# Properties

 Label 392.3.k.n Level 392 Weight 3 Character orbit 392.k Analytic conductor 10.681 Analytic rank 0 Dimension 16 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$392 = 2^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 392.k (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.6812263629$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{14}$$ Twist minimal: no (minimal twist has level 56) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -1 + \beta_{2} + \beta_{6} + \beta_{9} ) q^{3} + ( -1 + \beta_{2} + \beta_{3} - \beta_{12} ) q^{4} + ( \beta_{1} + \beta_{3} - \beta_{10} ) q^{5} + ( 3 - \beta_{2} - \beta_{7} - \beta_{8} - \beta_{10} - \beta_{14} ) q^{6} + ( 1 - \beta_{1} + \beta_{2} - \beta_{6} + \beta_{7} + \beta_{10} - \beta_{13} + \beta_{14} ) q^{8} + ( -\beta_{1} - 6 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{8} - \beta_{9} + \beta_{14} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -1 + \beta_{2} + \beta_{6} + \beta_{9} ) q^{3} + ( -1 + \beta_{2} + \beta_{3} - \beta_{12} ) q^{4} + ( \beta_{1} + \beta_{3} - \beta_{10} ) q^{5} + ( 3 - \beta_{2} - \beta_{7} - \beta_{8} - \beta_{10} - \beta_{14} ) q^{6} + ( 1 - \beta_{1} + \beta_{2} - \beta_{6} + \beta_{7} + \beta_{10} - \beta_{13} + \beta_{14} ) q^{8} + ( -\beta_{1} - 6 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{8} - \beta_{9} + \beta_{14} ) q^{9} + ( 2 - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} + 2 \beta_{9} + \beta_{11} - \beta_{12} - \beta_{13} ) q^{10} + ( 4 - 4 \beta_{2} - \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{11} - \beta_{13} + \beta_{15} ) q^{11} + ( 3 \beta_{1} + 3 \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{8} - \beta_{9} + \beta_{11} - 2 \beta_{14} ) q^{12} + ( -1 - \beta_{1} + \beta_{2} - \beta_{6} + \beta_{7} + \beta_{10} + \beta_{12} - 2 \beta_{14} + 2 \beta_{15} ) q^{13} + ( -1 + 3 \beta_{1} + 3 \beta_{6} - \beta_{8} - \beta_{12} + 2 \beta_{13} - 3 \beta_{14} + 3 \beta_{15} ) q^{15} + ( 2 \beta_{1} + 9 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{14} ) q^{16} + ( -10 + 10 \beta_{2} - 4 \beta_{6} - 4 \beta_{9} ) q^{17} + ( 3 - 3 \beta_{2} - \beta_{3} - 2 \beta_{5} + 6 \beta_{6} + 2 \beta_{7} + 3 \beta_{9} - \beta_{11} + \beta_{12} + \beta_{13} + 2 \beta_{15} ) q^{18} + ( \beta_{1} + 7 \beta_{2} - \beta_{8} - \beta_{9} ) q^{19} + ( 13 + \beta_{1} + 2 \beta_{2} + \beta_{6} + 2 \beta_{7} + 3 \beta_{8} + 2 \beta_{10} - \beta_{12} - \beta_{13} - 2 \beta_{15} ) q^{20} + ( 11 + 3 \beta_{1} + 3 \beta_{6} + 5 \beta_{8} + 3 \beta_{12} - \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{22} + ( 3 \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{8} + \beta_{9} + 2 \beta_{10} + 2 \beta_{11} + \beta_{14} ) q^{23} + ( 4 \beta_{3} + 4 \beta_{5} - 6 \beta_{6} + 4 \beta_{7} - 2 \beta_{9} - 2 \beta_{11} - 4 \beta_{12} + 2 \beta_{13} - 4 \beta_{15} ) q^{24} + ( 2 - 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{9} - \beta_{12} - \beta_{15} ) q^{25} + ( 3 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + 4 \beta_{5} - 2 \beta_{8} - 2 \beta_{9} - 7 \beta_{10} + \beta_{11} - \beta_{14} ) q^{26} + ( 4 - 8 \beta_{1} - 8 \beta_{6} + 4 \beta_{8} + 2 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} ) q^{27} + ( -2 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{6} - 2 \beta_{7} - 4 \beta_{8} - 2 \beta_{10} - 6 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{29} + ( 3 \beta_{1} - 15 \beta_{2} + 5 \beta_{3} + 6 \beta_{5} - 3 \beta_{8} - 3 \beta_{9} - 6 \beta_{10} - 3 \beta_{11} ) q^{30} + ( -6 \beta_{3} + 6 \beta_{7} - 6 \beta_{9} + 6 \beta_{12} ) q^{31} + ( 4 - 4 \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} - 6 \beta_{6} - 7 \beta_{7} + \beta_{9} - \beta_{12} + 2 \beta_{15} ) q^{32} + ( 8 \beta_{1} + 4 \beta_{2} + 4 \beta_{4} + 4 \beta_{5} - 4 \beta_{11} + 4 \beta_{14} ) q^{33} + ( -12 - 14 \beta_{1} + 4 \beta_{2} - 14 \beta_{6} + 4 \beta_{7} + 4 \beta_{8} + 4 \beta_{10} + 4 \beta_{14} ) q^{34} + ( 3 + 10 \beta_{1} - 10 \beta_{2} + 10 \beta_{6} - 10 \beta_{7} - 4 \beta_{8} - 10 \beta_{10} + 5 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} ) q^{36} + ( 6 \beta_{1} - 8 \beta_{2} - 2 \beta_{3} - 6 \beta_{4} + 6 \beta_{5} - 8 \beta_{8} - 8 \beta_{9} - 10 \beta_{10} + 2 \beta_{11} - 6 \beta_{14} ) q^{37} + ( -3 + 3 \beta_{2} - \beta_{4} - 8 \beta_{6} + \beta_{7} - \beta_{9} ) q^{38} + ( 7 - 7 \beta_{2} - 11 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} + 11 \beta_{6} - 4 \beta_{7} - 3 \beta_{9} - 2 \beta_{11} + 11 \beta_{12} + 2 \beta_{13} + 3 \beta_{15} ) q^{39} + ( 14 \beta_{1} + 14 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 4 \beta_{8} - 4 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{14} ) q^{40} + ( -16 + 14 \beta_{1} + 14 \beta_{6} - 2 \beta_{8} - 2 \beta_{12} - 4 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{41} + ( -6 \beta_{1} - 6 \beta_{6} - 4 \beta_{8} + 4 \beta_{12} + \beta_{13} + 3 \beta_{14} + 3 \beta_{15} ) q^{43} + ( 4 \beta_{1} + 2 \beta_{2} + 6 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} + 6 \beta_{8} + 6 \beta_{9} + 6 \beta_{10} + 4 \beta_{11} + 2 \beta_{14} ) q^{44} + ( 7 - 7 \beta_{2} - 13 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - 7 \beta_{7} + 4 \beta_{11} + 13 \beta_{12} - 4 \beta_{13} + 2 \beta_{15} ) q^{45} + ( 19 - 19 \beta_{2} + 3 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} - 5 \beta_{9} - 5 \beta_{11} - 3 \beta_{12} + 5 \beta_{13} + 2 \beta_{15} ) q^{46} + ( 14 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 4 \beta_{8} + 4 \beta_{9} - 2 \beta_{10} + 4 \beta_{11} + 2 \beta_{14} ) q^{47} + ( -22 + 6 \beta_{1} + 2 \beta_{2} + 6 \beta_{6} + 2 \beta_{7} - 16 \beta_{8} + 2 \beta_{10} - 8 \beta_{12} - 2 \beta_{13} + 2 \beta_{14} ) q^{48} + ( 3 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{6} + 2 \beta_{7} - 3 \beta_{8} + 2 \beta_{10} + \beta_{12} + \beta_{13} + 2 \beta_{15} ) q^{50} + ( 18 \beta_{1} + 46 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} - 10 \beta_{8} - 10 \beta_{9} - 4 \beta_{14} ) q^{51} + ( 17 - 17 \beta_{2} + 5 \beta_{3} + 2 \beta_{5} + 5 \beta_{6} - 6 \beta_{7} + 17 \beta_{9} - 3 \beta_{11} - 5 \beta_{12} + 3 \beta_{13} - 2 \beta_{15} ) q^{52} + ( 4 - 4 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 4 \beta_{7} + 2 \beta_{11} + 4 \beta_{12} - 2 \beta_{13} - 2 \beta_{15} ) q^{53} + ( 2 \beta_{1} - 30 \beta_{2} - 6 \beta_{3} + 4 \beta_{5} - 6 \beta_{8} - 6 \beta_{9} + 4 \beta_{10} + 2 \beta_{11} ) q^{54} + ( 4 + 12 \beta_{1} + 6 \beta_{2} + 12 \beta_{6} + 6 \beta_{7} + 10 \beta_{8} + 6 \beta_{10} + 6 \beta_{12} + 4 \beta_{13} + 4 \beta_{14} - 4 \beta_{15} ) q^{55} + ( 7 + 9 \beta_{1} + 9 \beta_{6} - 7 \beta_{8} - \beta_{12} - \beta_{14} - \beta_{15} ) q^{57} + ( -4 \beta_{2} + 2 \beta_{4} + 4 \beta_{5} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - 8 \beta_{11} + 2 \beta_{14} ) q^{58} + ( 13 - 13 \beta_{2} + 8 \beta_{3} + 6 \beta_{4} + 6 \beta_{5} + 7 \beta_{6} - 13 \beta_{9} + 2 \beta_{11} - 8 \beta_{12} - 2 \beta_{13} - 6 \beta_{15} ) q^{59} + ( -28 + 28 \beta_{2} + 6 \beta_{3} - 2 \beta_{4} + 18 \beta_{6} + 2 \beta_{7} + 24 \beta_{9} + 2 \beta_{11} - 6 \beta_{12} - 2 \beta_{13} ) q^{60} + ( 3 \beta_{1} - 8 \beta_{2} + 3 \beta_{3} - 8 \beta_{4} + 8 \beta_{5} - 8 \beta_{8} - 8 \beta_{9} - 11 \beta_{10} - 8 \beta_{14} ) q^{61} + ( -6 + 6 \beta_{1} - 12 \beta_{2} + 6 \beta_{6} - 12 \beta_{7} - 6 \beta_{8} - 12 \beta_{10} + 6 \beta_{12} + 6 \beta_{13} ) q^{62} + ( -14 - 8 \beta_{1} + 11 \beta_{2} - 8 \beta_{6} + 11 \beta_{7} + 15 \beta_{8} + 11 \beta_{10} - 5 \beta_{12} - 2 \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{64} + ( -21 \beta_{1} + 9 \beta_{2} + 5 \beta_{3} + 5 \beta_{4} + 5 \beta_{5} + 11 \beta_{8} + 11 \beta_{9} + 5 \beta_{14} ) q^{65} + ( -44 + 44 \beta_{2} + 12 \beta_{3} + 8 \beta_{4} - 8 \beta_{5} + 20 \beta_{9} - 4 \beta_{11} - 12 \beta_{12} + 4 \beta_{13} + 8 \beta_{15} ) q^{66} + ( -38 + 38 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} - 16 \beta_{6} - 6 \beta_{9} - \beta_{11} + 4 \beta_{12} + \beta_{13} + 3 \beta_{15} ) q^{67} + ( -12 \beta_{1} - 26 \beta_{2} - 10 \beta_{3} + 8 \beta_{4} - 8 \beta_{5} + 4 \beta_{8} + 4 \beta_{9} - 4 \beta_{11} + 8 \beta_{14} ) q^{68} + ( 4 - 20 \beta_{1} - 12 \beta_{2} - 20 \beta_{6} - 12 \beta_{7} - 8 \beta_{8} - 12 \beta_{10} + 2 \beta_{12} - 12 \beta_{13} + 6 \beta_{14} - 6 \beta_{15} ) q^{69} + ( 6 + 6 \beta_{1} - 10 \beta_{2} + 6 \beta_{6} - 10 \beta_{7} - 4 \beta_{8} - 10 \beta_{10} - 8 \beta_{12} + 10 \beta_{14} - 10 \beta_{15} ) q^{71} + ( -11 \beta_{1} + 32 \beta_{2} + 6 \beta_{3} - 11 \beta_{4} + 4 \beta_{5} + 14 \beta_{8} + 14 \beta_{9} + 9 \beta_{10} + 9 \beta_{11} - 11 \beta_{14} ) q^{72} + ( -14 + 14 \beta_{2} - 4 \beta_{4} - 4 \beta_{5} + 8 \beta_{6} + 4 \beta_{11} - 4 \beta_{13} + 4 \beta_{15} ) q^{73} + ( 2 - 2 \beta_{2} + 4 \beta_{3} - 6 \beta_{4} + 12 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} + 10 \beta_{9} - 4 \beta_{11} - 4 \beta_{12} + 4 \beta_{13} - 12 \beta_{15} ) q^{74} + ( -13 \beta_{1} + 9 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} + 9 \beta_{8} + 9 \beta_{9} + 2 \beta_{11} - 2 \beta_{14} ) q^{75} + ( -11 - 3 \beta_{1} - 3 \beta_{6} + \beta_{8} - 7 \beta_{12} - \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{76} + ( -29 - \beta_{1} - 6 \beta_{2} - \beta_{6} - 6 \beta_{7} + 9 \beta_{8} - 6 \beta_{10} + 17 \beta_{12} + 9 \beta_{13} - 8 \beta_{14} + 6 \beta_{15} ) q^{78} + ( -22 \beta_{1} - 4 \beta_{2} - 16 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 4 \beta_{8} - 4 \beta_{9} + 10 \beta_{10} - 4 \beta_{11} + 2 \beta_{14} ) q^{79} + ( 6 - 6 \beta_{2} + 10 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} - 18 \beta_{6} + 8 \beta_{7} - 6 \beta_{9} - 2 \beta_{11} - 10 \beta_{12} + 2 \beta_{13} + 4 \beta_{15} ) q^{80} + ( -6 + 6 \beta_{2} + 5 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 29 \beta_{6} + 3 \beta_{9} + 8 \beta_{11} - 5 \beta_{12} - 8 \beta_{13} + 3 \beta_{15} ) q^{81} + ( -16 \beta_{1} + 62 \beta_{2} + 14 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} + 10 \beta_{8} + 10 \beta_{9} - 2 \beta_{11} + 4 \beta_{14} ) q^{82} + ( -9 - 27 \beta_{1} - 27 \beta_{6} - \beta_{8} + 8 \beta_{12} + 6 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{83} + ( -10 - 26 \beta_{1} + 14 \beta_{2} - 26 \beta_{6} + 14 \beta_{7} + 4 \beta_{8} + 14 \beta_{10} - 10 \beta_{12} - 8 \beta_{13} + 12 \beta_{14} - 12 \beta_{15} ) q^{85} + ( \beta_{1} - 31 \beta_{2} - 7 \beta_{3} - 2 \beta_{4} - 6 \beta_{5} + 7 \beta_{8} + 7 \beta_{9} - 8 \beta_{10} - 3 \beta_{11} - 2 \beta_{14} ) q^{86} + ( 14 - 14 \beta_{2} - 18 \beta_{3} - 10 \beta_{4} + 10 \beta_{5} - 2 \beta_{6} - 8 \beta_{7} - 6 \beta_{9} + 8 \beta_{11} + 18 \beta_{12} - 8 \beta_{13} - 10 \beta_{15} ) q^{87} + ( 56 - 56 \beta_{2} + 6 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} + 6 \beta_{7} - 14 \beta_{9} + 4 \beta_{11} - 6 \beta_{12} - 4 \beta_{13} + 4 \beta_{15} ) q^{88} + ( 2 \beta_{1} - 64 \beta_{2} - 2 \beta_{3} - 6 \beta_{4} - 6 \beta_{5} - 6 \beta_{8} - 6 \beta_{9} + 4 \beta_{11} - 6 \beta_{14} ) q^{89} + ( 26 - 5 \beta_{1} - 3 \beta_{2} - 5 \beta_{6} - 3 \beta_{7} + 18 \beta_{8} - 3 \beta_{10} + \beta_{12} + 17 \beta_{13} - 13 \beta_{14} + 4 \beta_{15} ) q^{90} + ( -60 + 14 \beta_{1} + 6 \beta_{2} + 14 \beta_{6} + 6 \beta_{7} + 6 \beta_{10} + 6 \beta_{12} - 6 \beta_{13} + 6 \beta_{14} + 8 \beta_{15} ) q^{92} + ( 12 \beta_{2} + 12 \beta_{4} - 12 \beta_{5} + 12 \beta_{8} + 12 \beta_{9} + 12 \beta_{10} + 12 \beta_{14} ) q^{93} + ( 56 - 56 \beta_{2} + 16 \beta_{3} - 4 \beta_{5} + 4 \beta_{6} - 8 \beta_{7} + 4 \beta_{9} - 16 \beta_{12} + 4 \beta_{15} ) q^{94} + ( -7 + 7 \beta_{2} + 7 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} - 11 \beta_{6} + 8 \beta_{7} - \beta_{9} + 2 \beta_{11} - 7 \beta_{12} - 2 \beta_{13} - 3 \beta_{15} ) q^{95} + ( -4 \beta_{1} + 58 \beta_{2} - 10 \beta_{3} - 6 \beta_{4} - 4 \beta_{5} - 14 \beta_{8} - 14 \beta_{9} - 10 \beta_{10} - 8 \beta_{11} - 6 \beta_{14} ) q^{96} + ( -8 + 30 \beta_{1} + 30 \beta_{6} + 6 \beta_{8} - 10 \beta_{12} - 8 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} ) q^{97} + ( 32 - 26 \beta_{1} - 26 \beta_{6} + 12 \beta_{8} + 7 \beta_{13} - 7 \beta_{14} - 7 \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - q^{2} - 8q^{3} - 5q^{4} + 44q^{6} + 26q^{8} - 48q^{9} + O(q^{10})$$ $$16q - q^{2} - 8q^{3} - 5q^{4} + 44q^{6} + 26q^{8} - 48q^{9} + 16q^{10} + 32q^{11} + 30q^{12} + 71q^{16} - 80q^{17} + 29q^{18} + 56q^{19} + 216q^{20} + 132q^{22} + 22q^{24} + 16q^{25} + 24q^{26} + 64q^{27} - 96q^{30} + 19q^{32} + 32q^{33} - 148q^{34} - 66q^{36} - 14q^{38} + 84q^{40} - 256q^{41} - 50q^{44} + 152q^{46} - 268q^{48} + 66q^{50} + 368q^{51} + 132q^{52} - 228q^{54} + 112q^{57} - 24q^{58} + 104q^{59} - 192q^{60} - 240q^{62} - 110q^{64} + 72q^{65} - 276q^{66} - 304q^{67} - 190q^{68} + 209q^{72} - 112q^{73} - 8q^{74} + 72q^{75} - 140q^{76} - 608q^{78} + 124q^{80} - 48q^{81} + 450q^{82} - 144q^{83} - 210q^{86} + 486q^{88} - 512q^{89} + 368q^{90} - 944q^{92} + 472q^{94} + 558q^{96} - 128q^{97} + 512q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - x^{15} + 3 x^{14} + 6 x^{13} - 22 x^{12} + 44 x^{11} - 20 x^{10} - 112 x^{9} + 368 x^{8} - 448 x^{7} - 320 x^{6} + 2816 x^{5} - 5632 x^{4} + 6144 x^{3} + 12288 x^{2} - 16384 x + 65536$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-3 \nu^{15} - 33 \nu^{14} - 293 \nu^{13} + 642 \nu^{12} - 918 \nu^{11} - 1068 \nu^{10} + 3468 \nu^{9} - 9312 \nu^{8} + 6768 \nu^{7} + 19584 \nu^{6} - 73536 \nu^{5} + 98304 \nu^{4} - 66048 \nu^{3} - 565248 \nu^{2} + 1093632 \nu - 2064384$$$$)/802816$$ $$\beta_{2}$$ $$=$$ $$($$$$9 \nu^{15} + 71 \nu^{14} - 165 \nu^{13} + 246 \nu^{12} + 234 \nu^{11} - 852 \nu^{10} + 2412 \nu^{9} - 1968 \nu^{8} - 4560 \nu^{7} + 18624 \nu^{6} - 26688 \nu^{5} + 20736 \nu^{4} + 136704 \nu^{3} - 282624 \nu^{2} + 528384 \nu + 753664$$$$)/802816$$ $$\beta_{3}$$ $$=$$ $$($$$$10 \nu^{15} + 19 \nu^{14} + 29 \nu^{13} + 159 \nu^{12} - 34 \nu^{11} - 326 \nu^{10} + 1308 \nu^{9} - 1860 \nu^{8} + 160 \nu^{7} + 7888 \nu^{6} - 19200 \nu^{5} + 13632 \nu^{4} + 34816 \nu^{3} - 130048 \nu^{2} + 319488 \nu + 212992$$$$)/200704$$ $$\beta_{4}$$ $$=$$ $$($$$$-21 \nu^{15} - 387 \nu^{14} - 151 \nu^{13} + 1066 \nu^{12} - 3730 \nu^{11} + 5588 \nu^{10} + 1700 \nu^{9} - 35376 \nu^{8} + 44048 \nu^{7} - 53440 \nu^{6} - 80576 \nu^{5} + 361216 \nu^{4} - 552448 \nu^{3} + 178176 \nu^{2} + 2076672 \nu - 7618560$$$$)/802816$$ $$\beta_{5}$$ $$=$$ $$($$$$-57 \nu^{15} - 71 \nu^{14} - 251 \nu^{13} + 154 \nu^{12} - 682 \nu^{11} + 948 \nu^{10} - 620 \nu^{9} - 2960 \nu^{8} - 27952 \nu^{7} + 64 \nu^{6} - 9152 \nu^{5} + 42752 \nu^{4} - 128512 \nu^{3} - 290816 \nu^{2} + 348160 \nu - 3112960$$$$)/802816$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{15} - \nu^{14} + 3 \nu^{13} + 6 \nu^{12} - 22 \nu^{11} + 44 \nu^{10} - 20 \nu^{9} - 112 \nu^{8} + 368 \nu^{7} - 448 \nu^{6} - 320 \nu^{5} + 2816 \nu^{4} - 5632 \nu^{3} + 6144 \nu^{2} + 12288 \nu - 16384$$$$)/16384$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{15} - 103 \nu^{14} + 53 \nu^{13} + 178 \nu^{12} - 778 \nu^{11} + 1860 \nu^{10} - 716 \nu^{9} - 3824 \nu^{8} + 13968 \nu^{7} - 22080 \nu^{6} - 19776 \nu^{5} + 90368 \nu^{4} - 213504 \nu^{3} + 72704 \nu^{2} + 331776 \nu - 1753088$$$$)/200704$$ $$\beta_{8}$$ $$=$$ $$($$$$-33 \nu^{15} + 3 \nu^{14} + 3 \nu^{13} - 168 \nu^{12} + 58 \nu^{11} - 40 \nu^{10} - 2372 \nu^{9} + 424 \nu^{8} - 752 \nu^{7} + 7712 \nu^{6} + 2368 \nu^{5} - 7040 \nu^{4} - 26112 \nu^{3} + 5120 \nu^{2} - 45056 \nu - 770048$$$$)/401408$$ $$\beta_{9}$$ $$=$$ $$($$$$-43 \nu^{15} - 65 \nu^{14} - 85 \nu^{13} + 106 \nu^{12} - 454 \nu^{11} + 1012 \nu^{10} + 300 \nu^{9} - 96 \nu^{8} + 6448 \nu^{7} - 12992 \nu^{6} + 17088 \nu^{5} + 36096 \nu^{4} - 80384 \nu^{3} - 187392 \nu^{2} + 167936 \nu - 2277376$$$$)/401408$$ $$\beta_{10}$$ $$=$$ $$($$$$-65 \nu^{15} + 313 \nu^{14} + 53 \nu^{13} - 942 \nu^{12} + 3158 \nu^{11} - 6204 \nu^{10} - 2988 \nu^{9} + 19728 \nu^{8} - 52784 \nu^{7} + 14272 \nu^{6} + 132672 \nu^{5} - 404224 \nu^{4} + 629248 \nu^{3} + 192512 \nu^{2} - 2363392 \nu + 4882432$$$$)/802816$$ $$\beta_{11}$$ $$=$$ $$($$$$25 \nu^{15} - 14 \nu^{14} - 20 \nu^{13} - 29 \nu^{12} - 56 \nu^{11} - 214 \nu^{10} - 8 \nu^{9} - 140 \nu^{8} - 176 \nu^{7} - 5008 \nu^{6} - 2240 \nu^{5} - 8768 \nu^{4} - 32256 \nu^{3} + 150528 \nu^{2} + 563200 \nu + 606208$$$$)/200704$$ $$\beta_{12}$$ $$=$$ $$($$$$-63 \nu^{15} + 69 \nu^{14} - 123 \nu^{13} + 208 \nu^{12} + 102 \nu^{11} - 936 \nu^{10} + 3396 \nu^{9} + 120 \nu^{8} - 4560 \nu^{7} + 14688 \nu^{6} - 19008 \nu^{5} - 30336 \nu^{4} + 158208 \nu^{3} - 254976 \nu^{2} + 356352 \nu - 1556480$$$$)/401408$$ $$\beta_{13}$$ $$=$$ $$($$$$-41 \nu^{15} + 271 \nu^{14} - 337 \nu^{13} - 436 \nu^{12} + 1338 \nu^{11} - 3664 \nu^{10} + 2140 \nu^{9} + 7800 \nu^{8} - 27600 \nu^{7} + 24032 \nu^{6} - 18752 \nu^{5} - 219264 \nu^{4} + 372736 \nu^{3} - 717824 \nu^{2} + 339968 \nu + 1671168$$$$)/401408$$ $$\beta_{14}$$ $$=$$ $$($$$$-37 \nu^{15} + 391 \nu^{14} - 313 \nu^{13} - 368 \nu^{12} + 3042 \nu^{11} - 5272 \nu^{10} + 1772 \nu^{9} + 19944 \nu^{8} - 51312 \nu^{7} + 64672 \nu^{6} + 39232 \nu^{5} - 435072 \nu^{4} + 672256 \nu^{3} - 699392 \nu^{2} - 765952 \nu + 4939776$$$$)/401408$$ $$\beta_{15}$$ $$=$$ $$($$$$-23 \nu^{15} + 4 \nu^{14} + 92 \nu^{13} - 337 \nu^{12} + 642 \nu^{11} - 46 \nu^{10} - 2076 \nu^{9} + 4980 \nu^{8} - 6840 \nu^{7} - 7088 \nu^{6} + 36832 \nu^{5} - 69696 \nu^{4} - 31872 \nu^{3} + 177664 \nu^{2} - 698368 \nu + 436224$$$$)/100352$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{11} + \beta_{9} + \beta_{8} + \beta_{3} + \beta_{2} - \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{13} + 3 \beta_{12} + \beta_{11} - 3 \beta_{9} + \beta_{6} - 3 \beta_{3} + \beta_{2} - 1$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-4 \beta_{15} - \beta_{13} - \beta_{12} + 4 \beta_{10} + 3 \beta_{8} + 4 \beta_{7} - 11 \beta_{6} + 4 \beta_{2} - 11 \beta_{1} - 9$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$-8 \beta_{14} - \beta_{11} + 4 \beta_{10} + 3 \beta_{9} + 3 \beta_{8} + 4 \beta_{5} - 8 \beta_{4} - \beta_{3} + 35 \beta_{2} - 3 \beta_{1}$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$4 \beta_{15} - 3 \beta_{13} - 11 \beta_{12} + 3 \beta_{11} + 39 \beta_{9} - 28 \beta_{7} + 31 \beta_{6} - 4 \beta_{5} + 8 \beta_{4} + 11 \beta_{3} + 3 \beta_{2} - 3$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$4 \beta_{15} + 8 \beta_{14} - 15 \beta_{13} + 9 \beta_{12} - 44 \beta_{10} + 29 \beta_{8} - 44 \beta_{7} + 11 \beta_{6} - 44 \beta_{2} + 11 \beta_{1} + 1$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$8 \beta_{14} - 3 \beta_{11} - 12 \beta_{10} + 25 \beta_{9} + 25 \beta_{8} - 92 \beta_{5} + 8 \beta_{4} - 27 \beta_{3} + \beta_{2} + 31 \beta_{1}$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$-52 \beta_{15} - 41 \beta_{13} + 79 \beta_{12} + 41 \beta_{11} + 21 \beta_{9} + 92 \beta_{7} - 35 \beta_{6} + 52 \beta_{5} - 184 \beta_{4} - 79 \beta_{3} - 103 \beta_{2} + 103$$$$)/4$$ $$\nu^{9}$$ $$=$$ $$($$$$124 \beta_{15} - 104 \beta_{14} + 75 \beta_{13} + 259 \beta_{12} - 308 \beta_{10} - 497 \beta_{8} - 308 \beta_{7} + 57 \beta_{6} - 308 \beta_{2} + 57 \beta_{1} + 155$$$$)/4$$ $$\nu^{10}$$ $$=$$ $$($$$$248 \beta_{14} - 177 \beta_{11} - 548 \beta_{10} - 61 \beta_{9} - 61 \beta_{8} + 140 \beta_{5} + 248 \beta_{4} - 89 \beta_{3} + 507 \beta_{2} - 1115 \beta_{1}$$$$)/4$$ $$\nu^{11}$$ $$=$$ $$($$$$804 \beta_{15} - 227 \beta_{13} - 411 \beta_{12} + 227 \beta_{11} - 57 \beta_{9} + 788 \beta_{7} - 1217 \beta_{6} - 804 \beta_{5} + 280 \beta_{4} + 411 \beta_{3} + 6259 \beta_{2} - 6259$$$$)/4$$ $$\nu^{12}$$ $$=$$ $$($$$$-300 \beta_{15} + 1608 \beta_{14} - 1959 \beta_{13} + 257 \beta_{12} + 2436 \beta_{10} + 69 \beta_{8} + 2436 \beta_{7} + 1875 \beta_{6} + 2436 \beta_{2} + 1875 \beta_{1} + 1673$$$$)/4$$ $$\nu^{13}$$ $$=$$ $$($$$$-600 \beta_{14} - 219 \beta_{11} + 948 \beta_{10} + 1185 \beta_{9} + 1185 \beta_{8} + 708 \beta_{5} - 600 \beta_{4} + 7725 \beta_{3} - 7863 \beta_{2} - 2921 \beta_{1}$$$$)/4$$ $$\nu^{14}$$ $$=$$ $$($$$$1164 \beta_{15} + 4479 \beta_{13} + 3351 \beta_{12} - 4479 \beta_{11} - 1203 \beta_{9} - 4260 \beta_{7} + 28533 \beta_{6} - 1164 \beta_{5} + 1416 \beta_{4} - 3351 \beta_{3} + 3377 \beta_{2} - 3377$$$$)/4$$ $$\nu^{15}$$ $$=$$ $$($$$$-2148 \beta_{15} + 2328 \beta_{14} + 979 \beta_{13} - 16757 \beta_{12} - 3540 \beta_{10} - 11273 \beta_{8} - 3540 \beta_{7} + 17 \beta_{6} - 3540 \beta_{2} + 17 \beta_{1} - 115997$$$$)/4$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/392\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$295$$ $$297$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 −0.575587 + 1.91538i 0.288997 + 1.97901i −1.99898 − 0.0637211i 1.56937 + 1.23978i −1.78423 − 0.903622i 1.94657 + 0.459219i 0.109554 − 1.99700i 0.944308 − 1.76303i −0.575587 − 1.91538i 0.288997 − 1.97901i −1.99898 + 0.0637211i 1.56937 − 1.23978i −1.78423 + 0.903622i 1.94657 − 0.459219i 0.109554 + 1.99700i 0.944308 + 1.76303i
−1.94657 0.459219i −2.61182 + 4.52380i 3.57824 + 1.78780i 5.42814 3.13394i 7.16148 7.60647i 0 −6.14428 5.12327i −9.14316 15.8364i −12.0054 + 3.60771i
67.2 −1.56937 1.23978i 0.0487183 0.0843825i 0.925871 + 3.89137i −3.00119 + 1.73274i −0.181073 + 0.0720276i 0 3.37142 7.25490i 4.49525 + 7.78601i 6.85820 + 1.00151i
67.3 −0.944308 + 1.76303i −1.72064 + 2.98023i −2.21656 3.32969i −4.22869 + 2.44143i −3.62943 5.84780i 0 7.96347 0.763618i −1.42120 2.46158i −0.311142 9.76078i
67.4 −0.288997 1.97901i 0.0487183 0.0843825i −3.83296 + 1.14386i 3.00119 1.73274i −0.181073 0.0720276i 0 3.37142 + 7.25490i 4.49525 + 7.78601i −4.29644 5.43862i
67.5 −0.109554 + 1.99700i 2.28374 3.95555i −3.97600 0.437557i −4.96451 + 2.86626i 7.64902 + 4.99396i 0 1.30939 7.89212i −5.93090 10.2726i −5.18003 10.2281i
67.6 0.575587 1.91538i −2.61182 + 4.52380i −3.33740 2.20494i −5.42814 + 3.13394i 7.16148 + 7.60647i 0 −6.14428 + 5.12327i −9.14316 15.8364i 2.87833 + 12.2008i
67.7 1.78423 + 0.903622i 2.28374 3.95555i 2.36693 + 3.22453i 4.96451 2.86626i 7.64902 4.99396i 0 1.30939 + 7.89212i −5.93090 10.2726i 11.4478 0.628019i
67.8 1.99898 + 0.0637211i −1.72064 + 2.98023i 3.99188 + 0.254755i 4.22869 2.44143i −3.62943 + 5.84780i 0 7.96347 + 0.763618i −1.42120 2.46158i 8.60865 4.61093i
275.1 −1.94657 + 0.459219i −2.61182 4.52380i 3.57824 1.78780i 5.42814 + 3.13394i 7.16148 + 7.60647i 0 −6.14428 + 5.12327i −9.14316 + 15.8364i −12.0054 3.60771i
275.2 −1.56937 + 1.23978i 0.0487183 + 0.0843825i 0.925871 3.89137i −3.00119 1.73274i −0.181073 0.0720276i 0 3.37142 + 7.25490i 4.49525 7.78601i 6.85820 1.00151i
275.3 −0.944308 1.76303i −1.72064 2.98023i −2.21656 + 3.32969i −4.22869 2.44143i −3.62943 + 5.84780i 0 7.96347 + 0.763618i −1.42120 + 2.46158i −0.311142 + 9.76078i
275.4 −0.288997 + 1.97901i 0.0487183 + 0.0843825i −3.83296 1.14386i 3.00119 + 1.73274i −0.181073 + 0.0720276i 0 3.37142 7.25490i 4.49525 7.78601i −4.29644 + 5.43862i
275.5 −0.109554 1.99700i 2.28374 + 3.95555i −3.97600 + 0.437557i −4.96451 2.86626i 7.64902 4.99396i 0 1.30939 + 7.89212i −5.93090 + 10.2726i −5.18003 + 10.2281i
275.6 0.575587 + 1.91538i −2.61182 4.52380i −3.33740 + 2.20494i −5.42814 3.13394i 7.16148 7.60647i 0 −6.14428 5.12327i −9.14316 + 15.8364i 2.87833 12.2008i
275.7 1.78423 0.903622i 2.28374 + 3.95555i 2.36693 3.22453i 4.96451 + 2.86626i 7.64902 + 4.99396i 0 1.30939 7.89212i −5.93090 + 10.2726i 11.4478 + 0.628019i
275.8 1.99898 0.0637211i −1.72064 2.98023i 3.99188 0.254755i 4.22869 + 2.44143i −3.62943 5.84780i 0 7.96347 0.763618i −1.42120 + 2.46158i 8.60865 + 4.61093i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 275.8 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.c even 3 1 inner
8.d odd 2 1 inner
56.k odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 392.3.k.n 16
7.b odd 2 1 392.3.k.o 16
7.c even 3 1 392.3.g.m 8
7.c even 3 1 inner 392.3.k.n 16
7.d odd 6 1 56.3.g.b 8
7.d odd 6 1 392.3.k.o 16
8.d odd 2 1 inner 392.3.k.n 16
21.g even 6 1 504.3.g.b 8
28.f even 6 1 224.3.g.b 8
28.g odd 6 1 1568.3.g.m 8
56.e even 2 1 392.3.k.o 16
56.j odd 6 1 224.3.g.b 8
56.k odd 6 1 392.3.g.m 8
56.k odd 6 1 inner 392.3.k.n 16
56.m even 6 1 56.3.g.b 8
56.m even 6 1 392.3.k.o 16
56.p even 6 1 1568.3.g.m 8
84.j odd 6 1 2016.3.g.b 8
112.v even 12 2 1792.3.d.j 16
112.x odd 12 2 1792.3.d.j 16
168.ba even 6 1 2016.3.g.b 8
168.be odd 6 1 504.3.g.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.g.b 8 7.d odd 6 1
56.3.g.b 8 56.m even 6 1
224.3.g.b 8 28.f even 6 1
224.3.g.b 8 56.j odd 6 1
392.3.g.m 8 7.c even 3 1
392.3.g.m 8 56.k odd 6 1
392.3.k.n 16 1.a even 1 1 trivial
392.3.k.n 16 7.c even 3 1 inner
392.3.k.n 16 8.d odd 2 1 inner
392.3.k.n 16 56.k odd 6 1 inner
392.3.k.o 16 7.b odd 2 1
392.3.k.o 16 7.d odd 6 1
392.3.k.o 16 56.e even 2 1
392.3.k.o 16 56.m even 6 1
504.3.g.b 8 21.g even 6 1
504.3.g.b 8 168.be odd 6 1
1568.3.g.m 8 28.g odd 6 1
1568.3.g.m 8 56.p even 6 1
1792.3.d.j 16 112.v even 12 2
1792.3.d.j 16 112.x odd 12 2
2016.3.g.b 8 84.j odd 6 1
2016.3.g.b 8 168.ba even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(392, [\chi])$$:

 $$T_{3}^{8} + \cdots$$ $$T_{5}^{16} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + 3 T^{2} - 6 T^{3} - 22 T^{4} - 44 T^{5} - 20 T^{6} + 112 T^{7} + 368 T^{8} + 448 T^{9} - 320 T^{10} - 2816 T^{11} - 5632 T^{12} - 6144 T^{13} + 12288 T^{14} + 16384 T^{15} + 65536 T^{16}$$
$3$ $$( 1 + 4 T + 2 T^{2} - 14 T^{4} - 140 T^{5} + 672 T^{6} + 3140 T^{7} + 3439 T^{8} + 28260 T^{9} + 54432 T^{10} - 102060 T^{11} - 91854 T^{12} + 1062882 T^{14} + 19131876 T^{15} + 43046721 T^{16} )^{2}$$
$5$ $$1 + 92 T^{2} + 3000 T^{4} + 78776 T^{6} + 4124130 T^{8} + 144056708 T^{10} + 2959272640 T^{12} + 91907152084 T^{14} + 3067686313171 T^{16} + 57441970052500 T^{18} + 1155965875000000 T^{20} + 35170094726562500 T^{22} + 629292297363281250 T^{24} + 7512664794921875000 T^{26} +$$$$17\!\cdots\!00$$$$T^{28} +$$$$34\!\cdots\!00$$$$T^{30} +$$$$23\!\cdots\!25$$$$T^{32}$$
$7$ 1
$11$ $$( 1 - 16 T - 72 T^{2} + 4640 T^{3} - 20750 T^{4} - 644496 T^{5} + 8328608 T^{6} + 43860880 T^{7} - 1416483549 T^{8} + 5307166480 T^{9} + 121939149728 T^{10} - 1141763978256 T^{11} - 4447946780750 T^{12} + 120349650148640 T^{13} - 225966843123912 T^{14} - 6075997337331856 T^{15} + 45949729863572161 T^{16} )^{2}$$
$13$ $$( 1 - 444 T^{2} + 142936 T^{4} - 35361044 T^{6} + 6575436334 T^{8} - 1009946777684 T^{10} + 116597286336856 T^{12} - 10344349794381564 T^{14} + 665416609183179841 T^{16} )^{2}$$
$17$ $$( 1 + 40 T + 292 T^{2} - 10704 T^{3} - 137846 T^{4} + 3266264 T^{5} + 77813520 T^{6} + 47796952 T^{7} - 13592851501 T^{8} + 13813319128 T^{9} + 6499063003920 T^{10} + 78839672672216 T^{11} - 961580260212086 T^{12} - 21579198710406096 T^{13} + 170125693271090212 T^{14} + 6735113062376037160 T^{15} + 48661191875666868481 T^{16} )^{2}$$
$19$ $$( 1 - 28 T - 926 T^{2} + 14784 T^{3} + 1071538 T^{4} - 10273676 T^{5} - 557016672 T^{6} + 569025380 T^{7} + 272464898159 T^{8} + 205418162180 T^{9} - 72590969711712 T^{10} - 483334138528556 T^{11} + 18198533173827058 T^{12} + 90641683555329984 T^{13} - 2049529615055265086 T^{14} - 22372187201920755388 T^{15} +$$$$28\!\cdots\!81$$$$T^{16} )^{2}$$
$23$ $$1 + 1744 T^{2} + 1769380 T^{4} + 1394487904 T^{6} + 800412311114 T^{8} + 293154906699184 T^{10} + 41608418133156240 T^{12} - 40513926748661757392 T^{14} -$$$$37\!\cdots\!61$$$$T^{16} -$$$$11\!\cdots\!72$$$$T^{18} +$$$$32\!\cdots\!40$$$$T^{20} +$$$$64\!\cdots\!64$$$$T^{22} +$$$$49\!\cdots\!54$$$$T^{24} +$$$$23\!\cdots\!04$$$$T^{26} +$$$$84\!\cdots\!80$$$$T^{28} +$$$$23\!\cdots\!64$$$$T^{30} +$$$$37\!\cdots\!21$$$$T^{32}$$
$29$ $$( 1 - 3384 T^{2} + 6555580 T^{4} - 8754768776 T^{6} + 8490907402822 T^{8} - 6192081614658056 T^{10} + 3279405379878872380 T^{12} -$$$$11\!\cdots\!44$$$$T^{14} +$$$$25\!\cdots\!21$$$$T^{16} )^{2}$$
$31$ $$1 + 3944 T^{2} + 7116516 T^{4} + 8387060656 T^{6} + 8423414263178 T^{8} + 8514445544397720 T^{10} + 8434364772846028816 T^{12} +$$$$79\!\cdots\!12$$$$T^{14} +$$$$74\!\cdots\!23$$$$T^{16} +$$$$73\!\cdots\!52$$$$T^{18} +$$$$71\!\cdots\!56$$$$T^{20} +$$$$67\!\cdots\!20$$$$T^{22} +$$$$61\!\cdots\!18$$$$T^{24} +$$$$56\!\cdots\!56$$$$T^{26} +$$$$44\!\cdots\!36$$$$T^{28} +$$$$22\!\cdots\!04$$$$T^{30} +$$$$52\!\cdots\!61$$$$T^{32}$$
$37$ $$1 + 3512 T^{2} + 3145476 T^{4} - 4974960880 T^{6} - 8515936106358 T^{8} + 12549151402549448 T^{10} + 35685999046008816016 T^{12} +$$$$80\!\cdots\!88$$$$T^{14} -$$$$37\!\cdots\!25$$$$T^{16} +$$$$15\!\cdots\!68$$$$T^{18} +$$$$12\!\cdots\!36$$$$T^{20} +$$$$82\!\cdots\!88$$$$T^{22} -$$$$10\!\cdots\!78$$$$T^{24} -$$$$11\!\cdots\!80$$$$T^{26} +$$$$13\!\cdots\!36$$$$T^{28} +$$$$28\!\cdots\!52$$$$T^{30} +$$$$15\!\cdots\!81$$$$T^{32}$$
$41$ $$( 1 + 64 T + 4956 T^{2} + 221760 T^{3} + 11848326 T^{4} + 372778560 T^{5} + 14004471516 T^{6} + 304006671424 T^{7} + 7984925229121 T^{8} )^{4}$$
$43$ $$( 1 + 4680 T^{2} - 58016 T^{3} + 10251086 T^{4} - 107271584 T^{5} + 15999988680 T^{6} + 11688200277601 T^{8} )^{4}$$
$47$ $$1 + 8392 T^{2} + 30858916 T^{4} + 77629558512 T^{6} + 181309996781322 T^{8} + 338519134932094776 T^{10} +$$$$32\!\cdots\!52$$$$T^{12} +$$$$46\!\cdots\!92$$$$T^{14} -$$$$14\!\cdots\!37$$$$T^{16} +$$$$22\!\cdots\!52$$$$T^{18} +$$$$77\!\cdots\!72$$$$T^{20} +$$$$39\!\cdots\!16$$$$T^{22} +$$$$10\!\cdots\!62$$$$T^{24} +$$$$21\!\cdots\!12$$$$T^{26} +$$$$41\!\cdots\!96$$$$T^{28} +$$$$55\!\cdots\!12$$$$T^{30} +$$$$32\!\cdots\!41$$$$T^{32}$$
$53$ $$1 + 18920 T^{2} + 195169572 T^{4} + 1399932528560 T^{6} + 7743546865787466 T^{8} + 34991808675085792280 T^{10} +$$$$13\!\cdots\!12$$$$T^{12} +$$$$44\!\cdots\!80$$$$T^{14} +$$$$13\!\cdots\!79$$$$T^{16} +$$$$35\!\cdots\!80$$$$T^{18} +$$$$83\!\cdots\!32$$$$T^{20} +$$$$17\!\cdots\!80$$$$T^{22} +$$$$30\!\cdots\!86$$$$T^{24} +$$$$42\!\cdots\!60$$$$T^{26} +$$$$47\!\cdots\!32$$$$T^{28} +$$$$36\!\cdots\!20$$$$T^{30} +$$$$15\!\cdots\!41$$$$T^{32}$$
$59$ $$( 1 - 52 T + 450 T^{2} - 532224 T^{3} + 35666162 T^{4} - 868645540 T^{5} + 179864519968 T^{6} - 10977595733044 T^{7} + 261680209248047 T^{8} - 38213010746726164 T^{9} + 2179483319543964448 T^{10} - 36639932422074611140 T^{11} +$$$$52\!\cdots\!02$$$$T^{12} -$$$$27\!\cdots\!24$$$$T^{13} +$$$$80\!\cdots\!50$$$$T^{14} -$$$$32\!\cdots\!72$$$$T^{15} +$$$$21\!\cdots\!41$$$$T^{16} )^{2}$$
$61$ $$1 + 16316 T^{2} + 126039736 T^{4} + 653164234872 T^{6} + 2819894062793314 T^{8} + 11641021751267899300 T^{10} +$$$$46\!\cdots\!32$$$$T^{12} +$$$$16\!\cdots\!44$$$$T^{14} +$$$$61\!\cdots\!79$$$$T^{16} +$$$$23\!\cdots\!04$$$$T^{18} +$$$$88\!\cdots\!92$$$$T^{20} +$$$$30\!\cdots\!00$$$$T^{22} +$$$$10\!\cdots\!54$$$$T^{24} +$$$$33\!\cdots\!72$$$$T^{26} +$$$$88\!\cdots\!76$$$$T^{28} +$$$$15\!\cdots\!96$$$$T^{30} +$$$$13\!\cdots\!21$$$$T^{32}$$
$67$ $$( 1 + 152 T + 880 T^{2} - 696592 T^{3} + 25800514 T^{4} + 6259290472 T^{5} + 135213846720 T^{6} - 1900601864504 T^{7} + 289722470344099 T^{8} - 8531801769758456 T^{9} + 2724710586130173120 T^{10} +$$$$56\!\cdots\!68$$$$T^{11} +$$$$10\!\cdots\!74$$$$T^{12} -$$$$12\!\cdots\!08$$$$T^{13} +$$$$72\!\cdots\!80$$$$T^{14} +$$$$55\!\cdots\!08$$$$T^{15} +$$$$16\!\cdots\!81$$$$T^{16} )^{2}$$
$71$ $$( 1 - 9864 T^{2} + 51888284 T^{4} - 294531431096 T^{6} + 1789421441990854 T^{8} - 7484538771485032376 T^{10} +$$$$33\!\cdots\!24$$$$T^{12} -$$$$16\!\cdots\!24$$$$T^{14} +$$$$41\!\cdots\!21$$$$T^{16} )^{2}$$
$73$ $$( 1 + 56 T - 15324 T^{2} - 438256 T^{3} + 161331658 T^{4} + 2027862984 T^{5} - 1181077316848 T^{6} - 5241850793144 T^{7} + 6684367707557907 T^{8} - 27933822876664376 T^{9} - 33540518283482864368 T^{10} +$$$$30\!\cdots\!76$$$$T^{11} +$$$$13\!\cdots\!98$$$$T^{12} -$$$$18\!\cdots\!44$$$$T^{13} -$$$$35\!\cdots\!04$$$$T^{14} +$$$$68\!\cdots\!04$$$$T^{15} +$$$$65\!\cdots\!61$$$$T^{16} )^{2}$$
$79$ $$1 + 24968 T^{2} + 292531236 T^{4} + 2048900954480 T^{6} + 9731167232364042 T^{8} + 40635655541936341112 T^{10} +$$$$28\!\cdots\!16$$$$T^{12} +$$$$28\!\cdots\!32$$$$T^{14} +$$$$21\!\cdots\!95$$$$T^{16} +$$$$11\!\cdots\!92$$$$T^{18} +$$$$43\!\cdots\!76$$$$T^{20} +$$$$24\!\cdots\!92$$$$T^{22} +$$$$22\!\cdots\!82$$$$T^{24} +$$$$18\!\cdots\!80$$$$T^{26} +$$$$10\!\cdots\!16$$$$T^{28} +$$$$33\!\cdots\!48$$$$T^{30} +$$$$52\!\cdots\!41$$$$T^{32}$$
$83$ $$( 1 + 36 T + 16478 T^{2} + 177884 T^{3} + 135298114 T^{4} + 1225442876 T^{5} + 782018213438 T^{6} + 11769853441284 T^{7} + 2252292232139041 T^{8} )^{4}$$
$89$ $$( 1 + 256 T + 17284 T^{2} - 187392 T^{3} + 84842826 T^{4} + 25389871104 T^{5} + 1853625901584 T^{6} + 111590371439872 T^{7} + 11164291531924819 T^{8} + 883907332175226112 T^{9} +$$$$11\!\cdots\!44$$$$T^{10} +$$$$12\!\cdots\!44$$$$T^{11} +$$$$33\!\cdots\!06$$$$T^{12} -$$$$58\!\cdots\!92$$$$T^{13} +$$$$42\!\cdots\!64$$$$T^{14} +$$$$50\!\cdots\!96$$$$T^{15} +$$$$15\!\cdots\!61$$$$T^{16} )^{2}$$
$97$ $$( 1 + 32 T + 19484 T^{2} + 1437536 T^{3} + 199130566 T^{4} + 13525776224 T^{5} + 1724904511004 T^{6} + 26655104157728 T^{7} + 7837433594376961 T^{8} )^{4}$$