# Properties

 Label 74.5.d.a Level $74$ Weight $5$ Character orbit 74.d Analytic conductor $7.649$ Analytic rank $0$ Dimension $14$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$74 = 2 \cdot 37$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 74.d (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.64937726820$$ Analytic rank: $$0$$ Dimension: $$14$$ Relative dimension: $$7$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{14} + \cdots)$$ Defining polynomial: $$x^{14} + 727 x^{12} + 198453 x^{10} + 24875201 x^{8} + 1392846203 x^{6} + 29089700589 x^{4} + 220261242916 x^{2} + 446074380544$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{13}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 + 2 \beta_{5} ) q^{2} + ( \beta_{1} + \beta_{5} ) q^{3} -8 \beta_{5} q^{4} + ( -1 - \beta_{5} + \beta_{6} ) q^{5} + ( -2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{5} ) q^{6} + ( 3 - \beta_{4} ) q^{7} + ( 16 + 16 \beta_{5} ) q^{8} + ( -25 - \beta_{2} + \beta_{3} + \beta_{6} + \beta_{7} ) q^{9} +O(q^{10})$$ $$q + ( -2 + 2 \beta_{5} ) q^{2} + ( \beta_{1} + \beta_{5} ) q^{3} -8 \beta_{5} q^{4} + ( -1 - \beta_{5} + \beta_{6} ) q^{5} + ( -2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{5} ) q^{6} + ( 3 - \beta_{4} ) q^{7} + ( 16 + 16 \beta_{5} ) q^{8} + ( -25 - \beta_{2} + \beta_{3} + \beta_{6} + \beta_{7} ) q^{9} + ( 4 - 2 \beta_{6} - 2 \beta_{7} ) q^{10} + ( \beta_{1} + 10 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{10} + \beta_{11} + \beta_{13} ) q^{11} + ( 8 + 8 \beta_{2} ) q^{12} + ( -4 + \beta_{1} + \beta_{2} - 4 \beta_{5} - \beta_{6} + \beta_{9} ) q^{13} + ( -6 + 2 \beta_{4} + 6 \beta_{5} + 2 \beta_{12} ) q^{14} + ( -27 - 5 \beta_{1} + 5 \beta_{2} - \beta_{3} + 27 \beta_{5} - 5 \beta_{7} - \beta_{8} - 3 \beta_{10} + \beta_{13} ) q^{15} -64 q^{16} + ( -24 + 3 \beta_{1} + 3 \beta_{2} - \beta_{4} - 24 \beta_{5} - 9 \beta_{6} - 3 \beta_{11} + \beta_{12} ) q^{17} + ( 50 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 50 \beta_{5} - 4 \beta_{7} + 2 \beta_{13} ) q^{18} + ( -15 + \beta_{1} + \beta_{2} + \beta_{3} - 3 \beta_{4} - 15 \beta_{5} + \beta_{6} - \beta_{11} + 3 \beta_{12} + \beta_{13} ) q^{19} + ( -8 + 8 \beta_{5} + 8 \beta_{7} ) q^{20} + ( 6 \beta_{1} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + 5 \beta_{12} + 2 \beta_{13} ) q^{21} + ( -20 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 20 \beta_{5} + 4 \beta_{6} - 4 \beta_{11} - 2 \beta_{13} ) q^{22} + ( -30 - 16 \beta_{1} - 16 \beta_{2} + 4 \beta_{3} - 30 \beta_{5} + 2 \beta_{6} - \beta_{9} - 3 \beta_{11} + 4 \beta_{13} ) q^{23} + ( -16 + 16 \beta_{1} - 16 \beta_{2} + 16 \beta_{5} ) q^{24} + ( 15 \beta_{1} + 37 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} - 4 \beta_{10} + 4 \beta_{11} + 5 \beta_{12} ) q^{25} + ( 16 - 4 \beta_{2} + 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} ) q^{26} + ( -20 \beta_{1} + 22 \beta_{5} + 11 \beta_{6} - 11 \beta_{7} - 4 \beta_{10} + 4 \beta_{11} + \beta_{12} + 7 \beta_{13} ) q^{27} + ( -24 \beta_{5} - 8 \beta_{12} ) q^{28} + ( -42 - 13 \beta_{1} + 13 \beta_{2} - 5 \beta_{4} + 42 \beta_{5} - 3 \beta_{8} - \beta_{10} - 5 \beta_{12} ) q^{29} + ( 20 \beta_{1} - 108 \beta_{5} - 10 \beta_{6} + 10 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + 6 \beta_{10} - 6 \beta_{11} - 4 \beta_{13} ) q^{30} + ( -47 + 11 \beta_{1} - 11 \beta_{2} + 5 \beta_{3} - 5 \beta_{4} + 47 \beta_{5} + 16 \beta_{7} + 4 \beta_{8} + 3 \beta_{10} - 5 \beta_{12} - 5 \beta_{13} ) q^{31} + ( 128 - 128 \beta_{5} ) q^{32} + ( 38 + 32 \beta_{2} - 3 \beta_{3} + 5 \beta_{4} - 17 \beta_{6} - 17 \beta_{7} + 3 \beta_{8} + 3 \beta_{9} + 4 \beta_{10} + 4 \beta_{11} ) q^{33} + ( 96 - 12 \beta_{2} + 4 \beta_{4} + 18 \beta_{6} + 18 \beta_{7} + 6 \beta_{10} + 6 \beta_{11} ) q^{34} + ( 127 + \beta_{1} + \beta_{2} - 6 \beta_{3} - 5 \beta_{4} + 127 \beta_{5} + 23 \beta_{6} + 4 \beta_{9} + 3 \beta_{11} + 5 \beta_{12} - 6 \beta_{13} ) q^{35} + ( 8 \beta_{1} + 200 \beta_{5} - 8 \beta_{6} + 8 \beta_{7} - 8 \beta_{13} ) q^{36} + ( -181 - \beta_{1} + 9 \beta_{2} + \beta_{3} - 5 \beta_{4} + 17 \beta_{5} - 7 \beta_{6} - 6 \beta_{7} + 2 \beta_{8} - 3 \beta_{9} - 3 \beta_{10} - 10 \beta_{11} - 11 \beta_{12} - 10 \beta_{13} ) q^{37} + ( 60 - 4 \beta_{2} - 4 \beta_{3} + 12 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{10} + 2 \beta_{11} ) q^{38} + ( -59 + \beta_{1} - \beta_{2} + 10 \beta_{3} + 8 \beta_{4} + 59 \beta_{5} - 23 \beta_{7} + 4 \beta_{8} + 14 \beta_{10} + 8 \beta_{12} - 10 \beta_{13} ) q^{39} + ( -32 \beta_{5} + 16 \beta_{6} - 16 \beta_{7} ) q^{40} + ( 11 \beta_{1} - 353 \beta_{5} - 19 \beta_{6} + 19 \beta_{7} + 3 \beta_{8} - 3 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} - 10 \beta_{12} + 10 \beta_{13} ) q^{41} + ( -2 - 12 \beta_{1} - 12 \beta_{2} - 4 \beta_{3} + 10 \beta_{4} - 2 \beta_{5} - 4 \beta_{6} - 4 \beta_{9} - 8 \beta_{11} - 10 \beta_{12} - 4 \beta_{13} ) q^{42} + ( 160 - 75 \beta_{1} - 75 \beta_{2} + 3 \beta_{3} + 10 \beta_{4} + 160 \beta_{5} + 31 \beta_{6} + 21 \beta_{11} - 10 \beta_{12} + 3 \beta_{13} ) q^{43} + ( 80 + 8 \beta_{2} + 8 \beta_{3} - 8 \beta_{6} - 8 \beta_{7} + 8 \beta_{10} + 8 \beta_{11} ) q^{44} + ( 444 + 33 \beta_{1} + 33 \beta_{2} - 9 \beta_{3} + 15 \beta_{4} + 444 \beta_{5} - 48 \beta_{6} + 3 \beta_{9} - 15 \beta_{12} - 9 \beta_{13} ) q^{45} + ( 120 + 64 \beta_{2} - 16 \beta_{3} - 4 \beta_{6} - 4 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + 6 \beta_{10} + 6 \beta_{11} ) q^{46} + ( 397 + 24 \beta_{2} - 12 \beta_{3} - 31 \beta_{4} - 29 \beta_{6} - 29 \beta_{7} - 5 \beta_{8} - 5 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{47} + ( -64 \beta_{1} - 64 \beta_{5} ) q^{48} + ( 492 + 94 \beta_{2} + 20 \beta_{3} + 15 \beta_{4} + 42 \beta_{6} + 42 \beta_{7} - 4 \beta_{8} - 4 \beta_{9} + 15 \beta_{10} + 15 \beta_{11} ) q^{49} + ( -74 - 30 \beta_{1} - 30 \beta_{2} + 10 \beta_{4} - 74 \beta_{5} - 8 \beta_{6} - 4 \beta_{9} - 16 \beta_{11} - 10 \beta_{12} ) q^{50} + ( -205 - 12 \beta_{1} + 12 \beta_{2} + 13 \beta_{3} - \beta_{4} + 205 \beta_{5} + 103 \beta_{7} + 4 \beta_{8} + 17 \beta_{10} - \beta_{12} - 13 \beta_{13} ) q^{51} + ( -32 - 8 \beta_{1} + 8 \beta_{2} + 32 \beta_{5} - 8 \beta_{7} + 8 \beta_{8} ) q^{52} + ( -1359 - 158 \beta_{2} + 25 \beta_{4} + 33 \beta_{6} + 33 \beta_{7} + \beta_{8} + \beta_{9} - 19 \beta_{10} - 19 \beta_{11} ) q^{53} + ( -44 + 40 \beta_{1} + 40 \beta_{2} - 14 \beta_{3} + 2 \beta_{4} - 44 \beta_{5} - 44 \beta_{6} - 16 \beta_{11} - 2 \beta_{12} - 14 \beta_{13} ) q^{54} + ( 1033 + 104 \beta_{1} - 104 \beta_{2} + 9 \beta_{3} - 20 \beta_{4} - 1033 \beta_{5} - 37 \beta_{7} + \beta_{8} + 12 \beta_{10} - 20 \beta_{12} - 9 \beta_{13} ) q^{55} + ( 48 - 16 \beta_{4} + 48 \beta_{5} + 16 \beta_{12} ) q^{56} + ( -183 - 87 \beta_{1} + 87 \beta_{2} - 9 \beta_{3} + 14 \beta_{4} + 183 \beta_{5} - \beta_{7} - 6 \beta_{8} - 19 \beta_{10} + 14 \beta_{12} + 9 \beta_{13} ) q^{57} + ( 52 \beta_{1} - 168 \beta_{5} + 6 \beta_{8} - 6 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} + 20 \beta_{12} ) q^{58} + ( -316 - 42 \beta_{1} - 42 \beta_{2} - 4 \beta_{3} - 15 \beta_{4} - 316 \beta_{5} + 104 \beta_{6} + 4 \beta_{9} - 14 \beta_{11} + 15 \beta_{12} - 4 \beta_{13} ) q^{59} + ( 216 - 40 \beta_{1} - 40 \beta_{2} + 8 \beta_{3} + 216 \beta_{5} + 40 \beta_{6} + 8 \beta_{9} + 24 \beta_{11} + 8 \beta_{13} ) q^{60} + ( -784 + 200 \beta_{1} - 200 \beta_{2} - 4 \beta_{3} - 40 \beta_{4} + 784 \beta_{5} - 18 \beta_{7} - 6 \beta_{8} + 11 \beta_{10} - 40 \beta_{12} + 4 \beta_{13} ) q^{61} + ( -44 \beta_{1} - 188 \beta_{5} + 32 \beta_{6} - 32 \beta_{7} - 8 \beta_{8} + 8 \beta_{9} - 6 \beta_{10} + 6 \beta_{11} + 20 \beta_{12} + 20 \beta_{13} ) q^{62} + ( -212 + 90 \beta_{2} - 2 \beta_{3} - 30 \beta_{4} - 80 \beta_{6} - 80 \beta_{7} + \beta_{8} + \beta_{9} ) q^{63} + 512 \beta_{5} q^{64} + ( 226 \beta_{1} - 739 \beta_{5} + 21 \beta_{6} - 21 \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} + 52 \beta_{12} + 25 \beta_{13} ) q^{65} + ( -76 + 64 \beta_{1} - 64 \beta_{2} + 6 \beta_{3} - 10 \beta_{4} + 76 \beta_{5} + 68 \beta_{7} - 12 \beta_{8} - 16 \beta_{10} - 10 \beta_{12} - 6 \beta_{13} ) q^{66} + ( 169 \beta_{1} - 873 \beta_{5} + 54 \beta_{6} - 54 \beta_{7} + 4 \beta_{8} - 4 \beta_{9} - 9 \beta_{10} + 9 \beta_{11} + 15 \beta_{12} + 23 \beta_{13} ) q^{67} + ( -192 - 24 \beta_{1} + 24 \beta_{2} - 8 \beta_{4} + 192 \beta_{5} - 72 \beta_{7} - 24 \beta_{10} - 8 \beta_{12} ) q^{68} + ( 1412 - 338 \beta_{1} + 338 \beta_{2} - 39 \beta_{3} - 10 \beta_{4} - 1412 \beta_{5} + 2 \beta_{7} - 31 \beta_{10} - 10 \beta_{12} + 39 \beta_{13} ) q^{69} + ( -508 - 4 \beta_{2} + 24 \beta_{3} + 20 \beta_{4} - 46 \beta_{6} - 46 \beta_{7} - 8 \beta_{8} - 8 \beta_{9} - 6 \beta_{10} - 6 \beta_{11} ) q^{70} + ( -761 - 50 \beta_{2} - 6 \beta_{3} + 53 \beta_{4} + 112 \beta_{6} + 112 \beta_{7} - 4 \beta_{8} - 4 \beta_{9} - 33 \beta_{10} - 33 \beta_{11} ) q^{71} + ( -400 - 16 \beta_{1} - 16 \beta_{2} + 16 \beta_{3} - 400 \beta_{5} + 32 \beta_{6} + 16 \beta_{13} ) q^{72} + ( -133 \beta_{1} + 3592 \beta_{5} - 80 \beta_{6} + 80 \beta_{7} + 25 \beta_{10} - 25 \beta_{11} + 50 \beta_{12} - 49 \beta_{13} ) q^{73} + ( 328 + 20 \beta_{1} - 16 \beta_{2} + 18 \beta_{3} - 12 \beta_{4} - 396 \beta_{5} + 2 \beta_{6} + 26 \beta_{7} + 2 \beta_{8} + 10 \beta_{9} + 26 \beta_{10} + 14 \beta_{11} + 32 \beta_{12} + 22 \beta_{13} ) q^{74} + ( -1238 - 110 \beta_{2} + 9 \beta_{3} + 57 \beta_{4} - 102 \beta_{6} - 102 \beta_{7} + 3 \beta_{10} + 3 \beta_{11} ) q^{75} + ( -120 - 8 \beta_{1} + 8 \beta_{2} + 8 \beta_{3} - 24 \beta_{4} + 120 \beta_{5} + 8 \beta_{7} - 8 \beta_{10} - 24 \beta_{12} - 8 \beta_{13} ) q^{76} + ( 220 \beta_{1} - 215 \beta_{5} - 104 \beta_{6} + 104 \beta_{7} - \beta_{8} + \beta_{9} + 10 \beta_{10} - 10 \beta_{11} + 95 \beta_{12} + 30 \beta_{13} ) q^{77} + ( -4 \beta_{1} - 236 \beta_{5} - 46 \beta_{6} + 46 \beta_{7} - 8 \beta_{8} + 8 \beta_{9} - 28 \beta_{10} + 28 \beta_{11} - 32 \beta_{12} + 40 \beta_{13} ) q^{78} + ( 1599 - 256 \beta_{1} - 256 \beta_{2} - 30 \beta_{3} + 40 \beta_{4} + 1599 \beta_{5} - 7 \beta_{6} - 3 \beta_{9} + 10 \beta_{11} - 40 \beta_{12} - 30 \beta_{13} ) q^{79} + ( 64 + 64 \beta_{5} - 64 \beta_{6} ) q^{80} + ( -218 + 386 \beta_{2} + \beta_{3} + 28 \beta_{4} - 102 \beta_{6} - 102 \beta_{7} - \beta_{8} - \beta_{9} - 34 \beta_{10} - 34 \beta_{11} ) q^{81} + ( 706 - 22 \beta_{1} - 22 \beta_{2} - 20 \beta_{3} - 20 \beta_{4} + 706 \beta_{5} + 76 \beta_{6} + 12 \beta_{9} - 8 \beta_{11} + 20 \beta_{12} - 20 \beta_{13} ) q^{82} + ( -1599 + 398 \beta_{2} - 10 \beta_{3} - 65 \beta_{4} - 136 \beta_{6} - 136 \beta_{7} - \beta_{10} - \beta_{11} ) q^{83} + ( 8 + 48 \beta_{2} + 16 \beta_{3} - 40 \beta_{4} + 8 \beta_{6} + 8 \beta_{7} + 8 \beta_{8} + 8 \beta_{9} + 16 \beta_{10} + 16 \beta_{11} ) q^{84} + ( -496 \beta_{1} - 4930 \beta_{5} - 100 \beta_{6} + 100 \beta_{7} - \beta_{8} + \beta_{9} + 21 \beta_{10} - 21 \beta_{11} - 50 \beta_{12} ) q^{85} + ( -640 + 300 \beta_{2} - 12 \beta_{3} - 40 \beta_{4} - 62 \beta_{6} - 62 \beta_{7} - 42 \beta_{10} - 42 \beta_{11} ) q^{86} + ( 1411 - 39 \beta_{1} - 39 \beta_{2} + 20 \beta_{3} + \beta_{4} + 1411 \beta_{5} - 124 \beta_{6} + 20 \beta_{9} + 55 \beta_{11} - \beta_{12} + 20 \beta_{13} ) q^{87} + ( -160 + 16 \beta_{1} - 16 \beta_{2} - 16 \beta_{3} + 160 \beta_{5} + 32 \beta_{7} - 32 \beta_{10} + 16 \beta_{13} ) q^{88} + ( 1018 - 12 \beta_{1} + 12 \beta_{2} - 24 \beta_{3} + 30 \beta_{4} - 1018 \beta_{5} - 151 \beta_{7} - 16 \beta_{8} + 17 \beta_{10} + 30 \beta_{12} + 24 \beta_{13} ) q^{89} + ( -1776 - 132 \beta_{2} + 36 \beta_{3} - 60 \beta_{4} + 96 \beta_{6} + 96 \beta_{7} - 6 \beta_{8} - 6 \beta_{9} ) q^{90} + ( -558 + 242 \beta_{1} + 242 \beta_{2} + 9 \beta_{3} + 102 \beta_{4} - 558 \beta_{5} + 255 \beta_{6} + 4 \beta_{9} + 59 \beta_{11} - 102 \beta_{12} + 9 \beta_{13} ) q^{91} + ( -240 + 128 \beta_{1} - 128 \beta_{2} + 32 \beta_{3} + 240 \beta_{5} + 16 \beta_{7} - 8 \beta_{8} - 24 \beta_{10} - 32 \beta_{13} ) q^{92} + ( -913 - 405 \beta_{1} - 405 \beta_{2} + 28 \beta_{3} - 68 \beta_{4} - 913 \beta_{5} + 336 \beta_{6} + \beta_{9} + 28 \beta_{11} + 68 \beta_{12} + 28 \beta_{13} ) q^{93} + ( -794 + 48 \beta_{1} - 48 \beta_{2} + 24 \beta_{3} + 62 \beta_{4} + 794 \beta_{5} + 116 \beta_{7} + 20 \beta_{8} + 8 \beta_{10} + 62 \beta_{12} - 24 \beta_{13} ) q^{94} + ( -74 \beta_{1} + 1278 \beta_{5} - 17 \beta_{8} + 17 \beta_{9} - 12 \beta_{10} + 12 \beta_{11} - 10 \beta_{12} - 48 \beta_{13} ) q^{95} + ( 128 + 128 \beta_{1} + 128 \beta_{2} + 128 \beta_{5} ) q^{96} + ( 1299 - 287 \beta_{1} - 287 \beta_{2} + 11 \beta_{3} - 24 \beta_{4} + 1299 \beta_{5} - 10 \beta_{6} - 22 \beta_{9} - 20 \beta_{11} + 24 \beta_{12} + 11 \beta_{13} ) q^{97} + ( -984 + 188 \beta_{1} - 188 \beta_{2} - 40 \beta_{3} - 30 \beta_{4} + 984 \beta_{5} - 168 \beta_{7} + 16 \beta_{8} - 60 \beta_{10} - 30 \beta_{12} + 40 \beta_{13} ) q^{98} + ( 522 \beta_{1} + 2689 \beta_{5} - 56 \beta_{6} + 56 \beta_{7} + 32 \beta_{8} - 32 \beta_{9} + 24 \beta_{10} - 24 \beta_{11} + 36 \beta_{12} - 47 \beta_{13} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$14 q - 28 q^{2} - 12 q^{5} - 40 q^{6} + 48 q^{7} + 224 q^{8} - 346 q^{9} + O(q^{10})$$ $$14 q - 28 q^{2} - 12 q^{5} - 40 q^{6} + 48 q^{7} + 224 q^{8} - 346 q^{9} + 48 q^{10} + 160 q^{12} - 56 q^{13} - 96 q^{14} - 378 q^{15} - 896 q^{16} - 348 q^{17} + 692 q^{18} - 184 q^{19} - 96 q^{20} - 320 q^{22} - 502 q^{23} - 320 q^{24} + 224 q^{26} - 474 q^{29} - 630 q^{31} + 1792 q^{32} + 632 q^{33} + 1392 q^{34} + 1826 q^{35} - 2544 q^{37} + 736 q^{38} - 798 q^{39} - 224 q^{42} + 1936 q^{43} + 1280 q^{44} + 6162 q^{45} + 2008 q^{46} + 5716 q^{47} + 7862 q^{49} - 1372 q^{50} - 2422 q^{51} - 448 q^{52} - 20228 q^{53} - 656 q^{54} + 14006 q^{55} + 768 q^{56} - 2270 q^{57} - 4502 q^{59} + 3024 q^{60} - 11906 q^{61} - 2588 q^{63} - 1264 q^{66} - 2784 q^{68} + 21440 q^{69} - 7304 q^{70} - 11224 q^{71} - 5536 q^{72} + 4924 q^{74} - 18652 q^{75} - 1472 q^{76} + 20488 q^{79} + 768 q^{80} - 1706 q^{81} + 9808 q^{82} - 20224 q^{83} + 896 q^{84} - 7744 q^{86} + 19636 q^{87} - 2560 q^{88} + 13864 q^{89} - 24648 q^{90} - 6070 q^{91} - 4016 q^{92} - 13800 q^{93} - 11432 q^{94} + 2560 q^{96} + 16622 q^{97} - 15724 q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{14} + 727 x^{12} + 198453 x^{10} + 24875201 x^{8} + 1392846203 x^{6} + 29089700589 x^{4} + 220261242916 x^{2} + 446074380544$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-12304969 \nu^{12} - 9298449380 \nu^{10} - 2566638510769 \nu^{8} - 316719169778245 \nu^{6} - 17222871309727364 \nu^{4} - 359304590593552423 \nu^{2} - 1405169049727037712$$$$)/ 245491411879762542$$ $$\beta_{3}$$ $$=$$ $$($$$$-18801796537865 \nu^{12} - 13221092087830923 \nu^{10} - 3463428522829055922 \nu^{8} - 409063233179906607991 \nu^{6} - 20339986533613910443005 \nu^{4} - 271514716795853091050922 \nu^{2} + 498016406895908104442914$$$$)/$$$$12\!\cdots\!94$$ $$\beta_{4}$$ $$=$$ $$($$$$47409101752019 \nu^{12} + 33959975741576592 \nu^{10} + 9015319915234947162 \nu^{8} + 1067713286268304877383 \nu^{6} + 52231409979860816876526 \nu^{4} + 704342690025659384774094 \nu^{2} + 1975328585141289468427688$$$$)/$$$$12\!\cdots\!94$$ $$\beta_{5}$$ $$=$$ $$($$$$-2103899231199 \nu^{13} - 1521316399946201 \nu^{11} - 411314791369625707 \nu^{9} - 50620689198140110127 \nu^{7} - 2718875122805293790837 \nu^{5} - 49698849631695007788979 \nu^{3} - 223432235231562044243660 \nu$$$$)/$$$$16\!\cdots\!96$$ $$\beta_{6}$$ $$=$$ $$($$$$107943329899431209 \nu^{13} + 498550196745827824 \nu^{12} + 77937783551342473407 \nu^{11} + 351388870020091384104 \nu^{10} + 20885021306499043067733 \nu^{9} + 92216155062786469008840 \nu^{8} + 2501970302348076513949177 \nu^{7} + 10908765475237904000805728 \nu^{6} + 123920636224798546356960771 \nu^{5} + 544285911094862180322373464 \nu^{4} + 1636778736634476986040454173 \nu^{3} + 7739926782842929799450140776 \nu^{2} + 2700586260015201950850004196 \nu + 21600420243978182353964482720$$$$)/$$$$62\!\cdots\!44$$ $$\beta_{7}$$ $$=$$ $$($$$$-107943329899431209 \nu^{13} + 498550196745827824 \nu^{12} - 77937783551342473407 \nu^{11} + 351388870020091384104 \nu^{10} - 20885021306499043067733 \nu^{9} + 92216155062786469008840 \nu^{8} - 2501970302348076513949177 \nu^{7} + 10908765475237904000805728 \nu^{6} - 123920636224798546356960771 \nu^{5} + 544285911094862180322373464 \nu^{4} - 1636778736634476986040454173 \nu^{3} + 7739926782842929799450140776 \nu^{2} - 2700586260015201950850004196 \nu + 21600420243978182353964482720$$$$)/$$$$62\!\cdots\!44$$ $$\beta_{8}$$ $$=$$ $$($$$$33656521722517663 \nu^{13} + 274799469079399992 \nu^{12} + 24979512999147996593 \nu^{11} + 208413211039772617320 \nu^{10} + 6946966630395695644363 \nu^{9} + 59146846017587708334240 \nu^{8} + 884378082421354786970575 \nu^{7} + 7638077306455251237639720 \nu^{6} + 50281306431579765857488445 \nu^{5} + 429405681651349754101224552 \nu^{4} + 1108799581866980612071590499 \nu^{3} + 8132472668178365074821464016 \nu^{2} + 10460680008268085441324131692 \nu + 34772983231242980815028210208$$$$)/$$$$69\!\cdots\!16$$ $$\beta_{9}$$ $$=$$ $$($$$$-33656521722517663 \nu^{13} + 274799469079399992 \nu^{12} - 24979512999147996593 \nu^{11} + 208413211039772617320 \nu^{10} - 6946966630395695644363 \nu^{9} + 59146846017587708334240 \nu^{8} - 884378082421354786970575 \nu^{7} + 7638077306455251237639720 \nu^{6} - 50281306431579765857488445 \nu^{5} + 429405681651349754101224552 \nu^{4} - 1108799581866980612071590499 \nu^{3} + 8132472668178365074821464016 \nu^{2} - 10460680008268085441324131692 \nu + 34772983231242980815028210208$$$$)/$$$$69\!\cdots\!16$$ $$\beta_{10}$$ $$=$$ $$($$$$-265395398547017 \nu^{13} - 507721039196870 \nu^{12} - 191230379620914831 \nu^{11} - 354946558521410916 \nu^{10} - 51569748159940393263 \nu^{9} - 93045402770843938572 \nu^{8} - 6339968373134804507437 \nu^{7} - 11199851603697750612970 \nu^{6} - 341146202375384322760347 \nu^{5} - 602874082269251195103312 \nu^{4} - 6341451118994707007475987 \nu^{3} - 11959983531852379883492508 \nu^{2} - 36569923611273160153620068 \nu - 67128966349034414879860832$$$$)/$$$$30\!\cdots\!48$$ $$\beta_{11}$$ $$=$$ $$($$$$265395398547017 \nu^{13} - 507721039196870 \nu^{12} + 191230379620914831 \nu^{11} - 354946558521410916 \nu^{10} + 51569748159940393263 \nu^{9} - 93045402770843938572 \nu^{8} + 6339968373134804507437 \nu^{7} - 11199851603697750612970 \nu^{6} + 341146202375384322760347 \nu^{5} - 602874082269251195103312 \nu^{4} + 6341451118994707007475987 \nu^{3} - 11959983531852379883492508 \nu^{2} + 36569923611273160153620068 \nu - 67128966349034414879860832$$$$)/$$$$30\!\cdots\!48$$ $$\beta_{12}$$ $$=$$ $$($$$$-135690237002928139 \nu^{13} - 96160991710869586749 \nu^{11} - 25219857555996423409575 \nu^{9} - 2941883694364667506119419 \nu^{7} - 140144085116852535706083465 \nu^{5} - 1690405252672206307138150599 \nu^{3} - 2255614367703528923186463628 \nu$$$$)/$$$$15\!\cdots\!36$$ $$\beta_{13}$$ $$=$$ $$($$$$-1021773154872925165 \nu^{13} - 737592777303500015499 \nu^{11} - 198914369448739464567873 \nu^{9} - 24341630416105279031908829 \nu^{7} - 1285947471643293103204545759 \nu^{5} - 22136730179730163303202906265 \nu^{3} - 91357857871182401852709423652 \nu$$$$)/$$$$62\!\cdots\!44$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{7} + \beta_{6} + \beta_{3} + \beta_{2} - 105$$ $$\nu^{3}$$ $$=$$ $$4 \beta_{13} + \beta_{12} + 4 \beta_{11} - 4 \beta_{10} - 8 \beta_{7} + 8 \beta_{6} + 176 \beta_{5} - 182 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-18 \beta_{11} - 18 \beta_{10} - \beta_{9} - \beta_{8} - 307 \beta_{7} - 307 \beta_{6} + 24 \beta_{4} - 220 \beta_{3} - 97 \beta_{2} + 19052$$ $$\nu^{5}$$ $$=$$ $$-1245 \beta_{13} - 404 \beta_{12} - 1124 \beta_{11} + 1124 \beta_{10} - 89 \beta_{9} + 89 \beta_{8} + 2609 \beta_{7} - 2609 \beta_{6} - 28585 \beta_{5} + 36167 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$5529 \beta_{11} + 5529 \beta_{10} + 288 \beta_{9} + 288 \beta_{8} + 76134 \beta_{7} + 76134 \beta_{6} - 8781 \beta_{4} + 46752 \beta_{3} - 19158 \beta_{2} - 3761944$$ $$\nu^{7}$$ $$=$$ $$335694 \beta_{13} + 108600 \beta_{12} + 272961 \beta_{11} - 272961 \beta_{10} + 31482 \beta_{9} - 31482 \beta_{8} - 674397 \beta_{7} + 674397 \beta_{6} + 2308320 \beta_{5} - 7404553 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-1410900 \beta_{11} - 1410900 \beta_{10} - 48414 \beta_{9} - 48414 \beta_{8} - 17746969 \beta_{7} - 17746969 \beta_{6} + 2365650 \beta_{4} - 10019839 \beta_{3} + 13207433 \beta_{2} + 766346751$$ $$\nu^{9}$$ $$=$$ $$-85539364 \beta_{13} - 25900663 \beta_{12} - 63226792 \beta_{11} + 63226792 \beta_{10} - 8488224 \beta_{9} + 8488224 \beta_{8} + 163508774 \beta_{7} - 163508774 \beta_{6} + 406760308 \beta_{5} + 1543874144 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$344816172 \beta_{11} + 344816172 \beta_{10} + 6690385 \beta_{9} + 6690385 \beta_{8} + 4044235699 \beta_{7} + 4044235699 \beta_{6} - 577860768 \beta_{4} + 2169953110 \beta_{3} - 4754872415 \beta_{2} - 159038075618$$ $$\nu^{11}$$ $$=$$ $$21091510863 \beta_{13} + 5967707030 \beta_{12} + 14350338992 \beta_{11} - 14350338992 \beta_{10} + 2080565645 \beta_{9} - 2080565645 \beta_{8} - 38766844175 \beta_{7} + 38766844175 \beta_{6} - 280197571361 \beta_{5} - 326240246105 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$-83390218377 \beta_{11} - 83390218377 \beta_{10} - 970438770 \beta_{9} - 970438770 \beta_{8} - 913455955356 \beta_{7} - 913455955356 \beta_{6} + 135652745271 \beta_{4} - 474395508768 \beta_{3} + 1417944463104 \beta_{2} + 33445339123504$$ $$\nu^{13}$$ $$=$$ $$-5090601807096 \beta_{13} - 1366094383680 \beta_{12} - 3225223383777 \beta_{11} + 3225223383777 \beta_{10} - 487441218402 \beta_{9} + 487441218402 \beta_{8} + 9109535667525 \beta_{7} - 9109535667525 \beta_{6} + 100252884728790 \beta_{5} + 69684013855333 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/74\mathbb{Z}\right)^\times$$.

 $$n$$ $$39$$ $$\chi(n)$$ $$-\beta_{5}$$

## 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
 − 15.1052i − 9.51876i − 4.34447i − 3.21245i 1.77388i 13.3074i 14.0996i − 14.0996i − 13.3074i − 1.77388i 3.21245i 4.34447i 9.51876i 15.1052i
−2.00000 2.00000i 16.1052i 8.00000i −17.1794 + 17.1794i −32.2104 + 32.2104i −18.7264 16.0000 16.0000i −178.377 68.7178
31.2 −2.00000 2.00000i 10.5188i 8.00000i 30.2483 30.2483i −21.0375 + 21.0375i 69.0478 16.0000 16.0000i −29.6443 −120.993
31.3 −2.00000 2.00000i 5.34447i 8.00000i 10.1818 10.1818i −10.6889 + 10.6889i −94.0964 16.0000 16.0000i 52.4367 −40.7273
31.4 −2.00000 2.00000i 4.21245i 8.00000i −19.1769 + 19.1769i −8.42489 + 8.42489i 69.6480 16.0000 16.0000i 63.2553 76.7077
31.5 −2.00000 2.00000i 0.773882i 8.00000i 2.70834 2.70834i 1.54776 1.54776i −17.1599 16.0000 16.0000i 80.4011 −10.8334
31.6 −2.00000 2.00000i 12.3074i 8.00000i 10.7621 10.7621i 24.6148 24.6148i 35.0611 16.0000 16.0000i −70.4723 −43.0483
31.7 −2.00000 2.00000i 13.0996i 8.00000i −23.5442 + 23.5442i 26.1992 26.1992i −19.7742 16.0000 16.0000i −90.5991 94.1766
43.1 −2.00000 + 2.00000i 13.0996i 8.00000i −23.5442 23.5442i 26.1992 + 26.1992i −19.7742 16.0000 + 16.0000i −90.5991 94.1766
43.2 −2.00000 + 2.00000i 12.3074i 8.00000i 10.7621 + 10.7621i 24.6148 + 24.6148i 35.0611 16.0000 + 16.0000i −70.4723 −43.0483
43.3 −2.00000 + 2.00000i 0.773882i 8.00000i 2.70834 + 2.70834i 1.54776 + 1.54776i −17.1599 16.0000 + 16.0000i 80.4011 −10.8334
43.4 −2.00000 + 2.00000i 4.21245i 8.00000i −19.1769 19.1769i −8.42489 8.42489i 69.6480 16.0000 + 16.0000i 63.2553 76.7077
43.5 −2.00000 + 2.00000i 5.34447i 8.00000i 10.1818 + 10.1818i −10.6889 10.6889i −94.0964 16.0000 + 16.0000i 52.4367 −40.7273
43.6 −2.00000 + 2.00000i 10.5188i 8.00000i 30.2483 + 30.2483i −21.0375 21.0375i 69.0478 16.0000 + 16.0000i −29.6443 −120.993
43.7 −2.00000 + 2.00000i 16.1052i 8.00000i −17.1794 17.1794i −32.2104 32.2104i −18.7264 16.0000 + 16.0000i −178.377 68.7178
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 43.7 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.d odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.5.d.a 14
37.d odd 4 1 inner 74.5.d.a 14

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.5.d.a 14 1.a even 1 1 trivial
74.5.d.a 14 37.d odd 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{14} + 740 T_{3}^{12} + 207280 T_{3}^{10} + 27433122 T_{3}^{8} + 1725350400 T_{3}^{6} + 45141045588 T_{3}^{4} + 404504432361 T_{3}^{2} + 226431319104$$ acting on $$S_{5}^{\mathrm{new}}(74, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 8 + 4 T + T^{2} )^{7}$$
$3$ $$226431319104 + 404504432361 T^{2} + 45141045588 T^{4} + 1725350400 T^{6} + 27433122 T^{8} + 207280 T^{10} + 740 T^{12} + T^{14}$$
$5$ $$620588620524423168 - 273427801103532672 T + 60235363607807844 T^{2} - 3498264471565008 T^{3} + 74643350301216 T^{4} + 4872792793392 T^{5} + 467944112853 T^{6} - 20074895052 T^{7} + 496476072 T^{8} + 56651568 T^{9} + 2860027 T^{10} + 6720 T^{11} + 72 T^{12} + 12 T^{13} + T^{14}$$
$7$ $$( -100815033192 - 11645344668 T - 161084388 T^{2} + 18014378 T^{3} + 294002 T^{4} - 10081 T^{5} - 24 T^{6} + T^{7} )^{2}$$
$11$ $$35\!\cdots\!24$$$$+$$$$30\!\cdots\!49$$$$T^{2} +$$$$11\!\cdots\!08$$$$T^{4} + 1765871760294107680 T^{6} + 136813990467842 T^{8} + 5685544208 T^{10} + 119972 T^{12} + T^{14}$$
$13$ $$23\!\cdots\!88$$$$-$$$$61\!\cdots\!12$$$$T +$$$$81\!\cdots\!44$$$$T^{2} -$$$$32\!\cdots\!20$$$$T^{3} +$$$$30\!\cdots\!04$$$$T^{4} - 74224507890936394652 T^{5} + 14106130957302605221 T^{6} - 26340751501582612 T^{7} + 12641890448128 T^{8} + 259623679844 T^{9} + 7172640171 T^{10} - 4324272 T^{11} + 1568 T^{12} + 56 T^{13} + T^{14}$$
$17$ $$10\!\cdots\!68$$$$+$$$$14\!\cdots\!60$$$$T +$$$$92\!\cdots\!00$$$$T^{2} +$$$$38\!\cdots\!84$$$$T^{3} +$$$$61\!\cdots\!56$$$$T^{4} +$$$$47\!\cdots\!60$$$$T^{5} +$$$$58\!\cdots\!96$$$$T^{6} - 912226772476469792 T^{7} + 1377488646146624 T^{8} + 9293010049232 T^{9} + 42755476512 T^{10} - 38274032 T^{11} + 60552 T^{12} + 348 T^{13} + T^{14}$$
$19$ $$53\!\cdots\!88$$$$-$$$$41\!\cdots\!04$$$$T +$$$$15\!\cdots\!16$$$$T^{2} -$$$$69\!\cdots\!88$$$$T^{3} +$$$$12\!\cdots\!40$$$$T^{4} +$$$$91\!\cdots\!56$$$$T^{5} +$$$$10\!\cdots\!16$$$$T^{6} - 298547958083429056 T^{7} + 574513264458528 T^{8} + 5688400414208 T^{9} + 29086468352 T^{10} - 6359768 T^{11} + 16928 T^{12} + 184 T^{13} + T^{14}$$
$23$ $$66\!\cdots\!48$$$$-$$$$97\!\cdots\!60$$$$T +$$$$71\!\cdots\!00$$$$T^{2} -$$$$12\!\cdots\!12$$$$T^{3} +$$$$63\!\cdots\!38$$$$T^{4} +$$$$16\!\cdots\!66$$$$T^{5} +$$$$22\!\cdots\!69$$$$T^{6} -$$$$16\!\cdots\!32$$$$T^{7} + 68238471433958742 T^{8} + 253811633961330 T^{9} + 875547485155 T^{10} - 319830472 T^{11} + 126002 T^{12} + 502 T^{13} + T^{14}$$
$29$ $$14\!\cdots\!92$$$$-$$$$22\!\cdots\!60$$$$T +$$$$17\!\cdots\!00$$$$T^{2} -$$$$68\!\cdots\!92$$$$T^{3} +$$$$12\!\cdots\!22$$$$T^{4} +$$$$85\!\cdots\!70$$$$T^{5} +$$$$49\!\cdots\!21$$$$T^{6} -$$$$40\!\cdots\!72$$$$T^{7} + 178027246709927006 T^{8} + 779809485497234 T^{9} + 2036410149891 T^{10} - 272630864 T^{11} + 112338 T^{12} + 474 T^{13} + T^{14}$$
$31$ $$17\!\cdots\!88$$$$+$$$$11\!\cdots\!28$$$$T +$$$$35\!\cdots\!84$$$$T^{2} -$$$$85\!\cdots\!48$$$$T^{3} +$$$$11\!\cdots\!54$$$$T^{4} +$$$$37\!\cdots\!26$$$$T^{5} +$$$$63\!\cdots\!13$$$$T^{6} -$$$$27\!\cdots\!32$$$$T^{7} + 722245154104056158 T^{8} + 2832887379233394 T^{9} + 5689071333439 T^{10} - 364752824 T^{11} + 198450 T^{12} + 630 T^{13} + T^{14}$$
$37$ $$81\!\cdots\!21$$$$+$$$$11\!\cdots\!84$$$$T +$$$$16\!\cdots\!59$$$$T^{2} +$$$$16\!\cdots\!64$$$$T^{3} +$$$$16\!\cdots\!33$$$$T^{4} +$$$$13\!\cdots\!72$$$$T^{5} +$$$$11\!\cdots\!31$$$$T^{6} +$$$$80\!\cdots\!80$$$$T^{7} + 58989892954318336671 T^{8} + 37167143747442832 T^{9} + 25411378954193 T^{10} + 13160374304 T^{11} + 6957359 T^{12} + 2544 T^{13} + T^{14}$$
$41$ $$72\!\cdots\!36$$$$+$$$$10\!\cdots\!17$$$$T^{2} +$$$$41\!\cdots\!48$$$$T^{4} +$$$$62\!\cdots\!32$$$$T^{6} +$$$$42\!\cdots\!54$$$$T^{8} + 137732853458320 T^{10} + 20036256 T^{12} + T^{14}$$
$43$ $$97\!\cdots\!28$$$$-$$$$19\!\cdots\!60$$$$T +$$$$18\!\cdots\!00$$$$T^{2} +$$$$14\!\cdots\!16$$$$T^{3} +$$$$10\!\cdots\!48$$$$T^{4} -$$$$21\!\cdots\!40$$$$T^{5} +$$$$22\!\cdots\!76$$$$T^{6} +$$$$36\!\cdots\!12$$$$T^{7} +$$$$13\!\cdots\!84$$$$T^{8} - 158897944552397600 T^{9} + 101285827755248 T^{10} + 5937825624 T^{11} + 1874048 T^{12} - 1936 T^{13} + T^{14}$$
$47$ $$($$$$35\!\cdots\!52$$$$- 66447747299264662188 T - 171855164490193008 T^{2} + 82020699224100 T^{3} + 44808697206 T^{4} - 17455037 T^{5} - 2858 T^{6} + T^{7} )^{2}$$
$53$ $$($$$$15\!\cdots\!48$$$$+$$$$85\!\cdots\!16$$$$T - 159782108392779924 T^{2} - 598334419801778 T^{3} - 188622671030 T^{4} + 9018197 T^{5} + 10114 T^{6} + T^{7} )^{2}$$
$59$ $$18\!\cdots\!28$$$$+$$$$66\!\cdots\!52$$$$T +$$$$11\!\cdots\!84$$$$T^{2} +$$$$14\!\cdots\!16$$$$T^{3} +$$$$76\!\cdots\!36$$$$T^{4} +$$$$28\!\cdots\!60$$$$T^{5} +$$$$55\!\cdots\!08$$$$T^{6} +$$$$34\!\cdots\!28$$$$T^{7} +$$$$84\!\cdots\!24$$$$T^{8} + 3354145944778251312 T^{9} + 672858080714056 T^{10} + 16248593168 T^{11} + 10134002 T^{12} + 4502 T^{13} + T^{14}$$
$61$ $$49\!\cdots\!88$$$$+$$$$63\!\cdots\!92$$$$T +$$$$40\!\cdots\!64$$$$T^{2} -$$$$38\!\cdots\!08$$$$T^{3} +$$$$19\!\cdots\!26$$$$T^{4} +$$$$50\!\cdots\!18$$$$T^{5} -$$$$15\!\cdots\!27$$$$T^{6} +$$$$33\!\cdots\!40$$$$T^{7} +$$$$92\!\cdots\!70$$$$T^{8} + 12776968110218817698 T^{9} + 854663900936535 T^{10} + 38817306632 T^{11} + 70876418 T^{12} + 11906 T^{13} + T^{14}$$
$67$ $$15\!\cdots\!84$$$$+$$$$12\!\cdots\!96$$$$T^{2} +$$$$39\!\cdots\!44$$$$T^{4} +$$$$67\!\cdots\!49$$$$T^{6} +$$$$64\!\cdots\!98$$$$T^{8} + 3536450867395315 T^{10} + 97715094 T^{12} + T^{14}$$
$71$ $$( -$$$$29\!\cdots\!72$$$$-$$$$27\!\cdots\!16$$$$T + 3215160885128645688 T^{2} + 4279448363023780 T^{3} - 372011918398 T^{4} - 116981019 T^{5} + 5612 T^{6} + T^{7} )^{2}$$
$73$ $$19\!\cdots\!16$$$$+$$$$19\!\cdots\!45$$$$T^{2} +$$$$48\!\cdots\!32$$$$T^{4} +$$$$45\!\cdots\!64$$$$T^{6} +$$$$18\!\cdots\!06$$$$T^{8} + 34439410891823880 T^{10} + 302534824 T^{12} + T^{14}$$
$79$ $$21\!\cdots\!48$$$$+$$$$19\!\cdots\!08$$$$T +$$$$88\!\cdots\!84$$$$T^{2} -$$$$26\!\cdots\!16$$$$T^{3} +$$$$92\!\cdots\!88$$$$T^{4} +$$$$72\!\cdots\!80$$$$T^{5} +$$$$44\!\cdots\!37$$$$T^{6} -$$$$12\!\cdots\!88$$$$T^{7} +$$$$60\!\cdots\!72$$$$T^{8} - 44965322539290485904 T^{9} + 2419025219292139 T^{10} - 624799566984 T^{11} + 209879072 T^{12} - 20488 T^{13} + T^{14}$$
$83$ $$($$$$88\!\cdots\!88$$$$+$$$$11\!\cdots\!56$$$$T + 39001629908430900648 T^{2} + 903065800676956 T^{3} - 1281008043862 T^{4} - 106929595 T^{5} + 10112 T^{6} + T^{7} )^{2}$$
$89$ $$17\!\cdots\!08$$$$-$$$$24\!\cdots\!88$$$$T +$$$$17\!\cdots\!84$$$$T^{2} -$$$$67\!\cdots\!88$$$$T^{3} +$$$$16\!\cdots\!72$$$$T^{4} -$$$$24\!\cdots\!80$$$$T^{5} +$$$$25\!\cdots\!08$$$$T^{6} -$$$$36\!\cdots\!20$$$$T^{7} +$$$$87\!\cdots\!48$$$$T^{8} -$$$$12\!\cdots\!08$$$$T^{9} + 9893404705681932 T^{10} - 416116877312 T^{11} + 96105248 T^{12} - 13864 T^{13} + T^{14}$$
$97$ $$32\!\cdots\!88$$$$-$$$$39\!\cdots\!28$$$$T +$$$$23\!\cdots\!84$$$$T^{2} -$$$$56\!\cdots\!64$$$$T^{3} +$$$$73\!\cdots\!16$$$$T^{4} -$$$$41\!\cdots\!84$$$$T^{5} +$$$$94\!\cdots\!12$$$$T^{6} +$$$$25\!\cdots\!16$$$$T^{7} +$$$$53\!\cdots\!20$$$$T^{8} - 48712689037267508824 T^{9} + 2311131043175708 T^{10} + 292033537784 T^{11} + 138145442 T^{12} - 16622 T^{13} + T^{14}$$