# Properties

 Label 43.3.d.a Level $43$ Weight $3$ Character orbit 43.d Analytic conductor $1.172$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$43$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 43.d (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.17166513675$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Defining polynomial: $$x^{12} + 37 x^{10} + 483 x^{8} + 2718 x^{6} + 6923 x^{4} + 7253 x^{2} + 1849$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} - \beta_{2} ) q^{2} + \beta_{3} q^{3} + ( -2 + \beta_{1} + \beta_{5} + \beta_{9} + \beta_{11} ) q^{4} + ( -1 - \beta_{4} - \beta_{11} ) q^{5} + ( -2 - \beta_{4} - \beta_{5} + 2 \beta_{7} + \beta_{8} ) q^{6} + ( 1 - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} - \beta_{8} + \beta_{10} ) q^{7} + ( -2 - 5 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 4 \beta_{8} - \beta_{9} - 2 \beta_{10} - 3 \beta_{11} ) q^{8} + ( 1 + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - 3 \beta_{7} - \beta_{8} + \beta_{10} + 2 \beta_{11} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{1} - \beta_{2} ) q^{2} + \beta_{3} q^{3} + ( -2 + \beta_{1} + \beta_{5} + \beta_{9} + \beta_{11} ) q^{4} + ( -1 - \beta_{4} - \beta_{11} ) q^{5} + ( -2 - \beta_{4} - \beta_{5} + 2 \beta_{7} + \beta_{8} ) q^{6} + ( 1 - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} - \beta_{8} + \beta_{10} ) q^{7} + ( -2 - 5 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 4 \beta_{8} - \beta_{9} - 2 \beta_{10} - 3 \beta_{11} ) q^{8} + ( 1 + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - 3 \beta_{7} - \beta_{8} + \beta_{10} + 2 \beta_{11} ) q^{9} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{10} + ( 5 - \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{9} - 2 \beta_{10} ) q^{11} + ( -1 - 2 \beta_{1} + \beta_{3} - \beta_{5} + \beta_{8} - \beta_{11} ) q^{12} + ( 2 + \beta_{1} + 2 \beta_{4} - 3 \beta_{6} + 4 \beta_{7} - 2 \beta_{8} - \beta_{10} + \beta_{11} ) q^{13} + ( 2 + 6 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} + 5 \beta_{5} - 5 \beta_{7} - 5 \beta_{8} + 2 \beta_{10} + 4 \beta_{11} ) q^{14} + ( 2 + \beta_{2} - 4 \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - 3 \beta_{7} - 2 \beta_{8} + 2 \beta_{11} ) q^{15} + ( 6 - 3 \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{9} + 3 \beta_{10} - 4 \beta_{11} ) q^{16} + ( -2 - 5 \beta_{1} + \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{10} - 3 \beta_{11} ) q^{17} + ( 6 - \beta_{1} + 5 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - 3 \beta_{10} - \beta_{11} ) q^{18} + ( 4 - 7 \beta_{1} + \beta_{3} + 3 \beta_{4} - 3 \beta_{5} + 4 \beta_{7} + 3 \beta_{8} ) q^{19} + ( -7 + 3 \beta_{1} - \beta_{3} - 2 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} - 7 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{20} + ( -6 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{9} + \beta_{10} ) q^{21} + ( -4 + 3 \beta_{1} - 5 \beta_{2} + 7 \beta_{3} + \beta_{4} + 2 \beta_{5} - 4 \beta_{6} + 12 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - 4 \beta_{10} - 2 \beta_{11} ) q^{22} + ( -4 - 3 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} - 8 \beta_{5} + 5 \beta_{6} + \beta_{7} + \beta_{8} - 5 \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{23} + ( -3 + 2 \beta_{1} - 3 \beta_{3} - 3 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{9} + 5 \beta_{10} + 2 \beta_{11} ) q^{24} + ( -1 + 3 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} + 3 \beta_{6} - 3 \beta_{7} + 2 \beta_{8} + 2 \beta_{10} + \beta_{11} ) q^{25} + ( -1 + 3 \beta_{1} - 12 \beta_{2} + \beta_{3} + 3 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} + \beta_{8} + 6 \beta_{9} - 2 \beta_{10} + 3 \beta_{11} ) q^{26} + ( -2 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} - 8 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - \beta_{9} - 3 \beta_{10} - 4 \beta_{11} ) q^{27} + ( -17 + \beta_{1} + 6 \beta_{2} + 6 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - \beta_{6} + 13 \beta_{7} + 6 \beta_{8} + 2 \beta_{9} - 8 \beta_{10} + \beta_{11} ) q^{28} + ( -16 + 2 \beta_{1} - 4 \beta_{2} + 5 \beta_{3} - 6 \beta_{4} + 7 \beta_{5} - 2 \beta_{6} + 10 \beta_{7} + 5 \beta_{8} + 4 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{29} + ( 14 + \beta_{1} - \beta_{2} + 4 \beta_{4} - \beta_{6} - 5 \beta_{7} + 2 \beta_{9} - 3 \beta_{10} + \beta_{11} ) q^{30} + ( -3 - 2 \beta_{1} + 10 \beta_{2} + 4 \beta_{3} + \beta_{4} + 14 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + 4 \beta_{10} + 3 \beta_{11} ) q^{31} + ( 5 + 16 \beta_{1} - 12 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + 9 \beta_{5} + 4 \beta_{6} - 14 \beta_{7} - 10 \beta_{8} - 2 \beta_{9} + 6 \beta_{10} + 4 \beta_{11} ) q^{32} + ( -13 + 2 \beta_{1} + 6 \beta_{3} - 6 \beta_{4} - \beta_{5} + 2 \beta_{6} - 4 \beta_{7} + 3 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - 7 \beta_{11} ) q^{33} + ( 21 - 2 \beta_{1} - 3 \beta_{3} - \beta_{4} - 8 \beta_{5} + 2 \beta_{6} - 14 \beta_{7} - 3 \beta_{8} - 4 \beta_{9} + 5 \beta_{10} - 2 \beta_{11} ) q^{34} + ( 1 - 3 \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{35} + ( 1 - 6 \beta_{1} + 4 \beta_{3} + \beta_{4} - 9 \beta_{5} - 8 \beta_{6} + 32 \beta_{7} + 8 \beta_{8} - 7 \beta_{10} - 6 \beta_{11} ) q^{36} + ( 8 + 10 \beta_{1} - 3 \beta_{3} - \beta_{4} + 4 \beta_{5} + \beta_{6} + 6 \beta_{7} - 3 \beta_{8} + \beta_{9} + \beta_{10} + 3 \beta_{11} ) q^{37} + ( 25 + 4 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} - \beta_{4} + 14 \beta_{5} + 7 \beta_{6} - 35 \beta_{7} - 9 \beta_{8} - 7 \beta_{9} + 13 \beta_{10} + 3 \beta_{11} ) q^{38} + ( 5 - \beta_{1} - \beta_{2} + 4 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} - 8 \beta_{7} - 6 \beta_{8} + \beta_{9} - 3 \beta_{10} - 2 \beta_{11} ) q^{39} + ( -9 \beta_{1} + 14 \beta_{2} - 5 \beta_{3} + \beta_{4} - 13 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{9} - 3 \beta_{10} - 2 \beta_{11} ) q^{40} + ( 11 + 2 \beta_{1} - \beta_{2} + 7 \beta_{3} + 3 \beta_{4} - 7 \beta_{5} + \beta_{9} - 2 \beta_{10} + 3 \beta_{11} ) q^{41} + ( -9 - 7 \beta_{1} + 8 \beta_{2} + 5 \beta_{3} + 2 \beta_{4} + 6 \beta_{5} - 2 \beta_{6} + 20 \beta_{7} - 2 \beta_{8} + \beta_{9} + \beta_{11} ) q^{42} + ( -7 \beta_{1} - 5 \beta_{2} - 6 \beta_{5} - 7 \beta_{6} + 6 \beta_{7} + 6 \beta_{8} + \beta_{9} - 6 \beta_{10} + \beta_{11} ) q^{43} + ( -16 - 9 \beta_{1} - 13 \beta_{2} - 2 \beta_{3} + 7 \beta_{4} - \beta_{5} + 3 \beta_{9} - \beta_{10} + 4 \beta_{11} ) q^{44} + ( 14 - 2 \beta_{1} + 2 \beta_{2} + 6 \beta_{3} + 6 \beta_{5} - 28 \beta_{7} ) q^{45} + ( 16 + 10 \beta_{1} - 9 \beta_{3} + 4 \beta_{4} - 8 \beta_{5} - 6 \beta_{6} + 14 \beta_{7} + 2 \beta_{8} - 6 \beta_{9} - 6 \beta_{10} - 4 \beta_{11} ) q^{46} + ( 2 + 6 \beta_{1} + 2 \beta_{2} - 9 \beta_{3} + 2 \beta_{4} + 11 \beta_{5} + 3 \beta_{9} - \beta_{10} + 4 \beta_{11} ) q^{47} + ( -7 + 9 \beta_{1} - 9 \beta_{3} + 3 \beta_{4} - \beta_{6} - 11 \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} + 3 \beta_{11} ) q^{48} + ( -22 + 9 \beta_{1} - 16 \beta_{2} - 7 \beta_{3} - \beta_{4} - 6 \beta_{5} + 3 \beta_{6} + 18 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} + 5 \beta_{10} + \beta_{11} ) q^{49} + ( -18 - 2 \beta_{1} + 14 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} + 7 \beta_{5} + 2 \beta_{6} + 9 \beta_{7} + 2 \beta_{8} - 4 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{50} + ( -1 + \beta_{1} + 12 \beta_{2} - 13 \beta_{3} + 10 \beta_{4} + 4 \beta_{5} - 6 \beta_{6} + 8 \beta_{7} - 14 \beta_{8} + 3 \beta_{9} + 10 \beta_{10} + 13 \beta_{11} ) q^{51} + ( -1 - 19 \beta_{1} + 13 \beta_{2} - 7 \beta_{3} - \beta_{4} + 6 \beta_{5} + 9 \beta_{6} - 50 \beta_{7} - 3 \beta_{8} + 8 \beta_{10} + 7 \beta_{11} ) q^{52} + ( -11 - 2 \beta_{1} - 8 \beta_{2} - 5 \beta_{3} - 3 \beta_{4} - 21 \beta_{5} + 5 \beta_{6} + 12 \beta_{7} + 4 \beta_{8} - 5 \beta_{9} - 7 \beta_{10} - 6 \beta_{11} ) q^{53} + ( 11 - 3 \beta_{1} + \beta_{2} - 6 \beta_{3} + 6 \beta_{4} - 4 \beta_{5} - \beta_{9} + 3 \beta_{10} - 4 \beta_{11} ) q^{54} + ( -14 - \beta_{1} + 15 \beta_{3} - \beta_{4} - 6 \beta_{5} - 4 \beta_{6} - 11 \beta_{7} + 2 \beta_{8} - 4 \beta_{9} - 4 \beta_{10} - 7 \beta_{11} ) q^{55} + ( -8 - 47 \beta_{1} + 14 \beta_{2} + 17 \beta_{3} - 8 \beta_{4} - 12 \beta_{5} - 3 \beta_{6} + 56 \beta_{7} + 19 \beta_{8} - 11 \beta_{10} - 19 \beta_{11} ) q^{56} + ( -5 + 9 \beta_{1} - 5 \beta_{2} + 8 \beta_{3} - 5 \beta_{4} + 6 \beta_{5} + 9 \beta_{6} + 10 \beta_{7} + 4 \beta_{8} + 4 \beta_{10} - \beta_{11} ) q^{57} + ( -11 - 38 \beta_{1} + 13 \beta_{2} - 7 \beta_{3} - 11 \beta_{4} - 9 \beta_{5} + 10 \beta_{6} - 38 \beta_{7} + 9 \beta_{8} - \beta_{10} - 12 \beta_{11} ) q^{58} + ( 32 + 3 \beta_{1} + 5 \beta_{2} + \beta_{3} - 3 \beta_{4} - 6 \beta_{9} - 4 \beta_{10} - 2 \beta_{11} ) q^{59} + ( 3 - 8 \beta_{1} + 5 \beta_{2} - 4 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} - \beta_{10} + 2 \beta_{11} ) q^{60} + ( 20 + 7 \beta_{1} + 3 \beta_{2} - 7 \beta_{3} + 8 \beta_{4} + 8 \beta_{5} - 7 \beta_{6} - 13 \beta_{7} - 7 \beta_{8} + 14 \beta_{9} + 13 \beta_{10} + 7 \beta_{11} ) q^{61} + ( 14 - 4 \beta_{1} + 12 \beta_{3} - 4 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} + 12 \beta_{7} - 6 \beta_{8} - 3 \beta_{9} - 3 \beta_{10} - \beta_{11} ) q^{62} + ( -4 + 12 \beta_{1} - 3 \beta_{3} + 2 \beta_{4} + 5 \beta_{5} + \beta_{6} - 10 \beta_{7} - 4 \beta_{8} + \beta_{9} + \beta_{10} + 7 \beta_{11} ) q^{63} + ( -46 + 19 \beta_{1} + 16 \beta_{2} + 3 \beta_{3} - 10 \beta_{4} + 10 \beta_{5} + 2 \beta_{9} - \beta_{10} + 3 \beta_{11} ) q^{64} + ( -13 + 7 \beta_{1} - 9 \beta_{2} + 18 \beta_{3} - 5 \beta_{4} + 9 \beta_{5} - 2 \beta_{6} + 28 \beta_{7} + 12 \beta_{8} + \beta_{9} - 3 \beta_{10} - 2 \beta_{11} ) q^{65} + ( -33 + 4 \beta_{1} + 8 \beta_{2} - 11 \beta_{3} + 4 \beta_{4} - 3 \beta_{6} + 28 \beta_{7} - 9 \beta_{8} + 3 \beta_{9} + 13 \beta_{10} + 8 \beta_{11} ) q^{66} + ( 18 + 4 \beta_{1} - 24 \beta_{2} + 3 \beta_{3} + 4 \beta_{4} + 7 \beta_{6} - 17 \beta_{7} - 3 \beta_{8} - 7 \beta_{9} - 9 \beta_{10} - 8 \beta_{11} ) q^{67} + ( 4 + 42 \beta_{1} - 18 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} + 8 \beta_{5} - 2 \beta_{6} - 15 \beta_{7} - 14 \beta_{8} + 2 \beta_{10} + 6 \beta_{11} ) q^{68} + ( 46 - 2 \beta_{1} - 10 \beta_{2} + 9 \beta_{3} - 8 \beta_{4} + 7 \beta_{5} + 2 \beta_{6} - 18 \beta_{7} + 9 \beta_{8} - 4 \beta_{9} - 12 \beta_{10} - 2 \beta_{11} ) q^{69} + ( 13 + 17 \beta_{1} - 9 \beta_{2} - 9 \beta_{3} + 6 \beta_{4} + 5 \beta_{5} - 26 \beta_{7} - 12 \beta_{8} + 8 \beta_{10} + 8 \beta_{11} ) q^{70} + ( -7 + 5 \beta_{1} + 14 \beta_{2} - 16 \beta_{3} + 15 \beta_{4} + 2 \beta_{5} - 5 \beta_{6} - 5 \beta_{7} - 16 \beta_{8} + 10 \beta_{9} + 22 \beta_{10} + 5 \beta_{11} ) q^{71} + ( 21 + 4 \beta_{1} - 33 \beta_{2} - 7 \beta_{3} + 9 \beta_{4} + 13 \beta_{5} - 4 \beta_{6} - 13 \beta_{7} - 7 \beta_{8} + 8 \beta_{9} + 9 \beta_{10} + 4 \beta_{11} ) q^{72} + ( 23 + 5 \beta_{1} - 16 \beta_{2} + 7 \beta_{3} - 3 \beta_{4} - 9 \beta_{5} - 5 \beta_{6} - 6 \beta_{7} + 7 \beta_{8} + 10 \beta_{9} - 6 \beta_{10} + 5 \beta_{11} ) q^{73} + ( -36 - 3 \beta_{1} - 8 \beta_{2} + 12 \beta_{3} - 5 \beta_{4} - 10 \beta_{5} - 4 \beta_{6} + 47 \beta_{7} + 16 \beta_{8} + 4 \beta_{9} - 18 \beta_{10} - 7 \beta_{11} ) q^{74} + ( -8 - 5 \beta_{2} + 4 \beta_{3} - 11 \beta_{4} - 6 \beta_{5} + 6 \beta_{6} + 10 \beta_{7} + 16 \beta_{8} - 3 \beta_{9} - 2 \beta_{10} - 5 \beta_{11} ) q^{75} + ( 8 + 62 \beta_{1} + 6 \beta_{3} + 16 \beta_{4} + 11 \beta_{5} - 6 \beta_{6} - 25 \beta_{7} - 17 \beta_{8} - 6 \beta_{9} - 6 \beta_{10} + 27 \beta_{11} ) q^{76} + ( 14 - \beta_{1} - 2 \beta_{2} + 5 \beta_{3} - 4 \beta_{4} - 5 \beta_{5} + \beta_{6} - 4 \beta_{7} + 5 \beta_{8} - 2 \beta_{9} - 7 \beta_{10} - \beta_{11} ) q^{77} + ( 9 - \beta_{1} - 4 \beta_{2} + 12 \beta_{3} - 9 \beta_{5} + 4 \beta_{9} + \beta_{10} + 3 \beta_{11} ) q^{78} + ( -7 - 19 \beta_{1} + 6 \beta_{2} + 14 \beta_{3} - 7 \beta_{4} + 8 \beta_{5} + 7 \beta_{6} + 33 \beta_{7} - 2 \beta_{8} - 7 \beta_{11} ) q^{79} + ( 31 - 10 \beta_{1} - 2 \beta_{3} - 3 \beta_{4} - 8 \beta_{5} + 42 \beta_{7} + 8 \beta_{8} - 11 \beta_{11} ) q^{80} + ( 18 + 3 \beta_{1} - 22 \beta_{2} + 5 \beta_{3} + \beta_{4} - 21 \beta_{5} - 4 \beta_{6} - 6 \beta_{7} + 11 \beta_{8} + 4 \beta_{9} - 20 \beta_{10} - 8 \beta_{11} ) q^{81} + ( -22 - 11 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - 16 \beta_{4} - 17 \beta_{5} + 4 \beta_{6} + 40 \beta_{7} + 28 \beta_{8} - 2 \beta_{9} - 7 \beta_{10} - 9 \beta_{11} ) q^{82} + ( 6 - 13 \beta_{1} + 40 \beta_{2} - 6 \beta_{3} + \beta_{4} + 14 \beta_{5} + 2 \beta_{6} - 15 \beta_{7} - 10 \beta_{8} - 2 \beta_{9} + 16 \beta_{10} + 7 \beta_{11} ) q^{83} + ( 14 - 10 \beta_{1} - 5 \beta_{2} - \beta_{3} - 4 \beta_{5} - \beta_{9} + 4 \beta_{10} - 5 \beta_{11} ) q^{84} + ( -27 - 7 \beta_{1} + 11 \beta_{2} - 13 \beta_{3} - 6 \beta_{4} - 7 \beta_{5} + 8 \beta_{6} + 46 \beta_{7} + 4 \beta_{8} - 4 \beta_{9} + 8 \beta_{10} + 4 \beta_{11} ) q^{85} + ( 34 - 11 \beta_{1} - 11 \beta_{2} - 20 \beta_{3} + 5 \beta_{4} + 3 \beta_{5} + 8 \beta_{6} - 78 \beta_{7} - 4 \beta_{8} + 5 \beta_{9} + 12 \beta_{10} - 2 \beta_{11} ) q^{86} + ( -43 - 13 \beta_{1} - 6 \beta_{2} + 9 \beta_{3} - 13 \beta_{4} - 3 \beta_{5} - 9 \beta_{9} - 2 \beta_{10} - 7 \beta_{11} ) q^{87} + ( 43 - 20 \beta_{1} + 18 \beta_{2} + 21 \beta_{3} - 6 \beta_{4} + 29 \beta_{5} + 16 \beta_{6} - 102 \beta_{7} - 4 \beta_{8} - 8 \beta_{9} + 6 \beta_{10} - 2 \beta_{11} ) q^{88} + ( -4 + 38 \beta_{1} - 17 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} + 7 \beta_{6} + 3 \beta_{7} + 3 \beta_{8} + 7 \beta_{9} + 7 \beta_{10} ) q^{89} + ( 12 \beta_{1} + 14 \beta_{2} + 6 \beta_{3} - 6 \beta_{4} - 2 \beta_{5} - 2 \beta_{9} - 2 \beta_{11} ) q^{90} + ( -10 + 4 \beta_{1} - 15 \beta_{3} - 6 \beta_{5} - 4 \beta_{7} + 6 \beta_{8} - 6 \beta_{11} ) q^{91} + ( -86 + 28 \beta_{1} - 48 \beta_{2} - 6 \beta_{3} + 13 \beta_{4} - 11 \beta_{5} - 12 \beta_{6} + 94 \beta_{7} - 5 \beta_{8} + 12 \beta_{9} - 4 \beta_{10} + 4 \beta_{11} ) q^{92} + ( -60 - 2 \beta_{1} + 11 \beta_{2} - 8 \beta_{3} + 6 \beta_{4} + 15 \beta_{5} + 2 \beta_{6} + 25 \beta_{7} - 8 \beta_{8} - 4 \beta_{9} + 8 \beta_{10} - 2 \beta_{11} ) q^{93} + ( -7 - 29 \beta_{1} + 15 \beta_{2} + 25 \beta_{3} - 3 \beta_{4} + 10 \beta_{5} + 2 \beta_{6} + 12 \beta_{7} + 4 \beta_{8} - \beta_{9} - 13 \beta_{10} - 14 \beta_{11} ) q^{94} + ( 3 - 36 \beta_{1} + 13 \beta_{2} + 4 \beta_{3} + 3 \beta_{4} - 11 \beta_{5} - 16 \beta_{6} - 66 \beta_{7} - 13 \beta_{10} - 10 \beta_{11} ) q^{95} + ( -46 - 6 \beta_{1} + 22 \beta_{2} + 3 \beta_{3} - 7 \beta_{4} + 4 \beta_{5} - 5 \beta_{6} + 46 \beta_{7} + 7 \beta_{8} + 5 \beta_{9} + 5 \beta_{10} + 5 \beta_{11} ) q^{96} + ( -47 + 4 \beta_{1} + 6 \beta_{2} + 10 \beta_{3} - 26 \beta_{4} + 14 \beta_{5} + 5 \beta_{9} + 7 \beta_{10} - 2 \beta_{11} ) q^{97} + ( -38 + 19 \beta_{1} - 19 \beta_{3} + 7 \beta_{4} + 18 \beta_{5} + 5 \beta_{6} - 58 \beta_{7} - 13 \beta_{8} + 5 \beta_{9} + 5 \beta_{10} + 25 \beta_{11} ) q^{98} + ( -11 \beta_{1} + 11 \beta_{2} - 32 \beta_{3} - 2 \beta_{5} + 11 \beta_{6} + 14 \beta_{7} - 6 \beta_{8} + 11 \beta_{10} + 11 \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 26q^{4} - 3q^{5} - 9q^{6} + 9q^{7} - 12q^{9} + O(q^{10})$$ $$12q - 26q^{4} - 3q^{5} - 9q^{6} + 9q^{7} - 12q^{9} - q^{10} + 28q^{11} - 6q^{12} + 24q^{13} - 18q^{14} - 13q^{15} + 110q^{16} - 7q^{17} + 33q^{18} + 66q^{19} - 99q^{20} - 80q^{21} - 16q^{23} - 2q^{24} - 21q^{25} + 9q^{26} - 192q^{28} - 111q^{29} + 99q^{30} - 29q^{31} - 114q^{33} + 213q^{34} + 38q^{35} + 152q^{36} + 120q^{37} + 172q^{38} - 29q^{40} + 94q^{41} + 5q^{43} - 174q^{44} + 156q^{46} - 18q^{47} - 213q^{48} - 99q^{49} - 198q^{50} - 234q^{52} - 58q^{53} + 128q^{54} - 258q^{55} + 315q^{56} + 51q^{57} - 196q^{58} + 336q^{59} - 5q^{60} + 204q^{61} + 261q^{62} - 153q^{63} - 604q^{64} - 201q^{66} + 115q^{67} - 106q^{68} + 423q^{69} - 66q^{71} + 294q^{72} + 249q^{73} - 214q^{74} - 438q^{76} + 117q^{77} + 136q^{78} + 236q^{79} + 681q^{80} + 110q^{81} - 4q^{83} + 248q^{84} + 102q^{86} - 408q^{87} - 45q^{89} - 44q^{90} - 156q^{91} - 483q^{92} - 567q^{93} - 389q^{95} - 278q^{96} - 370q^{97} - 879q^{98} + 157q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 37 x^{10} + 483 x^{8} + 2718 x^{6} + 6923 x^{4} + 7253 x^{2} + 1849$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$4 \nu^{10} + 137 \nu^{8} + 1563 \nu^{6} + 6752 \nu^{4} + 10271 \nu^{2} + 124 \nu + 3053$$$$)/248$$ $$\beta_{2}$$ $$=$$ $$($$$$4 \nu^{10} + 137 \nu^{8} + 1563 \nu^{6} + 6752 \nu^{4} + 10271 \nu^{2} - 124 \nu + 3053$$$$)/248$$ $$\beta_{3}$$ $$=$$ $$($$$$-1485 \nu^{11} + 26101 \nu^{10} - 54558 \nu^{9} + 904290 \nu^{8} - 695669 \nu^{7} + 10473897 \nu^{6} - 3624075 \nu^{5} + 46125971 \nu^{4} - 7162682 \nu^{3} + 70377670 \nu^{2} - 3213627 \nu + 18940167$$$$)/810464$$ $$\beta_{4}$$ $$=$$ $$($$$$-1485 \nu^{11} + 25327 \nu^{10} - 54558 \nu^{9} + 861118 \nu^{8} - 695669 \nu^{7} + 9649587 \nu^{6} - 3624075 \nu^{5} + 39890025 \nu^{4} - 7162682 \nu^{3} + 55263514 \nu^{2} - 3213627 \nu + 11827709$$$$)/810464$$ $$\beta_{5}$$ $$=$$ $$($$$$-1485 \nu^{11} - 26101 \nu^{10} - 54558 \nu^{9} - 904290 \nu^{8} - 695669 \nu^{7} - 10473897 \nu^{6} - 3624075 \nu^{5} - 46125971 \nu^{4} - 7162682 \nu^{3} - 70377670 \nu^{2} - 3213627 \nu - 18940167$$$$)/810464$$ $$\beta_{6}$$ $$=$$ $$($$$$1370 \nu^{11} + 12857 \nu^{10} + 48798 \nu^{9} + 441352 \nu^{8} + 595576 \nu^{7} + 5030871 \nu^{6} + 2975030 \nu^{5} + 21503999 \nu^{4} + 6488098 \nu^{3} + 31612912 \nu^{2} + 5730264 \nu + 9110281$$$$)/405232$$ $$\beta_{7}$$ $$=$$ $$($$$$71 \nu^{11} + 2455 \nu^{9} + 28402 \nu^{7} + 125769 \nu^{5} + 201197 \nu^{3} + 73310 \nu + 5332$$$$)/10664$$ $$\beta_{8}$$ $$=$$ $$($$$$12339 \nu^{11} - 387 \nu^{10} + 422014 \nu^{9} - 21586 \nu^{8} + 4765455 \nu^{7} - 412155 \nu^{6} + 19760933 \nu^{5} - 3117973 \nu^{4} + 25470362 \nu^{3} - 7557078 \nu^{2} + 1369793 \nu - 3556229$$$$)/810464$$ $$\beta_{9}$$ $$=$$ $$($$$$-16407 \nu^{11} + 30315 \nu^{10} - 564102 \nu^{9} + 1049286 \nu^{8} - 6440307 \nu^{7} + 12125183 \nu^{6} - 27515185 \nu^{5} + 53241869 \nu^{4} - 39401330 \nu^{3} + 82695794 \nu^{2} - 7859677 \nu + 27766433$$$$)/810464$$ $$\beta_{10}$$ $$=$$ $$($$$$16407 \nu^{11} + 30315 \nu^{10} + 564102 \nu^{9} + 1049286 \nu^{8} + 6440307 \nu^{7} + 12125183 \nu^{6} + 27515185 \nu^{5} + 53241869 \nu^{4} + 39401330 \nu^{3} + 81885330 \nu^{2} + 7859677 \nu + 22903649$$$$)/810464$$ $$\beta_{11}$$ $$=$$ $$($$$$8946 \nu^{11} - 8643 \nu^{10} + 309330 \nu^{9} - 296356 \nu^{8} + 3567988 \nu^{7} - 3379585 \nu^{6} + 15569630 \nu^{5} - 14590717 \nu^{4} + 23282006 \nu^{3} - 22536644 \nu^{2} + 5334036 \nu - 6970343$$$$)/405232$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$-\beta_{2} + \beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{11} + \beta_{9} + \beta_{5} + \beta_{1} - 6$$ $$\nu^{3}$$ $$=$$ $$-3 \beta_{11} - 2 \beta_{10} - \beta_{9} + 4 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 3 \beta_{4} + 2 \beta_{3} + 10 \beta_{2} - 13 \beta_{1} - 2$$ $$\nu^{4}$$ $$=$$ $$-16 \beta_{11} + 3 \beta_{10} - 13 \beta_{9} - 12 \beta_{5} - \beta_{4} - 3 \beta_{3} + \beta_{2} - 15 \beta_{1} + 62$$ $$\nu^{5}$$ $$=$$ $$52 \beta_{11} + 38 \beta_{10} + 14 \beta_{9} - 74 \beta_{8} - 46 \beta_{7} - 28 \beta_{6} + 41 \beta_{5} + 51 \beta_{4} - 34 \beta_{3} - 124 \beta_{2} + 176 \beta_{1} + 37$$ $$\nu^{6}$$ $$=$$ $$227 \beta_{11} - 61 \beta_{10} + 166 \beta_{9} + 154 \beta_{5} + 10 \beta_{4} + 63 \beta_{3} - 4 \beta_{2} + 223 \beta_{1} - 774$$ $$\nu^{7}$$ $$=$$ $$-789 \beta_{11} - 619 \beta_{10} - 170 \beta_{9} + 1148 \beta_{8} + 934 \beta_{7} + 340 \beta_{6} - 678 \beta_{5} - 744 \beta_{4} + 515 \beta_{3} + 1631 \beta_{2} - 2420 \beta_{1} - 637$$ $$\nu^{8}$$ $$=$$ $$-3164 \beta_{11} + 1023 \beta_{10} - 2141 \beta_{9} - 2061 \beta_{5} - 66 \beta_{4} - 1037 \beta_{3} - 172 \beta_{2} - 3336 \beta_{1} + 10235$$ $$\nu^{9}$$ $$=$$ $$11644 \beta_{11} + 9675 \beta_{10} + 1969 \beta_{9} - 17120 \beta_{8} - 17240 \beta_{7} - 3938 \beta_{6} + 10488 \beta_{5} + 10529 \beta_{4} - 7747 \beta_{3} - 21934 \beta_{2} + 33578 \beta_{1} + 10589$$ $$\nu^{10}$$ $$=$$ $$44107 \beta_{11} - 16266 \beta_{10} + 27841 \beta_{9} + 28102 \beta_{5} + 41 \beta_{4} + 15964 \beta_{3} + 5797 \beta_{2} + 49904 \beta_{1} - 138121$$ $$\nu^{11}$$ $$=$$ $$-170609 \beta_{11} - 148565 \beta_{10} - 22044 \beta_{9} + 252482 \beta_{8} + 298456 \beta_{7} + 44088 \beta_{6} - 158389 \beta_{5} - 148285 \beta_{4} + 116417 \beta_{3} + 298324 \beta_{2} - 468933 \beta_{1} - 171272$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/43\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$\beta_{7}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 − 3.53204i − 1.61947i − 0.604188i 1.51156i 2.15301i 3.82317i − 3.82317i − 2.15301i − 1.51156i 0.604188i 1.61947i 3.53204i
3.53204i 0.357493 0.206399i −8.47532 2.82830 1.63292i −0.729010 1.26268i 0.813980 + 0.469951i 15.8070i −4.41480 + 7.64666i −5.76753 9.98966i
7.2 1.61947i 2.75697 1.59174i 1.37732 −6.40200 + 3.69620i −2.57777 4.46483i 1.18578 + 0.684612i 8.70840i 0.567259 0.982521i 5.98588 + 10.3678i
7.3 0.604188i −1.35447 + 0.782004i 3.63496 4.60389 2.65806i 0.472478 + 0.818356i −0.191259 0.110424i 4.61295i −3.27694 + 5.67582i −1.60597 2.78161i
7.4 1.51156i −3.66976 + 2.11873i 1.71518 −4.51092 + 2.60438i −3.20260 5.54707i 1.23619 + 0.713712i 8.63885i 4.47807 7.75625i −3.93669 6.81854i
7.5 2.15301i 2.77779 1.60376i −0.635471 −0.468965 + 0.270757i 3.45292 + 5.98063i −7.68536 4.43714i 7.24388i 0.644090 1.11560i −0.582944 1.00969i
7.6 3.82317i −0.868030 + 0.501157i −10.6167 2.44970 1.41434i −1.91601 3.31863i 9.14067 + 5.27737i 25.2966i −3.99768 + 6.92419i 5.40725 + 9.36563i
37.1 3.82317i −0.868030 0.501157i −10.6167 2.44970 + 1.41434i −1.91601 + 3.31863i 9.14067 5.27737i 25.2966i −3.99768 6.92419i 5.40725 9.36563i
37.2 2.15301i 2.77779 + 1.60376i −0.635471 −0.468965 0.270757i 3.45292 5.98063i −7.68536 + 4.43714i 7.24388i 0.644090 + 1.11560i −0.582944 + 1.00969i
37.3 1.51156i −3.66976 2.11873i 1.71518 −4.51092 2.60438i −3.20260 + 5.54707i 1.23619 0.713712i 8.63885i 4.47807 + 7.75625i −3.93669 + 6.81854i
37.4 0.604188i −1.35447 0.782004i 3.63496 4.60389 + 2.65806i 0.472478 0.818356i −0.191259 + 0.110424i 4.61295i −3.27694 5.67582i −1.60597 + 2.78161i
37.5 1.61947i 2.75697 + 1.59174i 1.37732 −6.40200 3.69620i −2.57777 + 4.46483i 1.18578 0.684612i 8.70840i 0.567259 + 0.982521i 5.98588 10.3678i
37.6 3.53204i 0.357493 + 0.206399i −8.47532 2.82830 + 1.63292i −0.729010 + 1.26268i 0.813980 0.469951i 15.8070i −4.41480 7.64666i −5.76753 + 9.98966i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 37.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 43.3.d.a 12
3.b odd 2 1 387.3.j.c 12
4.b odd 2 1 688.3.t.c 12
43.d odd 6 1 inner 43.3.d.a 12
129.h even 6 1 387.3.j.c 12
172.f even 6 1 688.3.t.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.3.d.a 12 1.a even 1 1 trivial
43.3.d.a 12 43.d odd 6 1 inner
387.3.j.c 12 3.b odd 2 1
387.3.j.c 12 129.h even 6 1
688.3.t.c 12 4.b odd 2 1
688.3.t.c 12 172.f even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(43, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1849 + 7253 T^{2} + 6923 T^{4} + 2718 T^{6} + 483 T^{8} + 37 T^{10} + T^{12}$$
$3$ $$784 - 1596 T - 1325 T^{2} + 4902 T^{3} + 6979 T^{4} + 774 T^{5} - 1723 T^{6} - 246 T^{7} + 355 T^{8} - 21 T^{10} + T^{12}$$
$5$ $$1048576 + 2408448 T + 815872 T^{2} - 2361408 T^{3} + 941152 T^{4} + 6084 T^{5} - 58849 T^{6} + 2163 T^{7} + 2968 T^{8} - 189 T^{9} - 60 T^{10} + 3 T^{11} + T^{12}$$
$7$ $$1444 + 5130 T - 7263 T^{2} - 47385 T^{3} + 151503 T^{4} - 195516 T^{5} + 133601 T^{6} - 47007 T^{7} + 5733 T^{8} + 756 T^{9} - 57 T^{10} - 9 T^{11} + T^{12}$$
$11$ $$( -178808 - 7404 T + 24970 T^{2} + 2423 T^{3} - 296 T^{4} - 14 T^{5} + T^{6} )^{2}$$
$13$ $$232783090576 - 147926659124 T + 98097238137 T^{2} - 4252254942 T^{3} + 2359306945 T^{4} - 211051428 T^{5} + 42132863 T^{6} - 2326308 T^{7} + 230487 T^{8} - 8758 T^{9} + 803 T^{10} - 24 T^{11} + T^{12}$$
$17$ $$98916610336489 - 2914293978343 T + 1738207243012 T^{2} - 1683308835 T^{3} + 20635527616 T^{4} - 36096115 T^{5} + 113224686 T^{6} - 70597 T^{7} + 452212 T^{8} - 361 T^{9} + 824 T^{10} + 7 T^{11} + T^{12}$$
$19$ $$102837203793424 + 88551255998424 T + 21886784781000 T^{2} - 3039484365558 T^{3} - 57995984711 T^{4} + 19530539304 T^{5} + 599184338 T^{6} - 49880202 T^{7} - 604685 T^{8} + 66660 T^{9} + 442 T^{10} - 66 T^{11} + T^{12}$$
$23$ $$485480586573376 + 299533435399496 T + 166731680678249 T^{2} + 12554785869778 T^{3} + 1148026207549 T^{4} + 26526985552 T^{5} + 2417063547 T^{6} + 48546244 T^{7} + 3386803 T^{8} + 32886 T^{9} + 2179 T^{10} + 16 T^{11} + T^{12}$$
$29$ $$595026727966619536 - 73485753953077284 T + 1776722810209863 T^{2} + 154182092639211 T^{3} - 5349260766317 T^{4} - 250609626834 T^{5} + 9527027147 T^{6} + 274114161 T^{7} - 6143915 T^{8} - 197358 T^{9} + 2329 T^{10} + 111 T^{11} + T^{12}$$
$31$ $$462743427606784 + 8173584947008 T + 15701942444080 T^{2} - 1334066228232 T^{3} + 469144709128 T^{4} - 20004092438 T^{5} + 2071982415 T^{6} + 8663251 T^{7} + 4279828 T^{8} - 10817 T^{9} + 2912 T^{10} + 29 T^{11} + T^{12}$$
$37$ $$4169976860828164 - 2233472362845702 T + 533822740906411 T^{2} - 72343281958332 T^{3} + 5932262917693 T^{4} - 287301400044 T^{5} + 6933643139 T^{6} - 505554 T^{7} - 3189467 T^{8} - 30600 T^{9} + 5055 T^{10} - 120 T^{11} + T^{12}$$
$41$ $$( -6575768 + 40179572 T + 4000442 T^{2} + 58539 T^{3} - 3315 T^{4} - 47 T^{5} + T^{6} )^{2}$$
$43$ $$39959630797262576401 - 108057411566421245 T + 2746727065236235 T^{2} + 196796674441468 T^{3} + 18161276039777 T^{4} + 101582834113 T^{5} + 786324230 T^{6} + 54939337 T^{7} + 5312177 T^{8} + 31132 T^{9} + 235 T^{10} - 5 T^{11} + T^{12}$$
$47$ $$( 1304641296 + 177319584 T + 4960944 T^{2} - 153972 T^{3} - 6093 T^{4} + 9 T^{5} + T^{6} )^{2}$$
$53$ $$9894647748760336 + 6965403174500748 T + 4832018449105121 T^{2} + 86393525855954 T^{3} + 13596474143055 T^{4} + 362579887068 T^{5} + 31714232165 T^{6} + 619627230 T^{7} + 21935749 T^{8} + 161988 T^{9} + 6843 T^{10} + 58 T^{11} + T^{12}$$
$59$ $$( 141472984 + 20709196 T - 1317590 T^{2} - 105745 T^{3} + 8200 T^{4} - 168 T^{5} + T^{6} )^{2}$$
$61$ $$24\!\cdots\!76$$$$+ 20041167457436810244 T + 416389760711392051 T^{2} - 9910949551557384 T^{3} - 380373932820225 T^{4} + 4005084351720 T^{5} + 207245809239 T^{6} - 1569649692 T^{7} - 61386067 T^{8} + 625668 T^{9} + 10805 T^{10} - 204 T^{11} + T^{12}$$
$67$ $$99962183356913089 - 39915682575764615 T + 14446026708910870 T^{2} - 951596748491013 T^{3} + 95098644514048 T^{4} + 1166786903365 T^{5} + 329471882268 T^{6} - 2273273063 T^{7} + 92998216 T^{8} - 463529 T^{9} + 18974 T^{10} - 115 T^{11} + T^{12}$$
$71$ $$88749356477152034596 - 42849058970408648694 T + 8236128843236797247 T^{2} - 647036100102141750 T^{3} + 21081787722302051 T^{4} - 59306103167526 T^{5} - 3178743204141 T^{6} + 14166467334 T^{7} + 392257881 T^{8} - 1538790 T^{9} - 21863 T^{10} + 66 T^{11} + T^{12}$$
$73$ $$85269096262549704961 - 5828278856791468857 T - 111699838858810390 T^{2} + 16711320112933641 T^{3} + 351883314548612 T^{4} - 47222785260993 T^{5} + 1222488188136 T^{6} - 8356066257 T^{7} - 105677664 T^{8} + 1196445 T^{9} + 15862 T^{10} - 249 T^{11} + T^{12}$$
$79$ $$85\!\cdots\!16$$$$- 16395555828758061736 T + 1605089335345382729 T^{2} - 11437207108221128 T^{3} + 1937616937247353 T^{4} - 20631307219262 T^{5} + 837719801037 T^{6} - 13951903190 T^{7} + 334577197 T^{8} - 4074570 T^{9} + 43657 T^{10} - 236 T^{11} + T^{12}$$
$83$ $$77061552355339876 - 196403534518176106 T + 513839487474130169 T^{2} + 33492241084045384 T^{3} + 2712841701436603 T^{4} - 1584959528774 T^{5} + 1301609338797 T^{6} + 12226810288 T^{7} + 334990729 T^{8} + 1142502 T^{9} + 19519 T^{10} + 4 T^{11} + T^{12}$$
$89$ $$15\!\cdots\!29$$$$+$$$$14\!\cdots\!53$$$$T + 19991046216612241126 T^{2} - 2611299862890770817 T^{3} + 43309385615521632 T^{4} + 76335572877693 T^{5} - 5821069469208 T^{6} + 2668285749 T^{7} + 605274452 T^{8} - 1288485 T^{9} - 27958 T^{10} + 45 T^{11} + T^{12}$$
$97$ $$( 170062861216 + 7831477408 T - 5140072 T^{2} - 2663940 T^{3} - 10644 T^{4} + 185 T^{5} + T^{6} )^{2}$$