# Properties

 Label 33.5.b.a Level 33 Weight 5 Character orbit 33.b Analytic conductor 3.411 Analytic rank 0 Dimension 14 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$33 = 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 33.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.41120878177$$ Analytic rank: $$0$$ Dimension: $$14$$ Coefficient field: $$\mathbb{Q}[x]/(x^{14} + \cdots)$$ Defining polynomial: $$x^{14} + 162 x^{12} + 10041 x^{10} + 298396 x^{8} + 4418856 x^{6} + 32113344 x^{4} + 102865552 x^{2} + 102193344$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{8}\cdot 3^{9}\cdot 11^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + \beta_{4} q^{3} + ( -7 + \beta_{2} ) q^{4} -\beta_{9} q^{5} -\beta_{7} q^{6} + ( 6 - \beta_{3} + \beta_{4} ) q^{7} + ( -6 \beta_{1} - \beta_{4} + \beta_{5} ) q^{8} + ( -5 + \beta_{12} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + \beta_{4} q^{3} + ( -7 + \beta_{2} ) q^{4} -\beta_{9} q^{5} -\beta_{7} q^{6} + ( 6 - \beta_{3} + \beta_{4} ) q^{7} + ( -6 \beta_{1} - \beta_{4} + \beta_{5} ) q^{8} + ( -5 + \beta_{12} ) q^{9} + ( -12 + \beta_{2} - 3 \beta_{4} - \beta_{6} ) q^{10} + ( \beta_{4} - \beta_{10} ) q^{11} + ( -9 - 4 \beta_{1} - \beta_{2} + \beta_{3} - 9 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{9} + 3 \beta_{10} + \beta_{13} ) q^{12} + ( -6 + 3 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + \beta_{6} + \beta_{9} - \beta_{11} - \beta_{12} - \beta_{13} ) q^{13} + ( -1 + 10 \beta_{1} + 8 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} + \beta_{8} + 2 \beta_{9} - 6 \beta_{10} + \beta_{11} - 2 \beta_{13} ) q^{14} + ( 11 + 10 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + 3 \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} ) q^{15} + ( 24 - 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{7} - \beta_{9} + \beta_{11} ) q^{16} + ( 4 + 9 \beta_{1} + 7 \beta_{4} - \beta_{5} + 2 \beta_{7} - 2 \beta_{8} + 3 \beta_{9} - \beta_{11} - \beta_{12} + 3 \beta_{13} ) q^{17} + ( -13 \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} - 2 \beta_{9} + 6 \beta_{10} - 2 \beta_{11} ) q^{18} + ( 76 - 7 \beta_{2} - \beta_{3} + 4 \beta_{4} + 3 \beta_{6} + 4 \beta_{7} - \beta_{9} + \beta_{11} - \beta_{12} - \beta_{13} ) q^{19} + ( 2 - 14 \beta_{1} + 12 \beta_{4} + 4 \beta_{7} - \beta_{8} + \beta_{9} - 6 \beta_{10} - 2 \beta_{11} + 4 \beta_{12} ) q^{20} + ( 48 + 23 \beta_{1} - 7 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - 5 \beta_{5} - \beta_{6} + 2 \beta_{9} - 2 \beta_{13} ) q^{21} + ( -3 \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{7} + \beta_{12} + \beta_{13} ) q^{22} + ( -7 + 14 \beta_{1} - 8 \beta_{4} - 4 \beta_{5} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - 6 \beta_{10} + \beta_{11} - 2 \beta_{12} ) q^{23} + ( 88 + 8 \beta_{1} - 7 \beta_{2} - 8 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} + 9 \beta_{9} + 6 \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} ) q^{24} + ( -86 - 4 \beta_{2} + 8 \beta_{3} - 12 \beta_{4} - 4 \beta_{6} + 10 \beta_{7} + 3 \beta_{8} - 3 \beta_{9} + 3 \beta_{11} - 2 \beta_{12} - 2 \beta_{13} ) q^{25} + ( -3 - 48 \beta_{1} - 18 \beta_{4} + 6 \beta_{5} - 6 \beta_{7} + \beta_{8} - 10 \beta_{9} + 3 \beta_{11} - 8 \beta_{12} + 2 \beta_{13} ) q^{26} + ( -84 - 56 \beta_{1} + 10 \beta_{2} + 5 \beta_{3} - 10 \beta_{4} + 5 \beta_{5} + \beta_{6} + 2 \beta_{7} + 14 \beta_{9} + 3 \beta_{10} + 2 \beta_{11} - 3 \beta_{12} - 3 \beta_{13} ) q^{27} + ( -132 + 11 \beta_{2} + 4 \beta_{3} - 29 \beta_{4} - \beta_{6} - 20 \beta_{7} - 6 \beta_{8} + 3 \beta_{9} - 3 \beta_{11} + 7 \beta_{12} + 7 \beta_{13} ) q^{28} + ( -12 + 29 \beta_{1} - 33 \beta_{4} + 3 \beta_{5} + 6 \beta_{7} + 2 \beta_{8} + 11 \beta_{9} - 3 \beta_{11} + 5 \beta_{12} + \beta_{13} ) q^{29} + ( -216 + 24 \beta_{1} + 27 \beta_{2} - 3 \beta_{3} - 11 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} - 21 \beta_{9} + 6 \beta_{10} + 3 \beta_{11} - 3 \beta_{12} - 3 \beta_{13} ) q^{30} + ( 240 - 30 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} - 16 \beta_{7} - 3 \beta_{8} + 6 \beta_{9} - 6 \beta_{11} + 2 \beta_{12} + 2 \beta_{13} ) q^{31} + ( 12 - 20 \beta_{1} + 41 \beta_{4} + 7 \beta_{5} - 2 \beta_{8} + 2 \beta_{9} - 12 \beta_{10} + 4 \beta_{12} - 4 \beta_{13} ) q^{32} + ( -41 - \beta_{1} + 8 \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} + 6 \beta_{9} - \beta_{11} + \beta_{12} + 2 \beta_{13} ) q^{33} + ( -134 + 22 \beta_{2} - 12 \beta_{3} + 76 \beta_{4} + 4 \beta_{6} - 4 \beta_{7} + 6 \beta_{8} + 2 \beta_{12} + 2 \beta_{13} ) q^{34} + ( -2 - 56 \beta_{1} + 8 \beta_{7} - 26 \beta_{9} - 4 \beta_{11} + 10 \beta_{12} - 2 \beta_{13} ) q^{35} + ( 183 - 38 \beta_{1} + \beta_{2} - 13 \beta_{3} - 7 \beta_{4} + 5 \beta_{5} + \beta_{6} + 8 \beta_{7} + 9 \beta_{8} - 28 \beta_{9} + 3 \beta_{10} + 8 \beta_{11} - 6 \beta_{12} - 9 \beta_{13} ) q^{36} + ( -35 + 4 \beta_{2} - 10 \beta_{4} - 2 \beta_{6} + 14 \beta_{7} + 3 \beta_{8} - 7 \beta_{9} + 7 \beta_{11} ) q^{37} + ( 18 + 168 \beta_{1} + 66 \beta_{4} - 12 \beta_{5} - 12 \beta_{7} - 2 \beta_{8} - 42 \beta_{9} - 6 \beta_{10} + 6 \beta_{11} - 8 \beta_{12} - 4 \beta_{13} ) q^{38} + ( 324 - 68 \beta_{1} - 35 \beta_{2} - 10 \beta_{3} + 9 \beta_{4} + 8 \beta_{5} + \beta_{6} - 6 \beta_{7} - 18 \beta_{8} - 17 \beta_{9} + 6 \beta_{10} - 9 \beta_{11} + 9 \beta_{12} + 17 \beta_{13} ) q^{39} + ( 164 + \beta_{2} + 2 \beta_{3} + 73 \beta_{4} - \beta_{6} - 16 \beta_{7} + 6 \beta_{8} + 7 \beta_{9} - 7 \beta_{11} + \beta_{12} + \beta_{13} ) q^{40} + ( 20 - 11 \beta_{1} + 55 \beta_{4} - \beta_{5} - 14 \beta_{7} - 2 \beta_{8} + 41 \beta_{9} + 7 \beta_{11} - 9 \beta_{12} - 5 \beta_{13} ) q^{41} + ( -503 + 164 \beta_{1} + 86 \beta_{2} + 16 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 8 \beta_{7} - 9 \beta_{8} + 24 \beta_{9} + 6 \beta_{10} - 13 \beta_{11} + 4 \beta_{12} + 8 \beta_{13} ) q^{42} + ( -476 + 67 \beta_{2} - 15 \beta_{3} - 20 \beta_{4} + 9 \beta_{6} + 12 \beta_{7} - 6 \beta_{8} - \beta_{9} + \beta_{11} - 5 \beta_{12} - 5 \beta_{13} ) q^{43} + ( -10 + 18 \beta_{1} - 41 \beta_{4} + 3 \beta_{5} + 4 \beta_{7} + \beta_{8} - 11 \beta_{9} + 8 \beta_{10} - 2 \beta_{11} + 4 \beta_{13} ) q^{44} + ( 230 + 82 \beta_{1} - 50 \beta_{2} + 20 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 4 \beta_{6} - 4 \beta_{7} + 18 \beta_{8} - \beta_{9} - 24 \beta_{10} + 2 \beta_{11} - \beta_{12} - 6 \beta_{13} ) q^{45} + ( -304 + 37 \beta_{2} + 16 \beta_{3} - 39 \beta_{4} - 5 \beta_{6} - 4 \beta_{9} + 4 \beta_{11} + 4 \beta_{12} + 4 \beta_{13} ) q^{46} + ( -37 - 6 \beta_{1} - 142 \beta_{4} + 4 \beta_{5} - 14 \beta_{7} + 11 \beta_{8} + 18 \beta_{9} + 42 \beta_{10} + 7 \beta_{11} - 14 \beta_{12} ) q^{47} + ( -225 + 190 \beta_{1} - 5 \beta_{2} + 8 \beta_{3} + 16 \beta_{4} - 10 \beta_{5} - 2 \beta_{6} + 9 \beta_{8} + 25 \beta_{9} - 18 \beta_{10} + 6 \beta_{11} - 6 \beta_{12} - 10 \beta_{13} ) q^{48} + ( 467 - 38 \beta_{2} - 4 \beta_{3} - 84 \beta_{4} + 32 \beta_{7} - 6 \beta_{8} - 4 \beta_{9} + 4 \beta_{11} - 12 \beta_{12} - 12 \beta_{13} ) q^{49} + ( 74 + 49 \beta_{1} + 204 \beta_{4} - 6 \beta_{5} + 22 \beta_{7} - 22 \beta_{8} + 55 \beta_{9} - 12 \beta_{10} - 11 \beta_{11} + 19 \beta_{12} + 3 \beta_{13} ) q^{50} + ( -530 - 256 \beta_{1} - 19 \beta_{2} - 5 \beta_{3} + 16 \beta_{4} - 14 \beta_{5} - 13 \beta_{6} - 14 \beta_{7} + 18 \beta_{8} - 27 \beta_{9} - 48 \beta_{10} + 5 \beta_{11} + 13 \beta_{12} + 5 \beta_{13} ) q^{51} + ( 858 - 99 \beta_{2} + 18 \beta_{3} - 157 \beta_{4} - 19 \beta_{6} + 24 \beta_{7} - 6 \beta_{8} - 5 \beta_{9} + 5 \beta_{11} - 7 \beta_{12} - 7 \beta_{13} ) q^{52} + ( 13 - 254 \beta_{1} - 34 \beta_{4} - 2 \beta_{5} + 14 \beta_{7} - 9 \beta_{8} + 2 \beta_{9} + 48 \beta_{10} - 7 \beta_{11} - 2 \beta_{12} + 16 \beta_{13} ) q^{53} + ( 1462 - 232 \beta_{1} - 136 \beta_{2} - 5 \beta_{3} + 4 \beta_{4} + 13 \beta_{5} + 8 \beta_{6} + 10 \beta_{7} - 18 \beta_{8} - 17 \beta_{9} + 6 \beta_{10} + \beta_{11} + \beta_{12} + 9 \beta_{13} ) q^{54} + ( 66 - 19 \beta_{2} + 2 \beta_{3} + 43 \beta_{4} + \beta_{6} - 8 \beta_{7} + 3 \beta_{8} + 5 \beta_{9} - 5 \beta_{11} - \beta_{12} - \beta_{13} ) q^{55} + ( -117 - 322 \beta_{1} - 368 \beta_{4} + 20 \beta_{5} + 18 \beta_{7} + 25 \beta_{8} + 2 \beta_{9} + 48 \beta_{10} - 9 \beta_{11} + 10 \beta_{12} + 8 \beta_{13} ) q^{56} + ( 276 + 249 \beta_{1} - 90 \beta_{2} + 6 \beta_{3} + 81 \beta_{4} - 3 \beta_{5} + 6 \beta_{6} - 12 \beta_{7} + 75 \beta_{9} - 48 \beta_{10} - 3 \beta_{11} + 3 \beta_{12} + 3 \beta_{13} ) q^{57} + ( -508 + 4 \beta_{2} - 16 \beta_{3} + 190 \beta_{4} + 30 \beta_{6} + 48 \beta_{7} + 18 \beta_{8} - 6 \beta_{9} + 6 \beta_{11} - 18 \beta_{12} - 18 \beta_{13} ) q^{58} + ( -12 - 184 \beta_{1} + 34 \beta_{4} - 22 \beta_{5} + 9 \beta_{8} - 24 \beta_{9} - 6 \beta_{10} + 24 \beta_{12} - 24 \beta_{13} ) q^{59} + ( -589 - 380 \beta_{1} + 58 \beta_{2} - 7 \beta_{3} + 11 \beta_{4} - 7 \beta_{5} - 11 \beta_{6} + 2 \beta_{7} - 18 \beta_{8} + 72 \beta_{9} - 15 \beta_{10} + 7 \beta_{11} + 2 \beta_{12} + 7 \beta_{13} ) q^{60} + ( -290 + 87 \beta_{2} - 2 \beta_{3} - 205 \beta_{4} + 9 \beta_{6} - 24 \beta_{7} - 36 \beta_{8} + 15 \beta_{9} - 15 \beta_{11} - 3 \beta_{12} - 3 \beta_{13} ) q^{61} + ( -100 + 680 \beta_{1} - 238 \beta_{4} - 20 \beta_{5} - 2 \beta_{7} + 24 \beta_{8} + 45 \beta_{9} + \beta_{11} - 7 \beta_{12} + 5 \beta_{13} ) q^{62} + ( -1222 + 40 \beta_{1} + 106 \beta_{2} + 17 \beta_{3} + 95 \beta_{4} + 8 \beta_{5} + 16 \beta_{6} + 2 \beta_{7} - 18 \beta_{8} - 70 \beta_{9} - 42 \beta_{10} + 2 \beta_{11} - 4 \beta_{12} ) q^{63} + ( 830 - 189 \beta_{2} - 18 \beta_{3} - 32 \beta_{4} + 10 \beta_{6} - 26 \beta_{7} - 12 \beta_{8} - 5 \beta_{9} + 5 \beta_{11} + 18 \beta_{12} + 18 \beta_{13} ) q^{64} + ( 74 + 352 \beta_{1} + 68 \beta_{4} - 8 \beta_{5} - 8 \beta_{7} - 22 \beta_{8} - 34 \beta_{9} + 96 \beta_{10} + 4 \beta_{11} - 26 \beta_{12} + 18 \beta_{13} ) q^{65} + ( 87 - 161 \beta_{1} + 22 \beta_{2} - \beta_{3} + 8 \beta_{5} + 7 \beta_{6} + 3 \beta_{7} + 9 \beta_{8} + 13 \beta_{9} - 4 \beta_{13} ) q^{66} + ( -1108 + 96 \beta_{2} + 12 \beta_{3} + 232 \beta_{4} - 28 \beta_{6} + 24 \beta_{7} + 45 \beta_{8} - 4 \beta_{9} + 4 \beta_{11} - 8 \beta_{12} - 8 \beta_{13} ) q^{67} + ( 10 - 346 \beta_{1} + 18 \beta_{4} - 6 \beta_{5} - 40 \beta_{7} - 32 \beta_{9} - 12 \beta_{10} + 20 \beta_{11} - 50 \beta_{12} + 10 \beta_{13} ) q^{68} + ( 707 + 124 \beta_{1} + 76 \beta_{2} + 20 \beta_{3} + 14 \beta_{4} - 16 \beta_{5} - 2 \beta_{6} - 10 \beta_{7} + 9 \beta_{8} - 18 \beta_{9} - 24 \beta_{10} - 11 \beta_{11} - 7 \beta_{12} + 16 \beta_{13} ) q^{69} + ( 1002 + 10 \beta_{2} + 138 \beta_{4} + 6 \beta_{6} + 4 \beta_{7} + 12 \beta_{8} + 14 \beta_{9} - 14 \beta_{11} - 16 \beta_{12} - 16 \beta_{13} ) q^{70} + ( 97 + 318 \beta_{1} + 160 \beta_{4} + 50 \beta_{5} + 14 \beta_{7} - 24 \beta_{8} - 50 \beta_{9} + 48 \beta_{10} - 7 \beta_{11} + 22 \beta_{12} - 8 \beta_{13} ) q^{71} + ( 568 + 106 \beta_{1} - 62 \beta_{2} + 32 \beta_{3} + 83 \beta_{4} - 31 \beta_{5} - 8 \beta_{6} + 2 \beta_{7} - 18 \beta_{8} - 22 \beta_{9} - 42 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} - 48 \beta_{13} ) q^{72} + ( 972 - 18 \beta_{2} + 20 \beta_{3} + 282 \beta_{4} + 2 \beta_{6} + 4 \beta_{7} + 42 \beta_{8} - 18 \beta_{9} + 18 \beta_{11} + 16 \beta_{12} + 16 \beta_{13} ) q^{73} + ( 96 - 64 \beta_{1} + 282 \beta_{4} - 12 \beta_{5} + 18 \beta_{7} - 24 \beta_{8} + 11 \beta_{9} - 12 \beta_{10} - 9 \beta_{11} + 27 \beta_{12} - 9 \beta_{13} ) q^{74} + ( -384 + 640 \beta_{1} + 166 \beta_{2} - 55 \beta_{3} - 20 \beta_{4} + 17 \beta_{5} - 11 \beta_{6} + 6 \beta_{7} + 9 \beta_{8} - 62 \beta_{9} - 45 \beta_{10} + 30 \beta_{11} - 3 \beta_{12} - 7 \beta_{13} ) q^{75} + ( -3212 + 190 \beta_{2} + 42 \beta_{3} - 366 \beta_{4} - 28 \beta_{6} - 44 \beta_{7} - 36 \beta_{8} + 2 \beta_{9} - 2 \beta_{11} + 20 \beta_{12} + 20 \beta_{13} ) q^{76} + ( -33 + 253 \beta_{1} - 35 \beta_{4} - 11 \beta_{5} + 11 \beta_{8} + 11 \beta_{9} - 20 \beta_{10} + 11 \beta_{12} - 11 \beta_{13} ) q^{77} + ( 1361 + 670 \beta_{1} - 98 \beta_{2} - 76 \beta_{3} - 22 \beta_{4} - 22 \beta_{5} + 4 \beta_{6} + 18 \beta_{7} + 45 \beta_{8} - 24 \beta_{9} + 66 \beta_{10} - 17 \beta_{11} + 8 \beta_{12} + 16 \beta_{13} ) q^{78} + ( 310 + 152 \beta_{2} - 53 \beta_{3} + 129 \beta_{4} - 48 \beta_{7} + 14 \beta_{9} - 14 \beta_{11} + 10 \beta_{12} + 10 \beta_{13} ) q^{79} + ( -85 - 62 \beta_{1} - 142 \beta_{4} + 22 \beta_{5} + 10 \beta_{7} + 23 \beta_{8} + 16 \beta_{9} - 72 \beta_{10} - 5 \beta_{11} + 22 \beta_{12} - 12 \beta_{13} ) q^{80} + ( -826 + 310 \beta_{1} - 38 \beta_{2} - 28 \beta_{3} - 82 \beta_{4} + 62 \beta_{5} - 20 \beta_{6} + 38 \beta_{7} - 9 \beta_{8} + 11 \beta_{9} + 12 \beta_{10} - 25 \beta_{11} + 23 \beta_{12} ) q^{81} + ( 646 - 92 \beta_{2} + 36 \beta_{3} - 266 \beta_{4} + 2 \beta_{6} - 84 \beta_{7} - 42 \beta_{8} + 8 \beta_{9} - 8 \beta_{11} + 34 \beta_{12} + 34 \beta_{13} ) q^{82} + ( 206 + 52 \beta_{1} + 376 \beta_{4} + 44 \beta_{5} - 20 \beta_{7} - 48 \beta_{8} - 48 \beta_{9} + 96 \beta_{10} + 10 \beta_{11} - 16 \beta_{12} - 4 \beta_{13} ) q^{83} + ( -2802 - 1554 \beta_{1} + 111 \beta_{2} + 24 \beta_{3} - 211 \beta_{4} + 60 \beta_{5} + 21 \beta_{6} + 24 \beta_{7} + 36 \beta_{8} - 9 \beta_{9} + 126 \beta_{10} - 15 \beta_{11} - 39 \beta_{12} + 3 \beta_{13} ) q^{84} + ( 2490 - 84 \beta_{2} - 80 \beta_{3} - 92 \beta_{4} + 32 \beta_{6} - 60 \beta_{7} - 42 \beta_{8} + 4 \beta_{9} - 4 \beta_{11} + 26 \beta_{12} + 26 \beta_{13} ) q^{85} + ( 48 - 1448 \beta_{1} + 246 \beta_{4} + 24 \beta_{5} - 36 \beta_{7} - 2 \beta_{8} - 62 \beta_{9} - 126 \beta_{10} + 18 \beta_{11} - 14 \beta_{12} - 22 \beta_{13} ) q^{86} + ( 2660 - 644 \beta_{1} + \beta_{2} - 13 \beta_{3} + 14 \beta_{4} + 14 \beta_{5} - 23 \beta_{6} - 44 \beta_{7} - 18 \beta_{8} + 69 \beta_{9} - 42 \beta_{10} + 7 \beta_{11} - 25 \beta_{12} - 17 \beta_{13} ) q^{87} + ( -506 - 24 \beta_{2} - 18 \beta_{3} + 85 \beta_{4} - 5 \beta_{6} + 34 \beta_{7} + 18 \beta_{8} - 14 \beta_{9} + 14 \beta_{11} - 3 \beta_{12} - 3 \beta_{13} ) q^{88} + ( -235 - 440 \beta_{1} - 360 \beta_{4} - 48 \beta_{5} - 62 \beta_{7} + 79 \beta_{8} + 15 \beta_{9} - 120 \beta_{10} + 31 \beta_{11} - 12 \beta_{12} - 50 \beta_{13} ) q^{89} + ( -1988 + 936 \beta_{1} - 27 \beta_{2} + 45 \beta_{3} - 297 \beta_{4} - 9 \beta_{5} - 9 \beta_{6} + 9 \beta_{7} - 144 \beta_{9} + 126 \beta_{10} + 36 \beta_{11} - 38 \beta_{12} + 18 \beta_{13} ) q^{90} + ( -3228 - 142 \beta_{2} - 14 \beta_{3} + 56 \beta_{4} - 70 \beta_{6} - 200 \beta_{7} + 26 \beta_{9} - 26 \beta_{11} + 74 \beta_{12} + 74 \beta_{13} ) q^{91} + ( -74 - 730 \beta_{1} - 284 \beta_{4} + 8 \beta_{5} + 44 \beta_{7} + 3 \beta_{8} + 41 \beta_{9} + 42 \beta_{10} - 22 \beta_{11} + 4 \beta_{12} + 40 \beta_{13} ) q^{92} + ( 300 - 824 \beta_{1} + 46 \beta_{2} - 28 \beta_{3} + 294 \beta_{4} - 64 \beta_{5} + 46 \beta_{6} + 12 \beta_{7} - 36 \beta_{8} - 86 \beta_{9} - 12 \beta_{10} - 30 \beta_{11} + 3 \beta_{12} - 34 \beta_{13} ) q^{93} + ( 254 - 13 \beta_{2} - 36 \beta_{3} - 403 \beta_{4} - 31 \beta_{6} + 196 \beta_{7} - 6 \beta_{8} - 51 \beta_{9} + 51 \beta_{11} - 47 \beta_{12} - 47 \beta_{13} ) q^{94} + ( -246 + 1464 \beta_{1} - 648 \beta_{4} + 48 \beta_{5} - 48 \beta_{7} + 70 \beta_{8} - 90 \beta_{9} - 12 \beta_{10} + 24 \beta_{11} - 38 \beta_{12} - 10 \beta_{13} ) q^{95} + ( -2670 + 72 \beta_{1} + 99 \beta_{2} - 54 \beta_{3} - 12 \beta_{4} + 9 \beta_{5} - 9 \beta_{6} - 20 \beta_{7} - 36 \beta_{8} + 183 \beta_{9} + 72 \beta_{10} - 21 \beta_{11} + 39 \beta_{12} + 3 \beta_{13} ) q^{96} + ( -2109 - 26 \beta_{2} + 140 \beta_{3} - 96 \beta_{4} + 4 \beta_{6} + 178 \beta_{7} + 33 \beta_{8} - 27 \beta_{9} + 27 \beta_{11} - 62 \beta_{12} - 62 \beta_{13} ) q^{97} + ( 196 + 1301 \beta_{1} + 858 \beta_{4} - 102 \beta_{5} + 20 \beta_{7} - 48 \beta_{8} + 126 \beta_{9} - 240 \beta_{10} - 10 \beta_{11} + 34 \beta_{12} - 14 \beta_{13} ) q^{98} + ( 274 - 348 \beta_{1} + 30 \beta_{2} + 33 \beta_{3} - 54 \beta_{4} - 3 \beta_{5} - 15 \beta_{6} + 12 \beta_{7} + 9 \beta_{8} - 42 \beta_{9} + \beta_{12} - 3 \beta_{13} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$14q - 5q^{3} - 100q^{4} - 2q^{6} + 76q^{7} - 67q^{9} + O(q^{10})$$ $$14q - 5q^{3} - 100q^{4} - 2q^{6} + 76q^{7} - 67q^{9} - 156q^{10} - 100q^{12} - 104q^{13} + 151q^{15} + 356q^{16} - 34q^{18} + 1072q^{19} + 718q^{21} + 1200q^{24} - 1060q^{25} - 1154q^{27} - 1808q^{28} - 3026q^{30} + 3310q^{31} - 605q^{33} - 2304q^{34} + 2644q^{36} - 362q^{37} + 4264q^{39} + 1896q^{40} - 7364q^{42} - 6740q^{43} + 3611q^{45} - 4068q^{46} - 2956q^{48} + 7074q^{49} - 7046q^{51} + 13072q^{52} + 20512q^{54} + 726q^{55} + 3876q^{57} - 7848q^{58} - 8416q^{60} - 3560q^{61} - 17662q^{63} + 12020q^{64} + 1210q^{66} - 16514q^{67} + 9833q^{69} + 13320q^{70} + 8160q^{72} + 12664q^{73} - 5386q^{75} - 43736q^{76} + 19096q^{78} + 3052q^{79} - 11611q^{81} + 10200q^{82} - 39184q^{84} + 34884q^{85} + 37068q^{87} - 7260q^{88} - 26686q^{90} - 45856q^{91} + 2719q^{93} + 6120q^{94} - 38368q^{96} - 27854q^{97} + 4235q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{14} + 162 x^{12} + 10041 x^{10} + 298396 x^{8} + 4418856 x^{6} + 32113344 x^{4} + 102865552 x^{2} + 102193344$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 23$$ $$\beta_{3}$$ $$=$$ $$($$$$21871 \nu^{13} + 664464 \nu^{12} + 3156768 \nu^{11} + 100588064 \nu^{10} + 162076727 \nu^{9} + 5517357520 \nu^{8} + 3389369462 \nu^{7} + 130053551424 \nu^{6} + 21692762028 \nu^{5} + 1188387849280 \nu^{4} - 30858950936 \nu^{3} + 3004965621248 \nu^{2} - 290266305120 \nu - 890860271616$$$$)/ 57508005888$$ $$\beta_{4}$$ $$=$$ $$($$$$21871 \nu^{13} - 130810 \nu^{12} + 3156768 \nu^{11} - 21010880 \nu^{10} + 162076727 \nu^{9} - 1256063274 \nu^{8} + 3389369462 \nu^{7} - 34111801220 \nu^{6} + 21692762028 \nu^{5} - 414055729672 \nu^{4} - 30858950936 \nu^{3} - 2052588372336 \nu^{2} - 290266305120 \nu - 2931692294592$$$$)/ 57508005888$$ $$\beta_{5}$$ $$=$$ $$($$$$21871 \nu^{13} - 130810 \nu^{12} + 3156768 \nu^{11} - 21010880 \nu^{10} + 162076727 \nu^{9} - 1256063274 \nu^{8} + 3389369462 \nu^{7} - 34111801220 \nu^{6} + 21692762028 \nu^{5} - 414055729672 \nu^{4} + 26649054952 \nu^{3} - 2052588372336 \nu^{2} + 1895037918624 \nu - 2931692294592$$$$)/ 57508005888$$ $$\beta_{6}$$ $$=$$ $$($$$$-65613 \nu^{13} + 2338070 \nu^{12} - 9470304 \nu^{11} + 343879424 \nu^{10} - 486230181 \nu^{9} + 18680585702 \nu^{8} - 10168108386 \nu^{7} + 457905222364 \nu^{6} - 65078286084 \nu^{5} + 5001787695160 \nu^{4} + 92576852808 \nu^{3} + 22603474823440 \nu^{2} + 870798915360 \nu + 31256572000320$$$$)/ 57508005888$$ $$\beta_{7}$$ $$=$$ $$($$$$515 \nu^{13} + 1521 \nu^{12} + 82720 \nu^{11} + 226496 \nu^{10} + 4945131 \nu^{9} + 12349801 \nu^{8} + 134298430 \nu^{7} + 295086762 \nu^{6} + 1630140668 \nu^{5} + 2886653140 \nu^{4} + 8081056584 \nu^{3} + 10000152728 \nu^{2} + 11542095648 \nu + 8799490656$$$$)/ 226409472$$ $$\beta_{8}$$ $$=$$ $$($$$$-43742 \nu^{13} - 327025 \nu^{12} - 6313536 \nu^{11} - 52527200 \nu^{10} - 324153454 \nu^{9} - 3140158185 \nu^{8} - 6778738924 \nu^{7} - 85279503050 \nu^{6} - 43385524056 \nu^{5} - 1035139324180 \nu^{4} + 61717901872 \nu^{3} - 5131470930840 \nu^{2} + 580532610240 \nu - 7271722730592$$$$)/ 14377001472$$ $$\beta_{9}$$ $$=$$ $$($$$$-53401 \nu^{13} - 8164552 \nu^{11} - 465987745 \nu^{9} - 12206545826 \nu^{7} - 147078874580 \nu^{5} - 774979556408 \nu^{3} - 1396072917216 \nu$$$$)/ 14377001472$$ $$\beta_{10}$$ $$=$$ $$($$$$-131008 \nu^{13} - 65405 \nu^{12} - 20665512 \nu^{11} - 10505440 \nu^{10} - 1227959504 \nu^{9} - 628031637 \nu^{8} - 33873891080 \nu^{7} - 17055900610 \nu^{6} - 433102614624 \nu^{5} - 207027864836 \nu^{4} - 2322165220192 \nu^{3} - 1026294186168 \nu^{2} - 3777106151808 \nu - 1465846147296$$$$)/ 28754002944$$ $$\beta_{11}$$ $$=$$ $$($$$$-96935 \nu^{13} + 73660 \nu^{12} - 15513224 \nu^{11} + 11023600 \nu^{10} - 931942655 \nu^{9} + 562222396 \nu^{8} - 25873076974 \nu^{7} + 10494856328 \nu^{6} - 332413977388 \nu^{5} + 34938112496 \nu^{4} - 1832132693512 \nu^{3} - 60603696928 \nu^{2} - 3152185369632 \nu + 1298665846656$$$$)/ 14377001472$$ $$\beta_{12}$$ $$=$$ $$($$$$-101413 \nu^{13} - 92456 \nu^{12} - 15308004 \nu^{11} - 14123416 \nu^{10} - 855129509 \nu^{9} - 817116984 \nu^{8} - 21451912838 \nu^{7} - 22018531528 \nu^{6} - 231554899284 \nu^{5} - 269347350560 \nu^{4} - 862863157864 \nu^{3} - 1178458639776 \nu^{2} - 159689778336 \nu - 635169306240$$$$)/ 14377001472$$ $$\beta_{13}$$ $$=$$ $$($$$$311765 \nu^{13} - 446532 \nu^{12} + 48470120 \nu^{11} - 70268592 \nu^{10} + 2804245565 \nu^{9} - 4146360516 \nu^{8} + 73626257434 \nu^{7} - 112260665496 \nu^{6} + 855472766212 \nu^{5} - 1366806160464 \nu^{4} + 3809173639000 \nu^{3} - 6462094024224 \nu^{2} + 3541338156384 \nu - 7104969198720$$$$)/ 28754002944$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 23$$ $$\nu^{3}$$ $$=$$ $$\beta_{5} - \beta_{4} - 38 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{11} - \beta_{9} + 2 \beta_{7} - 2 \beta_{4} - 2 \beta_{3} - 51 \beta_{2} + 872$$ $$\nu^{5}$$ $$=$$ $$-4 \beta_{13} + 4 \beta_{12} - 12 \beta_{10} + 2 \beta_{9} - 2 \beta_{8} - 57 \beta_{5} + 105 \beta_{4} + 1644 \beta_{1} + 12$$ $$\nu^{6}$$ $$=$$ $$18 \beta_{13} + 18 \beta_{12} - 75 \beta_{11} + 75 \beta_{9} - 12 \beta_{8} - 186 \beta_{7} + 10 \beta_{6} + 128 \beta_{4} + 142 \beta_{3} + 2355 \beta_{2} - 37698$$ $$\nu^{7}$$ $$=$$ $$356 \beta_{13} - 304 \beta_{12} - 26 \beta_{11} + 1236 \beta_{10} - 430 \beta_{9} + 180 \beta_{8} + 52 \beta_{7} + 2871 \beta_{5} - 7383 \beta_{4} - 74108 \beta_{1} - 1102$$ $$\nu^{8}$$ $$=$$ $$-1880 \beta_{13} - 1880 \beta_{12} + 4427 \beta_{11} - 4427 \beta_{9} + 1176 \beta_{8} + 12614 \beta_{7} - 960 \beta_{6} - 6486 \beta_{4} - 8454 \beta_{3} - 107771 \beta_{2} + 1695974$$ $$\nu^{9}$$ $$=$$ $$-24196 \beta_{13} + 18020 \beta_{12} + 3088 \beta_{11} - 90324 \beta_{10} + 37766 \beta_{9} - 11622 \beta_{8} - 6176 \beta_{7} - 142013 \beta_{5} + 451517 \beta_{4} + 3402352 \beta_{1} + 73772$$ $$\nu^{10}$$ $$=$$ $$137542 \beta_{13} + 137542 \beta_{12} - 243291 \beta_{11} + 243291 \beta_{9} - 83988 \beta_{8} - 761666 \beta_{7} + 64542 \beta_{6} + 292660 \beta_{4} + 477310 \beta_{3} + 4960383 \beta_{2} - 77693886$$ $$\nu^{11}$$ $$=$$ $$1481668 \beta_{13} - 979368 \beta_{12} - 251150 \beta_{11} + 5726868 \beta_{10} - 2551166 \beta_{9} + 667504 \beta_{8} + 502300 \beta_{7} + 7016295 \beta_{5} - 25847175 \beta_{4} - 158104044 \beta_{1} - 4402834$$ $$\nu^{12}$$ $$=$$ $$-8733980 \beta_{13} - 8733980 \beta_{12} + 12961511 \beta_{11} - 12961511 \beta_{9} + 5302944 \beta_{8} + 43390982 \beta_{7} - 3756452 \beta_{6} - 11971378 \beta_{4} - 26188430 \beta_{3} - 230287095 \beta_{2} + 3603294598$$ $$\nu^{13}$$ $$=$$ $$-85753652 \beta_{13} + 50963020 \beta_{12} + 17395316 \beta_{11} - 336880932 \beta_{10} + 153010510 \beta_{9} - 36276138 \beta_{8} - 34790632 \beta_{7} - 347279349 \beta_{5} + 1424726901 \beta_{4} + 7420320424 \beta_{1} + 248253520$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/33\mathbb{Z}\right)^\times$$.

 $$n$$ $$13$$ $$23$$ $$\chi(n)$$ $$1$$ $$-1$$

## Embeddings

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

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

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
23.1
 − 7.08422i − 6.53421i − 6.24203i − 3.64872i − 3.08869i − 2.36054i − 1.31515i 1.31515i 2.36054i 3.08869i 3.64872i 6.24203i 6.53421i 7.08422i
7.08422i 1.47928 8.87760i −34.1862 5.51941i −62.8909 10.4795i 86.8582 128.835i −76.6235 26.2648i 39.1007
23.2 6.53421i −8.21241 + 3.68190i −26.6960 12.2301i 24.0583 + 53.6616i −55.6823 69.8897i 53.8873 60.4745i 79.9138
23.3 6.24203i 7.09522 + 5.53695i −22.9629 42.0693i 34.5618 44.2886i −11.7748 43.4628i 19.6843 + 78.5718i −262.598
23.4 3.64872i 1.72713 + 8.83273i 2.68684 34.1690i 32.2281 6.30180i 45.8062 68.1830i −75.0341 + 30.5105i 124.673
23.5 3.08869i −4.49864 7.79501i 6.45997 9.71602i −24.0764 + 13.8949i −66.8890 69.3720i −40.5244 + 70.1340i −30.0098
23.6 2.36054i 7.81446 4.46478i 10.4279 9.92884i −10.5393 18.4463i −13.3317 62.3839i 41.1315 69.7797i 23.4374
23.7 1.31515i −7.90503 + 4.30239i 14.2704 39.9329i 5.65827 + 10.3963i 53.0133 39.8100i 43.9789 68.0210i −52.5176
23.8 1.31515i −7.90503 4.30239i 14.2704 39.9329i 5.65827 10.3963i 53.0133 39.8100i 43.9789 + 68.0210i −52.5176
23.9 2.36054i 7.81446 + 4.46478i 10.4279 9.92884i −10.5393 + 18.4463i −13.3317 62.3839i 41.1315 + 69.7797i 23.4374
23.10 3.08869i −4.49864 + 7.79501i 6.45997 9.71602i −24.0764 13.8949i −66.8890 69.3720i −40.5244 70.1340i −30.0098
23.11 3.64872i 1.72713 8.83273i 2.68684 34.1690i 32.2281 + 6.30180i 45.8062 68.1830i −75.0341 30.5105i 124.673
23.12 6.24203i 7.09522 5.53695i −22.9629 42.0693i 34.5618 + 44.2886i −11.7748 43.4628i 19.6843 78.5718i −262.598
23.13 6.53421i −8.21241 3.68190i −26.6960 12.2301i 24.0583 53.6616i −55.6823 69.8897i 53.8873 + 60.4745i 79.9138
23.14 7.08422i 1.47928 + 8.87760i −34.1862 5.51941i −62.8909 + 10.4795i 86.8582 128.835i −76.6235 + 26.2648i 39.1007
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 23.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.5.b.a 14
3.b odd 2 1 inner 33.5.b.a 14
4.b odd 2 1 528.5.i.d 14
12.b even 2 1 528.5.i.d 14

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.5.b.a 14 1.a even 1 1 trivial
33.5.b.a 14 3.b odd 2 1 inner
528.5.i.d 14 4.b odd 2 1
528.5.i.d 14 12.b even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{5}^{\mathrm{new}}(33, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 62 T^{2} + 2233 T^{4} - 61956 T^{6} + 1516584 T^{8} - 32463936 T^{10} + 606487184 T^{12} - 10126457152 T^{14} + 155260719104 T^{16} - 2127556509696 T^{18} + 25444057350144 T^{20} - 266098993790976 T^{22} + 2455209464823808 T^{24} - 17451448556060672 T^{26} + 72057594037927936 T^{28}$$
$3$ $$1 + 5 T + 46 T^{2} + 573 T^{3} + 5832 T^{4} + 23355 T^{5} + 839241 T^{6} + 6874470 T^{7} + 67978521 T^{8} + 153232155 T^{9} + 3099363912 T^{10} + 24665771133 T^{11} + 160392082446 T^{12} + 1412147682405 T^{13} + 22876792454961 T^{14}$$
$5$ $$1 - 3845 T^{2} + 7248802 T^{4} - 9433228797 T^{6} + 9906146923044 T^{8} - 8848175303763135 T^{10} + 6779542001980760729 T^{12} -$$$$45\!\cdots\!74$$$$T^{14} +$$$$26\!\cdots\!25$$$$T^{16} -$$$$13\!\cdots\!75$$$$T^{18} +$$$$59\!\cdots\!00$$$$T^{20} -$$$$21\!\cdots\!25$$$$T^{22} +$$$$65\!\cdots\!50$$$$T^{24} -$$$$13\!\cdots\!25$$$$T^{26} +$$$$13\!\cdots\!25$$$$T^{28}$$
$7$ $$( 1 - 38 T + 7357 T^{2} - 311528 T^{3} + 32742715 T^{4} - 1355111066 T^{5} + 103333215303 T^{6} - 4090706001456 T^{7} + 248103049942503 T^{8} - 7811945628387866 T^{9} + 453201322055490715 T^{10} - 10352988394486660328 T^{11} +$$$$58\!\cdots\!57$$$$T^{12} -$$$$72\!\cdots\!38$$$$T^{13} +$$$$45\!\cdots\!01$$$$T^{14} )^{2}$$
$11$ $$( 1 + 1331 T^{2} )^{7}$$
$13$ $$( 1 + 52 T + 59035 T^{2} - 3104360 T^{3} + 2755858153 T^{4} - 145996022516 T^{5} + 86746134227931 T^{6} - 8990566727995824 T^{7} + 2477556339683937291 T^{8} -$$$$11\!\cdots\!36$$$$T^{9} +$$$$64\!\cdots\!93$$$$T^{10} -$$$$20\!\cdots\!60$$$$T^{11} +$$$$11\!\cdots\!35$$$$T^{12} +$$$$28\!\cdots\!72$$$$T^{13} +$$$$15\!\cdots\!21$$$$T^{14} )^{2}$$
$17$ $$1 - 485846 T^{2} + 129742709311 T^{4} - 25028073501120084 T^{6} +$$$$38\!\cdots\!61$$$$T^{8} -$$$$47\!\cdots\!70$$$$T^{10} +$$$$50\!\cdots\!59$$$$T^{12} -$$$$45\!\cdots\!68$$$$T^{14} +$$$$35\!\cdots\!19$$$$T^{16} -$$$$23\!\cdots\!70$$$$T^{18} +$$$$12\!\cdots\!81$$$$T^{20} -$$$$59\!\cdots\!24$$$$T^{22} +$$$$21\!\cdots\!11$$$$T^{24} -$$$$55\!\cdots\!86$$$$T^{26} +$$$$80\!\cdots\!81$$$$T^{28}$$
$19$ $$( 1 - 536 T + 545053 T^{2} - 216080112 T^{3} + 137229356139 T^{4} - 44238459488520 T^{5} + 22701454646567351 T^{6} - 6335223407637576544 T^{7} +$$$$29\!\cdots\!71$$$$T^{8} -$$$$75\!\cdots\!20$$$$T^{9} +$$$$30\!\cdots\!79$$$$T^{10} -$$$$62\!\cdots\!72$$$$T^{11} +$$$$20\!\cdots\!53$$$$T^{12} -$$$$26\!\cdots\!56$$$$T^{13} +$$$$63\!\cdots\!41$$$$T^{14} )^{2}$$
$23$ $$1 - 2990093 T^{2} + 4317529073506 T^{4} - 3991111390473390741 T^{6} +$$$$26\!\cdots\!84$$$$T^{8} -$$$$13\!\cdots\!23$$$$T^{10} +$$$$52\!\cdots\!57$$$$T^{12} -$$$$16\!\cdots\!26$$$$T^{14} +$$$$40\!\cdots\!17$$$$T^{16} -$$$$81\!\cdots\!03$$$$T^{18} +$$$$12\!\cdots\!44$$$$T^{20} -$$$$15\!\cdots\!61$$$$T^{22} +$$$$12\!\cdots\!06$$$$T^{24} -$$$$68\!\cdots\!33$$$$T^{26} +$$$$18\!\cdots\!61$$$$T^{28}$$
$29$ $$1 - 5650778 T^{2} + 15779875706959 T^{4} - 28943806993315163500 T^{6} +$$$$39\!\cdots\!65$$$$T^{8} -$$$$42\!\cdots\!78$$$$T^{10} +$$$$37\!\cdots\!95$$$$T^{12} -$$$$28\!\cdots\!04$$$$T^{14} +$$$$18\!\cdots\!95$$$$T^{16} -$$$$10\!\cdots\!38$$$$T^{18} +$$$$49\!\cdots\!65$$$$T^{20} -$$$$18\!\cdots\!00$$$$T^{22} +$$$$49\!\cdots\!59$$$$T^{24} -$$$$88\!\cdots\!58$$$$T^{26} +$$$$78\!\cdots\!21$$$$T^{28}$$
$31$ $$( 1 - 1655 T + 4105918 T^{2} - 5731077719 T^{3} + 8813615578576 T^{4} - 10505503670622809 T^{5} + 11979775957570276833 T^{6} -$$$$11\!\cdots\!34$$$$T^{7} +$$$$11\!\cdots\!93$$$$T^{8} -$$$$89\!\cdots\!69$$$$T^{9} +$$$$69\!\cdots\!36$$$$T^{10} -$$$$41\!\cdots\!39$$$$T^{11} +$$$$27\!\cdots\!18$$$$T^{12} -$$$$10\!\cdots\!55$$$$T^{13} +$$$$57\!\cdots\!41$$$$T^{14} )^{2}$$
$37$ $$( 1 + 181 T + 10236082 T^{2} + 2678247153 T^{3} + 48626479267980 T^{4} + 13313899617346179 T^{5} +$$$$13\!\cdots\!17$$$$T^{6} +$$$$33\!\cdots\!38$$$$T^{7} +$$$$26\!\cdots\!37$$$$T^{8} +$$$$46\!\cdots\!59$$$$T^{9} +$$$$32\!\cdots\!80$$$$T^{10} +$$$$33\!\cdots\!73$$$$T^{11} +$$$$23\!\cdots\!82$$$$T^{12} +$$$$78\!\cdots\!41$$$$T^{13} +$$$$81\!\cdots\!21$$$$T^{14} )^{2}$$
$41$ $$1 - 20067002 T^{2} + 193946959226095 T^{4} -$$$$11\!\cdots\!60$$$$T^{6} +$$$$52\!\cdots\!85$$$$T^{8} -$$$$18\!\cdots\!34$$$$T^{10} +$$$$53\!\cdots\!31$$$$T^{12} -$$$$14\!\cdots\!76$$$$T^{14} +$$$$42\!\cdots\!51$$$$T^{16} -$$$$11\!\cdots\!94$$$$T^{18} +$$$$26\!\cdots\!85$$$$T^{20} -$$$$48\!\cdots\!60$$$$T^{22} +$$$$62\!\cdots\!95$$$$T^{24} -$$$$52\!\cdots\!42$$$$T^{26} +$$$$20\!\cdots\!41$$$$T^{28}$$
$43$ $$( 1 + 3370 T + 17487001 T^{2} + 41215605264 T^{3} + 116672249476671 T^{4} + 222115030094308806 T^{5} +$$$$47\!\cdots\!39$$$$T^{6} +$$$$82\!\cdots\!12$$$$T^{7} +$$$$16\!\cdots\!39$$$$T^{8} +$$$$25\!\cdots\!06$$$$T^{9} +$$$$46\!\cdots\!71$$$$T^{10} +$$$$56\!\cdots\!64$$$$T^{11} +$$$$81\!\cdots\!01$$$$T^{12} +$$$$53\!\cdots\!70$$$$T^{13} +$$$$54\!\cdots\!01$$$$T^{14} )^{2}$$
$47$ $$1 - 28348958 T^{2} + 434446812146299 T^{4} -$$$$47\!\cdots\!88$$$$T^{6} +$$$$41\!\cdots\!25$$$$T^{8} -$$$$29\!\cdots\!10$$$$T^{10} +$$$$17\!\cdots\!35$$$$T^{12} -$$$$94\!\cdots\!84$$$$T^{14} +$$$$42\!\cdots\!35$$$$T^{16} -$$$$16\!\cdots\!10$$$$T^{18} +$$$$55\!\cdots\!25$$$$T^{20} -$$$$15\!\cdots\!08$$$$T^{22} +$$$$33\!\cdots\!99$$$$T^{24} -$$$$51\!\cdots\!38$$$$T^{26} +$$$$43\!\cdots\!21$$$$T^{28}$$
$53$ $$1 - 68012342 T^{2} + 2294298247507435 T^{4} -$$$$50\!\cdots\!52$$$$T^{6} +$$$$83\!\cdots\!05$$$$T^{8} -$$$$10\!\cdots\!18$$$$T^{10} +$$$$11\!\cdots\!95$$$$T^{12} -$$$$97\!\cdots\!24$$$$T^{14} +$$$$70\!\cdots\!95$$$$T^{16} -$$$$41\!\cdots\!78$$$$T^{18} +$$$$20\!\cdots\!05$$$$T^{20} -$$$$76\!\cdots\!32$$$$T^{22} +$$$$21\!\cdots\!35$$$$T^{24} -$$$$39\!\cdots\!62$$$$T^{26} +$$$$36\!\cdots\!21$$$$T^{28}$$
$59$ $$1 - 94058909 T^{2} + 4263490602643738 T^{4} -$$$$12\!\cdots\!69$$$$T^{6} +$$$$26\!\cdots\!12$$$$T^{8} -$$$$43\!\cdots\!23$$$$T^{10} +$$$$60\!\cdots\!13$$$$T^{12} -$$$$76\!\cdots\!30$$$$T^{14} +$$$$88\!\cdots\!73$$$$T^{16} -$$$$93\!\cdots\!43$$$$T^{18} +$$$$82\!\cdots\!32$$$$T^{20} -$$$$57\!\cdots\!89$$$$T^{22} +$$$$29\!\cdots\!38$$$$T^{24} -$$$$94\!\cdots\!89$$$$T^{26} +$$$$14\!\cdots\!41$$$$T^{28}$$
$61$ $$( 1 + 1780 T + 41912563 T^{2} + 97658312472 T^{3} + 1128391765815297 T^{4} + 2332428541262115660 T^{5} +$$$$21\!\cdots\!39$$$$T^{6} +$$$$39\!\cdots\!20$$$$T^{7} +$$$$29\!\cdots\!99$$$$T^{8} +$$$$44\!\cdots\!60$$$$T^{9} +$$$$29\!\cdots\!37$$$$T^{10} +$$$$35\!\cdots\!92$$$$T^{11} +$$$$21\!\cdots\!63$$$$T^{12} +$$$$12\!\cdots\!80$$$$T^{13} +$$$$97\!\cdots\!81$$$$T^{14} )^{2}$$
$67$ $$( 1 + 8257 T + 78448894 T^{2} + 418710202913 T^{3} + 2726921864628272 T^{4} + 11796880355207077727 T^{5} +$$$$64\!\cdots\!45$$$$T^{6} +$$$$25\!\cdots\!86$$$$T^{7} +$$$$13\!\cdots\!45$$$$T^{8} +$$$$47\!\cdots\!07$$$$T^{9} +$$$$22\!\cdots\!92$$$$T^{10} +$$$$69\!\cdots\!53$$$$T^{11} +$$$$26\!\cdots\!94$$$$T^{12} +$$$$55\!\cdots\!97$$$$T^{13} +$$$$13\!\cdots\!41$$$$T^{14} )^{2}$$
$71$ $$1 - 203956901 T^{2} + 19113840255144106 T^{4} -$$$$10\!\cdots\!93$$$$T^{6} +$$$$40\!\cdots\!72$$$$T^{8} -$$$$10\!\cdots\!75$$$$T^{10} +$$$$19\!\cdots\!77$$$$T^{12} -$$$$39\!\cdots\!18$$$$T^{14} +$$$$12\!\cdots\!97$$$$T^{16} -$$$$42\!\cdots\!75$$$$T^{18} +$$$$10\!\cdots\!32$$$$T^{20} -$$$$18\!\cdots\!13$$$$T^{22} +$$$$21\!\cdots\!06$$$$T^{24} -$$$$14\!\cdots\!61$$$$T^{26} +$$$$46\!\cdots\!21$$$$T^{28}$$
$73$ $$( 1 - 6332 T + 112811287 T^{2} - 949500906536 T^{3} + 7258163516936341 T^{4} - 54948310613542514372 T^{5} +$$$$33\!\cdots\!35$$$$T^{6} -$$$$18\!\cdots\!16$$$$T^{7} +$$$$93\!\cdots\!35$$$$T^{8} -$$$$44\!\cdots\!32$$$$T^{9} +$$$$16\!\cdots\!61$$$$T^{10} -$$$$61\!\cdots\!96$$$$T^{11} +$$$$20\!\cdots\!87$$$$T^{12} -$$$$33\!\cdots\!12$$$$T^{13} +$$$$14\!\cdots\!81$$$$T^{14} )^{2}$$
$79$ $$( 1 - 1526 T + 204404437 T^{2} - 322950507640 T^{3} + 19831551969918467 T^{4} - 29336944829432468842 T^{5} +$$$$11\!\cdots\!79$$$$T^{6} -$$$$14\!\cdots\!04$$$$T^{7} +$$$$45\!\cdots\!99$$$$T^{8} -$$$$44\!\cdots\!62$$$$T^{9} +$$$$11\!\cdots\!47$$$$T^{10} -$$$$74\!\cdots\!40$$$$T^{11} +$$$$18\!\cdots\!37$$$$T^{12} -$$$$53\!\cdots\!06$$$$T^{13} +$$$$13\!\cdots\!61$$$$T^{14} )^{2}$$
$83$ $$1 - 343468394 T^{2} + 60970942911104059 T^{4} -$$$$73\!\cdots\!96$$$$T^{6} +$$$$67\!\cdots\!53$$$$T^{8} -$$$$50\!\cdots\!62$$$$T^{10} +$$$$30\!\cdots\!83$$$$T^{12} -$$$$15\!\cdots\!72$$$$T^{14} +$$$$69\!\cdots\!03$$$$T^{16} -$$$$25\!\cdots\!22$$$$T^{18} +$$$$77\!\cdots\!13$$$$T^{20} -$$$$18\!\cdots\!56$$$$T^{22} +$$$$35\!\cdots\!59$$$$T^{24} -$$$$44\!\cdots\!54$$$$T^{26} +$$$$29\!\cdots\!81$$$$T^{28}$$
$89$ $$1 - 392958917 T^{2} + 74853447914451754 T^{4} -$$$$92\!\cdots\!25$$$$T^{6} +$$$$84\!\cdots\!84$$$$T^{8} -$$$$63\!\cdots\!11$$$$T^{10} +$$$$41\!\cdots\!65$$$$T^{12} -$$$$26\!\cdots\!18$$$$T^{14} +$$$$16\!\cdots\!65$$$$T^{16} -$$$$98\!\cdots\!71$$$$T^{18} +$$$$51\!\cdots\!44$$$$T^{20} -$$$$22\!\cdots\!25$$$$T^{22} +$$$$70\!\cdots\!54$$$$T^{24} -$$$$14\!\cdots\!77$$$$T^{26} +$$$$14\!\cdots\!61$$$$T^{28}$$
$97$ $$( 1 + 13927 T + 376447066 T^{2} + 3249505605443 T^{3} + 55136591519145860 T^{4} +$$$$32\!\cdots\!61$$$$T^{5} +$$$$52\!\cdots\!65$$$$T^{6} +$$$$26\!\cdots\!26$$$$T^{7} +$$$$46\!\cdots\!65$$$$T^{8} +$$$$25\!\cdots\!21$$$$T^{9} +$$$$38\!\cdots\!60$$$$T^{10} +$$$$19\!\cdots\!03$$$$T^{11} +$$$$20\!\cdots\!66$$$$T^{12} +$$$$67\!\cdots\!87$$$$T^{13} +$$$$42\!\cdots\!61$$$$T^{14} )^{2}$$