# Properties

 Label 29.2.e.a Level 29 Weight 2 Character orbit 29.e Analytic conductor 0.232 Analytic rank 0 Dimension 12 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$29$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 29.e (of order $$14$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.231566165862$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$2$$ over $$\Q(\zeta_{14})$$ Coefficient field: 12.0.7877952219361.1 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 3 x^{11} + 13 x^{9} - 18 x^{8} - 14 x^{7} + 57 x^{6} - 28 x^{5} - 72 x^{4} + 104 x^{3} - 96 x + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{11} + 7 \nu^{10} - 10 \nu^{9} - 7 \nu^{8} + 40 \nu^{7} - 14 \nu^{6} - 83 \nu^{5} + 102 \nu^{4} + 68 \nu^{3} - 176 \nu^{2} + 32 \nu + 96$$$$)/128$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{11} + 5 \nu^{10} - 10 \nu^{9} + 7 \nu^{8} + 4 \nu^{7} - 58 \nu^{6} + 59 \nu^{5} + 38 \nu^{4} - 196 \nu^{3} + 96 \nu^{2} + 256 \nu - 288$$$$)/128$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{11} + 5 \nu^{10} - 10 \nu^{9} + 7 \nu^{8} + 36 \nu^{7} - 58 \nu^{6} - 5 \nu^{5} + 134 \nu^{4} - 100 \nu^{3} - 128 \nu^{2} + 192 \nu - 32$$$$)/128$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{11} - 3 \nu^{10} + 4 \nu^{9} + \nu^{8} - 18 \nu^{7} + 22 \nu^{6} + 17 \nu^{5} - 52 \nu^{4} + 44 \nu^{3} + 40 \nu^{2} - 80 \nu + 64$$$$)/64$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{11} + 5 \nu^{10} - 10 \nu^{9} - 25 \nu^{8} + 68 \nu^{7} + 6 \nu^{6} - 165 \nu^{5} + 134 \nu^{4} + 220 \nu^{3} - 288 \nu^{2} + 224$$$$)/128$$ $$\beta_{6}$$ $$=$$ $$($$$$3 \nu^{11} + \nu^{10} - 18 \nu^{9} + 11 \nu^{8} + 36 \nu^{7} - 50 \nu^{6} - 49 \nu^{5} + 94 \nu^{4} + 12 \nu^{3} - 128 \nu^{2} + 96$$$$)/128$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{10} + 3 \nu^{9} - 9 \nu^{7} + 10 \nu^{6} + 6 \nu^{5} - 29 \nu^{4} + 16 \nu^{3} + 20 \nu^{2} - 40 \nu + 16$$$$)/16$$ $$\beta_{8}$$ $$=$$ $$($$$$\nu^{11} - 5 \nu^{10} + 10 \nu^{9} + 9 \nu^{8} - 36 \nu^{7} + 26 \nu^{6} + 53 \nu^{5} - 86 \nu^{4} - 12 \nu^{3} + 96 \nu^{2} - 64 \nu + 32$$$$)/64$$ $$\beta_{9}$$ $$=$$ $$($$$$\nu^{11} - 5 \nu^{9} + 5 \nu^{8} + 9 \nu^{7} - 16 \nu^{6} - 5 \nu^{5} + 31 \nu^{4} - 12 \nu^{2} + 8 \nu + 16$$$$)/32$$ $$\beta_{10}$$ $$=$$ $$($$$$3 \nu^{11} - 11 \nu^{10} - 14 \nu^{9} + 59 \nu^{8} - 40 \nu^{7} - 122 \nu^{6} + 199 \nu^{5} + 82 \nu^{4} - 420 \nu^{3} + 176 \nu^{2} + 352 \nu - 352$$$$)/128$$ $$\beta_{11}$$ $$=$$ $$($$$$-15 \nu^{11} + 31 \nu^{10} + 30 \nu^{9} - 151 \nu^{8} + 80 \nu^{7} + 290 \nu^{6} - 435 \nu^{5} - 146 \nu^{4} + 740 \nu^{3} - 368 \nu^{2} - 416 \nu + 480$$$$)/128$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{8} + \beta_{5} + \beta_{2}$$ $$\nu^{2}$$ $$=$$ $$\beta_{9} - \beta_{6} - \beta_{4} - \beta_{3} + 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{9} + \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{10} + \beta_{9} - 2 \beta_{8} - 3 \beta_{6} - \beta_{5} - \beta_{3} - 2 \beta_{2} + 2 \beta_{1} + 1$$ $$\nu^{5}$$ $$=$$ $$-\beta_{10} - \beta_{8} - \beta_{7} - \beta_{5} + 5 \beta_{4} + 4 \beta_{3} - 2 \beta_{1} - 2$$ $$\nu^{6}$$ $$=$$ $$2 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} + 3 \beta_{8} - 3 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 2 \beta_{1} - 4$$ $$\nu^{7}$$ $$=$$ $$-5 \beta_{10} + \beta_{9} + 6 \beta_{8} - 5 \beta_{7} + 5 \beta_{6} + 5 \beta_{3} + \beta_{2} - 10 \beta_{1} - 5$$ $$\nu^{8}$$ $$=$$ $$4 \beta_{11} + 4 \beta_{10} + 10 \beta_{9} + 11 \beta_{8} + 4 \beta_{7} + 4 \beta_{6} + \beta_{5} - 6 \beta_{4} + 8 \beta_{3} + \beta_{2} - 4 \beta_{1} - 10$$ $$\nu^{9}$$ $$=$$ $$-6 \beta_{11} - 10 \beta_{10} - 3 \beta_{9} + 4 \beta_{8} - 11 \beta_{6} + 4 \beta_{5} - 15 \beta_{4} - 5 \beta_{3} + 6 \beta_{1} + 5$$ $$\nu^{10}$$ $$=$$ $$2 \beta_{11} + 9 \beta_{9} + 9 \beta_{7} - 7 \beta_{6} - 9 \beta_{5} + \beta_{4} + 9 \beta_{3} + 5 \beta_{2} + 18 \beta_{1} - 1$$ $$\nu^{11}$$ $$=$$ $$-18 \beta_{11} - 29 \beta_{10} - 29 \beta_{9} + 8 \beta_{8} - 28 \beta_{7} - 7 \beta_{6} + \beta_{5} - 39 \beta_{3} + 8 \beta_{2} + 36 \beta_{1} + 11$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/29\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$\beta_{11}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 0.639551 + 1.26134i 1.38491 − 0.286410i 1.23295 + 0.692694i −1.41140 + 0.0891373i 1.23295 − 0.692694i −1.41140 − 0.0891373i −1.25719 − 0.647667i 0.911180 + 1.08155i −1.25719 + 0.647667i 0.911180 − 1.08155i 0.639551 − 1.26134i 1.38491 + 0.286410i
−2.21089 0.504621i −1.23248 2.55926i 2.83146 + 1.36356i 0.0128801 0.0564316i 1.43341 + 6.28018i 1.40728 0.677709i −2.02596 1.61565i −3.16036 + 3.96297i −0.0569531 + 0.118264i
4.2 −0.536089 0.122359i 0.855966 + 1.77743i −1.52952 0.736577i 0.610610 2.67526i −0.241390 1.05760i −4.03077 + 1.94112i 1.58965 + 1.26771i −0.556117 + 0.697349i −0.654683 + 1.35946i
5.1 −0.909335 + 0.725171i 0.960118 0.219141i −0.144024 + 0.631009i −1.18424 1.48499i −0.714155 + 0.895521i −0.339509 1.48749i −1.33591 2.77404i −1.82910 + 0.880850i 2.15374 + 0.491577i
5.2 1.21127 0.965958i −2.86109 + 0.653024i 0.0890656 0.390222i 0.283269 + 0.355208i −2.83476 + 3.55468i −0.759522 3.32768i 1.07536 + 2.23300i 5.05647 2.43507i 0.686232 + 0.156628i
6.1 −0.909335 0.725171i 0.960118 + 0.219141i −0.144024 0.631009i −1.18424 + 1.48499i −0.714155 0.895521i −0.339509 + 1.48749i −1.33591 + 2.77404i −1.82910 0.880850i 2.15374 0.491577i
6.2 1.21127 + 0.965958i −2.86109 0.653024i 0.0890656 + 0.390222i 0.283269 0.355208i −2.83476 3.55468i −0.759522 + 3.32768i 1.07536 2.23300i 5.05647 + 2.43507i 0.686232 0.156628i
9.1 −1.12916 2.34472i −0.343489 + 0.273923i −2.97573 + 3.73144i 2.32488 1.11960i 1.03013 + 0.496082i 0.0468435 + 0.0587399i 7.03485 + 1.60566i −0.624612 + 2.73660i −5.25031 4.18698i
9.2 0.0741982 + 0.154074i −0.879032 + 0.701005i 1.22875 1.54080i −2.54740 + 1.22676i −0.173229 0.0834229i −1.82432 2.28763i 0.662012 + 0.151100i −0.386273 + 1.69237i −0.378025 0.301465i
13.1 −1.12916 + 2.34472i −0.343489 0.273923i −2.97573 3.73144i 2.32488 + 1.11960i 1.03013 0.496082i 0.0468435 0.0587399i 7.03485 1.60566i −0.624612 2.73660i −5.25031 + 4.18698i
13.2 0.0741982 0.154074i −0.879032 0.701005i 1.22875 + 1.54080i −2.54740 1.22676i −0.173229 + 0.0834229i −1.82432 + 2.28763i 0.662012 0.151100i −0.386273 1.69237i −0.378025 + 0.301465i
22.1 −2.21089 + 0.504621i −1.23248 + 2.55926i 2.83146 1.36356i 0.0128801 + 0.0564316i 1.43341 6.28018i 1.40728 + 0.677709i −2.02596 + 1.61565i −3.16036 3.96297i −0.0569531 0.118264i
22.2 −0.536089 + 0.122359i 0.855966 1.77743i −1.52952 + 0.736577i 0.610610 + 2.67526i −0.241390 + 1.05760i −4.03077 1.94112i 1.58965 1.26771i −0.556117 0.697349i −0.654683 1.35946i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 22.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.e even 14 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 29.2.e.a 12
3.b odd 2 1 261.2.o.a 12
4.b odd 2 1 464.2.y.d 12
5.b even 2 1 725.2.q.a 12
5.c odd 4 2 725.2.p.a 24
29.b even 2 1 841.2.e.i 12
29.c odd 4 2 841.2.d.m 24
29.d even 7 1 841.2.b.e 12
29.d even 7 1 841.2.e.a 12
29.d even 7 1 841.2.e.e 12
29.d even 7 1 841.2.e.f 12
29.d even 7 1 841.2.e.h 12
29.d even 7 1 841.2.e.i 12
29.e even 14 1 inner 29.2.e.a 12
29.e even 14 1 841.2.b.e 12
29.e even 14 1 841.2.e.a 12
29.e even 14 1 841.2.e.e 12
29.e even 14 1 841.2.e.f 12
29.e even 14 1 841.2.e.h 12
29.f odd 28 2 841.2.a.k 12
29.f odd 28 4 841.2.d.k 24
29.f odd 28 4 841.2.d.l 24
29.f odd 28 2 841.2.d.m 24
87.h odd 14 1 261.2.o.a 12
87.k even 28 2 7569.2.a.bp 12
116.h odd 14 1 464.2.y.d 12
145.l even 14 1 725.2.q.a 12
145.q odd 28 2 725.2.p.a 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.2.e.a 12 1.a even 1 1 trivial
29.2.e.a 12 29.e even 14 1 inner
261.2.o.a 12 3.b odd 2 1
261.2.o.a 12 87.h odd 14 1
464.2.y.d 12 4.b odd 2 1
464.2.y.d 12 116.h odd 14 1
725.2.p.a 24 5.c odd 4 2
725.2.p.a 24 145.q odd 28 2
725.2.q.a 12 5.b even 2 1
725.2.q.a 12 145.l even 14 1
841.2.a.k 12 29.f odd 28 2
841.2.b.e 12 29.d even 7 1
841.2.b.e 12 29.e even 14 1
841.2.d.k 24 29.f odd 28 4
841.2.d.l 24 29.f odd 28 4
841.2.d.m 24 29.c odd 4 2
841.2.d.m 24 29.f odd 28 2
841.2.e.a 12 29.d even 7 1
841.2.e.a 12 29.e even 14 1
841.2.e.e 12 29.d even 7 1
841.2.e.e 12 29.e even 14 1
841.2.e.f 12 29.d even 7 1
841.2.e.f 12 29.e even 14 1
841.2.e.h 12 29.d even 7 1
841.2.e.h 12 29.e even 14 1
841.2.e.i 12 29.b even 2 1
841.2.e.i 12 29.d even 7 1
7569.2.a.bp 12 87.k even 28 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(29, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 7 T + 27 T^{2} + 70 T^{3} + 130 T^{4} + 161 T^{5} + 84 T^{6} - 147 T^{7} - 401 T^{8} - 301 T^{9} + 646 T^{10} + 2534 T^{11} + 4577 T^{12} + 5068 T^{13} + 2584 T^{14} - 2408 T^{15} - 6416 T^{16} - 4704 T^{17} + 5376 T^{18} + 20608 T^{19} + 33280 T^{20} + 35840 T^{21} + 27648 T^{22} + 14336 T^{23} + 4096 T^{24}$$
$3$ $$1 + 7 T + 29 T^{2} + 91 T^{3} + 244 T^{4} + 546 T^{5} + 1012 T^{6} + 1491 T^{7} + 1497 T^{8} - 259 T^{9} - 5693 T^{10} - 16660 T^{11} - 32900 T^{12} - 49980 T^{13} - 51237 T^{14} - 6993 T^{15} + 121257 T^{16} + 362313 T^{17} + 737748 T^{18} + 1194102 T^{19} + 1600884 T^{20} + 1791153 T^{21} + 1712421 T^{22} + 1240029 T^{23} + 531441 T^{24}$$
$5$ $$1 + T - 11 T^{2} - 11 T^{3} + 66 T^{4} + 76 T^{5} - 174 T^{6} - 69 T^{7} + 399 T^{8} - 1417 T^{9} - 1605 T^{10} + 5700 T^{11} + 14536 T^{12} + 28500 T^{13} - 40125 T^{14} - 177125 T^{15} + 249375 T^{16} - 215625 T^{17} - 2718750 T^{18} + 5937500 T^{19} + 25781250 T^{20} - 21484375 T^{21} - 107421875 T^{22} + 48828125 T^{23} + 244140625 T^{24}$$
$7$ $$1 + 11 T + 47 T^{2} + 123 T^{3} + 458 T^{4} + 2166 T^{5} + 7378 T^{6} + 20319 T^{7} + 61291 T^{8} + 197341 T^{9} + 577023 T^{10} + 1543704 T^{11} + 4058132 T^{12} + 10805928 T^{13} + 28274127 T^{14} + 67687963 T^{15} + 147159691 T^{16} + 341501433 T^{17} + 868014322 T^{18} + 1783794138 T^{19} + 2640278858 T^{20} + 4963493661 T^{21} + 13276336703 T^{22} + 21750594173 T^{23} + 13841287201 T^{24}$$
$11$ $$1 - 7 T + 17 T^{2} + 7 T^{3} - 14 T^{5} - 1448 T^{6} + 5383 T^{7} + 13177 T^{8} - 74137 T^{9} + 167363 T^{10} - 476952 T^{11} + 2425844 T^{12} - 5246472 T^{13} + 20250923 T^{14} - 98676347 T^{15} + 192924457 T^{16} + 866937533 T^{17} - 2565220328 T^{18} - 272820394 T^{19} + 16505633837 T^{21} + 440936218217 T^{22} - 1997181694277 T^{23} + 3138428376721 T^{24}$$
$13$ $$1 - 9 T - 9 T^{2} + 235 T^{3} + 440 T^{4} - 7052 T^{5} + 904 T^{6} + 93021 T^{7} + 10465 T^{8} - 1306603 T^{9} + 1744795 T^{10} + 3749024 T^{11} - 11467928 T^{12} + 48737312 T^{13} + 294870355 T^{14} - 2870606791 T^{15} + 298890865 T^{16} + 34538046153 T^{17} + 4363435336 T^{18} - 442502541884 T^{19} + 358921517240 T^{20} + 2492057352655 T^{21} - 1240726426641 T^{22} - 16129443546333 T^{23} + 23298085122481 T^{24}$$
$17$ $$1 - 133 T^{2} + 8874 T^{4} - 389130 T^{6} + 12426957 T^{8} - 303621745 T^{10} + 5811079504 T^{12} - 87746684305 T^{14} + 1037911875597 T^{16} - 9392652224970 T^{18} + 61902871531434 T^{20} - 268127188759717 T^{22} + 582622237229761 T^{24}$$
$19$ $$1 + 7 T + 31 T^{2} + 189 T^{3} + 1426 T^{4} + 8162 T^{5} + 37516 T^{6} + 183911 T^{7} + 989579 T^{8} + 5075581 T^{9} + 23151601 T^{10} + 103810854 T^{11} + 453309164 T^{12} + 1972406226 T^{13} + 8357727961 T^{14} + 34813410079 T^{15} + 128962924859 T^{16} + 455381843189 T^{17} + 1764973271596 T^{18} + 7295781133718 T^{19} + 24218560896466 T^{20} + 60987974880231 T^{21} + 190063053991831 T^{22} + 815431812287533 T^{23} + 2213314919066161 T^{24}$$
$23$ $$1 + 5 T - 43 T^{2} - 391 T^{3} - 308 T^{4} + 10696 T^{5} + 58590 T^{6} - 14345 T^{7} - 1422557 T^{8} - 5436135 T^{9} + 1966389 T^{10} + 79674994 T^{11} + 420307216 T^{12} + 1832524862 T^{13} + 1040219781 T^{14} - 66141454545 T^{15} - 398089773437 T^{16} - 92329340335 T^{17} + 8673422736510 T^{18} + 36418012981112 T^{19} - 24119783466548 T^{20} - 704250690632033 T^{21} - 1781339982186907 T^{22} + 4764048789569635 T^{23} + 21914624432020321 T^{24}$$
$29$ $$1 + 15 T + 126 T^{2} + 622 T^{3} + 1665 T^{4} - 4109 T^{5} - 52800 T^{6} - 119161 T^{7} + 1400265 T^{8} + 15169958 T^{9} + 89117406 T^{10} + 307667235 T^{11} + 594823321 T^{12}$$
$31$ $$1 + 21 T + 269 T^{2} + 2261 T^{3} + 14784 T^{4} + 77658 T^{5} + 355230 T^{6} + 1212127 T^{7} - 208667 T^{8} - 52988005 T^{9} - 588286643 T^{10} - 4352370134 T^{11} - 26495076348 T^{12} - 134923474154 T^{13} - 565343463923 T^{14} - 1578565656955 T^{15} - 192708356507 T^{16} + 34702166914177 T^{17} + 315267932601630 T^{18} + 2136574586632038 T^{19} + 12609141097527744 T^{20} + 59779985705277131 T^{21} + 220480009197835469 T^{22} + 533578014824501451 T^{23} + 787662783788549761 T^{24}$$
$37$ $$1 - 7 T - 3 T^{2} + 833 T^{3} - 5462 T^{4} - 9800 T^{5} + 411870 T^{6} - 2163693 T^{7} - 6602285 T^{8} + 140542059 T^{9} - 562585377 T^{10} - 2805854072 T^{11} + 36132816832 T^{12} - 103816600664 T^{13} - 770179381113 T^{14} + 7118876914527 T^{15} - 12373745057885 T^{16} - 150039034353201 T^{17} + 1056745736074830 T^{18} - 930332395903400 T^{19} - 19185162777316502 T^{20} + 108258129249299141 T^{21} - 14425753117253547 T^{22} - 1245423352456222891 T^{23} + 6582952005840035281 T^{24}$$
$41$ $$1 - 393 T^{2} + 73710 T^{4} - 8727326 T^{6} + 726625577 T^{8} - 44820888217 T^{10} + 2100378900880 T^{12} - 75343913092777 T^{14} + 2053270217089097 T^{16} - 41455708245189566 T^{18} + 588568838638508910 T^{20} - 5275105108889893593 T^{22} + 22563490300366186081 T^{24}$$
$43$ $$1 - 7 T + 139 T^{2} - 525 T^{3} + 4746 T^{4} + 19698 T^{5} - 227814 T^{6} + 3326155 T^{7} - 15690613 T^{8} + 60933523 T^{9} + 454409879 T^{10} - 6181766668 T^{11} + 56674975708 T^{12} - 265815966724 T^{13} + 840203866271 T^{14} + 4844641613161 T^{15} - 53643083415013 T^{16} + 488972867726665 T^{17} - 1440095001644886 T^{18} + 5354283001585686 T^{19} + 55472198517494346 T^{20} - 263861121266842575 T^{21} + 3003996041546510611 T^{22} - 6505056176298558949 T^{23} + 39959630797262576401 T^{24}$$
$47$ $$1 + 7 T - 9 T^{2} - 497 T^{3} - 5250 T^{4} - 33334 T^{5} - 12592 T^{6} + 1824669 T^{7} + 19098315 T^{8} + 97990109 T^{9} + 127809129 T^{10} - 4616139010 T^{11} - 56042219764 T^{12} - 216958533470 T^{13} + 282330365961 T^{14} + 10173627086707 T^{15} + 93193684837515 T^{16} + 418478724577683 T^{17} - 135731879422768 T^{18} - 16887775097513642 T^{19} - 125009254974245250 T^{20} - 556207845132075199 T^{21} - 473392190122470441 T^{22} + 17305114505588086121 T^{23} +$$$$11\!\cdots\!41$$$$T^{24}$$
$53$ $$1 + 10 T + 25 T^{2} - 278 T^{3} + 2512 T^{4} + 20294 T^{5} - 44832 T^{6} - 1725298 T^{7} + 3740845 T^{8} + 15939776 T^{9} - 71501305 T^{10} - 3189845416 T^{11} + 681878948 T^{12} - 169061807048 T^{13} - 200847165745 T^{14} + 2373066031552 T^{15} + 29517066396445 T^{16} - 721511847681914 T^{17} - 993672638135328 T^{18} + 23839587871852078 T^{19} + 156396342313338832 T^{20} - 917334278520992974 T^{21} + 4372186759137826225 T^{22} + 92690359293721915970 T^{23} +$$$$49\!\cdots\!41$$$$T^{24}$$
$59$ $$( 1 - 22 T + 446 T^{2} - 6050 T^{3} + 72311 T^{4} - 688228 T^{5} + 5840260 T^{6} - 40605452 T^{7} + 251714591 T^{8} - 1242542950 T^{9} + 5404343006 T^{10} - 15728334578 T^{11} + 42180533641 T^{12} )^{2}$$
$61$ $$1 + 7 T + 159 T^{2} + 105 T^{3} + 5012 T^{4} - 64120 T^{5} + 241116 T^{6} - 1456693 T^{7} + 53389705 T^{8} + 83034021 T^{9} + 2548004823 T^{10} - 10608027248 T^{11} + 41335695248 T^{12} - 647089662128 T^{13} + 9481125946383 T^{14} + 18847145120601 T^{15} + 739225366466905 T^{16} - 1230317519492593 T^{17} + 12422386584426876 T^{18} - 201512670645666520 T^{19} + 960837052742372372 T^{20} + 1227885339747584805 T^{21} +$$$$11\!\cdots\!59$$$$T^{22} +$$$$30\!\cdots\!27$$$$T^{23} +$$$$26\!\cdots\!21$$$$T^{24}$$
$67$ $$1 + 37 T + 513 T^{2} + 3025 T^{3} - 450 T^{4} - 212780 T^{5} - 3203838 T^{6} - 28238209 T^{7} - 156569483 T^{8} - 254147545 T^{9} + 8082101273 T^{10} + 110845067132 T^{11} + 930583608704 T^{12} + 7426619497844 T^{13} + 36280552614497 T^{14} - 76438178076835 T^{15} - 3155050596840443 T^{16} - 38125114947613363 T^{17} - 289814002211564622 T^{18} - 1289598215380627940 T^{19} - 182730454900488450 T^{20} + 82299766548792214675 T^{21} +$$$$93\!\cdots\!37$$$$T^{22} +$$$$45\!\cdots\!71$$$$T^{23} +$$$$81\!\cdots\!61$$$$T^{24}$$
$71$ $$1 + 21 T + 173 T^{2} + 721 T^{3} - 4778 T^{4} - 187446 T^{5} - 2306210 T^{6} - 16620177 T^{7} - 82259473 T^{8} + 61660641 T^{9} + 7528298205 T^{10} + 91816724300 T^{11} + 804638755620 T^{12} + 6518987425300 T^{13} + 37950151251405 T^{14} + 22069021680951 T^{15} - 2090351487104113 T^{16} - 29986611162215127 T^{17} - 295426155781449410 T^{18} - 1704843893209759386 T^{19} - 3085410372292246058 T^{20} + 33056769018001751351 T^{21} +$$$$56\!\cdots\!73$$$$T^{22} +$$$$48\!\cdots\!91$$$$T^{23} +$$$$16\!\cdots\!41$$$$T^{24}$$
$73$ $$1 - 14 T + 223 T^{2} - 1638 T^{3} + 14902 T^{4} - 24080 T^{5} - 61436 T^{6} + 7903616 T^{7} - 50738367 T^{8} + 656032846 T^{9} - 446390355 T^{10} + 13646495926 T^{11} + 186938570672 T^{12} + 996194202598 T^{13} - 2378814201795 T^{14} + 255207929652382 T^{15} - 1440880374012447 T^{16} + 16384761811580288 T^{17} - 9297369526291004 T^{18} - 266021356339855760 T^{19} + 12017868289405595062 T^{20} - 96431659028142841494 T^{21} +$$$$95\!\cdots\!27$$$$T^{22} -$$$$43\!\cdots\!78$$$$T^{23} +$$$$22\!\cdots\!21$$$$T^{24}$$
$79$ $$1 - 49 T + 1401 T^{2} - 29659 T^{3} + 510220 T^{4} - 7424550 T^{5} + 93952026 T^{6} - 1051207157 T^{7} + 10574913669 T^{8} - 97442137415 T^{9} + 846233526101 T^{10} - 7200350040954 T^{11} + 62681377749836 T^{12} - 568827653235366 T^{13} + 5281343436396341 T^{14} - 48042773988954185 T^{15} + 411893743975557189 T^{16} - 3234623709121447643 T^{17} + 22838558941382835546 T^{18} -$$$$14\!\cdots\!50$$$$T^{19} +$$$$77\!\cdots\!20$$$$T^{20} -$$$$35\!\cdots\!21$$$$T^{21} +$$$$13\!\cdots\!01$$$$T^{22} -$$$$36\!\cdots\!71$$$$T^{23} +$$$$59\!\cdots\!41$$$$T^{24}$$
$83$ $$1 - 5 T + 19 T^{2} - 291 T^{3} + 6964 T^{4} - 112648 T^{5} + 680674 T^{6} - 2772355 T^{7} + 73849055 T^{8} - 851751547 T^{9} + 7543184631 T^{10} - 61831617966 T^{11} + 242863289344 T^{12} - 5132024291178 T^{13} + 51964998922959 T^{14} - 487020461804489 T^{15} + 3504752157736655 T^{16} - 10920419021824265 T^{17} + 222539811702570706 T^{18} - 3056821871879502296 T^{19} + 15684963104616281524 T^{20} - 54399614282854257273 T^{21} +$$$$29\!\cdots\!31$$$$T^{22} -$$$$64\!\cdots\!35$$$$T^{23} +$$$$10\!\cdots\!61$$$$T^{24}$$
$89$ $$1 - 7 T + 319 T^{2} - 2107 T^{3} + 41592 T^{4} - 268912 T^{5} + 2519080 T^{6} - 17316453 T^{7} - 32462455 T^{8} + 662290951 T^{9} - 28551782921 T^{10} + 290592922368 T^{11} - 3660702381280 T^{12} + 25862770090752 T^{13} - 226158672517241 T^{14} + 466894589435519 T^{15} - 2036767175061655 T^{16} - 96696102997814397 T^{17} + 1251935630434035880 T^{18} - 11894336729426494448 T^{19} +$$$$16\!\cdots\!52$$$$T^{20} -$$$$73\!\cdots\!63$$$$T^{21} +$$$$99\!\cdots\!19$$$$T^{22} -$$$$19\!\cdots\!23$$$$T^{23} +$$$$24\!\cdots\!21$$$$T^{24}$$
$97$ $$1 - 14 T + 441 T^{2} - 5068 T^{3} + 83194 T^{4} - 784252 T^{5} + 9216214 T^{6} - 73602242 T^{7} + 720487153 T^{8} - 5423464648 T^{9} + 47440764385 T^{10} - 407534937824 T^{11} + 3645124614984 T^{12} - 39530888968928 T^{13} + 446370152098465 T^{14} - 4949849750684104 T^{15} + 63784209624826993 T^{16} - 632047495732056194 T^{17} + 7676848253434718806 T^{18} - 63366216198529076476 T^{19} +$$$$65\!\cdots\!34$$$$T^{20} -$$$$38\!\cdots\!56$$$$T^{21} +$$$$32\!\cdots\!09$$$$T^{22} -$$$$10\!\cdots\!42$$$$T^{23} +$$$$69\!\cdots\!41$$$$T^{24}$$