - 4 \beta_{11} ) q^{46} + ( 21 - 3 \beta_{1} - 4 \beta_{2} - 10 \beta_{3} - \beta_{5} + \beta_{6} - 2 \beta_{8} - 7 \beta_{9} - \beta_{11} ) q^{47} + ( -22 - 8 \beta_{1} - 4 \beta_{2} + 44 \beta_{3} + 4 \beta_{4} + 10 \beta_{5} - 4 \beta_{6} - 3 \beta_{7} + 2 \beta_{8} + 4 \beta_{9} - 6 \beta_{10} + 3 \beta_{11} ) q^{48} + ( 6 - 15 \beta_{1} - \beta_{2} - 3 \beta_{3} - 5 \beta_{4} - 3 \beta_{5} + 3 \beta_{7} - 3 \beta_{8} + 9 \beta_{10} - 8 \beta_{11} ) q^{49} + 5 \beta_{2} q^{50} + ( -4 + 10 \beta_{1} + 10 \beta_{2} - 15 \beta_{3} - 3 \beta_{5} + \beta_{6} + 8 \beta_{7} + \beta_{10} - 4 \beta_{11} ) q^{51} + ( 13 + 7 \beta_{1} + 11 \beta_{2} + 16 \beta_{3} + 3 \beta_{4} - 13 \beta_{5} + \beta_{6} + 3 \beta_{7} + \beta_{8} ) q^{52} + ( 33 + \beta_{1} + 3 \beta_{2} - 30 \beta_{3} + 3 \beta_{4} + 7 \beta_{5} - 7 \beta_{6} + 3 \beta_{7} + 7 \beta_{8} + 6 \beta_{9} - 6 \beta_{11} ) q^{53} + ( 11 + 15 \beta_{1} - 10 \beta_{2} - 10 \beta_{3} - 3 \beta_{5} + 3 \beta_{6} - 6 \beta_{8} + 5 \beta_{9} + 3 \beta_{10} + 9 \beta_{11} ) q^{54} + ( -4 + 10 \beta_{1} + 5 \beta_{2} + 8 \beta_{3} - 4 \beta_{5} + 4 \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{11} ) q^{55} + ( -46 + 19 \beta_{1} + 6 \beta_{2} + 5 \beta_{3} + 8 \beta_{4} + 12 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + 6 \beta_{8} + 3 \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{56} + ( -4 + 6 \beta_{2} + \beta_{4} - 8 \beta_{5} + 2 \beta_{7} - \beta_{8} - \beta_{9} - 8 \beta_{10} + 2 \beta_{11} ) q^{57} + ( 3 + 26 \beta_{1} + 23 \beta_{2} + 30 \beta_{3} + 6 \beta_{4} - 17 \beta_{5} - 3 \beta_{6} - 6 \beta_{7} + 3 \beta_{9} + 10 \beta_{10} + 3 \beta_{11} ) q^{58} + ( 1 + 6 \beta_{1} + 16 \beta_{2} + 11 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} - \beta_{6} + 10 \beta_{7} - \beta_{8} ) q^{59} + ( 24 + 2 \beta_{1} - \beta_{2} - 21 \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} + 3 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - 6 \beta_{11} ) q^{60} + ( 13 - 3 \beta_{1} + 8 \beta_{2} - 6 \beta_{3} + \beta_{5} - \beta_{6} + 2 \beta_{8} + 5 \beta_{9} + 15 \beta_{10} - \beta_{11} ) q^{61} + ( 20 - 40 \beta_{3} - 5 \beta_{4} + 12 \beta_{5} + 6 \beta_{6} + 2 \beta_{7} - 3 \beta_{8} - 5 \beta_{9} - 18 \beta_{10} - 2 \beta_{11} ) q^{62} + ( 43 - \beta_{1} - 11 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} + 24 \beta_{5} - 2 \beta_{6} - 14 \beta_{7} + 7 \beta_{8} - 7 \beta_{9} + 4 \beta_{10} + 9 \beta_{11} ) q^{63} + ( -16 + 2 \beta_{2} - 4 \beta_{4} + 7 \beta_{5} + 6 \beta_{7} + \beta_{8} + 4 \beta_{9} + 7 \beta_{10} + 6 \beta_{11} ) q^{64} + ( -1 - 5 \beta_{1} - 5 \beta_{2} + 3 \beta_{3} - 2 \beta_{5} + 4 \beta_{6} + 2 \beta_{7} - \beta_{10} - \beta_{11} ) q^{65} + ( 38 - 2 \beta_{1} - 10 \beta_{2} + 42 \beta_{3} + 6 \beta_{4} - 6 \beta_{5} - 6 \beta_{6} + 4 \beta_{7} - 6 \beta_{8} ) q^{66} + ( -25 + 8 \beta_{1} + \beta_{2} + 26 \beta_{3} + \beta_{4} + 11 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} - 20 \beta_{10} - 2 \beta_{11} ) q^{67} + ( -36 - 18 \beta_{1} + 9 \beta_{2} + 18 \beta_{3} + 5 \beta_{5} - 5 \beta_{6} + 10 \beta_{8} - 9 \beta_{9} + 24 \beta_{10} ) q^{68} + ( -6 \beta_{1} - 3 \beta_{2} - 9 \beta_{4} - 9 \beta_{5} - 6 \beta_{6} + \beta_{7} + 3 \beta_{8} - 9 \beta_{9} + 15 \beta_{10} - \beta_{11} ) q^{69} + ( 23 - 8 \beta_{1} + \beta_{2} - \beta_{3} - 4 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} - \beta_{8} - 4 \beta_{9} + \beta_{10} + 3 \beta_{11} ) q^{70} + ( -7 - 10 \beta_{2} - 4 \beta_{4} - 9 \beta_{5} + 2 \beta_{7} + 7 \beta_{8} + 4 \beta_{9} - 9 \beta_{10} + 2 \beta_{11} ) q^{71} + ( -1 + 13 \beta_{1} + 12 \beta_{2} - 30 \beta_{3} + 2 \beta_{4} - 29 \beta_{5} + 5 \beta_{6} + 2 \beta_{7} + \beta_{9} + 12 \beta_{10} - \beta_{11} ) q^{72} + ( -22 - 10 \beta_{1} - 19 \beta_{2} - 36 \beta_{3} - \beta_{4} - 15 \beta_{5} - 3 \beta_{6} - 14 \beta_{7} - 3 \beta_{8} ) q^{73} + ( 8 + 9 \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} - 2 \beta_{5} + 5 \beta_{6} + 5 \beta_{7} - 5 \beta_{8} + 2 \beta_{9} - 6 \beta_{10} - 10 \beta_{11} ) q^{74} + ( -5 + 5 \beta_{11} ) q^{75} + ( 28 - 30 \beta_{1} - 15 \beta_{2} - 56 \beta_{3} - 9 \beta_{4} + 22 \beta_{5} - 4 \beta_{6} + 3 \beta_{7} + 2 \beta_{8} - 9 \beta_{9} - 18 \beta_{10} - 3 \beta_{11} ) q^{76} + ( -35 + 13 \beta_{1} + 10 \beta_{2} + 40 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} - 10 \beta_{6} + 7 \beta_{7} - 6 \beta_{9} + 18 \beta_{10} - 4 \beta_{11} ) q^{77} + ( -26 - 4 \beta_{2} - 3 \beta_{4} + 4 \beta_{5} - 2 \beta_{7} + 5 \beta_{8} + 3 \beta_{9} + 4 \beta_{10} - 2 \beta_{11} ) q^{78} + ( -8 \beta_{1} - 14 \beta_{2} - 11 \beta_{3} + 12 \beta_{4} - 3 \beta_{5} + \beta_{6} + 6 \beta_{9} + \beta_{10} ) q^{79} + ( -10 + 2 \beta_{1} + 2 \beta_{2} - 16 \beta_{3} + 2 \beta_{4} + 5 \beta_{5} + 2 \beta_{6} - 6 \beta_{7} + 2 \beta_{8} ) q^{80} + ( -46 + 2 \beta_{2} + 42 \beta_{3} + 2 \beta_{4} + 6 \beta_{5} + 11 \beta_{6} - 4 \beta_{7} - 11 \beta_{8} + 4 \beta_{9} - 34 \beta_{10} + 8 \beta_{11} ) q^{81} + ( -36 + \beta_{1} + 14 \beta_{2} + 18 \beta_{3} + 5 \beta_{5} - 5 \beta_{6} + 10 \beta_{8} + 15 \beta_{9} + 30 \beta_{10} ) q^{82} + ( -44 - 4 \beta_{1} - 2 \beta_{2} + 88 \beta_{3} - \beta_{4} - 14 \beta_{5} + 2 \beta_{6} + 7 \beta_{7} - \beta_{8} - \beta_{9} + 12 \beta_{10} - 7 \beta_{11} ) q^{83} + ( -25 - 16 \beta_{1} + 7 \beta_{2} - 5 \beta_{3} + 5 \beta_{4} + 21 \beta_{5} + 9 \beta_{6} - \beta_{7} - \beta_{9} - 27 \beta_{10} + 15 \beta_{11} ) q^{84} + ( -6 \beta_{2} - \beta_{4} - 4 \beta_{5} - 5 \beta_{8} + \beta_{9} - 4 \beta_{10} ) q^{85} + ( 9 + \beta_{1} - 15 \beta_{3} + 2 \beta_{4} - 28 \beta_{5} - 4 \beta_{6} - 18 \beta_{7} + \beta_{9} + 16 \beta_{10} + 9 \beta_{11} ) q^{86} + ( 7 - 10 \beta_{1} - 23 \beta_{2} + 3 \beta_{4} + 27 \beta_{5} + 3 \beta_{6} - 7 \beta_{7} + 3 \beta_{8} ) q^{87} + ( 105 + 5 \beta_{1} + 2 \beta_{2} - 108 \beta_{3} + 2 \beta_{4} + 8 \beta_{5} + 12 \beta_{6} - 3 \beta_{7} - 12 \beta_{8} + 4 \beta_{9} - 40 \beta_{10} + 6 \beta_{11} ) q^{88} + ( 21 + 11 \beta_{1} - 16 \beta_{2} - 2 \beta_{3} - 5 \beta_{5} + 5 \beta_{6} - 10 \beta_{8} - 5 \beta_{9} + 3 \beta_{10} - 17 \beta_{11} ) q^{89} + ( -15 - 10 \beta_{1} - 5 \beta_{2} + 30 \beta_{3} + 9 \beta_{5} + 5 \beta_{7} - 9 \beta_{10} - 5 \beta_{11} ) q^{90} + ( -28 - 10 \beta_{1} - 7 \beta_{2} - 13 \beta_{3} - 3 \beta_{4} + \beta_{5} + 5 \beta_{6} + \beta_{7} - 10 \beta_{8} + \beta_{9} + 3 \beta_{10} - 7 \beta_{11} ) q^{91} + ( 66 - 23 \beta_{2} - 4 \beta_{4} - 10 \beta_{5} - 6 \beta_{7} - 16 \beta_{8} + 4 \beta_{9} - 10 \beta_{10} - 6 \beta_{11} ) q^{92} + ( -8 + 14 \beta_{1} + 11 \beta_{2} + 24 \beta_{3} + 6 \beta_{4} - 3 \beta_{5} - \beta_{6} + 16 \beta_{7} + 3 \beta_{9} + 2 \beta_{10} - 8 \beta_{11} ) q^{93} + ( 11 + 13 \beta_{1} + 25 \beta_{2} - 2 \beta_{3} + \beta_{4} - 28 \beta_{5} - 2 \beta_{6} - 13 \beta_{7} - 2 \beta_{8} ) q^{94} + ( 10 + 7 \beta_{1} + 4 \beta_{2} - 15 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} - 5 \beta_{6} - 5 \beta_{7} + 5 \beta_{8} + 8 \beta_{9} + 6 \beta_{10} + 10 \beta_{11} ) q^{95} + ( 57 + 3 \beta_{1} - 10 \beta_{2} - 25 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} - 4 \beta_{8} - 7 \beta_{9} + 12 \beta_{10} - 7 \beta_{11} ) q^{96} + ( 28 - 28 \beta_{1} - 14 \beta_{2} - 56 \beta_{3} + 4 \beta_{4} + 6 \beta_{5} + 12 \beta_{6} - 16 \beta_{7} - 6 \beta_{8} + 4 \beta_{9} - 18 \beta_{10} + 16 \beta_{11} ) q^{97} + ( 78 + 3 \beta_{1} + 6 \beta_{2} - 4 \beta_{3} - \beta_{4} + 3 \beta_{5} + 9 \beta_{6} + 15 \beta_{7} - 15 \beta_{8} - 36 \beta_{10} - 7 \beta_{11} ) q^{98} + ( 9 + 11 \beta_{2} + \beta_{4} - 3 \beta_{5} - 15 \beta_{7} + 17 \beta_{8} - \beta_{9} - 3 \beta_{10} - 15 \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 2q^{2} - 6q^{3} - 10q^{4} - 2q^{7} - 4q^{8} + 14q^{9} + O(q^{10})$$ $$12q - 2q^{2} - 6q^{3} - 10q^{4} - 2q^{7} - 4q^{8} + 14q^{9} - 14q^{11} + 18q^{12} - 2q^{14} - 20q^{15} - 22q^{16} + 48q^{17} + 64q^{18} - 30q^{19} - 84q^{21} - 88q^{22} - 14q^{23} - 36q^{24} + 30q^{25} + 66q^{26} + 202q^{28} + 64q^{29} + 20q^{30} + 132q^{31} - 54q^{32} - 192q^{33} + 30q^{35} + 156q^{36} + 44q^{37} - 300q^{38} - 24q^{39} - 138q^{42} - 4q^{43} + 6q^{44} - 180q^{45} - 214q^{46} + 204q^{47} - 24q^{49} - 20q^{50} - 132q^{51} + 252q^{52} + 196q^{53} + 168q^{54} - 460q^{56} - 48q^{57} + 158q^{58} + 72q^{59} + 150q^{60} + 72q^{61} + 536q^{63} - 140q^{64} + 30q^{65} + 744q^{66} - 138q^{67} - 348q^{68} + 240q^{70} - 8q^{71} - 196q^{72} - 528q^{73} + 50q^{74} - 30q^{75} - 176q^{77} - 312q^{78} - 12q^{79} - 240q^{80} - 310q^{81} - 378q^{82} - 276q^{84} - 40q^{86} + 138q^{87} + 604q^{88} + 204q^{89} - 480q^{91} + 732q^{92} + 84q^{93} - 42q^{94} + 60q^{95} + 540q^{96} + 898q^{98} - 44q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 2 x^{11} + 19 x^{10} - 26 x^{9} + 244 x^{8} - 338 x^{7} + 1249 x^{6} - 986 x^{5} + 3532 x^{4} - 3618 x^{3} + 3579 x^{2} - 1386 x + 441$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$115287976 \nu^{11} + 155566808 \nu^{10} + 1678212789 \nu^{9} + 3282242448 \nu^{8} + 24792751384 \nu^{7} + 39686977172 \nu^{6} + 96119820008 \nu^{5} + 175815191664 \nu^{4} + 682627769512 \nu^{3} + 320782558968 \nu^{2} - 130699075752 \nu - 423574271316$$$$)/ 1182297257421$$ $$\beta_{3}$$ $$=$$ $$($$$$-960485876 \nu^{11} + 1805683776 \nu^{10} - 18404798452 \nu^{9} + 23294419987 \nu^{8} - 237640796192 \nu^{7} + 299851474704 \nu^{6} - 1239333836296 \nu^{5} + 850919253728 \nu^{4} - 3568251305696 \nu^{3} + 2792410129856 \nu^{2} - 3758361509172 \nu + 1461932499888$$$$)/ 1182297257421$$ $$\beta_{4}$$ $$=$$ $$($$$$-6740525471 \nu^{11} + 28434054878 \nu^{10} - 139462837212 \nu^{9} + 440457052261 \nu^{8} - 1761800596603 \nu^{7} + 5766095751342 \nu^{6} - 10108276564454 \nu^{5} + 22821466101119 \nu^{4} - 28741675806183 \nu^{3} + 73697517310302 \nu^{2} - 49990933980288 \nu + 26654089921641$$$$)/ 7093783544526$$ $$\beta_{5}$$ $$=$$ $$($$$$-12593073845 \nu^{11} + 52875316152 \nu^{10} - 270584479006 \nu^{9} + 806911234849 \nu^{8} - 3348774190467 \nu^{7} + 10391062513814 \nu^{6} - 19436416368422 \nu^{5} + 39012412161279 \nu^{4} - 44802142308001 \nu^{3} + 119747155506090 \nu^{2} - 73606163842938 \nu + 41846762790189$$$$)/ 7093783544526$$ $$\beta_{6}$$ $$=$$ $$($$$$20099706409 \nu^{11} - 38300982339 \nu^{10} + 346926560957 \nu^{9} - 453614466560 \nu^{8} + 4326017128719 \nu^{7} - 6073195727767 \nu^{6} + 17671584512275 \nu^{5} - 13747066395156 \nu^{4} + 40786458885581 \nu^{3} - 65555644756557 \nu^{2} - 15233535549147 \nu + 20528318534472$$$$)/ 7093783544526$$ $$\beta_{7}$$ $$=$$ $$($$$$22979355133 \nu^{11} - 77959111894 \nu^{10} + 482194495722 \nu^{9} - 1167796047734 \nu^{8} + 6080191620431 \nu^{7} - 15135872139180 \nu^{6} + 34907225764618 \nu^{5} - 56837597110324 \nu^{4} + 88150693819461 \nu^{3} - 183879227117652 \nu^{2} + 130924993629384 \nu - 71882680944312$$$$)/ 7093783544526$$ $$\beta_{8}$$ $$=$$ $$($$$$-9058064565 \nu^{11} - 1839528853 \nu^{10} - 143550656091 \nu^{9} - 113877990722 \nu^{8} - 1903870510779 \nu^{7} - 1400508460871 \nu^{6} - 7225945022029 \nu^{5} - 10664026987838 \nu^{4} - 25070618179273 \nu^{3} - 20828074314297 \nu^{2} + 8520712695117 \nu - 3111541612074$$$$)/ 2364594514842$$ $$\beta_{9}$$ $$=$$ $$($$$$-27689168462 \nu^{11} + 31316075951 \nu^{10} - 479491314879 \nu^{9} + 276064172287 \nu^{8} - 6134603554204 \nu^{7} + 3707667136017 \nu^{6} - 26595641350109 \nu^{5} + 323405487461 \nu^{4} - 74185414334880 \nu^{3} + 28535552551683 \nu^{2} - 31486545985545 \nu - 5553545565645$$$$)/ 7093783544526$$ $$\beta_{10}$$ $$=$$ $$($$$$42084124100 \nu^{11} - 50430282123 \nu^{10} + 738914826223 \nu^{9} - 485302351243 \nu^{8} + 9532311601164 \nu^{7} - 6476786303297 \nu^{6} + 42851186250509 \nu^{5} - 5367747368085 \nu^{4} + 124441363582636 \nu^{3} - 53429037469809 \nu^{2} + 46461314938107 \nu + 9745511328693$$$$)/ 7093783544526$$ $$\beta_{11}$$ $$=$$ $$($$$$-65452485346 \nu^{11} + 87736926802 \nu^{10} - 1160252919867 \nu^{9} + 889059660188 \nu^{8} - 14929298435582 \nu^{7} + 11712787187742 \nu^{6} - 68211597317071 \nu^{5} + 12416532499534 \nu^{4} - 196334842415514 \nu^{3} + 93971894939196 \nu^{2} - 94070009770551 \nu - 14009629724496$$$$)/ 7093783544526$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{10} + \beta_{6} + \beta_{5} - 6 \beta_{3}$$ $$\nu^{3}$$ $$=$$ $$-\beta_{9} + \beta_{8} + \beta_{4} + 9 \beta_{2}$$ $$\nu^{4}$$ $$=$$ $$4 \beta_{11} + 28 \beta_{10} + 12 \beta_{8} - 2 \beta_{7} - 12 \beta_{6} - 2 \beta_{5} + 57 \beta_{3} - 2 \beta_{1} - 59$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{11} + 2 \beta_{10} - 16 \beta_{9} - 4 \beta_{7} - 14 \beta_{6} + 10 \beta_{5} - 32 \beta_{4} + 6 \beta_{3} - 113 \beta_{2} - 129 \beta_{1} + 2$$ $$\nu^{6}$$ $$=$$ $$-34 \beta_{11} - 177 \beta_{10} + 4 \beta_{9} - 143 \beta_{8} - 34 \beta_{7} - 177 \beta_{5} - 4 \beta_{4} - 38 \beta_{2} + 692$$ $$\nu^{7}$$ $$=$$ $$-84 \beta_{11} - 116 \beta_{10} + 422 \beta_{9} - 189 \beta_{8} + 42 \beta_{7} + 189 \beta_{6} - 131 \beta_{5} + 211 \beta_{4} - 138 \beta_{3} + 211 \beta_{2} + 1543 \beta_{1} + 180$$ $$\nu^{8}$$ $$=$$ $$-464 \beta_{11} - 2176 \beta_{10} + 100 \beta_{9} + 928 \beta_{7} + 1732 \beta_{6} + 2620 \beta_{5} + 200 \beta_{4} - 7233 \beta_{3} + 716 \beta_{2} + 816 \beta_{1} - 464$$ $$\nu^{9}$$ $$=$$ $$664 \beta_{11} + 1116 \beta_{10} - 2640 \beta_{9} + 2548 \beta_{8} + 664 \beta_{7} + 1116 \beta_{5} + 2640 \beta_{4} + 13337 \beta_{2} - 3932$$ $$\nu^{10}$$ $$=$$ $$11888 \beta_{11} + 53074 \beta_{10} - 3560 \beta_{9} + 21165 \beta_{8} - 5944 \beta_{7} - 21165 \beta_{6} - 5372 \beta_{5} - 1780 \beta_{4} + 85950 \beta_{3} - 1780 \beta_{2} - 13040 \beta_{1} - 91894$$ $$\nu^{11}$$ $$=$$ $$9504 \beta_{11} + 18380 \beta_{10} - 32481 \beta_{9} - 19008 \beta_{7} - 34205 \beta_{6} - 2555 \beta_{5} - 64962 \beta_{4} + 44388 \beta_{3} - 193242 \beta_{2} - 225723 \beta_{1} + 9504$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
26.1
 1.77870 + 3.08079i 1.18241 + 2.04800i 0.410701 + 0.711354i 0.242987 + 0.420865i −0.925400 − 1.60284i −1.68940 − 2.92612i 1.77870 − 3.08079i 1.18241 − 2.04800i 0.410701 − 0.711354i 0.242987 − 0.420865i −0.925400 + 1.60284i −1.68940 + 2.92612i
−1.77870 + 3.08079i −2.10717 + 1.21658i −4.32752 7.49548i 1.93649 + 1.11803i 8.65567i −5.39402 + 4.46146i 16.5598 −1.53989 + 2.66717i −6.88886 + 3.97728i
26.2 −1.18241 + 2.04800i 4.45439 2.57174i −0.796202 1.37906i −1.93649 1.11803i 12.1634i −1.42520 + 6.85338i −5.69355 8.72772 15.1168i 4.57947 2.64396i
26.3 −0.410701 + 0.711354i 0.507487 0.292998i 1.66265 + 2.87979i 1.93649 + 1.11803i 0.481337i 1.91172 6.73389i −6.01701 −4.32830 + 7.49684i −1.59064 + 0.918354i
26.4 −0.242987 + 0.420865i −4.74894 + 2.74180i 1.88192 + 3.25957i −1.93649 1.11803i 2.66488i 5.87843 + 3.80053i −3.77301 10.5350 18.2471i 0.941083 0.543334i
26.5 0.925400 1.60284i 0.731043 0.422068i 0.287270 + 0.497567i −1.93649 1.11803i 1.56233i −6.88972 1.23763i 8.46656 −4.14372 + 7.17713i −3.58406 + 2.06926i
26.6 1.68940 2.92612i −1.83681 + 1.06048i −3.70812 6.42265i 1.93649 + 1.11803i 7.16630i 4.91879 + 4.98051i −11.5428 −2.25076 + 3.89842i 6.54300 3.77760i
31.1 −1.77870 3.08079i −2.10717 1.21658i −4.32752 + 7.49548i 1.93649 1.11803i 8.65567i −5.39402 4.46146i 16.5598 −1.53989 2.66717i −6.88886 3.97728i
31.2 −1.18241 2.04800i 4.45439 + 2.57174i −0.796202 + 1.37906i −1.93649 + 1.11803i 12.1634i −1.42520 6.85338i −5.69355 8.72772 + 15.1168i 4.57947 + 2.64396i
31.3 −0.410701 0.711354i 0.507487 + 0.292998i 1.66265 2.87979i 1.93649 1.11803i 0.481337i 1.91172 + 6.73389i −6.01701 −4.32830 7.49684i −1.59064 0.918354i
31.4 −0.242987 0.420865i −4.74894 2.74180i 1.88192 3.25957i −1.93649 + 1.11803i 2.66488i 5.87843 3.80053i −3.77301 10.5350 + 18.2471i 0.941083 + 0.543334i
31.5 0.925400 + 1.60284i 0.731043 + 0.422068i 0.287270 0.497567i −1.93649 + 1.11803i 1.56233i −6.88972 + 1.23763i 8.46656 −4.14372 7.17713i −3.58406 2.06926i
31.6 1.68940 + 2.92612i −1.83681 1.06048i −3.70812 + 6.42265i 1.93649 1.11803i 7.16630i 4.91879 4.98051i −11.5428 −2.25076 3.89842i 6.54300 + 3.77760i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 31.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 35.3.h.a 12
3.b odd 2 1 315.3.w.c 12
4.b odd 2 1 560.3.bx.c 12
5.b even 2 1 175.3.i.d 12
5.c odd 4 2 175.3.j.b 24
7.b odd 2 1 245.3.h.c 12
7.c even 3 1 245.3.d.a 12
7.c even 3 1 245.3.h.c 12
7.d odd 6 1 inner 35.3.h.a 12
7.d odd 6 1 245.3.d.a 12
21.g even 6 1 315.3.w.c 12
28.f even 6 1 560.3.bx.c 12
35.i odd 6 1 175.3.i.d 12
35.k even 12 2 175.3.j.b 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.3.h.a 12 1.a even 1 1 trivial
35.3.h.a 12 7.d odd 6 1 inner
175.3.i.d 12 5.b even 2 1
175.3.i.d 12 35.i odd 6 1
175.3.j.b 24 5.c odd 4 2
175.3.j.b 24 35.k even 12 2
245.3.d.a 12 7.c even 3 1
245.3.d.a 12 7.d odd 6 1
245.3.h.c 12 7.b odd 2 1
245.3.h.c 12 7.c even 3 1
315.3.w.c 12 3.b odd 2 1
315.3.w.c 12 21.g even 6 1
560.3.bx.c 12 4.b odd 2 1
560.3.bx.c 12 28.f even 6 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T - 5 T^{2} - 6 T^{3} + 20 T^{4} + 10 T^{5} + 25 T^{6} + 58 T^{7} - 436 T^{8} - 470 T^{9} + 1107 T^{10} - 102 T^{11} - 5463 T^{12} - 408 T^{13} + 17712 T^{14} - 30080 T^{15} - 111616 T^{16} + 59392 T^{17} + 102400 T^{18} + 163840 T^{19} + 1310720 T^{20} - 1572864 T^{21} - 5242880 T^{22} + 8388608 T^{23} + 16777216 T^{24}$$
$3$ $$1 + 6 T + 38 T^{2} + 156 T^{3} + 660 T^{4} + 2070 T^{5} + 5744 T^{6} + 10314 T^{7} - 68 T^{8} - 114228 T^{9} - 708426 T^{10} - 2725974 T^{11} - 9235782 T^{12} - 24533766 T^{13} - 57382506 T^{14} - 83272212 T^{15} - 446148 T^{16} + 609031386 T^{17} + 3052597104 T^{18} + 9900745830 T^{19} + 28410835860 T^{20} + 60437596284 T^{21} + 132497807238 T^{22} + 188286357654 T^{23} + 282429536481 T^{24}$$
$5$ $$( 1 - 5 T^{2} + 25 T^{4} )^{3}$$
$7$ $$1 + 2 T + 14 T^{2} + 348 T^{3} + 1588 T^{4} + 14434 T^{5} + 136318 T^{6} + 707266 T^{7} + 3812788 T^{8} + 40941852 T^{9} + 80707214 T^{10} + 564950498 T^{11} + 13841287201 T^{12}$$
$11$ $$1 + 14 T - 179 T^{2} - 3378 T^{3} - 10510 T^{4} - 101510 T^{5} - 1940909 T^{6} + 49596994 T^{7} + 1461472640 T^{8} + 7377470974 T^{9} - 78163118391 T^{10} - 1070296414578 T^{11} - 9257941759776 T^{12} - 129505866163938 T^{13} - 1144386216362631 T^{14} + 13069639856170414 T^{15} + 313279639722515840 T^{16} + 1286418292311249394 T^{17} - 6091403882233179389 T^{18} - 38548405607034793910 T^{19} -$$$$48\!\cdots\!10$$$$T^{20} -$$$$18\!\cdots\!18$$$$T^{21} -$$$$12\!\cdots\!79$$$$T^{22} +$$$$11\!\cdots\!94$$$$T^{23} +$$$$98\!\cdots\!41$$$$T^{24}$$
$13$ $$1 - 1482 T^{2} + 1042167 T^{4} - 464849386 T^{6} + 148108329555 T^{8} - 35857522447788 T^{10} + 6808838187253242 T^{12} - 1024126698631273068 T^{14} +$$$$12\!\cdots\!55$$$$T^{16} -$$$$10\!\cdots\!66$$$$T^{18} +$$$$69\!\cdots\!47$$$$T^{20} -$$$$28\!\cdots\!82$$$$T^{22} +$$$$54\!\cdots\!61$$$$T^{24}$$
$17$ $$1 - 48 T + 2382 T^{2} - 77472 T^{3} + 2498625 T^{4} - 67135632 T^{5} + 1727446034 T^{6} - 40136784912 T^{7} + 880931070942 T^{8} - 18174432672288 T^{9} + 351810098240190 T^{10} - 6513275972033712 T^{11} + 112870694757995013 T^{12} - 1882336755917742768 T^{13} + 29383531215118908990 T^{14} -$$$$43\!\cdots\!72$$$$T^{15} +$$$$61\!\cdots\!22$$$$T^{16} -$$$$80\!\cdots\!88$$$$T^{17} +$$$$10\!\cdots\!74$$$$T^{18} -$$$$11\!\cdots\!28$$$$T^{19} +$$$$12\!\cdots\!25$$$$T^{20} -$$$$10\!\cdots\!48$$$$T^{21} +$$$$96\!\cdots\!82$$$$T^{22} -$$$$56\!\cdots\!72$$$$T^{23} +$$$$33\!\cdots\!21$$$$T^{24}$$
$19$ $$1 + 30 T + 1401 T^{2} + 33030 T^{3} + 884826 T^{4} + 20001714 T^{5} + 344601839 T^{6} + 6453829386 T^{7} + 66537113172 T^{8} + 653188957326 T^{9} - 1006933743831 T^{10} - 384405046252650 T^{11} - 5242806750443784 T^{12} - 138770221697206650 T^{13} - 131224612429799751 T^{14} + 30729849956873074206 T^{15} +$$$$11\!\cdots\!52$$$$T^{16} +$$$$39\!\cdots\!86$$$$T^{17} +$$$$76\!\cdots\!79$$$$T^{18} +$$$$15\!\cdots\!94$$$$T^{19} +$$$$25\!\cdots\!06$$$$T^{20} +$$$$34\!\cdots\!30$$$$T^{21} +$$$$52\!\cdots\!01$$$$T^{22} +$$$$40\!\cdots\!30$$$$T^{23} +$$$$48\!\cdots\!21$$$$T^{24}$$
$23$ $$1 + 14 T - 1631 T^{2} - 3594 T^{3} + 1665797 T^{4} - 8196524 T^{5} - 890248094 T^{6} + 13524083944 T^{7} + 223091831501 T^{8} - 7643356730246 T^{9} + 95428737941649 T^{10} + 2006796517495686 T^{11} - 92105247923668542 T^{12} + 1061595357755217894 T^{13} + 26704873454328997809 T^{14} -$$$$11\!\cdots\!94$$$$T^{15} +$$$$17\!\cdots\!81$$$$T^{16} +$$$$56\!\cdots\!56$$$$T^{17} -$$$$19\!\cdots\!74$$$$T^{18} -$$$$95\!\cdots\!16$$$$T^{19} +$$$$10\!\cdots\!17$$$$T^{20} -$$$$11\!\cdots\!86$$$$T^{21} -$$$$27\!\cdots\!31$$$$T^{22} +$$$$12\!\cdots\!06$$$$T^{23} +$$$$48\!\cdots\!41$$$$T^{24}$$
$29$ $$( 1 - 32 T + 3552 T^{2} - 122256 T^{3} + 5916460 T^{4} - 196641728 T^{5} + 6117094942 T^{6} - 165375693248 T^{7} + 4184599745260 T^{8} - 72720719932176 T^{9} + 1776875258837472 T^{10} - 13462631465606432 T^{11} + 353814783205469041 T^{12} )^{2}$$
$31$ $$1 - 132 T + 12894 T^{2} - 935352 T^{3} + 58065489 T^{4} - 3050976288 T^{5} + 143999803298 T^{6} - 6064589395908 T^{7} + 236052211330302 T^{8} - 8471733336563436 T^{9} + 289433514790600350 T^{10} - 9384030507705444000 T^{11} +$$$$29\!\cdots\!29$$$$T^{12} -$$$$90\!\cdots\!00$$$$T^{13} +$$$$26\!\cdots\!50$$$$T^{14} -$$$$75\!\cdots\!16$$$$T^{15} +$$$$20\!\cdots\!82$$$$T^{16} -$$$$49\!\cdots\!08$$$$T^{17} +$$$$11\!\cdots\!78$$$$T^{18} -$$$$23\!\cdots\!48$$$$T^{19} +$$$$42\!\cdots\!09$$$$T^{20} -$$$$65\!\cdots\!32$$$$T^{21} +$$$$86\!\cdots\!94$$$$T^{22} -$$$$85\!\cdots\!52$$$$T^{23} +$$$$62\!\cdots\!21$$$$T^{24}$$
$37$ $$1 - 44 T - 4291 T^{2} + 241124 T^{3} + 9668742 T^{4} - 689019636 T^{5} - 12024976949 T^{6} + 1280625867436 T^{7} + 4009330536896 T^{8} - 1538422889903564 T^{9} + 15464652696139049 T^{10} + 849812472525660964 T^{11} - 34333638656751212332 T^{12} +$$$$11\!\cdots\!16$$$$T^{13} +$$$$28\!\cdots\!89$$$$T^{14} -$$$$39\!\cdots\!76$$$$T^{15} +$$$$14\!\cdots\!16$$$$T^{16} +$$$$61\!\cdots\!64$$$$T^{17} -$$$$79\!\cdots\!69$$$$T^{18} -$$$$62\!\cdots\!04$$$$T^{19} +$$$$11\!\cdots\!22$$$$T^{20} +$$$$40\!\cdots\!96$$$$T^{21} -$$$$99\!\cdots\!91$$$$T^{22} -$$$$13\!\cdots\!36$$$$T^{23} +$$$$43\!\cdots\!61$$$$T^{24}$$
$41$ $$1 - 6978 T^{2} + 29395173 T^{4} - 89601849670 T^{6} + 218675773246986 T^{8} - 448976563577597898 T^{10} +$$$$80\!\cdots\!77$$$$T^{12} -$$$$12\!\cdots\!78$$$$T^{14} +$$$$17\!\cdots\!06$$$$T^{16} -$$$$20\!\cdots\!70$$$$T^{18} +$$$$18\!\cdots\!93$$$$T^{20} -$$$$12\!\cdots\!78$$$$T^{22} +$$$$50\!\cdots\!61$$$$T^{24}$$
$43$ $$( 1 + 2 T + 7538 T^{2} + 1596 T^{3} + 27008008 T^{4} - 11306606 T^{5} + 60967237342 T^{6} - 20905914494 T^{7} + 92335004758408 T^{8} + 10088895426204 T^{9} + 88105653692556338 T^{10} + 43222964626568498 T^{11} + 39959630797262576401 T^{12} )^{2}$$
$47$ $$1 - 204 T + 29397 T^{2} - 3167100 T^{3} + 290418402 T^{4} - 23370921444 T^{5} + 1705985151887 T^{6} - 114387377070276 T^{7} + 7121727778488432 T^{8} - 413844362647971516 T^{9} + 22564123030969504557 T^{10} -$$$$11\!\cdots\!96$$$$T^{11} +$$$$56\!\cdots\!04$$$$T^{12} -$$$$25\!\cdots\!64$$$$T^{13} +$$$$11\!\cdots\!17$$$$T^{14} -$$$$44\!\cdots\!64$$$$T^{15} +$$$$16\!\cdots\!52$$$$T^{16} -$$$$60\!\cdots\!24$$$$T^{17} +$$$$19\!\cdots\!67$$$$T^{18} -$$$$59\!\cdots\!36$$$$T^{19} +$$$$16\!\cdots\!42$$$$T^{20} -$$$$39\!\cdots\!00$$$$T^{21} +$$$$81\!\cdots\!97$$$$T^{22} -$$$$12\!\cdots\!36$$$$T^{23} +$$$$13\!\cdots\!81$$$$T^{24}$$
$53$ $$1 - 196 T + 13213 T^{2} - 531828 T^{3} + 46624490 T^{4} - 3460227932 T^{5} + 114727083151 T^{6} - 3187972234364 T^{7} + 109323786787880 T^{8} + 2196815143901836 T^{9} - 301463170054474635 T^{10} + 37355670780781789692 T^{11} -$$$$32\!\cdots\!76$$$$T^{12} +$$$$10\!\cdots\!28$$$$T^{13} -$$$$23\!\cdots\!35$$$$T^{14} +$$$$48\!\cdots\!44$$$$T^{15} +$$$$68\!\cdots\!80$$$$T^{16} -$$$$55\!\cdots\!36$$$$T^{17} +$$$$56\!\cdots\!91$$$$T^{18} -$$$$47\!\cdots\!08$$$$T^{19} +$$$$18\!\cdots\!90$$$$T^{20} -$$$$57\!\cdots\!92$$$$T^{21} +$$$$40\!\cdots\!13$$$$T^{22} -$$$$16\!\cdots\!64$$$$T^{23} +$$$$24\!\cdots\!81$$$$T^{24}$$
$59$ $$1 - 72 T + 17682 T^{2} - 1148688 T^{3} + 164787345 T^{4} - 10566780840 T^{5} + 1093761678926 T^{6} - 70159293997992 T^{7} + 5748831154849182 T^{8} - 368559907967058624 T^{9} + 25324575120331389906 T^{10} -$$$$15\!\cdots\!28$$$$T^{11} +$$$$95\!\cdots\!93$$$$T^{12} -$$$$54\!\cdots\!68$$$$T^{13} +$$$$30\!\cdots\!66$$$$T^{14} -$$$$15\!\cdots\!84$$$$T^{15} +$$$$84\!\cdots\!22$$$$T^{16} -$$$$35\!\cdots\!92$$$$T^{17} +$$$$19\!\cdots\!06$$$$T^{18} -$$$$65\!\cdots\!40$$$$T^{19} +$$$$35\!\cdots\!45$$$$T^{20} -$$$$86\!\cdots\!48$$$$T^{21} +$$$$46\!\cdots\!82$$$$T^{22} -$$$$65\!\cdots\!32$$$$T^{23} +$$$$31\!\cdots\!61$$$$T^{24}$$
$61$ $$1 - 72 T + 19176 T^{2} - 1256256 T^{3} + 196053300 T^{4} - 13507864560 T^{5} + 1475851108916 T^{6} - 102662314431264 T^{7} + 8714109428620692 T^{8} - 590990079276027360 T^{9} + 42479838876985802160 T^{10} -$$$$27\!\cdots\!84$$$$T^{11} +$$$$17\!\cdots\!82$$$$T^{12} -$$$$10\!\cdots\!64$$$$T^{13} +$$$$58\!\cdots\!60$$$$T^{14} -$$$$30\!\cdots\!60$$$$T^{15} +$$$$16\!\cdots\!52$$$$T^{16} -$$$$73\!\cdots\!64$$$$T^{17} +$$$$39\!\cdots\!36$$$$T^{18} -$$$$13\!\cdots\!60$$$$T^{19} +$$$$72\!\cdots\!00$$$$T^{20} -$$$$17\!\cdots\!36$$$$T^{21} +$$$$97\!\cdots\!76$$$$T^{22} -$$$$13\!\cdots\!12$$$$T^{23} +$$$$70\!\cdots\!41$$$$T^{24}$$
$67$ $$1 + 138 T - 9270 T^{2} - 1437548 T^{3} + 115886436 T^{4} + 10557292578 T^{5} - 1139954744456 T^{6} - 58817128757610 T^{7} + 8471104062197196 T^{8} + 229786064617296868 T^{9} - 50182738990417249110 T^{10} -$$$$37\!\cdots\!50$$$$T^{11} +$$$$25\!\cdots\!50$$$$T^{12} -$$$$16\!\cdots\!50$$$$T^{13} -$$$$10\!\cdots\!10$$$$T^{14} +$$$$20\!\cdots\!92$$$$T^{15} +$$$$34\!\cdots\!36$$$$T^{16} -$$$$10\!\cdots\!90$$$$T^{17} -$$$$93\!\cdots\!16$$$$T^{18} +$$$$38\!\cdots\!62$$$$T^{19} +$$$$19\!\cdots\!16$$$$T^{20} -$$$$10\!\cdots\!32$$$$T^{21} -$$$$30\!\cdots\!70$$$$T^{22} +$$$$20\!\cdots\!82$$$$T^{23} +$$$$66\!\cdots\!21$$$$T^{24}$$
$71$ $$( 1 + 4 T + 14826 T^{2} - 442692 T^{3} + 122154931 T^{4} - 3956292368 T^{5} + 781022121076 T^{6} - 19943669827088 T^{7} + 3104162139149011 T^{8} - 56708970889555332 T^{9} + 9573941854249652586 T^{10} + 13020974204039524804 T^{11} +$$$$16\!\cdots\!41$$$$T^{12} )^{2}$$
$73$ $$1 + 528 T + 147318 T^{2} + 28717920 T^{3} + 4362863697 T^{4} + 544924287120 T^{5} + 57462036213962 T^{6} + 5186233581298704 T^{7} + 403863565075955790 T^{8} + 27374688378441994752 T^{9} +$$$$16\!\cdots\!86$$$$T^{10} +$$$$95\!\cdots\!24$$$$T^{11} +$$$$62\!\cdots\!73$$$$T^{12} +$$$$51\!\cdots\!96$$$$T^{13} +$$$$46\!\cdots\!26$$$$T^{14} +$$$$41\!\cdots\!28$$$$T^{15} +$$$$32\!\cdots\!90$$$$T^{16} +$$$$22\!\cdots\!96$$$$T^{17} +$$$$13\!\cdots\!02$$$$T^{18} +$$$$66\!\cdots\!80$$$$T^{19} +$$$$28\!\cdots\!17$$$$T^{20} +$$$$99\!\cdots\!80$$$$T^{21} +$$$$27\!\cdots\!18$$$$T^{22} +$$$$51\!\cdots\!12$$$$T^{23} +$$$$52\!\cdots\!41$$$$T^{24}$$
$79$ $$1 + 12 T - 26574 T^{2} - 366848 T^{3} + 390887217 T^{4} + 5621569728 T^{5} - 3680496208466 T^{6} - 54811745357412 T^{7} + 24735046819850526 T^{8} + 328584508359816652 T^{9} -$$$$13\!\cdots\!54$$$$T^{10} -$$$$88\!\cdots\!28$$$$T^{11} +$$$$71\!\cdots\!73$$$$T^{12} -$$$$55\!\cdots\!48$$$$T^{13} -$$$$50\!\cdots\!74$$$$T^{14} +$$$$79\!\cdots\!92$$$$T^{15} +$$$$37\!\cdots\!86$$$$T^{16} -$$$$51\!\cdots\!12$$$$T^{17} -$$$$21\!\cdots\!06$$$$T^{18} +$$$$20\!\cdots\!68$$$$T^{19} +$$$$89\!\cdots\!57$$$$T^{20} -$$$$52\!\cdots\!28$$$$T^{21} -$$$$23\!\cdots\!74$$$$T^{22} +$$$$67\!\cdots\!92$$$$T^{23} +$$$$34\!\cdots\!81$$$$T^{24}$$
$83$ $$1 - 43200 T^{2} + 987057204 T^{4} - 15370870045228 T^{6} + 180607989502531968 T^{8} -$$$$16\!\cdots\!36$$$$T^{10} +$$$$12\!\cdots\!66$$$$T^{12} -$$$$79\!\cdots\!56$$$$T^{14} +$$$$40\!\cdots\!88$$$$T^{16} -$$$$16\!\cdots\!08$$$$T^{18} +$$$$50\!\cdots\!24$$$$T^{20} -$$$$10\!\cdots\!00$$$$T^{22} +$$$$11\!\cdots\!21$$$$T^{24}$$
$89$ $$1 - 204 T + 49104 T^{2} - 7187328 T^{3} + 1045283496 T^{4} - 127643879244 T^{5} + 15255527568428 T^{6} - 1684635845896692 T^{7} + 183174646224057864 T^{8} - 18563328978636389184 T^{9} +$$$$18\!\cdots\!76$$$$T^{10} -$$$$17\!\cdots\!52$$$$T^{11} +$$$$16\!\cdots\!74$$$$T^{12} -$$$$13\!\cdots\!92$$$$T^{13} +$$$$11\!\cdots\!16$$$$T^{14} -$$$$92\!\cdots\!24$$$$T^{15} +$$$$72\!\cdots\!84$$$$T^{16} -$$$$52\!\cdots\!92$$$$T^{17} +$$$$37\!\cdots\!88$$$$T^{18} -$$$$24\!\cdots\!04$$$$T^{19} +$$$$16\!\cdots\!56$$$$T^{20} -$$$$88\!\cdots\!68$$$$T^{21} +$$$$47\!\cdots\!04$$$$T^{22} -$$$$15\!\cdots\!84$$$$T^{23} +$$$$61\!\cdots\!41$$$$T^{24}$$
$97$ $$1 - 64284 T^{2} + 2051576418 T^{4} - 44019752488108 T^{6} + 711686325387159663 T^{8} -$$$$91\!\cdots\!92$$$$T^{10} +$$$$94\!\cdots\!68$$$$T^{12} -$$$$80\!\cdots\!52$$$$T^{14} +$$$$55\!\cdots\!43$$$$T^{16} -$$$$30\!\cdots\!28$$$$T^{18} +$$$$12\!\cdots\!78$$$$T^{20} -$$$$34\!\cdots\!84$$$$T^{22} +$$$$48\!\cdots\!81$$$$T^{24}$$