# Properties

 Label 108.3.f.c Level 108 Weight 3 Character orbit 108.f Analytic conductor 2.943 Analytic rank 0 Dimension 16 CM no Inner twists 4

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$108 = 2^{2} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 108.f (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.94278685509$$ 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: $$2^{8}\cdot 3^{4}$$ Twist minimal: no (minimal twist has level 36) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -\beta_{2} + \beta_{3} ) q^{4} + ( -\beta_{2} - \beta_{6} + \beta_{8} - \beta_{13} ) q^{5} + ( \beta_{6} - \beta_{10} - \beta_{11} + \beta_{14} ) q^{7} + ( 4 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} - \beta_{9} - \beta_{11} - \beta_{13} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -\beta_{2} + \beta_{3} ) q^{4} + ( -\beta_{2} - \beta_{6} + \beta_{8} - \beta_{13} ) q^{5} + ( \beta_{6} - \beta_{10} - \beta_{11} + \beta_{14} ) q^{7} + ( 4 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} - \beta_{9} - \beta_{11} - \beta_{13} ) q^{8} + ( 2 - \beta_{3} + 2 \beta_{5} - \beta_{7} + 2 \beta_{8} + 2 \beta_{10} - \beta_{11} - \beta_{13} - \beta_{15} ) q^{10} + ( \beta_{1} + \beta_{6} - \beta_{10} + \beta_{11} + \beta_{12} - \beta_{14} - \beta_{15} ) q^{11} + ( \beta_{1} - 5 \beta_{2} - \beta_{3} + \beta_{5} + \beta_{7} + \beta_{12} ) q^{13} + ( \beta_{1} + 3 \beta_{2} - \beta_{3} + \beta_{5} - 4 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - 4 \beta_{12} - 2 \beta_{14} ) q^{14} + ( -4 + 3 \beta_{1} + 4 \beta_{2} + \beta_{6} + 2 \beta_{10} - \beta_{11} - 3 \beta_{12} + \beta_{14} + \beta_{15} ) q^{16} + ( \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{4} + \beta_{5} + 3 \beta_{7} + \beta_{9} - 2 \beta_{15} ) q^{17} + ( 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + 4 \beta_{5} - 2 \beta_{7} - \beta_{8} + 3 \beta_{9} - \beta_{10} + 2 \beta_{11} - \beta_{13} + \beta_{15} ) q^{19} + ( -5 + \beta_{1} + 5 \beta_{2} - \beta_{6} + 2 \beta_{10} - 3 \beta_{11} + 3 \beta_{12} + 3 \beta_{14} ) q^{20} + ( 2 \beta_{1} + 6 \beta_{2} + 2 \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} + 3 \beta_{9} - \beta_{12} + \beta_{13} - 5 \beta_{14} ) q^{22} + ( -5 \beta_{1} - 2 \beta_{2} - \beta_{3} - 5 \beta_{5} + 2 \beta_{6} - \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - \beta_{12} + 2 \beta_{13} + \beta_{14} ) q^{23} + ( -7 \beta_{1} + 3 \beta_{4} - 3 \beta_{6} - 3 \beta_{10} + 2 \beta_{12} - \beta_{15} ) q^{25} + ( -4 + 3 \beta_{1} + 3 \beta_{2} - \beta_{3} - 3 \beta_{4} + 6 \beta_{5} - \beta_{7} + 3 \beta_{9} + \beta_{11} + \beta_{13} + 2 \beta_{15} ) q^{26} + ( 3 - 9 \beta_{1} - 9 \beta_{2} + 6 \beta_{3} + 9 \beta_{4} - 7 \beta_{5} + 5 \beta_{7} - 9 \beta_{9} + 3 \beta_{11} + 3 \beta_{13} - 3 \beta_{15} ) q^{28} + ( -7 + 3 \beta_{1} + 7 \beta_{2} - 2 \beta_{6} - 2 \beta_{10} + 3 \beta_{12} + 3 \beta_{15} ) q^{29} + ( -11 \beta_{1} - 6 \beta_{2} + \beta_{3} - 11 \beta_{5} + \beta_{7} - 6 \beta_{9} + \beta_{12} + 3 \beta_{14} ) q^{31} + ( -9 \beta_{1} - 18 \beta_{2} + 7 \beta_{3} - 9 \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} - 6 \beta_{9} + \beta_{12} + \beta_{13} + 5 \beta_{14} ) q^{32} + ( -2 + 6 \beta_{1} + 2 \beta_{2} - 9 \beta_{4} - 3 \beta_{6} + 3 \beta_{11} + 3 \beta_{12} - 3 \beta_{14} + 4 \beta_{15} ) q^{34} + ( -\beta_{1} - \beta_{2} + 4 \beta_{3} + \beta_{4} + 2 \beta_{5} + 4 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} + 5 \beta_{11} + \beta_{13} + 3 \beta_{15} ) q^{35} + ( -4 + 6 \beta_{1} + 6 \beta_{2} - \beta_{3} - 6 \beta_{4} - 11 \beta_{5} + \beta_{7} - 3 \beta_{8} + 6 \beta_{9} - 3 \beta_{10} + 3 \beta_{13} + 5 \beta_{15} ) q^{37} + ( 12 - \beta_{1} - 12 \beta_{2} + 4 \beta_{4} + \beta_{6} + 2 \beta_{10} + 7 \beta_{11} + \beta_{12} - 7 \beta_{14} + \beta_{15} ) q^{38} + ( -4 \beta_{1} + 7 \beta_{2} - 4 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + 3 \beta_{9} + 2 \beta_{12} - 4 \beta_{13} + 8 \beta_{14} ) q^{40} + ( -7 \beta_{1} - 13 \beta_{2} + 3 \beta_{3} - 7 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} - 3 \beta_{8} + 2 \beta_{9} - 3 \beta_{12} + 3 \beta_{13} ) q^{41} + ( 6 \beta_{1} + 6 \beta_{4} - 2 \beta_{6} + 2 \beta_{10} + 5 \beta_{11} - 5 \beta_{14} - 6 \beta_{15} ) q^{43} + ( 13 - 3 \beta_{1} - 3 \beta_{2} + 5 \beta_{3} + 3 \beta_{4} - 6 \beta_{5} + 8 \beta_{7} - 2 \beta_{8} - 3 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} + 6 \beta_{13} + 2 \beta_{15} ) q^{44} + ( -15 + 3 \beta_{1} + 3 \beta_{2} - 7 \beta_{3} - 3 \beta_{4} + \beta_{5} - 8 \beta_{7} - 4 \beta_{8} + 3 \beta_{9} - 4 \beta_{10} - 4 \beta_{11} + 2 \beta_{13} - 4 \beta_{15} ) q^{46} + ( 2 \beta_{1} - 4 \beta_{4} - 3 \beta_{6} + 3 \beta_{10} - 3 \beta_{11} + 6 \beta_{12} + 3 \beta_{14} - 2 \beta_{15} ) q^{47} + ( 17 \beta_{1} + 11 \beta_{2} + \beta_{3} + 17 \beta_{5} - \beta_{7} - 9 \beta_{9} - \beta_{12} ) q^{49} + ( 5 \beta_{1} + 34 \beta_{2} - 11 \beta_{3} + 5 \beta_{5} - \beta_{6} - 5 \beta_{7} + 6 \beta_{8} + 6 \beta_{9} - 5 \beta_{12} - \beta_{13} - \beta_{14} ) q^{50} + ( 13 - 3 \beta_{1} - 13 \beta_{2} - \beta_{6} - 2 \beta_{10} + \beta_{11} + 3 \beta_{12} - \beta_{14} + 6 \beta_{15} ) q^{52} + ( 8 - 4 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} + 4 \beta_{4} + 5 \beta_{5} - 3 \beta_{7} + \beta_{8} - 4 \beta_{9} + \beta_{10} - \beta_{13} - \beta_{15} ) q^{53} + ( 9 \beta_{3} + 9 \beta_{5} + 9 \beta_{7} + 2 \beta_{8} + 2 \beta_{10} - 7 \beta_{11} + 2 \beta_{13} + 9 \beta_{15} ) q^{55} + ( -36 + 5 \beta_{1} + 36 \beta_{2} - 7 \beta_{4} + 6 \beta_{6} - 6 \beta_{10} - 6 \beta_{11} + 6 \beta_{14} + \beta_{15} ) q^{56} + ( -2 \beta_{1} + \beta_{2} - 5 \beta_{3} - 2 \beta_{5} + \beta_{6} - 5 \beta_{7} + 4 \beta_{8} + 9 \beta_{9} - 5 \beta_{12} + \beta_{13} + \beta_{14} ) q^{58} + ( 10 \beta_{1} + 8 \beta_{2} - 6 \beta_{3} + 10 \beta_{5} - 6 \beta_{6} - 6 \beta_{7} - 6 \beta_{8} + 8 \beta_{9} - 6 \beta_{12} - 6 \beta_{13} - 3 \beta_{14} ) q^{59} + ( 7 - 25 \beta_{1} - 7 \beta_{2} + 6 \beta_{4} + 12 \beta_{6} + 12 \beta_{10} - 7 \beta_{12} - 13 \beta_{15} ) q^{61} + ( -29 + \beta_{1} + \beta_{2} - 11 \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{7} + \beta_{9} + 2 \beta_{11} - 4 \beta_{13} - 10 \beta_{15} ) q^{62} + ( -8 - 6 \beta_{1} - 6 \beta_{2} - \beta_{3} + 6 \beta_{4} + 11 \beta_{5} + 5 \beta_{7} - 2 \beta_{8} - 6 \beta_{9} - 2 \beta_{10} + \beta_{11} - 11 \beta_{13} - 7 \beta_{15} ) q^{64} + ( 3 - 3 \beta_{2} - \beta_{4} + 6 \beta_{6} + 6 \beta_{10} - 3 \beta_{12} - 2 \beta_{15} ) q^{65} + ( 3 \beta_{1} + 6 \beta_{2} - 9 \beta_{3} + 3 \beta_{5} + \beta_{6} - 9 \beta_{7} + \beta_{8} + 6 \beta_{9} - 9 \beta_{12} + \beta_{13} - 11 \beta_{14} ) q^{67} + ( \beta_{1} - 48 \beta_{2} - \beta_{3} + \beta_{5} - 5 \beta_{6} + 11 \beta_{7} + 6 \beta_{8} - 2 \beta_{9} + 11 \beta_{12} - 5 \beta_{13} - 5 \beta_{14} ) q^{68} + ( 15 + \beta_{1} - 15 \beta_{2} - 6 \beta_{4} + 8 \beta_{6} - 2 \beta_{10} - 2 \beta_{11} + 4 \beta_{12} + 2 \beta_{14} + 7 \beta_{15} ) q^{70} + ( 4 \beta_{1} + 4 \beta_{2} - 5 \beta_{3} - 4 \beta_{4} + 3 \beta_{5} - 5 \beta_{7} - 5 \beta_{8} + 4 \beta_{9} - 5 \beta_{10} - 22 \beta_{11} - 5 \beta_{13} - \beta_{15} ) q^{71} + ( -4 + 3 \beta_{1} + 3 \beta_{2} + 5 \beta_{3} - 3 \beta_{4} - 11 \beta_{5} - 5 \beta_{7} + 12 \beta_{8} + 3 \beta_{9} + 12 \beta_{10} - 12 \beta_{13} + 8 \beta_{15} ) q^{73} + ( 50 + \beta_{1} - 50 \beta_{2} - 3 \beta_{4} - 4 \beta_{6} - 6 \beta_{10} + 4 \beta_{11} - 2 \beta_{12} - 4 \beta_{14} - 7 \beta_{15} ) q^{74} + ( 11 \beta_{1} + 6 \beta_{2} - \beta_{3} + 11 \beta_{5} + 13 \beta_{6} + 5 \beta_{7} - 2 \beta_{8} + 6 \beta_{9} + 5 \beta_{12} + 13 \beta_{13} + \beta_{14} ) q^{76} + ( 16 \beta_{1} + 45 \beta_{2} + 16 \beta_{5} - \beta_{6} + \beta_{8} - 8 \beta_{9} - \beta_{13} ) q^{77} + ( 11 \beta_{1} + 6 \beta_{4} - 4 \beta_{6} + 4 \beta_{10} - 11 \beta_{11} + 5 \beta_{12} + 11 \beta_{14} - 11 \beta_{15} ) q^{79} + ( 44 + 2 \beta_{1} + 2 \beta_{2} - 8 \beta_{3} - 2 \beta_{4} - 8 \beta_{7} + 8 \beta_{8} + 2 \beta_{9} + 8 \beta_{10} + 8 \beta_{11} - 12 \beta_{13} - 6 \beta_{15} ) q^{80} + ( 26 - 9 \beta_{1} - 9 \beta_{2} + 2 \beta_{3} + 9 \beta_{4} + 9 \beta_{5} + 6 \beta_{7} - 6 \beta_{8} - 9 \beta_{9} - 6 \beta_{10} - 7 \beta_{15} ) q^{82} + ( -5 \beta_{1} + 21 \beta_{4} + \beta_{6} - \beta_{10} + \beta_{11} - 26 \beta_{12} - \beta_{14} + 5 \beta_{15} ) q^{83} + ( 3 \beta_{1} - 22 \beta_{2} + 9 \beta_{3} + 3 \beta_{5} - 3 \beta_{6} - 9 \beta_{7} + 3 \beta_{8} - 6 \beta_{9} - 9 \beta_{12} - 3 \beta_{13} ) q^{85} + ( -2 \beta_{1} + 27 \beta_{2} + 14 \beta_{3} - 2 \beta_{5} + 3 \beta_{6} + 11 \beta_{7} + 4 \beta_{8} - 4 \beta_{9} + 11 \beta_{12} + 3 \beta_{13} + \beta_{14} ) q^{86} + ( 22 \beta_{1} - 15 \beta_{4} - 3 \beta_{6} - 12 \beta_{10} - 9 \beta_{11} - 5 \beta_{12} + 9 \beta_{14} + 8 \beta_{15} ) q^{88} + ( 12 + 4 \beta_{1} + 4 \beta_{2} + 9 \beta_{3} - 4 \beta_{4} - 17 \beta_{5} - 9 \beta_{7} - 3 \beta_{8} + 4 \beta_{9} - 3 \beta_{10} + 3 \beta_{13} + 13 \beta_{15} ) q^{89} + ( 15 \beta_{1} + 15 \beta_{2} - 11 \beta_{3} - 15 \beta_{4} + 19 \beta_{5} - 11 \beta_{7} + 4 \beta_{8} + 15 \beta_{9} + 4 \beta_{10} + \beta_{11} + 4 \beta_{13} + 4 \beta_{15} ) q^{91} + ( -23 - 14 \beta_{1} + 23 \beta_{2} + 11 \beta_{4} - 13 \beta_{6} - 4 \beta_{10} + 9 \beta_{11} - 3 \beta_{12} - 9 \beta_{14} - 5 \beta_{15} ) q^{92} + ( 3 \beta_{1} + 3 \beta_{2} + \beta_{3} + 3 \beta_{5} - 12 \beta_{6} + 12 \beta_{7} + 6 \beta_{8} + 12 \beta_{12} - 12 \beta_{13} + 6 \beta_{14} ) q^{94} + ( 6 \beta_{1} - 8 \beta_{2} + 22 \beta_{3} + 6 \beta_{5} - 2 \beta_{6} + 22 \beta_{7} - 2 \beta_{8} - 8 \beta_{9} + 22 \beta_{12} - 2 \beta_{13} - 4 \beta_{14} ) q^{95} + ( -17 + \beta_{1} + 17 \beta_{2} - 9 \beta_{6} - 9 \beta_{10} + \beta_{12} + \beta_{15} ) q^{97} + ( -86 - 3 \beta_{1} - 3 \beta_{2} + 19 \beta_{3} + 3 \beta_{4} - 21 \beta_{5} + \beta_{7} - 3 \beta_{9} - \beta_{11} - \beta_{13} + 16 \beta_{15} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 3q^{2} - 5q^{4} - 6q^{5} + 54q^{8} + O(q^{10})$$ $$16q + 3q^{2} - 5q^{4} - 6q^{5} + 54q^{8} + 20q^{10} - 46q^{13} + 12q^{14} - 17q^{16} - 12q^{17} - 36q^{20} + 33q^{22} - 30q^{25} - 72q^{26} + 12q^{28} - 42q^{29} - 87q^{32} + 11q^{34} + 56q^{37} + 99q^{38} + 68q^{40} - 84q^{41} + 222q^{44} - 264q^{46} + 58q^{49} + 219q^{50} + 110q^{52} + 72q^{53} - 270q^{56} - 16q^{58} - 34q^{61} - 516q^{62} - 254q^{64} + 30q^{65} - 375q^{68} + 150q^{70} + 116q^{73} + 372q^{74} - 15q^{76} + 330q^{77} + 720q^{80} + 254q^{82} - 140q^{85} + 273q^{86} + 75q^{88} + 384q^{89} - 258q^{92} + 36q^{94} - 148q^{97} - 1170q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 3 x^{15} + 7 x^{14} - 30 x^{13} + 76 x^{12} - 144 x^{11} + 424 x^{10} - 912 x^{9} + 1552 x^{8} - 3648 x^{7} + 6784 x^{6} - 9216 x^{5} + 19456 x^{4} - 30720 x^{3} + 28672 x^{2} - 49152 x + 65536$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{15} + 17 \nu^{14} + 83 \nu^{13} - 394 \nu^{12} + 204 \nu^{11} - 2224 \nu^{10} + 6280 \nu^{9} - 7664 \nu^{8} + 26384 \nu^{7} - 63104 \nu^{6} + 58368 \nu^{5} - 187904 \nu^{4} + 372736 \nu^{3} - 258048 \nu^{2} + 634880 \nu - 1228800$$$$)/540672$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{15} + 17 \nu^{14} + 83 \nu^{13} - 394 \nu^{12} + 204 \nu^{11} - 2224 \nu^{10} + 6280 \nu^{9} - 7664 \nu^{8} + 26384 \nu^{7} - 63104 \nu^{6} + 58368 \nu^{5} - 187904 \nu^{4} + 372736 \nu^{3} + 282624 \nu^{2} + 634880 \nu - 1228800$$$$)/540672$$ $$\beta_{4}$$ $$=$$ $$($$$$13 \nu^{15} + 45 \nu^{14} + 375 \nu^{13} + 158 \nu^{12} - 692 \nu^{11} - 2512 \nu^{10} - 3544 \nu^{9} + 11600 \nu^{8} + 20560 \nu^{7} + 41344 \nu^{6} - 83200 \nu^{5} - 111104 \nu^{4} - 178176 \nu^{3} + 385024 \nu^{2} + 1224704 \nu + 786432$$$$)/540672$$ $$\beta_{5}$$ $$=$$ $$($$$$-5 \nu^{15} - 19 \nu^{14} + 91 \nu^{13} - 32 \nu^{12} + 520 \nu^{11} - 1464 \nu^{10} + 1688 \nu^{9} - 6208 \nu^{8} + 14864 \nu^{7} - 12896 \nu^{6} + 44672 \nu^{5} - 88320 \nu^{4} + 56832 \nu^{3} - 151552 \nu^{2} + 294912 \nu + 16384$$$$)/135168$$ $$\beta_{6}$$ $$=$$ $$($$$$7 \nu^{15} + 97 \nu^{14} - 211 \nu^{13} + 366 \nu^{12} - 1454 \nu^{11} + 3572 \nu^{10} - 6112 \nu^{9} + 19392 \nu^{8} - 33024 \nu^{7} + 60224 \nu^{6} - 130336 \nu^{5} + 209536 \nu^{4} - 226560 \nu^{3} + 553472 \nu^{2} - 647168 \nu + 679936$$$$)/135168$$ $$\beta_{7}$$ $$=$$ $$($$$$-25 \nu^{15} - 7 \nu^{14} + 15 \nu^{13} - 72 \nu^{12} + 136 \nu^{11} - 280 \nu^{10} + 1752 \nu^{9} - 1472 \nu^{8} + 2512 \nu^{7} - 18016 \nu^{6} + 9344 \nu^{5} - 13568 \nu^{4} + 120832 \nu^{3} - 36864 \nu^{2} + 32768 \nu - 819200$$$$)/270336$$ $$\beta_{8}$$ $$=$$ $$($$$$-61 \nu^{15} + 371 \nu^{14} - 839 \nu^{13} + 1506 \nu^{12} - 5404 \nu^{11} + 13168 \nu^{10} - 25448 \nu^{9} + 59184 \nu^{8} - 128208 \nu^{7} + 177280 \nu^{6} - 429056 \nu^{5} + 648704 \nu^{4} - 749568 \nu^{3} + 1638400 \nu^{2} - 2682880 \nu - 16384$$$$)/540672$$ $$\beta_{9}$$ $$=$$ $$($$$$-67 \nu^{15} + 181 \nu^{14} - 545 \nu^{13} + 2374 \nu^{12} - 5220 \nu^{11} + 11728 \nu^{10} - 34264 \nu^{9} + 67856 \nu^{8} - 128816 \nu^{7} + 303872 \nu^{6} - 506112 \nu^{5} + 796160 \nu^{4} - 1656832 \nu^{3} + 2285568 \nu^{2} - 2527232 \nu + 4472832$$$$)/540672$$ $$\beta_{10}$$ $$=$$ $$($$$$-7 \nu^{15} - 23 \nu^{14} + 11 \nu^{13} + 118 \nu^{12} + 156 \nu^{11} - 176 \nu^{10} - 1336 \nu^{9} - 1264 \nu^{8} + 1040 \nu^{7} + 10112 \nu^{6} + 8448 \nu^{5} + 5120 \nu^{4} - 83968 \nu^{3} - 49152 \nu^{2} + 20480 \nu + 393216$$$$)/49152$$ $$\beta_{11}$$ $$=$$ $$($$$$-47 \nu^{15} + 125 \nu^{14} - 73 \nu^{13} + 786 \nu^{12} - 1844 \nu^{11} + 2272 \nu^{10} - 8632 \nu^{9} + 17712 \nu^{8} - 13680 \nu^{7} + 74560 \nu^{6} - 92032 \nu^{5} + 42752 \nu^{4} - 374784 \nu^{3} + 458752 \nu^{2} + 212992 \nu + 892928$$$$)/270336$$ $$\beta_{12}$$ $$=$$ $$($$$$-25 \nu^{15} + 37 \nu^{14} - 95 \nu^{13} + 610 \nu^{12} - 942 \nu^{11} + 1612 \nu^{10} - 7312 \nu^{9} + 10496 \nu^{8} - 14912 \nu^{7} + 54848 \nu^{6} - 62112 \nu^{5} + 51200 \nu^{4} - 270592 \nu^{3} + 182784 \nu^{2} + 10240 \nu + 712704$$$$)/135168$$ $$\beta_{13}$$ $$=$$ $$($$$$26 \nu^{15} - 119 \nu^{14} + 365 \nu^{13} - 1015 \nu^{12} + 2642 \nu^{11} - 6300 \nu^{10} + 14560 \nu^{9} - 29864 \nu^{8} + 60304 \nu^{7} - 124816 \nu^{6} + 201088 \nu^{5} - 343296 \nu^{4} + 539136 \nu^{3} - 739328 \nu^{2} + 1007616 \nu - 1220608$$$$)/135168$$ $$\beta_{14}$$ $$=$$ $$($$$$-135 \nu^{15} + 257 \nu^{14} - 205 \nu^{13} + 2414 \nu^{12} - 4484 \nu^{11} + 4912 \nu^{10} - 26936 \nu^{9} + 44112 \nu^{8} - 51696 \nu^{7} + 218880 \nu^{6} - 231424 \nu^{5} + 237056 \nu^{4} - 1140736 \nu^{3} + 999424 \nu^{2} + 77824 \nu + 2965504$$$$)/540672$$ $$\beta_{15}$$ $$=$$ $$($$$$-137 \nu^{15} + 487 \nu^{14} - 811 \nu^{13} + 3994 \nu^{12} - 8940 \nu^{11} + 17456 \nu^{10} - 49352 \nu^{9} + 98160 \nu^{8} - 150928 \nu^{7} + 377472 \nu^{6} - 595968 \nu^{5} + 804352 \nu^{4} - 1593344 \nu^{3} + 2011136 \nu^{2} - 1552384 \nu + 3080192$$$$)/540672$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$-\beta_{13} - \beta_{11} - \beta_{9} + \beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - \beta_{1} + 4$$ $$\nu^{4}$$ $$=$$ $$\beta_{15} + \beta_{14} - 3 \beta_{12} - \beta_{11} + 2 \beta_{10} + \beta_{6} + 4 \beta_{2} + 3 \beta_{1} - 4$$ $$\nu^{5}$$ $$=$$ $$5 \beta_{14} + \beta_{13} + \beta_{12} - 6 \beta_{9} - 2 \beta_{8} + \beta_{7} + \beta_{6} - 9 \beta_{5} + 7 \beta_{3} - 18 \beta_{2} - 9 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-7 \beta_{15} - 11 \beta_{13} + \beta_{11} - 2 \beta_{10} - 6 \beta_{9} - 2 \beta_{8} + 5 \beta_{7} + 11 \beta_{5} + 6 \beta_{4} - \beta_{3} - 6 \beta_{2} - 6 \beta_{1} - 8$$ $$\nu^{7}$$ $$=$$ $$5 \beta_{15} - 17 \beta_{14} - 13 \beta_{12} + 17 \beta_{11} + 22 \beta_{10} + 11 \beta_{6} + 2 \beta_{4} + 48 \beta_{2} - 13 \beta_{1} - 48$$ $$\nu^{8}$$ $$=$$ $$23 \beta_{14} + 35 \beta_{13} + 43 \beta_{12} - 30 \beta_{9} - 22 \beta_{8} + 43 \beta_{7} + 35 \beta_{6} - 83 \beta_{5} + 17 \beta_{3} - 94 \beta_{2} - 83 \beta_{1}$$ $$\nu^{9}$$ $$=$$ $$-5 \beta_{15} - 5 \beta_{13} + 15 \beta_{11} - 70 \beta_{10} + 82 \beta_{9} - 70 \beta_{8} + 3 \beta_{7} + 85 \beta_{5} - 82 \beta_{4} - 87 \beta_{3} + 82 \beta_{2} + 82 \beta_{1} - 80$$ $$\nu^{10}$$ $$=$$ $$83 \beta_{15} - 247 \beta_{14} - 43 \beta_{12} + 247 \beta_{11} + 10 \beta_{10} - 67 \beta_{6} - 18 \beta_{4} - 176 \beta_{2} + 5 \beta_{1} + 176$$ $$\nu^{11}$$ $$=$$ $$-95 \beta_{14} + 245 \beta_{13} + 269 \beta_{12} - 98 \beta_{9} + 134 \beta_{8} + 269 \beta_{7} + 245 \beta_{6} + 123 \beta_{5} - 25 \beta_{3} - 722 \beta_{2} + 123 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$637 \beta_{15} + 365 \beta_{13} - 583 \beta_{11} - 490 \beta_{10} + 942 \beta_{9} - 490 \beta_{8} - 443 \beta_{7} + 1027 \beta_{5} - 942 \beta_{4} - 305 \beta_{3} + 942 \beta_{2} + 942 \beta_{1} - 944$$ $$\nu^{13}$$ $$=$$ $$949 \beta_{15} - 337 \beta_{14} - 893 \beta_{12} + 337 \beta_{11} - 730 \beta_{10} - 1253 \beta_{6} + 706 \beta_{4} - 3408 \beta_{2} - 717 \beta_{1} + 3408$$ $$\nu^{14}$$ $$=$$ $$1111 \beta_{14} + 739 \beta_{13} - 1525 \beta_{12} - 654 \beta_{9} + 2506 \beta_{8} - 1525 \beta_{7} + 739 \beta_{6} + 3245 \beta_{5} - 1503 \beta_{3} - 862 \beta_{2} + 3245 \beta_{1}$$ $$\nu^{15}$$ $$=$$ $$2459 \beta_{15} + 1131 \beta_{13} - 3137 \beta_{11} - 1478 \beta_{10} + 1698 \beta_{9} - 1478 \beta_{8} - 8365 \beta_{7} + 1189 \beta_{5} - 1698 \beta_{4} + 761 \beta_{3} + 1698 \beta_{2} + 1698 \beta_{1} - 13136$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$55$$ $$\chi(n)$$ $$-\beta_{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 −1.59523 − 1.20633i −1.26364 + 1.55023i −0.710719 + 1.86946i −0.523926 − 1.93016i 0.186266 − 1.99131i 1.63139 − 1.15696i 1.84233 + 0.778342i 1.93353 − 0.511345i −1.59523 + 1.20633i −1.26364 − 1.55023i −0.710719 − 1.86946i −0.523926 + 1.93016i 0.186266 + 1.99131i 1.63139 + 1.15696i 1.84233 − 0.778342i 1.93353 + 0.511345i
−1.59523 1.20633i 0 1.08951 + 3.84876i −1.10093 1.90686i 0 −7.23844 4.17912i 2.90487 7.45397i 0 −0.544081 + 4.36996i
19.2 −1.26364 + 1.55023i 0 −0.806428 3.91787i −1.35609 2.34881i 0 10.0431 + 5.79837i 7.09263 + 3.70062i 0 5.35481 + 0.865806i
19.3 −0.710719 + 1.86946i 0 −2.98976 2.65732i −1.35609 2.34881i 0 −10.0431 5.79837i 7.09263 3.70062i 0 5.35481 0.865806i
19.4 −0.523926 1.93016i 0 −3.45100 + 2.02252i 4.03104 + 6.98197i 0 3.90254 + 2.25313i 5.71184 + 5.60133i 0 11.3643 11.4386i
19.5 0.186266 1.99131i 0 −3.93061 0.741826i −3.07403 5.32438i 0 0.511543 + 0.295340i −2.20934 + 7.68888i 0 −11.1751 + 5.12959i
19.6 1.63139 1.15696i 0 1.32286 3.77492i −3.07403 5.32438i 0 −0.511543 0.295340i −2.20934 7.68888i 0 −11.1751 5.12959i
19.7 1.84233 + 0.778342i 0 2.78837 + 2.86793i −1.10093 1.90686i 0 7.23844 + 4.17912i 2.90487 + 7.45397i 0 −0.544081 4.36996i
19.8 1.93353 0.511345i 0 3.47705 1.97740i 4.03104 + 6.98197i 0 −3.90254 2.25313i 5.71184 5.60133i 0 11.3643 + 11.4386i
91.1 −1.59523 + 1.20633i 0 1.08951 3.84876i −1.10093 + 1.90686i 0 −7.23844 + 4.17912i 2.90487 + 7.45397i 0 −0.544081 4.36996i
91.2 −1.26364 1.55023i 0 −0.806428 + 3.91787i −1.35609 + 2.34881i 0 10.0431 5.79837i 7.09263 3.70062i 0 5.35481 0.865806i
91.3 −0.710719 1.86946i 0 −2.98976 + 2.65732i −1.35609 + 2.34881i 0 −10.0431 + 5.79837i 7.09263 + 3.70062i 0 5.35481 + 0.865806i
91.4 −0.523926 + 1.93016i 0 −3.45100 2.02252i 4.03104 6.98197i 0 3.90254 2.25313i 5.71184 5.60133i 0 11.3643 + 11.4386i
91.5 0.186266 + 1.99131i 0 −3.93061 + 0.741826i −3.07403 + 5.32438i 0 0.511543 0.295340i −2.20934 7.68888i 0 −11.1751 5.12959i
91.6 1.63139 + 1.15696i 0 1.32286 + 3.77492i −3.07403 + 5.32438i 0 −0.511543 + 0.295340i −2.20934 + 7.68888i 0 −11.1751 + 5.12959i
91.7 1.84233 0.778342i 0 2.78837 2.86793i −1.10093 + 1.90686i 0 7.23844 4.17912i 2.90487 7.45397i 0 −0.544081 + 4.36996i
91.8 1.93353 + 0.511345i 0 3.47705 + 1.97740i 4.03104 6.98197i 0 −3.90254 + 2.25313i 5.71184 + 5.60133i 0 11.3643 11.4386i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 91.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.c even 3 1 inner
36.f odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.3.f.c 16
3.b odd 2 1 36.3.f.c 16
4.b odd 2 1 inner 108.3.f.c 16
8.b even 2 1 1728.3.o.g 16
8.d odd 2 1 1728.3.o.g 16
9.c even 3 1 inner 108.3.f.c 16
9.c even 3 1 324.3.d.g 8
9.d odd 6 1 36.3.f.c 16
9.d odd 6 1 324.3.d.i 8
12.b even 2 1 36.3.f.c 16
24.f even 2 1 576.3.o.g 16
24.h odd 2 1 576.3.o.g 16
36.f odd 6 1 inner 108.3.f.c 16
36.f odd 6 1 324.3.d.g 8
36.h even 6 1 36.3.f.c 16
36.h even 6 1 324.3.d.i 8
72.j odd 6 1 576.3.o.g 16
72.l even 6 1 576.3.o.g 16
72.n even 6 1 1728.3.o.g 16
72.p odd 6 1 1728.3.o.g 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.3.f.c 16 3.b odd 2 1
36.3.f.c 16 9.d odd 6 1
36.3.f.c 16 12.b even 2 1
36.3.f.c 16 36.h even 6 1
108.3.f.c 16 1.a even 1 1 trivial
108.3.f.c 16 4.b odd 2 1 inner
108.3.f.c 16 9.c even 3 1 inner
108.3.f.c 16 36.f odd 6 1 inner
324.3.d.g 8 9.c even 3 1
324.3.d.g 8 36.f odd 6 1
324.3.d.i 8 9.d odd 6 1
324.3.d.i 8 36.h even 6 1
576.3.o.g 16 24.f even 2 1
576.3.o.g 16 24.h odd 2 1
576.3.o.g 16 72.j odd 6 1
576.3.o.g 16 72.l even 6 1
1728.3.o.g 16 8.b even 2 1
1728.3.o.g 16 8.d odd 2 1
1728.3.o.g 16 72.n even 6 1
1728.3.o.g 16 72.p odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(108, [\chi])$$:

 $$T_{5}^{8} + \cdots$$ $$T_{7}^{16} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 3 T + 7 T^{2} - 30 T^{3} + 76 T^{4} - 144 T^{5} + 424 T^{6} - 912 T^{7} + 1552 T^{8} - 3648 T^{9} + 6784 T^{10} - 9216 T^{11} + 19456 T^{12} - 30720 T^{13} + 28672 T^{14} - 49152 T^{15} + 65536 T^{16}$$
$3$ 1
$5$ $$( 1 + 3 T - 38 T^{2} + 201 T^{3} + 1495 T^{4} - 6984 T^{5} + 24112 T^{6} + 181380 T^{7} - 866084 T^{8} + 4534500 T^{9} + 15070000 T^{10} - 109125000 T^{11} + 583984375 T^{12} + 1962890625 T^{13} - 9277343750 T^{14} + 18310546875 T^{15} + 152587890625 T^{16} )^{2}$$
$7$ $$1 + 167 T^{2} + 13188 T^{4} + 535927 T^{6} + 4595825 T^{8} - 715412784 T^{10} - 48284089874 T^{12} - 2073288609886 T^{14} - 88693520972328 T^{16} - 4977965952336286 T^{18} - 278348169589725074 T^{20} - 9902233810610977584 T^{22} +$$$$15\!\cdots\!25$$$$T^{24} +$$$$42\!\cdots\!27$$$$T^{26} +$$$$25\!\cdots\!88$$$$T^{28} +$$$$76\!\cdots\!67$$$$T^{30} +$$$$11\!\cdots\!01$$$$T^{32}$$
$11$ $$1 + 524 T^{2} + 140094 T^{4} + 24486904 T^{6} + 2972860745 T^{8} + 215973352008 T^{10} - 2384433396482 T^{12} - 3310557806176204 T^{14} - 541778612591523276 T^{16} - 48469876840225802764 T^{18} -$$$$51\!\cdots\!42$$$$T^{20} +$$$$67\!\cdots\!68$$$$T^{22} +$$$$13\!\cdots\!45$$$$T^{24} +$$$$16\!\cdots\!04$$$$T^{26} +$$$$13\!\cdots\!54$$$$T^{28} +$$$$75\!\cdots\!44$$$$T^{30} +$$$$21\!\cdots\!21$$$$T^{32}$$
$13$ $$( 1 + 23 T - 276 T^{2} - 5009 T^{3} + 162641 T^{4} + 1572720 T^{5} - 34351730 T^{6} - 33339262 T^{7} + 8566506504 T^{8} - 5634335278 T^{9} - 981119760530 T^{10} + 7591219050480 T^{11} + 132671260194161 T^{12} - 690533185671641 T^{13} - 6430271493804756 T^{14} + 90559656871083647 T^{15} + 665416609183179841 T^{16} )^{2}$$
$17$ $$( 1 + 3 T + 578 T^{2} + 6549 T^{3} + 169242 T^{4} + 1892661 T^{5} + 48275138 T^{6} + 72412707 T^{7} + 6975757441 T^{8} )^{4}$$
$19$ $$( 1 - 1673 T^{2} + 1559890 T^{4} - 937716023 T^{6} + 401371470970 T^{8} - 122204089833383 T^{10} + 26492490152025490 T^{12} - 3702875859597687353 T^{14} +$$$$28\!\cdots\!81$$$$T^{16} )^{2}$$
$23$ $$1 + 2687 T^{2} + 3719652 T^{4} + 3461634655 T^{6} + 2442367009985 T^{8} + 1429403163693456 T^{10} + 759315200173466974 T^{12} +$$$$39\!\cdots\!18$$$$T^{14} +$$$$20\!\cdots\!24$$$$T^{16} +$$$$11\!\cdots\!38$$$$T^{18} +$$$$59\!\cdots\!94$$$$T^{20} +$$$$31\!\cdots\!76$$$$T^{22} +$$$$14\!\cdots\!85$$$$T^{24} +$$$$59\!\cdots\!55$$$$T^{26} +$$$$17\!\cdots\!32$$$$T^{28} +$$$$36\!\cdots\!47$$$$T^{30} +$$$$37\!\cdots\!21$$$$T^{32}$$
$29$ $$( 1 + 21 T - 2432 T^{2} - 19167 T^{3} + 4062037 T^{4} + 6623136 T^{5} - 4703569190 T^{6} - 3469324686 T^{7} + 4129311885376 T^{8} - 2917702060926 T^{9} - 3326745120272390 T^{10} + 3939595750954656 T^{11} + 2032019438564861557 T^{12} - 8063695540664952567 T^{13} -$$$$86\!\cdots\!12$$$$T^{14} +$$$$62\!\cdots\!01$$$$T^{15} +$$$$25\!\cdots\!21$$$$T^{16} )^{2}$$
$31$ $$1 + 4715 T^{2} + 10729584 T^{4} + 17585852527 T^{6} + 25170831884477 T^{8} + 32193274685973408 T^{10} + 37136398859226646834 T^{12} +$$$$40\!\cdots\!98$$$$T^{14} +$$$$41\!\cdots\!28$$$$T^{16} +$$$$37\!\cdots\!58$$$$T^{18} +$$$$31\!\cdots\!94$$$$T^{20} +$$$$25\!\cdots\!88$$$$T^{22} +$$$$18\!\cdots\!37$$$$T^{24} +$$$$11\!\cdots\!27$$$$T^{26} +$$$$66\!\cdots\!64$$$$T^{28} +$$$$27\!\cdots\!15$$$$T^{30} +$$$$52\!\cdots\!61$$$$T^{32}$$
$37$ $$( 1 - 14 T + 3040 T^{2} - 66050 T^{3} + 4581118 T^{4} - 90422450 T^{5} + 5697449440 T^{6} - 35920169726 T^{7} + 3512479453921 T^{8} )^{4}$$
$41$ $$( 1 + 42 T - 4424 T^{2} - 125040 T^{3} + 14943835 T^{4} + 232367040 T^{5} - 36025798940 T^{6} - 132403432026 T^{7} + 71285616374608 T^{8} - 222570169235706 T^{9} - 101800297638493340 T^{10} + 1103767662172616640 T^{11} +$$$$11\!\cdots\!35$$$$T^{12} -$$$$16\!\cdots\!40$$$$T^{13} -$$$$99\!\cdots\!44$$$$T^{14} +$$$$15\!\cdots\!62$$$$T^{15} +$$$$63\!\cdots\!41$$$$T^{16} )^{2}$$
$43$ $$1 + 10292 T^{2} + 52819302 T^{4} + 202099368136 T^{6} + 665872758097265 T^{8} + 1877680374529631208 T^{10} +$$$$45\!\cdots\!90$$$$T^{12} +$$$$10\!\cdots\!64$$$$T^{14} +$$$$19\!\cdots\!28$$$$T^{16} +$$$$34\!\cdots\!64$$$$T^{18} +$$$$53\!\cdots\!90$$$$T^{20} +$$$$75\!\cdots\!08$$$$T^{22} +$$$$90\!\cdots\!65$$$$T^{24} +$$$$94\!\cdots\!36$$$$T^{26} +$$$$84\!\cdots\!02$$$$T^{28} +$$$$56\!\cdots\!92$$$$T^{30} +$$$$18\!\cdots\!01$$$$T^{32}$$
$47$ $$1 + 12983 T^{2} + 90223140 T^{4} + 428939892679 T^{6} + 1551988218895697 T^{8} + 4556649670813575888 T^{10} +$$$$11\!\cdots\!54$$$$T^{12} +$$$$26\!\cdots\!06$$$$T^{14} +$$$$58\!\cdots\!04$$$$T^{16} +$$$$12\!\cdots\!86$$$$T^{18} +$$$$27\!\cdots\!94$$$$T^{20} +$$$$52\!\cdots\!08$$$$T^{22} +$$$$87\!\cdots\!37$$$$T^{24} +$$$$11\!\cdots\!79$$$$T^{26} +$$$$12\!\cdots\!40$$$$T^{28} +$$$$85\!\cdots\!63$$$$T^{30} +$$$$32\!\cdots\!41$$$$T^{32}$$
$53$ $$( 1 - 18 T + 10016 T^{2} - 126558 T^{3} + 40472766 T^{4} - 355501422 T^{5} + 79031057696 T^{6} - 398958500322 T^{7} + 62259690411361 T^{8} )^{4}$$
$59$ $$1 + 18092 T^{2} + 175903662 T^{4} + 1133035156984 T^{6} + 5240167912552121 T^{8} + 17175194950853605128 T^{10} +$$$$35\!\cdots\!18$$$$T^{12} +$$$$21\!\cdots\!16$$$$T^{14} -$$$$83\!\cdots\!68$$$$T^{16} +$$$$26\!\cdots\!76$$$$T^{18} +$$$$52\!\cdots\!78$$$$T^{20} +$$$$30\!\cdots\!68$$$$T^{22} +$$$$11\!\cdots\!61$$$$T^{24} +$$$$29\!\cdots\!84$$$$T^{26} +$$$$55\!\cdots\!82$$$$T^{28} +$$$$69\!\cdots\!32$$$$T^{30} +$$$$46\!\cdots\!81$$$$T^{32}$$
$61$ $$( 1 + 17 T - 3582 T^{2} - 125549 T^{3} - 18305593 T^{4} - 350178696 T^{5} - 12273736208 T^{6} + 2060229775052 T^{7} + 599123836260396 T^{8} + 7666114992968492 T^{9} - 169940200011910928 T^{10} - 18041337511166813256 T^{11} -$$$$35\!\cdots\!33$$$$T^{12} -$$$$89\!\cdots\!49$$$$T^{13} -$$$$95\!\cdots\!22$$$$T^{14} +$$$$16\!\cdots\!97$$$$T^{15} +$$$$36\!\cdots\!61$$$$T^{16} )^{2}$$
$67$ $$1 + 25148 T^{2} + 330341646 T^{4} + 3017899283800 T^{6} + 21559886998912025 T^{8} +$$$$12\!\cdots\!60$$$$T^{10} +$$$$66\!\cdots\!50$$$$T^{12} +$$$$31\!\cdots\!76$$$$T^{14} +$$$$14\!\cdots\!68$$$$T^{16} +$$$$63\!\cdots\!96$$$$T^{18} +$$$$27\!\cdots\!50$$$$T^{20} +$$$$10\!\cdots\!60$$$$T^{22} +$$$$35\!\cdots\!25$$$$T^{24} +$$$$10\!\cdots\!00$$$$T^{26} +$$$$22\!\cdots\!66$$$$T^{28} +$$$$33\!\cdots\!68$$$$T^{30} +$$$$27\!\cdots\!61$$$$T^{32}$$
$71$ $$( 1 - 12968 T^{2} + 137492380 T^{4} - 912038324888 T^{6} + 5454839368725190 T^{8} - 23176426971828216728 T^{10} +$$$$88\!\cdots\!80$$$$T^{12} -$$$$21\!\cdots\!88$$$$T^{14} +$$$$41\!\cdots\!21$$$$T^{16} )^{2}$$
$73$ $$( 1 - 29 T + 7774 T^{2} - 620747 T^{3} + 46170850 T^{4} - 3307960763 T^{5} + 220767925534 T^{6} - 4388692562381 T^{7} + 806460091894081 T^{8} )^{4}$$
$79$ $$1 + 23147 T^{2} + 328058736 T^{4} + 3136165224559 T^{6} + 21315067551811709 T^{8} + 91269506056145418912 T^{10} +$$$$59\!\cdots\!06$$$$T^{12} -$$$$27\!\cdots\!10$$$$T^{14} -$$$$25\!\cdots\!44$$$$T^{16} -$$$$10\!\cdots\!10$$$$T^{18} +$$$$90\!\cdots\!66$$$$T^{20} +$$$$53\!\cdots\!92$$$$T^{22} +$$$$49\!\cdots\!89$$$$T^{24} +$$$$28\!\cdots\!59$$$$T^{26} +$$$$11\!\cdots\!16$$$$T^{28} +$$$$31\!\cdots\!67$$$$T^{30} +$$$$52\!\cdots\!41$$$$T^{32}$$
$83$ $$1 + 17795 T^{2} + 28452672 T^{4} - 239952438809 T^{6} + 12519030554664557 T^{8} + 90364909803409302048 T^{10} -$$$$32\!\cdots\!46$$$$T^{12} +$$$$11\!\cdots\!22$$$$T^{14} +$$$$52\!\cdots\!80$$$$T^{16} +$$$$55\!\cdots\!62$$$$T^{18} -$$$$73\!\cdots\!86$$$$T^{20} +$$$$96\!\cdots\!28$$$$T^{22} +$$$$63\!\cdots\!17$$$$T^{24} -$$$$57\!\cdots\!09$$$$T^{26} +$$$$32\!\cdots\!12$$$$T^{28} +$$$$96\!\cdots\!95$$$$T^{30} +$$$$25\!\cdots\!61$$$$T^{32}$$
$89$ $$( 1 - 96 T + 27716 T^{2} - 1940016 T^{3} + 308634966 T^{4} - 15366866736 T^{5} + 1738963951556 T^{6} - 47710203932256 T^{7} + 3936588805702081 T^{8} )^{4}$$
$97$ $$( 1 + 74 T - 29868 T^{2} - 1540328 T^{3} + 607324823 T^{4} + 20751693936 T^{5} - 8122914917000 T^{6} - 74589814314322 T^{7} + 88320904907559480 T^{8} - 701815562883455698 T^{9} -$$$$71\!\cdots\!00$$$$T^{10} +$$$$17\!\cdots\!44$$$$T^{11} +$$$$47\!\cdots\!03$$$$T^{12} -$$$$11\!\cdots\!72$$$$T^{13} -$$$$20\!\cdots\!88$$$$T^{14} +$$$$48\!\cdots\!06$$$$T^{15} +$$$$61\!\cdots\!21$$$$T^{16} )^{2}$$
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