# Properties

 Label 210.3.o.b Level 210 Weight 3 Character orbit 210.o Analytic conductor 5.722 Analytic rank 0 Dimension 16 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$210 = 2 \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 210.o (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.72208555157$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{8}\cdot 7$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{9} q^{2} + ( -1 + \beta_{5} ) q^{3} + ( -2 - 2 \beta_{5} ) q^{4} + \beta_{7} q^{5} + ( -\beta_{1} - 2 \beta_{9} ) q^{6} + ( 2 + \beta_{2} + 2 \beta_{5} - \beta_{6} - \beta_{12} ) q^{7} + 2 \beta_{1} q^{8} -3 \beta_{5} q^{9} +O(q^{10})$$ $$q + \beta_{9} q^{2} + ( -1 + \beta_{5} ) q^{3} + ( -2 - 2 \beta_{5} ) q^{4} + \beta_{7} q^{5} + ( -\beta_{1} - 2 \beta_{9} ) q^{6} + ( 2 + \beta_{2} + 2 \beta_{5} - \beta_{6} - \beta_{12} ) q^{7} + 2 \beta_{1} q^{8} -3 \beta_{5} q^{9} -\beta_{11} q^{10} + ( \beta_{1} + \beta_{3} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{15} ) q^{11} + ( 4 + 2 \beta_{5} ) q^{12} + ( -1 + \beta_{2} + \beta_{3} - 3 \beta_{5} - \beta_{7} - \beta_{8} + \beta_{10} ) q^{13} + ( 1 - 2 \beta_{1} + \beta_{3} + \beta_{5} - \beta_{9} ) q^{14} + ( -\beta_{6} - \beta_{7} ) q^{15} + 4 \beta_{5} q^{16} + ( -\beta_{2} - \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{9} - 5 \beta_{11} ) q^{17} + ( 3 \beta_{1} + 3 \beta_{9} ) q^{18} + ( -5 + 2 \beta_{1} + \beta_{3} - 3 \beta_{5} - \beta_{6} + 6 \beta_{7} + \beta_{9} - \beta_{12} - \beta_{15} ) q^{19} + ( 2 \beta_{6} - 2 \beta_{7} ) q^{20} + ( -4 - 2 \beta_{2} - 3 \beta_{5} + 2 \beta_{6} + \beta_{12} ) q^{21} + ( -3 + \beta_{1} - \beta_{4} + \beta_{11} + \beta_{12} - \beta_{15} ) q^{22} + ( -\beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{23} + ( -2 \beta_{1} + 2 \beta_{9} ) q^{24} + ( 5 + 5 \beta_{5} ) q^{25} + ( -1 + 4 \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} + 2 \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{26} + ( 3 + 6 \beta_{5} ) q^{27} + ( -2 \beta_{5} + 2 \beta_{12} ) q^{28} + ( 5 + 4 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - 6 \beta_{6} - 4 \beta_{7} + \beta_{8} - \beta_{9} - 2 \beta_{10} - 2 \beta_{12} - \beta_{13} - \beta_{14} ) q^{29} + ( \beta_{10} + 2 \beta_{11} ) q^{30} + ( 6 - \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - 4 \beta_{5} - 2 \beta_{6} + \beta_{9} - 7 \beta_{11} - \beta_{12} + \beta_{14} + \beta_{15} ) q^{31} + ( -4 \beta_{1} - 4 \beta_{9} ) q^{32} + ( -3 \beta_{1} - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{14} + 2 \beta_{15} ) q^{33} + ( \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{5} + 11 \beta_{6} - 10 \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} ) q^{34} + ( -2 - \beta_{4} - 3 \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} + \beta_{11} - \beta_{15} ) q^{35} -6 q^{36} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - 7 \beta_{5} + 2 \beta_{6} - \beta_{7} - 2 \beta_{8} + 7 \beta_{9} + 7 \beta_{10} + 12 \beta_{11} - 2 \beta_{12} - \beta_{13} - \beta_{15} ) q^{37} + ( -3 + 3 \beta_{1} + \beta_{2} + \beta_{3} + 4 \beta_{5} - \beta_{6} + \beta_{8} - 3 \beta_{9} - 5 \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} ) q^{38} + ( 4 - \beta_{1} - \beta_{2} - \beta_{3} + 5 \beta_{5} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{10} - \beta_{12} + \beta_{14} + \beta_{15} ) q^{39} -2 \beta_{10} q^{40} + ( 1 - 8 \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{4} + 4 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} + \beta_{8} - 13 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} + 3 \beta_{13} + \beta_{14} ) q^{41} + ( -2 + 2 \beta_{1} - \beta_{3} - \beta_{5} + \beta_{14} ) q^{42} + ( -4 + 6 \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{4} - 2 \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} - 2 \beta_{11} + \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{43} + ( 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{14} ) q^{44} + 3 \beta_{6} q^{45} + ( -1 + \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{7} - \beta_{9} - 2 \beta_{10} - \beta_{11} + 3 \beta_{12} - 2 \beta_{13} - \beta_{14} + \beta_{15} ) q^{46} + ( -2 + 2 \beta_{1} - 2 \beta_{5} + 4 \beta_{7} - 2 \beta_{8} - 10 \beta_{10} + 2 \beta_{12} - 2 \beta_{14} - 2 \beta_{15} ) q^{47} + ( -4 - 8 \beta_{5} ) q^{48} + ( -1 - 19 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + 3 \beta_{5} - 18 \beta_{6} + 13 \beta_{7} + \beta_{8} - 6 \beta_{9} - \beta_{10} - \beta_{11} - 2 \beta_{12} + \beta_{15} ) q^{49} -5 \beta_{1} q^{50} + ( \beta_{1} + \beta_{2} + \beta_{4} + 2 \beta_{5} - 3 \beta_{6} + \beta_{7} + 2 \beta_{9} + 5 \beta_{10} + 10 \beta_{11} + \beta_{12} - \beta_{13} ) q^{51} + ( -4 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} + 2 \beta_{12} - 2 \beta_{14} - 2 \beta_{15} ) q^{52} + ( 2 + 16 \beta_{1} + 2 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 14 \beta_{9} + 2 \beta_{10} + 2 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{53} + ( -6 \beta_{1} - 3 \beta_{9} ) q^{54} + ( -1 - 2 \beta_{3} - \beta_{4} - 5 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{9} - \beta_{10} - 2 \beta_{12} - \beta_{13} - \beta_{15} ) q^{55} + ( 4 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} + 4 \beta_{9} - 2 \beta_{14} ) q^{56} + ( 7 - 4 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - 3 \beta_{6} - 7 \beta_{7} - \beta_{8} + \beta_{10} + 2 \beta_{12} + \beta_{14} + \beta_{15} ) q^{57} + ( -3 - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 11 \beta_{5} - \beta_{6} + \beta_{8} + 3 \beta_{9} + 6 \beta_{10} + 11 \beta_{11} + 5 \beta_{12} + \beta_{13} + 2 \beta_{14} ) q^{58} + ( 8 - 5 \beta_{1} - 3 \beta_{3} - 5 \beta_{5} + 14 \beta_{6} + 3 \beta_{7} + \beta_{8} + 9 \beta_{9} - 3 \beta_{10} - 14 \beta_{11} - 4 \beta_{12} - 2 \beta_{13} - 6 \beta_{14} - \beta_{15} ) q^{59} + ( -2 \beta_{6} + 4 \beta_{7} ) q^{60} + ( 10 + 4 \beta_{1} + 3 \beta_{2} - \beta_{3} + 3 \beta_{4} + 7 \beta_{5} - 2 \beta_{6} + 6 \beta_{7} + 3 \beta_{8} + 5 \beta_{9} - 12 \beta_{10} - 5 \beta_{12} + 3 \beta_{13} + 3 \beta_{14} + 4 \beta_{15} ) q^{61} + ( 5 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} + 12 \beta_{6} - 14 \beta_{7} - 2 \beta_{8} + 10 \beta_{9} + 2 \beta_{10} ) q^{62} + ( 6 + 3 \beta_{2} + 3 \beta_{5} - 3 \beta_{6} ) q^{63} + 8 q^{64} + ( \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - \beta_{4} - 3 \beta_{5} + 3 \beta_{6} + \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} - 2 \beta_{14} - \beta_{15} ) q^{65} + ( 4 - \beta_{1} + \beta_{2} + \beta_{4} - 3 \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} - 2 \beta_{11} - 2 \beta_{12} - \beta_{13} + \beta_{15} ) q^{66} + ( -16 - 28 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} - \beta_{4} - 18 \beta_{5} - 3 \beta_{6} + \beta_{8} - 25 \beta_{9} + 2 \beta_{11} - 5 \beta_{12} + \beta_{13} + \beta_{14} ) q^{67} + ( -2 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} - 10 \beta_{10} - 2 \beta_{12} + 2 \beta_{13} ) q^{68} + ( -1 + \beta_{1} - \beta_{2} - 3 \beta_{3} - \beta_{4} - 3 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + \beta_{8} + 2 \beta_{9} - 3 \beta_{10} - \beta_{11} - \beta_{12} - 2 \beta_{13} - \beta_{15} ) q^{69} + ( 3 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{10} + \beta_{11} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{70} + ( -6 + 4 \beta_{2} + 2 \beta_{5} - 11 \beta_{6} - 8 \beta_{7} + 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} + 2 \beta_{13} + \beta_{14} - 3 \beta_{15} ) q^{71} -6 \beta_{9} q^{72} + ( -16 + 10 \beta_{1} + 2 \beta_{3} + \beta_{4} + 13 \beta_{5} + 4 \beta_{6} - 3 \beta_{7} - 3 \beta_{8} - 14 \beta_{9} + 3 \beta_{10} - 7 \beta_{11} + \beta_{12} + 5 \beta_{14} + 2 \beta_{15} ) q^{73} + ( -12 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - 15 \beta_{5} - 13 \beta_{6} + 24 \beta_{7} - \beta_{8} + 4 \beta_{9} - 2 \beta_{12} - \beta_{13} + 2 \beta_{14} - 3 \beta_{15} ) q^{74} + ( -10 - 5 \beta_{5} ) q^{75} + ( 6 + 2 \beta_{2} - 2 \beta_{3} + 14 \beta_{5} + 10 \beta_{6} - 10 \beta_{7} + 2 \beta_{8} - 4 \beta_{9} - 2 \beta_{10} - 2 \beta_{14} + 2 \beta_{15} ) q^{76} + ( 16 + 23 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} + 23 \beta_{5} - 17 \beta_{6} + 11 \beta_{7} - 7 \beta_{9} + 19 \beta_{10} + 5 \beta_{11} + \beta_{12} + \beta_{13} + 2 \beta_{15} ) q^{77} + ( 2 - 6 \beta_{1} + 3 \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} + \beta_{8} + \beta_{9} - \beta_{11} - 3 \beta_{12} + \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{78} + ( 1 + 2 \beta_{1} - 3 \beta_{3} - 35 \beta_{5} + 9 \beta_{6} - 4 \beta_{7} + \beta_{9} + 10 \beta_{10} + 22 \beta_{11} - \beta_{12} - 2 \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{79} -4 \beta_{6} q^{80} + ( -9 - 9 \beta_{5} ) q^{81} + ( 21 - 6 \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 13 \beta_{5} + \beta_{6} + 4 \beta_{7} + 3 \beta_{8} - \beta_{9} - 4 \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} + 4 \beta_{14} + 4 \beta_{15} ) q^{82} + ( -7 + 3 \beta_{1} + 7 \beta_{2} + \beta_{3} - \beta_{4} - 17 \beta_{5} - 6 \beta_{6} + \beta_{7} - 3 \beta_{8} - \beta_{10} - 3 \beta_{11} + \beta_{12} - 4 \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{83} + ( 4 + 2 \beta_{2} + 6 \beta_{5} - 2 \beta_{6} - 4 \beta_{12} ) q^{84} + ( 8 - 27 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{7} + \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{11} - 3 \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{85} + ( 3 - 4 \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 9 \beta_{5} + 5 \beta_{6} - 4 \beta_{7} - 5 \beta_{8} - 4 \beta_{9} + 4 \beta_{10} + \beta_{11} - 5 \beta_{12} - \beta_{13} + 2 \beta_{14} ) q^{86} + ( -7 - 5 \beta_{1} - \beta_{2} + 2 \beta_{4} + 3 \beta_{5} + 19 \beta_{6} - 3 \beta_{7} - 2 \beta_{8} + 5 \beta_{9} + 3 \beta_{10} - \beta_{11} + 5 \beta_{12} + \beta_{13} + 3 \beta_{14} - \beta_{15} ) q^{87} + ( 4 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} + 6 \beta_{5} + 2 \beta_{6} + 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{13} + 2 \beta_{15} ) q^{88} + ( -26 - 31 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} - \beta_{4} - 13 \beta_{5} + \beta_{6} + 5 \beta_{7} + 3 \beta_{8} - 16 \beta_{9} - 29 \beta_{10} - \beta_{11} - 3 \beta_{12} - 2 \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{89} + ( -3 \beta_{10} - 3 \beta_{11} ) q^{90} + ( -27 + 21 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - 20 \beta_{5} - 21 \beta_{6} + 11 \beta_{7} + 38 \beta_{9} + 19 \beta_{10} + 5 \beta_{11} + \beta_{13} + 3 \beta_{14} + 2 \beta_{15} ) q^{91} + ( 2 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} - 4 \beta_{14} - 2 \beta_{15} ) q^{92} + ( -1 - 3 \beta_{2} - 2 \beta_{3} - \beta_{4} + 14 \beta_{5} + 3 \beta_{6} + \beta_{8} - 2 \beta_{9} + 6 \beta_{10} + 14 \beta_{11} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{93} + ( -2 + 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 20 \beta_{6} - 2 \beta_{9} - 4 \beta_{11} + 2 \beta_{12} - 2 \beta_{14} - 2 \beta_{15} ) q^{94} + ( 27 + 2 \beta_{1} + \beta_{2} - 2 \beta_{4} + 25 \beta_{5} + \beta_{6} - 5 \beta_{7} - \beta_{8} + 3 \beta_{9} - \beta_{10} - \beta_{12} - 2 \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{95} + ( 8 \beta_{1} + 4 \beta_{9} ) q^{96} + ( 10 + 29 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + 23 \beta_{5} + 3 \beta_{7} + 4 \beta_{8} + 57 \beta_{9} - 21 \beta_{10} - 17 \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{97} + ( 13 - 7 \beta_{1} + \beta_{2} + 3 \beta_{3} + \beta_{4} - 24 \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} - 5 \beta_{9} + 20 \beta_{10} + 6 \beta_{11} - 2 \beta_{12} + \beta_{13} + 4 \beta_{14} + \beta_{15} ) q^{98} + ( 6 \beta_{1} - 3 \beta_{6} - 3 \beta_{14} - 3 \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 24q^{3} - 16q^{4} + 4q^{7} + 24q^{9} + O(q^{10})$$ $$16q - 24q^{3} - 16q^{4} + 4q^{7} + 24q^{9} - 4q^{11} + 48q^{12} + 8q^{14} - 32q^{16} + 12q^{17} - 72q^{19} - 24q^{21} - 48q^{22} - 12q^{23} + 40q^{25} + 32q^{28} + 72q^{29} + 120q^{31} + 12q^{33} - 20q^{35} - 96q^{36} + 44q^{37} - 72q^{38} + 36q^{39} - 24q^{42} - 56q^{43} - 8q^{44} + 8q^{46} - 24q^{47} - 40q^{49} - 12q^{51} - 72q^{52} + 32q^{53} + 16q^{56} + 144q^{57} - 88q^{58} + 132q^{59} + 96q^{61} + 60q^{63} + 128q^{64} + 20q^{65} + 72q^{66} - 164q^{67} - 24q^{68} - 136q^{71} - 348q^{73} - 112q^{74} - 120q^{75} + 96q^{77} + 280q^{79} - 72q^{81} + 264q^{82} - 24q^{84} + 120q^{85} - 88q^{86} - 108q^{87} + 48q^{88} - 300q^{89} - 272q^{91} + 48q^{92} - 120q^{93} + 200q^{95} + 384q^{98} - 24q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 92 x^{14} - 112 x^{13} + 5846 x^{12} - 7728 x^{11} + 197216 x^{10} - 298200 x^{9} + 4836403 x^{8} - 6808704 x^{7} + 64376800 x^{6} - 91953512 x^{5} + 595763862 x^{4} - 630430976 x^{3} + 1087013404 x^{2} + 294123256 x + 101626561$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$57754008957283349 \nu^{15} + 676562680999836294 \nu^{14} + 8239103585403158159 \nu^{13} + 54078590776251682475 \nu^{12} + 455975368108340251668 \nu^{11} + 2793391292439344623383 \nu^{10} + 16545566601055515429559 \nu^{9} + 76261143494901216401580 \nu^{8} + 306572240997688184788012 \nu^{7} + 1467974756910134628134465 \nu^{6} + 3497231496721962844392765 \nu^{5} + 10591214719874231223793348 \nu^{4} - 4685579583089306764764583 \nu^{3} + 35874681437029057265441973 \nu^{2} + 9955887688266404429946106 \nu - 1202728311133779179554909109$$$$)/$$$$85\!\cdots\!24$$ $$\beta_{2}$$ $$=$$ $$($$$$-$$$$27\!\cdots\!19$$$$\nu^{15} +$$$$25\!\cdots\!52$$$$\nu^{14} -$$$$30\!\cdots\!02$$$$\nu^{13} +$$$$26\!\cdots\!96$$$$\nu^{12} -$$$$22\!\cdots\!35$$$$\nu^{11} +$$$$17\!\cdots\!62$$$$\nu^{10} -$$$$98\!\cdots\!99$$$$\nu^{9} +$$$$62\!\cdots\!58$$$$\nu^{8} -$$$$28\!\cdots\!05$$$$\nu^{7} +$$$$15\!\cdots\!18$$$$\nu^{6} -$$$$52\!\cdots\!69$$$$\nu^{5} +$$$$21\!\cdots\!26$$$$\nu^{4} -$$$$58\!\cdots\!34$$$$\nu^{3} +$$$$20\!\cdots\!34$$$$\nu^{2} -$$$$31\!\cdots\!69$$$$\nu +$$$$20\!\cdots\!70$$$$)/$$$$31\!\cdots\!48$$ $$\beta_{3}$$ $$=$$ $$($$$$86\!\cdots\!45$$$$\nu^{15} +$$$$16\!\cdots\!16$$$$\nu^{14} +$$$$78\!\cdots\!33$$$$\nu^{13} +$$$$14\!\cdots\!37$$$$\nu^{12} +$$$$32\!\cdots\!90$$$$\nu^{11} +$$$$88\!\cdots\!13$$$$\nu^{10} +$$$$38\!\cdots\!59$$$$\nu^{9} +$$$$29\!\cdots\!12$$$$\nu^{8} -$$$$10\!\cdots\!18$$$$\nu^{7} +$$$$69\!\cdots\!23$$$$\nu^{6} -$$$$72\!\cdots\!31$$$$\nu^{5} +$$$$85\!\cdots\!64$$$$\nu^{4} -$$$$13\!\cdots\!69$$$$\nu^{3} +$$$$71\!\cdots\!81$$$$\nu^{2} -$$$$11\!\cdots\!44$$$$\nu +$$$$56\!\cdots\!21$$$$)/$$$$95\!\cdots\!44$$ $$\beta_{4}$$ $$=$$ $$($$$$-$$$$28\!\cdots\!85$$$$\nu^{15} -$$$$52\!\cdots\!90$$$$\nu^{14} -$$$$22\!\cdots\!43$$$$\nu^{13} -$$$$41\!\cdots\!63$$$$\nu^{12} -$$$$74\!\cdots\!28$$$$\nu^{11} -$$$$25\!\cdots\!91$$$$\nu^{10} +$$$$17\!\cdots\!47$$$$\nu^{9} -$$$$78\!\cdots\!44$$$$\nu^{8} +$$$$68\!\cdots\!48$$$$\nu^{7} -$$$$18\!\cdots\!85$$$$\nu^{6} +$$$$27\!\cdots\!01$$$$\nu^{5} -$$$$20\!\cdots\!96$$$$\nu^{4} +$$$$41\!\cdots\!75$$$$\nu^{3} -$$$$19\!\cdots\!33$$$$\nu^{2} +$$$$30\!\cdots\!48$$$$\nu -$$$$18\!\cdots\!31$$$$)/$$$$31\!\cdots\!48$$ $$\beta_{5}$$ $$=$$ $$($$$$17\!\cdots\!52$$$$\nu^{15} -$$$$54\!\cdots\!68$$$$\nu^{14} +$$$$15\!\cdots\!20$$$$\nu^{13} -$$$$24\!\cdots\!26$$$$\nu^{12} +$$$$10\!\cdots\!84$$$$\nu^{11} -$$$$16\!\cdots\!08$$$$\nu^{10} +$$$$34\!\cdots\!20$$$$\nu^{9} -$$$$61\!\cdots\!59$$$$\nu^{8} +$$$$84\!\cdots\!08$$$$\nu^{7} -$$$$14\!\cdots\!20$$$$\nu^{6} +$$$$11\!\cdots\!24$$$$\nu^{5} -$$$$18\!\cdots\!30$$$$\nu^{4} +$$$$10\!\cdots\!24$$$$\nu^{3} -$$$$13\!\cdots\!92$$$$\nu^{2} +$$$$18\!\cdots\!72$$$$\nu -$$$$13\!\cdots\!85$$$$)/$$$$62\!\cdots\!01$$ $$\beta_{6}$$ $$=$$ $$($$$$-$$$$15\!\cdots\!96$$$$\nu^{15} +$$$$38\!\cdots\!85$$$$\nu^{14} -$$$$13\!\cdots\!72$$$$\nu^{13} +$$$$20\!\cdots\!38$$$$\nu^{12} -$$$$88\!\cdots\!08$$$$\nu^{11} +$$$$14\!\cdots\!26$$$$\nu^{10} -$$$$29\!\cdots\!68$$$$\nu^{9} +$$$$53\!\cdots\!48$$$$\nu^{8} -$$$$73\!\cdots\!36$$$$\nu^{7} +$$$$12\!\cdots\!58$$$$\nu^{6} -$$$$99\!\cdots\!36$$$$\nu^{5} +$$$$16\!\cdots\!18$$$$\nu^{4} -$$$$93\!\cdots\!40$$$$\nu^{3} +$$$$12\!\cdots\!43$$$$\nu^{2} -$$$$16\!\cdots\!76$$$$\nu +$$$$68\!\cdots\!16$$$$)/$$$$43\!\cdots\!02$$ $$\beta_{7}$$ $$=$$ $$($$$$15\!\cdots\!20$$$$\nu^{15} -$$$$67\!\cdots\!15$$$$\nu^{14} +$$$$14\!\cdots\!96$$$$\nu^{13} -$$$$23\!\cdots\!76$$$$\nu^{12} +$$$$92\!\cdots\!60$$$$\nu^{11} -$$$$15\!\cdots\!89$$$$\nu^{10} +$$$$31\!\cdots\!16$$$$\nu^{9} -$$$$59\!\cdots\!84$$$$\nu^{8} +$$$$77\!\cdots\!16$$$$\nu^{7} -$$$$13\!\cdots\!95$$$$\nu^{6} +$$$$10\!\cdots\!04$$$$\nu^{5} -$$$$17\!\cdots\!02$$$$\nu^{4} +$$$$94\!\cdots\!28$$$$\nu^{3} -$$$$12\!\cdots\!24$$$$\nu^{2} +$$$$16\!\cdots\!36$$$$\nu +$$$$10\!\cdots\!40$$$$)/$$$$43\!\cdots\!02$$ $$\beta_{8}$$ $$=$$ $$($$$$-$$$$40\!\cdots\!63$$$$\nu^{15} +$$$$23\!\cdots\!27$$$$\nu^{14} -$$$$37\!\cdots\!65$$$$\nu^{13} +$$$$25\!\cdots\!21$$$$\nu^{12} -$$$$26\!\cdots\!94$$$$\nu^{11} +$$$$16\!\cdots\!94$$$$\nu^{10} -$$$$10\!\cdots\!69$$$$\nu^{9} +$$$$57\!\cdots\!77$$$$\nu^{8} -$$$$28\!\cdots\!42$$$$\nu^{7} +$$$$13\!\cdots\!66$$$$\nu^{6} -$$$$45\!\cdots\!51$$$$\nu^{5} +$$$$17\!\cdots\!11$$$$\nu^{4} -$$$$51\!\cdots\!97$$$$\nu^{3} +$$$$15\!\cdots\!21$$$$\nu^{2} -$$$$25\!\cdots\!62$$$$\nu +$$$$12\!\cdots\!22$$$$)/$$$$95\!\cdots\!44$$ $$\beta_{9}$$ $$=$$ $$($$$$-$$$$13\!\cdots\!02$$$$\nu^{15} +$$$$21\!\cdots\!67$$$$\nu^{14} -$$$$12\!\cdots\!27$$$$\nu^{13} +$$$$17\!\cdots\!98$$$$\nu^{12} -$$$$77\!\cdots\!07$$$$\nu^{11} +$$$$11\!\cdots\!47$$$$\nu^{10} -$$$$26\!\cdots\!76$$$$\nu^{9} +$$$$45\!\cdots\!39$$$$\nu^{8} -$$$$63\!\cdots\!65$$$$\nu^{7} +$$$$10\!\cdots\!73$$$$\nu^{6} -$$$$83\!\cdots\!68$$$$\nu^{5} +$$$$14\!\cdots\!09$$$$\nu^{4} -$$$$76\!\cdots\!85$$$$\nu^{3} +$$$$10\!\cdots\!02$$$$\nu^{2} -$$$$13\!\cdots\!99$$$$\nu +$$$$98\!\cdots\!05$$$$)/$$$$31\!\cdots\!48$$ $$\beta_{10}$$ $$=$$ $$($$$$37\!\cdots\!08$$$$\nu^{15} -$$$$23\!\cdots\!47$$$$\nu^{14} +$$$$34\!\cdots\!47$$$$\nu^{13} -$$$$63\!\cdots\!32$$$$\nu^{12} +$$$$22\!\cdots\!79$$$$\nu^{11} -$$$$42\!\cdots\!09$$$$\nu^{10} +$$$$74\!\cdots\!82$$$$\nu^{9} -$$$$15\!\cdots\!65$$$$\nu^{8} +$$$$18\!\cdots\!33$$$$\nu^{7} -$$$$35\!\cdots\!35$$$$\nu^{6} +$$$$24\!\cdots\!50$$$$\nu^{5} -$$$$45\!\cdots\!11$$$$\nu^{4} +$$$$22\!\cdots\!31$$$$\nu^{3} -$$$$30\!\cdots\!16$$$$\nu^{2} +$$$$38\!\cdots\!01$$$$\nu +$$$$23\!\cdots\!35$$$$)/$$$$73\!\cdots\!88$$ $$\beta_{11}$$ $$=$$ $$($$$$38\!\cdots\!37$$$$\nu^{15} -$$$$24\!\cdots\!57$$$$\nu^{14} +$$$$35\!\cdots\!20$$$$\nu^{13} -$$$$47\!\cdots\!39$$$$\nu^{12} +$$$$22\!\cdots\!83$$$$\nu^{11} -$$$$32\!\cdots\!10$$$$\nu^{10} +$$$$75\!\cdots\!95$$$$\nu^{9} -$$$$12\!\cdots\!37$$$$\nu^{8} +$$$$18\!\cdots\!93$$$$\nu^{7} -$$$$28\!\cdots\!18$$$$\nu^{6} +$$$$24\!\cdots\!61$$$$\nu^{5} -$$$$41\!\cdots\!79$$$$\nu^{4} +$$$$22\!\cdots\!80$$$$\nu^{3} -$$$$29\!\cdots\!03$$$$\nu^{2} +$$$$39\!\cdots\!71$$$$\nu -$$$$16\!\cdots\!40$$$$)/$$$$73\!\cdots\!88$$ $$\beta_{12}$$ $$=$$ $$($$$$-$$$$12\!\cdots\!55$$$$\nu^{15} -$$$$75\!\cdots\!74$$$$\nu^{14} -$$$$11\!\cdots\!05$$$$\nu^{13} +$$$$13\!\cdots\!02$$$$\nu^{12} -$$$$75\!\cdots\!00$$$$\nu^{11} +$$$$97\!\cdots\!32$$$$\nu^{10} -$$$$25\!\cdots\!13$$$$\nu^{9} +$$$$38\!\cdots\!46$$$$\nu^{8} -$$$$63\!\cdots\!12$$$$\nu^{7} +$$$$89\!\cdots\!84$$$$\nu^{6} -$$$$86\!\cdots\!63$$$$\nu^{5} +$$$$12\!\cdots\!74$$$$\nu^{4} -$$$$80\!\cdots\!43$$$$\nu^{3} +$$$$85\!\cdots\!86$$$$\nu^{2} -$$$$19\!\cdots\!38$$$$\nu -$$$$11\!\cdots\!88$$$$)/$$$$95\!\cdots\!44$$ $$\beta_{13}$$ $$=$$ $$($$$$-$$$$54\!\cdots\!54$$$$\nu^{15} +$$$$64\!\cdots\!51$$$$\nu^{14} -$$$$51\!\cdots\!97$$$$\nu^{13} +$$$$64\!\cdots\!74$$$$\nu^{12} -$$$$32\!\cdots\!63$$$$\nu^{11} +$$$$44\!\cdots\!07$$$$\nu^{10} -$$$$11\!\cdots\!18$$$$\nu^{9} +$$$$16\!\cdots\!07$$$$\nu^{8} -$$$$27\!\cdots\!69$$$$\nu^{7} +$$$$38\!\cdots\!69$$$$\nu^{6} -$$$$38\!\cdots\!30$$$$\nu^{5} +$$$$51\!\cdots\!69$$$$\nu^{4} -$$$$35\!\cdots\!13$$$$\nu^{3} +$$$$39\!\cdots\!86$$$$\nu^{2} -$$$$83\!\cdots\!95$$$$\nu -$$$$62\!\cdots\!31$$$$)/$$$$31\!\cdots\!48$$ $$\beta_{14}$$ $$=$$ $$($$$$16\!\cdots\!09$$$$\nu^{15} -$$$$44\!\cdots\!19$$$$\nu^{14} +$$$$15\!\cdots\!46$$$$\nu^{13} -$$$$22\!\cdots\!83$$$$\nu^{12} +$$$$99\!\cdots\!85$$$$\nu^{11} -$$$$15\!\cdots\!40$$$$\nu^{10} +$$$$34\!\cdots\!09$$$$\nu^{9} -$$$$56\!\cdots\!53$$$$\nu^{8} +$$$$85\!\cdots\!27$$$$\nu^{7} -$$$$12\!\cdots\!16$$$$\nu^{6} +$$$$11\!\cdots\!35$$$$\nu^{5} -$$$$17\!\cdots\!59$$$$\nu^{4} +$$$$11\!\cdots\!26$$$$\nu^{3} -$$$$12\!\cdots\!35$$$$\nu^{2} +$$$$27\!\cdots\!75$$$$\nu +$$$$11\!\cdots\!92$$$$)/$$$$95\!\cdots\!44$$ $$\beta_{15}$$ $$=$$ $$($$$$-$$$$28\!\cdots\!68$$$$\nu^{15} -$$$$22\!\cdots\!11$$$$\nu^{14} -$$$$26\!\cdots\!91$$$$\nu^{13} +$$$$10\!\cdots\!23$$$$\nu^{12} -$$$$16\!\cdots\!61$$$$\nu^{11} +$$$$89\!\cdots\!56$$$$\nu^{10} -$$$$55\!\cdots\!42$$$$\nu^{9} +$$$$41\!\cdots\!93$$$$\nu^{8} -$$$$13\!\cdots\!35$$$$\nu^{7} +$$$$90\!\cdots\!88$$$$\nu^{6} -$$$$17\!\cdots\!66$$$$\nu^{5} +$$$$13\!\cdots\!07$$$$\nu^{4} -$$$$16\!\cdots\!05$$$$\nu^{3} +$$$$67\!\cdots\!25$$$$\nu^{2} -$$$$24\!\cdots\!73$$$$\nu -$$$$15\!\cdots\!84$$$$)/$$$$95\!\cdots\!44$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2 \beta_{15} - 3 \beta_{14} - 3 \beta_{13} - 7 \beta_{12} - 2 \beta_{11} - 2 \beta_{10} + 11 \beta_{9} - \beta_{8} + 2 \beta_{7} - 6 \beta_{6} + 3 \beta_{5} - \beta_{4} - \beta_{3} + 7 \beta_{2} + 2 \beta_{1} + 6$$$$)/14$$ $$\nu^{2}$$ $$=$$ $$($$$$-4 \beta_{15} - \beta_{14} - \beta_{13} + 7 \beta_{12} - 3 \beta_{11} + 4 \beta_{10} - 106 \beta_{9} - 5 \beta_{8} + 80 \beta_{7} - 37 \beta_{6} - 328 \beta_{5} + 2 \beta_{4} + 9 \beta_{3} - 7 \beta_{2} - 102 \beta_{1} - 327$$$$)/14$$ $$\nu^{3}$$ $$=$$ $$($$$$-7 \beta_{15} + 13 \beta_{14} + 3 \beta_{13} + 33 \beta_{12} + 5 \beta_{11} + 23 \beta_{10} + 3 \beta_{9} - 8 \beta_{8} - 17 \beta_{7} + 14 \beta_{6} - 5 \beta_{5} - 14 \beta_{4} - 8 \beta_{3} - 2 \beta_{2} + 81 \beta_{1} + 28$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-34 \beta_{15} - 306 \beta_{14} - 124 \beta_{13} - 238 \beta_{12} + 622 \beta_{11} + 104 \beta_{10} + 2550 \beta_{9} + 108 \beta_{8} - 818 \beta_{7} + 1852 \beta_{6} + 5731 \beta_{5} - 74 \beta_{4} - 340 \beta_{3} + 56 \beta_{2} + 232 \beta_{1} + 164$$$$)/7$$ $$\nu^{5}$$ $$=$$ $$($$$$-1117 \beta_{15} + 1560 \beta_{14} + 4836 \beta_{13} - 406 \beta_{12} - 7045 \beta_{11} - 8543 \beta_{10} - 34280 \beta_{9} + 443 \beta_{8} + 6863 \beta_{7} + 362 \beta_{6} - 26382 \beta_{5} + 5413 \beta_{4} + 7107 \beta_{3} - 8127 \beta_{2} - 35263 \beta_{1} - 25842$$$$)/14$$ $$\nu^{6}$$ $$=$$ $$($$$$2061 \beta_{15} + 5494 \beta_{14} + 811 \beta_{13} + 1876 \beta_{12} - 4770 \beta_{11} + 7830 \beta_{10} + 811 \beta_{9} + 373 \beta_{8} - 15516 \beta_{7} - 12763 \beta_{6} + 1184 \beta_{5} - 1811 \beta_{4} + 373 \beta_{3} + 1995 \beta_{2} + 32084 \beta_{1} + 64523$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$46482 \beta_{15} - 354315 \beta_{14} - 259745 \beta_{13} - 345261 \beta_{12} + 559452 \beta_{11} + 65042 \beta_{10} + 1750445 \beta_{9} + 75053 \beta_{8} - 101204 \beta_{7} + 250496 \beta_{6} + 1678491 \beta_{5} - 19517 \beta_{4} - 307833 \beta_{3} + 352709 \beta_{2} + 334798 \beta_{1} + 325744$$$$)/14$$ $$\nu^{8}$$ $$=$$ $$($$$$-315240 \beta_{15} - 42060 \beta_{14} + 478908 \beta_{13} + 254436 \beta_{12} - 1679676 \beta_{11} - 2310768 \beta_{10} - 6757440 \beta_{9} - 357300 \beta_{8} + 4565496 \beta_{7} - 2009568 \beta_{6} - 10803691 \beta_{5} + 585096 \beta_{4} + 1006524 \beta_{3} - 755412 \beta_{2} - 6609192 \beta_{1} - 10594639$$$$)/7$$ $$\nu^{9}$$ $$=$$ $$($$$$413535 \beta_{15} + 2372935 \beta_{14} + 112891 \beta_{13} + 2269825 \beta_{12} - 1436811 \beta_{11} + 3607869 \beta_{10} + 112891 \beta_{9} - 211658 \beta_{8} - 4336941 \beta_{7} - 1554814 \beta_{6} - 98767 \beta_{5} - 1733618 \beta_{4} - 211658 \beta_{3} + 14124 \beta_{2} + 10864805 \beta_{1} + 11318974$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$-2265457 \beta_{15} - 111398577 \beta_{14} - 69460730 \beta_{13} - 83848639 \beta_{12} + 294771937 \beta_{11} + 74881098 \beta_{10} + 698069887 \beta_{9} + 33611164 \beta_{8} - 177400620 \beta_{7} + 399004544 \beta_{6} + 1028061492 \beta_{5} - 8326683 \beta_{4} - 113664034 \beta_{3} + 64929816 \beta_{2} + 103071894 \beta_{1} + 75521956$$$$)/14$$ $$\nu^{11}$$ $$=$$ $$($$$$-243360247 \beta_{15} + 50569868 \beta_{14} + 623548862 \beta_{13} + 22383382 \beta_{12} - 1317124749 \beta_{11} - 1744957007 \beta_{10} - 4918793818 \beta_{9} - 192790379 \beta_{8} + 2514880763 \beta_{7} - 969107890 \beta_{6} - 5623279626 \beta_{5} + 660025487 \beta_{4} + 939862359 \beta_{3} - 783548605 \beta_{2} - 4932887061 \beta_{1} - 5416396004$$$$)/14$$ $$\nu^{12}$$ $$=$$ $$131425990 \beta_{15} + 439882708 \beta_{14} + 25153442 \beta_{13} + 277613336 \beta_{12} - 415417460 \beta_{11} + 721029208 \beta_{10} + 25153442 \beta_{9} + 2844970 \beta_{8} - 1073956258 \beta_{7} - 692915344 \beta_{6} + 27998412 \beta_{5} - 258149834 \beta_{4} + 2844970 \beta_{3} + 53151854 \beta_{2} + 2226950764 \beta_{1} + 3311145295$$ $$\nu^{13}$$ $$=$$ $$($$$$1858626996 \beta_{15} - 49415739999 \beta_{14} - 34314451069 \beta_{13} - 39818081499 \beta_{12} + 108116449488 \beta_{11} + 22710961138 \beta_{10} + 261580508011 \beta_{9} + 13278787213 \beta_{8} - 47123346394 \beta_{7} + 107489354722 \beta_{6} + 315616721213 \beta_{5} - 1822501717 \beta_{4} - 47557113003 \beta_{3} + 38031705061 \beta_{2} + 47593238282 \beta_{1} + 37995579782$$$$)/14$$ $$\nu^{14}$$ $$=$$ $$($$$$-96789315536 \beta_{15} + 697022381 \beta_{14} + 205388683541 \beta_{13} + 31472309085 \beta_{12} - 547413084613 \beta_{11} - 761175861064 \beta_{10} - 1987359475830 \beta_{9} - 96092293155 \beta_{8} + 1256838728428 \beta_{7} - 541601837555 \beta_{6} - 2683523511612 \beta_{5} + 221473349274 \beta_{4} + 334347330543 \beta_{3} - 254339703121 \beta_{2} - 1971971832478 \beta_{1} - 2602818861809$$$$)/14$$ $$\nu^{15}$$ $$=$$ $$($$$$92116109945 \beta_{15} + 363675325983 \beta_{14} + 13560958727 \beta_{13} + 279122846147 \beta_{12} - 289138472617 \beta_{11} + 571259983073 \beta_{10} + 13560958727 \beta_{9} - 10562294418 \beta_{8} - 776330541935 \beta_{7} - 408090250152 \beta_{6} + 2998664309 \beta_{5} - 244437298584 \beta_{4} - 10562294418 \beta_{3} + 16559623036 \beta_{2} + 1660680609871 \beta_{1} + 2122889016574$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/210\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$71$$ $$127$$ $$\chi(n)$$ $$-\beta_{5}$$ $$1$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 −3.67087 − 6.35814i 2.96377 + 5.13339i 1.92573 + 3.33546i −2.63284 − 4.56021i −2.10711 − 3.64962i 2.81422 + 4.87437i 0.848921 + 1.47037i −0.141814 − 0.245629i −3.67087 + 6.35814i 2.96377 − 5.13339i 1.92573 − 3.33546i −2.63284 + 4.56021i −2.10711 + 3.64962i 2.81422 − 4.87437i 0.848921 − 1.47037i −0.141814 + 0.245629i
−0.707107 1.22474i −1.50000 0.866025i −1.00000 + 1.73205i −1.93649 + 1.11803i 2.44949i −2.59373 6.50174i 2.82843 1.50000 + 2.59808i 2.73861 + 1.58114i
31.2 −0.707107 1.22474i −1.50000 0.866025i −1.00000 + 1.73205i −1.93649 + 1.11803i 2.44949i 5.73733 + 4.01037i 2.82843 1.50000 + 2.59808i 2.73861 + 1.58114i
31.3 −0.707107 1.22474i −1.50000 0.866025i −1.00000 + 1.73205i 1.93649 1.11803i 2.44949i −2.67372 + 6.46925i 2.82843 1.50000 + 2.59808i −2.73861 1.58114i
31.4 −0.707107 1.22474i −1.50000 0.866025i −1.00000 + 1.73205i 1.93649 1.11803i 2.44949i 1.94434 6.72455i 2.82843 1.50000 + 2.59808i −2.73861 1.58114i
31.5 0.707107 + 1.22474i −1.50000 0.866025i −1.00000 + 1.73205i −1.93649 + 1.11803i 2.44949i −5.26304 4.61524i −2.82843 1.50000 + 2.59808i −2.73861 1.58114i
31.6 0.707107 + 1.22474i −1.50000 0.866025i −1.00000 + 1.73205i −1.93649 + 1.11803i 2.44949i 6.99242 0.325616i −2.82843 1.50000 + 2.59808i −2.73861 1.58114i
31.7 0.707107 + 1.22474i −1.50000 0.866025i −1.00000 + 1.73205i 1.93649 1.11803i 2.44949i −6.38854 + 2.86123i −2.82843 1.50000 + 2.59808i 2.73861 + 1.58114i
31.8 0.707107 + 1.22474i −1.50000 0.866025i −1.00000 + 1.73205i 1.93649 1.11803i 2.44949i 4.24494 5.56601i −2.82843 1.50000 + 2.59808i 2.73861 + 1.58114i
61.1 −0.707107 + 1.22474i −1.50000 + 0.866025i −1.00000 1.73205i −1.93649 1.11803i 2.44949i −2.59373 + 6.50174i 2.82843 1.50000 2.59808i 2.73861 1.58114i
61.2 −0.707107 + 1.22474i −1.50000 + 0.866025i −1.00000 1.73205i −1.93649 1.11803i 2.44949i 5.73733 4.01037i 2.82843 1.50000 2.59808i 2.73861 1.58114i
61.3 −0.707107 + 1.22474i −1.50000 + 0.866025i −1.00000 1.73205i 1.93649 + 1.11803i 2.44949i −2.67372 6.46925i 2.82843 1.50000 2.59808i −2.73861 + 1.58114i
61.4 −0.707107 + 1.22474i −1.50000 + 0.866025i −1.00000 1.73205i 1.93649 + 1.11803i 2.44949i 1.94434 + 6.72455i 2.82843 1.50000 2.59808i −2.73861 + 1.58114i
61.5 0.707107 1.22474i −1.50000 + 0.866025i −1.00000 1.73205i −1.93649 1.11803i 2.44949i −5.26304 + 4.61524i −2.82843 1.50000 2.59808i −2.73861 + 1.58114i
61.6 0.707107 1.22474i −1.50000 + 0.866025i −1.00000 1.73205i −1.93649 1.11803i 2.44949i 6.99242 + 0.325616i −2.82843 1.50000 2.59808i −2.73861 + 1.58114i
61.7 0.707107 1.22474i −1.50000 + 0.866025i −1.00000 1.73205i 1.93649 + 1.11803i 2.44949i −6.38854 2.86123i −2.82843 1.50000 2.59808i 2.73861 1.58114i
61.8 0.707107 1.22474i −1.50000 + 0.866025i −1.00000 1.73205i 1.93649 + 1.11803i 2.44949i 4.24494 + 5.56601i −2.82843 1.50000 2.59808i 2.73861 1.58114i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 61.8 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 210.3.o.b 16
3.b odd 2 1 630.3.v.c 16
5.b even 2 1 1050.3.p.i 16
5.c odd 4 2 1050.3.q.e 32
7.c even 3 1 1470.3.f.d 16
7.d odd 6 1 inner 210.3.o.b 16
7.d odd 6 1 1470.3.f.d 16
21.g even 6 1 630.3.v.c 16
35.i odd 6 1 1050.3.p.i 16
35.k even 12 2 1050.3.q.e 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.o.b 16 1.a even 1 1 trivial
210.3.o.b 16 7.d odd 6 1 inner
630.3.v.c 16 3.b odd 2 1
630.3.v.c 16 21.g even 6 1
1050.3.p.i 16 5.b even 2 1
1050.3.p.i 16 35.i odd 6 1
1050.3.q.e 32 5.c odd 4 2
1050.3.q.e 32 35.k even 12 2
1470.3.f.d 16 7.c even 3 1
1470.3.f.d 16 7.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{16} + \cdots$$ acting on $$S_{3}^{\mathrm{new}}(210, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 2 T^{2} + 4 T^{4} )^{4}$$
$3$ $$( 1 + 3 T + 3 T^{2} )^{8}$$
$5$ $$( 1 - 5 T^{2} + 25 T^{4} )^{4}$$
$7$ $$1 - 4 T + 28 T^{2} - 352 T^{3} + 2898 T^{4} - 16836 T^{5} + 70272 T^{6} - 1158948 T^{7} + 7794283 T^{8} - 56788452 T^{9} + 168723072 T^{10} - 1980738564 T^{11} + 16706393298 T^{12} - 99431287648 T^{13} + 387556041628 T^{14} - 2712892291396 T^{15} + 33232930569601 T^{16}$$
$11$ $$1 + 4 T - 364 T^{2} - 2552 T^{3} + 54108 T^{4} + 655724 T^{5} - 3422504 T^{6} - 97911028 T^{7} - 146055574 T^{8} + 11147160032 T^{9} + 52178728220 T^{10} - 1271915177612 T^{11} - 12776124769680 T^{12} + 140823145942564 T^{13} + 3084355376442156 T^{14} - 7698059096283072 T^{15} - 472018303966569005 T^{16} - 931465150650251712 T^{17} + 45158047066489605996 T^{18} +$$$$24\!\cdots\!04$$$$T^{19} -$$$$27\!\cdots\!80$$$$T^{20} -$$$$32\!\cdots\!12$$$$T^{21} +$$$$16\!\cdots\!20$$$$T^{22} +$$$$42\!\cdots\!12$$$$T^{23} -$$$$67\!\cdots\!14$$$$T^{24} -$$$$54\!\cdots\!68$$$$T^{25} -$$$$23\!\cdots\!04$$$$T^{26} +$$$$53\!\cdots\!04$$$$T^{27} +$$$$53\!\cdots\!28$$$$T^{28} -$$$$30\!\cdots\!72$$$$T^{29} -$$$$52\!\cdots\!84$$$$T^{30} +$$$$69\!\cdots\!04$$$$T^{31} +$$$$21\!\cdots\!21$$$$T^{32}$$
$13$ $$1 - 1336 T^{2} + 822548 T^{4} - 307870064 T^{6} + 77337463770 T^{8} - 13349067637096 T^{10} + 1484717408181296 T^{12} - 75519988778385192 T^{14} - 843289248891793565 T^{16} -$$$$21\!\cdots\!12$$$$T^{18} +$$$$12\!\cdots\!16$$$$T^{20} -$$$$31\!\cdots\!76$$$$T^{22} +$$$$51\!\cdots\!70$$$$T^{24} -$$$$58\!\cdots\!64$$$$T^{26} +$$$$44\!\cdots\!28$$$$T^{28} -$$$$20\!\cdots\!56$$$$T^{30} +$$$$44\!\cdots\!81$$$$T^{32}$$
$17$ $$1 - 12 T + 420 T^{2} - 4464 T^{3} + 92044 T^{4} - 785292 T^{5} + 23989560 T^{6} + 227912772 T^{7} - 3236850998 T^{8} + 177940653384 T^{9} - 2004860299572 T^{10} + 43692613137348 T^{11} - 466147839356624 T^{12} + 8985915182672820 T^{13} - 29797959923559588 T^{14} - 769960022203743096 T^{15} + 34516652247342574675 T^{16} -$$$$22\!\cdots\!44$$$$T^{17} -$$$$24\!\cdots\!48$$$$T^{18} +$$$$21\!\cdots\!80$$$$T^{19} -$$$$32\!\cdots\!84$$$$T^{20} +$$$$88\!\cdots\!52$$$$T^{21} -$$$$11\!\cdots\!92$$$$T^{22} +$$$$29\!\cdots\!36$$$$T^{23} -$$$$15\!\cdots\!38$$$$T^{24} +$$$$32\!\cdots\!48$$$$T^{25} +$$$$97\!\cdots\!60$$$$T^{26} -$$$$92\!\cdots\!88$$$$T^{27} +$$$$31\!\cdots\!24$$$$T^{28} -$$$$43\!\cdots\!16$$$$T^{29} +$$$$11\!\cdots\!20$$$$T^{30} -$$$$98\!\cdots\!88$$$$T^{31} +$$$$23\!\cdots\!61$$$$T^{32}$$
$19$ $$1 + 72 T + 3812 T^{2} + 150048 T^{3} + 4883342 T^{4} + 136887912 T^{5} + 3384543112 T^{6} + 75691203624 T^{7} + 1567163213049 T^{8} + 30524546336184 T^{9} + 578634766673960 T^{10} + 10979211701479464 T^{11} + 213751725988675694 T^{12} + 4327693265152902096 T^{13} + 88720456587214840044 T^{14} +$$$$17\!\cdots\!80$$$$T^{15} +$$$$34\!\cdots\!48$$$$T^{16} +$$$$64\!\cdots\!80$$$$T^{17} +$$$$11\!\cdots\!24$$$$T^{18} +$$$$20\!\cdots\!76$$$$T^{19} +$$$$36\!\cdots\!54$$$$T^{20} +$$$$67\!\cdots\!64$$$$T^{21} +$$$$12\!\cdots\!60$$$$T^{22} +$$$$24\!\cdots\!64$$$$T^{23} +$$$$45\!\cdots\!69$$$$T^{24} +$$$$78\!\cdots\!84$$$$T^{25} +$$$$12\!\cdots\!12$$$$T^{26} +$$$$18\!\cdots\!32$$$$T^{27} +$$$$23\!\cdots\!82$$$$T^{28} +$$$$26\!\cdots\!88$$$$T^{29} +$$$$24\!\cdots\!92$$$$T^{30} +$$$$16\!\cdots\!72$$$$T^{31} +$$$$83\!\cdots\!61$$$$T^{32}$$
$23$ $$1 + 12 T - 1804 T^{2} - 41256 T^{3} + 1078172 T^{4} + 48109188 T^{5} - 226519400 T^{6} - 29579538300 T^{7} + 68149425194 T^{8} + 16885870838304 T^{9} + 1241401844668 T^{10} - 10296857200481412 T^{11} - 100210062266615184 T^{12} + 3750005340988049388 T^{13} + 80231845600155444876 T^{14} -$$$$51\!\cdots\!40$$$$T^{15} -$$$$38\!\cdots\!85$$$$T^{16} -$$$$27\!\cdots\!60$$$$T^{17} +$$$$22\!\cdots\!16$$$$T^{18} +$$$$55\!\cdots\!32$$$$T^{19} -$$$$78\!\cdots\!04$$$$T^{20} -$$$$42\!\cdots\!88$$$$T^{21} +$$$$27\!\cdots\!28$$$$T^{22} +$$$$19\!\cdots\!36$$$$T^{23} +$$$$41\!\cdots\!34$$$$T^{24} -$$$$95\!\cdots\!00$$$$T^{25} -$$$$38\!\cdots\!00$$$$T^{26} +$$$$43\!\cdots\!52$$$$T^{27} +$$$$51\!\cdots\!52$$$$T^{28} -$$$$10\!\cdots\!84$$$$T^{29} -$$$$24\!\cdots\!24$$$$T^{30} +$$$$85\!\cdots\!88$$$$T^{31} +$$$$37\!\cdots\!21$$$$T^{32}$$
$29$ $$( 1 - 36 T + 1444 T^{2} + 19116 T^{3} - 1274860 T^{4} + 68800452 T^{5} + 84102156 T^{6} - 30035473452 T^{7} + 2426711187974 T^{8} - 25259833173132 T^{9} + 59483856997836 T^{10} + 40924113344941092 T^{11} - 637744142027460460 T^{12} + 8042239471766642316 T^{13} +$$$$51\!\cdots\!04$$$$T^{14} -$$$$10\!\cdots\!16$$$$T^{15} +$$$$25\!\cdots\!21$$$$T^{16} )^{2}$$
$31$ $$1 - 120 T + 11588 T^{2} - 814560 T^{3} + 49123694 T^{4} - 2488742424 T^{5} + 113293093576 T^{6} - 4558808688792 T^{7} + 170270490089625 T^{8} - 5869377562140936 T^{9} + 196357797692622056 T^{10} - 6377368910762118360 T^{11} +$$$$21\!\cdots\!82$$$$T^{12} -$$$$69\!\cdots\!28$$$$T^{13} +$$$$23\!\cdots\!88$$$$T^{14} -$$$$74\!\cdots\!40$$$$T^{15} +$$$$23\!\cdots\!16$$$$T^{16} -$$$$71\!\cdots\!40$$$$T^{17} +$$$$21\!\cdots\!48$$$$T^{18} -$$$$61\!\cdots\!68$$$$T^{19} +$$$$17\!\cdots\!62$$$$T^{20} -$$$$52\!\cdots\!60$$$$T^{21} +$$$$15\!\cdots\!16$$$$T^{22} -$$$$44\!\cdots\!56$$$$T^{23} +$$$$12\!\cdots\!25$$$$T^{24} -$$$$31\!\cdots\!72$$$$T^{25} +$$$$76\!\cdots\!76$$$$T^{26} -$$$$16\!\cdots\!64$$$$T^{27} +$$$$30\!\cdots\!74$$$$T^{28} -$$$$48\!\cdots\!60$$$$T^{29} +$$$$66\!\cdots\!08$$$$T^{30} -$$$$66\!\cdots\!20$$$$T^{31} +$$$$52\!\cdots\!61$$$$T^{32}$$
$37$ $$1 - 44 T - 1956 T^{2} - 11728 T^{3} + 3652806 T^{4} + 242885524 T^{5} - 1534355576 T^{6} - 317275895964 T^{7} - 9605253544351 T^{8} + 75954637404076 T^{9} + 6072970847630504 T^{10} + 248299820626414612 T^{11} + 4168440621293518230 T^{12} +$$$$29\!\cdots\!60$$$$T^{13} +$$$$10\!\cdots\!28$$$$T^{14} -$$$$69\!\cdots\!04$$$$T^{15} -$$$$27\!\cdots\!16$$$$T^{16} -$$$$95\!\cdots\!76$$$$T^{17} +$$$$20\!\cdots\!08$$$$T^{18} +$$$$75\!\cdots\!40$$$$T^{19} +$$$$14\!\cdots\!30$$$$T^{20} +$$$$11\!\cdots\!88$$$$T^{21} +$$$$39\!\cdots\!24$$$$T^{22} +$$$$68\!\cdots\!64$$$$T^{23} -$$$$11\!\cdots\!91$$$$T^{24} -$$$$53\!\cdots\!56$$$$T^{25} -$$$$35\!\cdots\!76$$$$T^{26} +$$$$76\!\cdots\!56$$$$T^{27} +$$$$15\!\cdots\!66$$$$T^{28} -$$$$69\!\cdots\!52$$$$T^{29} -$$$$15\!\cdots\!76$$$$T^{30} -$$$$48\!\cdots\!56$$$$T^{31} +$$$$15\!\cdots\!81$$$$T^{32}$$
$41$ $$1 - 11416 T^{2} + 66218264 T^{4} - 265318066664 T^{6} + 835818107018940 T^{8} - 2208514761298495288 T^{10} +$$$$50\!\cdots\!56$$$$T^{12} -$$$$10\!\cdots\!32$$$$T^{14} +$$$$18\!\cdots\!14$$$$T^{16} -$$$$28\!\cdots\!52$$$$T^{18} +$$$$40\!\cdots\!76$$$$T^{20} -$$$$49\!\cdots\!28$$$$T^{22} +$$$$53\!\cdots\!40$$$$T^{24} -$$$$47\!\cdots\!64$$$$T^{26} +$$$$33\!\cdots\!04$$$$T^{28} -$$$$16\!\cdots\!36$$$$T^{30} +$$$$40\!\cdots\!81$$$$T^{32}$$
$43$ $$( 1 + 28 T + 5732 T^{2} + 194288 T^{3} + 21755610 T^{4} + 646615948 T^{5} + 59916131504 T^{6} + 1611056221164 T^{7} + 122468758030675 T^{8} + 2978842952932236 T^{9} + 204841330302006704 T^{10} + 4087494160581305452 T^{11} +$$$$25\!\cdots\!10$$$$T^{12} +$$$$41\!\cdots\!12$$$$T^{13} +$$$$22\!\cdots\!32$$$$T^{14} +$$$$20\!\cdots\!72$$$$T^{15} +$$$$13\!\cdots\!01$$$$T^{16} )^{2}$$
$47$ $$1 + 24 T + 10040 T^{2} + 236352 T^{3} + 48191588 T^{4} + 529473000 T^{5} + 142018937488 T^{6} - 1533668366952 T^{7} + 289429105887690 T^{8} - 11114511768111792 T^{9} + 616929826339364168 T^{10} - 28449623988679606104 T^{11} +$$$$19\!\cdots\!48$$$$T^{12} -$$$$39\!\cdots\!52$$$$T^{13} +$$$$60\!\cdots\!84$$$$T^{14} -$$$$28\!\cdots\!36$$$$T^{15} +$$$$14\!\cdots\!27$$$$T^{16} -$$$$62\!\cdots\!24$$$$T^{17} +$$$$29\!\cdots\!04$$$$T^{18} -$$$$42\!\cdots\!08$$$$T^{19} +$$$$46\!\cdots\!28$$$$T^{20} -$$$$14\!\cdots\!96$$$$T^{21} +$$$$71\!\cdots\!88$$$$T^{22} -$$$$28\!\cdots\!48$$$$T^{23} +$$$$16\!\cdots\!90$$$$T^{24} -$$$$19\!\cdots\!28$$$$T^{25} +$$$$39\!\cdots\!88$$$$T^{26} +$$$$32\!\cdots\!00$$$$T^{27} +$$$$65\!\cdots\!28$$$$T^{28} +$$$$70\!\cdots\!08$$$$T^{29} +$$$$66\!\cdots\!40$$$$T^{30} +$$$$34\!\cdots\!76$$$$T^{31} +$$$$32\!\cdots\!41$$$$T^{32}$$
$53$ $$1 - 32 T - 12072 T^{2} + 345280 T^{3} + 71529764 T^{4} - 1574568864 T^{5} - 290206976304 T^{6} + 2770742930720 T^{7} + 1006589576083018 T^{8} + 3889510628997504 T^{9} - 3360408915215254232 T^{10} - 25194864185334307040 T^{11} +$$$$10\!\cdots\!32$$$$T^{12} +$$$$34\!\cdots\!88$$$$T^{13} -$$$$28\!\cdots\!00$$$$T^{14} -$$$$63\!\cdots\!04$$$$T^{15} +$$$$75\!\cdots\!55$$$$T^{16} -$$$$17\!\cdots\!36$$$$T^{17} -$$$$22\!\cdots\!00$$$$T^{18} +$$$$75\!\cdots\!52$$$$T^{19} +$$$$64\!\cdots\!52$$$$T^{20} -$$$$44\!\cdots\!60$$$$T^{21} -$$$$16\!\cdots\!12$$$$T^{22} +$$$$53\!\cdots\!76$$$$T^{23} +$$$$39\!\cdots\!78$$$$T^{24} +$$$$30\!\cdots\!80$$$$T^{25} -$$$$88\!\cdots\!04$$$$T^{26} -$$$$13\!\cdots\!76$$$$T^{27} +$$$$17\!\cdots\!84$$$$T^{28} +$$$$23\!\cdots\!20$$$$T^{29} -$$$$22\!\cdots\!92$$$$T^{30} -$$$$17\!\cdots\!68$$$$T^{31} +$$$$15\!\cdots\!41$$$$T^{32}$$
$59$ $$1 - 132 T + 9660 T^{2} - 508464 T^{3} + 17070268 T^{4} - 1368116580 T^{5} + 85370423112 T^{6} - 5735724829716 T^{7} + 274189008275050 T^{8} - 2776853544451464 T^{9} + 245103035882958612 T^{10} + 4667418467579246988 T^{11} +$$$$23\!\cdots\!68$$$$T^{12} -$$$$15\!\cdots\!44$$$$T^{13} +$$$$45\!\cdots\!16$$$$T^{14} -$$$$18\!\cdots\!68$$$$T^{15} -$$$$23\!\cdots\!33$$$$T^{16} -$$$$62\!\cdots\!08$$$$T^{17} +$$$$55\!\cdots\!76$$$$T^{18} -$$$$65\!\cdots\!04$$$$T^{19} +$$$$34\!\cdots\!28$$$$T^{20} +$$$$23\!\cdots\!88$$$$T^{21} +$$$$43\!\cdots\!72$$$$T^{22} -$$$$17\!\cdots\!04$$$$T^{23} +$$$$59\!\cdots\!50$$$$T^{24} -$$$$43\!\cdots\!36$$$$T^{25} +$$$$22\!\cdots\!12$$$$T^{26} -$$$$12\!\cdots\!80$$$$T^{27} +$$$$54\!\cdots\!48$$$$T^{28} -$$$$56\!\cdots\!24$$$$T^{29} +$$$$37\!\cdots\!60$$$$T^{30} -$$$$17\!\cdots\!32$$$$T^{31} +$$$$46\!\cdots\!81$$$$T^{32}$$
$61$ $$1 - 96 T + 17624 T^{2} - 1396992 T^{3} + 159354980 T^{4} - 12864972960 T^{5} + 1094820209872 T^{6} - 90213896412000 T^{7} + 6336184490820426 T^{8} - 516522392725304640 T^{9} + 33635567216910203432 T^{10} -$$$$25\!\cdots\!96$$$$T^{11} +$$$$16\!\cdots\!12$$$$T^{12} -$$$$10\!\cdots\!72$$$$T^{13} +$$$$72\!\cdots\!12$$$$T^{14} -$$$$43\!\cdots\!24$$$$T^{15} +$$$$28\!\cdots\!43$$$$T^{16} -$$$$16\!\cdots\!04$$$$T^{17} +$$$$10\!\cdots\!92$$$$T^{18} -$$$$56\!\cdots\!92$$$$T^{19} +$$$$31\!\cdots\!72$$$$T^{20} -$$$$18\!\cdots\!96$$$$T^{21} +$$$$89\!\cdots\!72$$$$T^{22} -$$$$51\!\cdots\!40$$$$T^{23} +$$$$23\!\cdots\!86$$$$T^{24} -$$$$12\!\cdots\!00$$$$T^{25} +$$$$55\!\cdots\!72$$$$T^{26} -$$$$24\!\cdots\!60$$$$T^{27} +$$$$11\!\cdots\!80$$$$T^{28} -$$$$36\!\cdots\!12$$$$T^{29} +$$$$17\!\cdots\!44$$$$T^{30} -$$$$34\!\cdots\!96$$$$T^{31} +$$$$13\!\cdots\!21$$$$T^{32}$$
$67$ $$1 + 164 T - 6340 T^{2} - 2597968 T^{3} - 31915482 T^{4} + 22697073892 T^{5} + 967634972296 T^{6} - 123392891205548 T^{7} - 9455087268413407 T^{8} + 333882071018564764 T^{9} + 51639925158822741896 T^{10} +$$$$26\!\cdots\!32$$$$T^{11} -$$$$16\!\cdots\!86$$$$T^{12} -$$$$60\!\cdots\!36$$$$T^{13} +$$$$18\!\cdots\!60$$$$T^{14} +$$$$15\!\cdots\!28$$$$T^{15} +$$$$54\!\cdots\!88$$$$T^{16} +$$$$70\!\cdots\!92$$$$T^{17} +$$$$36\!\cdots\!60$$$$T^{18} -$$$$55\!\cdots\!84$$$$T^{19} -$$$$67\!\cdots\!26$$$$T^{20} +$$$$47\!\cdots\!68$$$$T^{21} +$$$$42\!\cdots\!56$$$$T^{22} +$$$$12\!\cdots\!56$$$$T^{23} -$$$$15\!\cdots\!67$$$$T^{24} -$$$$91\!\cdots\!32$$$$T^{25} +$$$$32\!\cdots\!96$$$$T^{26} +$$$$33\!\cdots\!88$$$$T^{27} -$$$$21\!\cdots\!22$$$$T^{28} -$$$$78\!\cdots\!92$$$$T^{29} -$$$$85\!\cdots\!40$$$$T^{30} +$$$$99\!\cdots\!36$$$$T^{31} +$$$$27\!\cdots\!61$$$$T^{32}$$
$71$ $$( 1 + 68 T + 26092 T^{2} + 2195028 T^{3} + 351820772 T^{4} + 28773766364 T^{5} + 3188424927972 T^{6} + 217589962697708 T^{7} + 19655929349959526 T^{8} + 1096871001959146028 T^{9} + 81023237162072440932 T^{10} +$$$$36\!\cdots\!44$$$$T^{11} +$$$$22\!\cdots\!92$$$$T^{12} +$$$$71\!\cdots\!28$$$$T^{13} +$$$$42\!\cdots\!72$$$$T^{14} +$$$$56\!\cdots\!08$$$$T^{15} +$$$$41\!\cdots\!21$$$$T^{16} )^{2}$$
$73$ $$1 + 348 T + 82708 T^{2} + 14734320 T^{3} + 2214323678 T^{4} + 290251205388 T^{5} + 34312832193656 T^{6} + 3733330194438684 T^{7} + 379285162573239545 T^{8} + 36467064302261366820 T^{9} +$$$$33\!\cdots\!56$$$$T^{10} +$$$$29\!\cdots\!16$$$$T^{11} +$$$$24\!\cdots\!70$$$$T^{12} +$$$$20\!\cdots\!16$$$$T^{13} +$$$$16\!\cdots\!72$$$$T^{14} +$$$$12\!\cdots\!12$$$$T^{15} +$$$$91\!\cdots\!12$$$$T^{16} +$$$$65\!\cdots\!48$$$$T^{17} +$$$$45\!\cdots\!52$$$$T^{18} +$$$$30\!\cdots\!24$$$$T^{19} +$$$$20\!\cdots\!70$$$$T^{20} +$$$$12\!\cdots\!84$$$$T^{21} +$$$$76\!\cdots\!76$$$$T^{22} +$$$$44\!\cdots\!80$$$$T^{23} +$$$$24\!\cdots\!45$$$$T^{24} +$$$$12\!\cdots\!96$$$$T^{25} +$$$$63\!\cdots\!56$$$$T^{26} +$$$$28\!\cdots\!52$$$$T^{27} +$$$$11\!\cdots\!98$$$$T^{28} +$$$$41\!\cdots\!80$$$$T^{29} +$$$$12\!\cdots\!48$$$$T^{30} +$$$$27\!\cdots\!52$$$$T^{31} +$$$$42\!\cdots\!21$$$$T^{32}$$
$79$ $$1 - 280 T + 14532 T^{2} + 1622944 T^{3} + 35326638 T^{4} - 30655524856 T^{5} + 55399514824 T^{6} + 161497193433480 T^{7} + 10355496367899353 T^{8} - 1123575678148193512 T^{9} - 61265347082641448728 T^{10} +$$$$14\!\cdots\!40$$$$T^{11} +$$$$44\!\cdots\!62$$$$T^{12} -$$$$73\!\cdots\!60$$$$T^{13} +$$$$53\!\cdots\!92$$$$T^{14} -$$$$10\!\cdots\!56$$$$T^{15} +$$$$35\!\cdots\!48$$$$T^{16} -$$$$66\!\cdots\!96$$$$T^{17} +$$$$20\!\cdots\!52$$$$T^{18} -$$$$17\!\cdots\!60$$$$T^{19} +$$$$67\!\cdots\!82$$$$T^{20} +$$$$13\!\cdots\!40$$$$T^{21} -$$$$36\!\cdots\!48$$$$T^{22} -$$$$41\!\cdots\!72$$$$T^{23} +$$$$23\!\cdots\!13$$$$T^{24} +$$$$23\!\cdots\!80$$$$T^{25} +$$$$49\!\cdots\!24$$$$T^{26} -$$$$17\!\cdots\!96$$$$T^{27} +$$$$12\!\cdots\!78$$$$T^{28} +$$$$35\!\cdots\!24$$$$T^{29} +$$$$19\!\cdots\!52$$$$T^{30} -$$$$23\!\cdots\!80$$$$T^{31} +$$$$52\!\cdots\!41$$$$T^{32}$$
$83$ $$1 - 66184 T^{2} + 2113671608 T^{4} - 43986302935160 T^{6} + 678322851950602236 T^{8} -$$$$83\!\cdots\!72$$$$T^{10} +$$$$84\!\cdots\!60$$$$T^{12} -$$$$72\!\cdots\!04$$$$T^{14} +$$$$53\!\cdots\!06$$$$T^{16} -$$$$34\!\cdots\!84$$$$T^{18} +$$$$19\!\cdots\!60$$$$T^{20} -$$$$88\!\cdots\!92$$$$T^{22} +$$$$34\!\cdots\!16$$$$T^{24} -$$$$10\!\cdots\!60$$$$T^{26} +$$$$24\!\cdots\!68$$$$T^{28} -$$$$35\!\cdots\!44$$$$T^{30} +$$$$25\!\cdots\!61$$$$T^{32}$$
$89$ $$1 + 300 T + 54668 T^{2} + 7400400 T^{3} + 757620572 T^{4} + 53632718700 T^{5} + 870152589352 T^{6} - 480156239482500 T^{7} - 98736006237054870 T^{8} - 12632355432189921000 T^{9} -$$$$12\!\cdots\!92$$$$T^{10} -$$$$89\!\cdots\!00$$$$T^{11} -$$$$37\!\cdots\!44$$$$T^{12} +$$$$17\!\cdots\!40$$$$T^{13} +$$$$60\!\cdots\!12$$$$T^{14} +$$$$83\!\cdots\!20$$$$T^{15} +$$$$83\!\cdots\!23$$$$T^{16} +$$$$66\!\cdots\!20$$$$T^{17} +$$$$37\!\cdots\!92$$$$T^{18} +$$$$84\!\cdots\!40$$$$T^{19} -$$$$14\!\cdots\!64$$$$T^{20} -$$$$27\!\cdots\!00$$$$T^{21} -$$$$30\!\cdots\!32$$$$T^{22} -$$$$24\!\cdots\!00$$$$T^{23} -$$$$15\!\cdots\!70$$$$T^{24} -$$$$58\!\cdots\!00$$$$T^{25} +$$$$84\!\cdots\!52$$$$T^{26} +$$$$41\!\cdots\!00$$$$T^{27} +$$$$46\!\cdots\!52$$$$T^{28} +$$$$35\!\cdots\!00$$$$T^{29} +$$$$20\!\cdots\!08$$$$T^{30} +$$$$90\!\cdots\!00$$$$T^{31} +$$$$24\!\cdots\!21$$$$T^{32}$$
$97$ $$1 - 70448 T^{2} + 2604307832 T^{4} - 66719407643536 T^{6} + 1323480529449253148 T^{8} -$$$$21\!\cdots\!72$$$$T^{10} +$$$$29\!\cdots\!68$$$$T^{12} -$$$$34\!\cdots\!84$$$$T^{14} +$$$$34\!\cdots\!38$$$$T^{16} -$$$$30\!\cdots\!04$$$$T^{18} +$$$$23\!\cdots\!48$$$$T^{20} -$$$$14\!\cdots\!52$$$$T^{22} +$$$$81\!\cdots\!08$$$$T^{24} -$$$$36\!\cdots\!36$$$$T^{26} +$$$$12\!\cdots\!92$$$$T^{28} -$$$$30\!\cdots\!28$$$$T^{30} +$$$$37\!\cdots\!41$$$$T^{32}$$