Properties

 Label 1350.2.e Level 1350 Weight 2 Character orbit e Rep. character $$\chi_{1350}(451,\cdot)$$ Character field $$\Q(\zeta_{3})$$ Dimension 38 Newform subspaces 14 Sturm bound 540 Trace bound 17

Related objects

Defining parameters

 Level: $$N$$ = $$1350 = 2 \cdot 3^{3} \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 1350.e (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$9$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$14$$ Sturm bound: $$540$$ Trace bound: $$17$$ Distinguishing $$T_p$$: $$7$$, $$11$$, $$17$$

Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(1350, [\chi])$$.

Total New Old
Modular forms 612 38 574
Cusp forms 468 38 430
Eisenstein series 144 0 144

Trace form

 $$38q + q^{2} - 19q^{4} - 2q^{7} - 2q^{8} + O(q^{10})$$ $$38q + q^{2} - 19q^{4} - 2q^{7} - 2q^{8} + q^{11} - 2q^{13} - 2q^{14} - 19q^{16} - 18q^{17} - 2q^{19} + 3q^{22} - 6q^{23} + 20q^{26} + 4q^{28} - 18q^{29} - 8q^{31} + q^{32} + 3q^{34} + 16q^{37} + 11q^{38} + 13q^{41} + q^{43} - 2q^{44} + 12q^{46} + 18q^{47} - 27q^{49} - 2q^{52} + 48q^{53} - 2q^{56} + 6q^{58} - 43q^{59} - 20q^{61} - 16q^{62} + 38q^{64} - 17q^{67} + 9q^{68} + 24q^{71} + 22q^{73} - 8q^{74} + q^{76} + 42q^{77} - 8q^{79} - 30q^{82} - 24q^{83} - 35q^{86} + 3q^{88} - 92q^{89} + 56q^{91} - 6q^{92} - 6q^{94} + 7q^{97} - 42q^{98} + O(q^{100})$$

Decomposition of $$S_{2}^{\mathrm{new}}(1350, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
1350.2.e.a $$2$$ $$10.780$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$0$$ $$-1$$ $$q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}+(-1+\zeta_{6})q^{7}+\cdots$$
1350.2.e.b $$2$$ $$10.780$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$0$$ $$-1$$ $$q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}+(-1+\zeta_{6})q^{7}+\cdots$$
1350.2.e.c $$2$$ $$10.780$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$0$$ $$2$$ $$q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}+(2-2\zeta_{6})q^{7}+\cdots$$
1350.2.e.d $$2$$ $$10.780$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$0$$ $$2$$ $$q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}+(2-2\zeta_{6})q^{7}+\cdots$$
1350.2.e.e $$2$$ $$10.780$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$0$$ $$4$$ $$q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}+(4-4\zeta_{6})q^{7}+\cdots$$
1350.2.e.f $$2$$ $$10.780$$ $$\Q(\sqrt{-3})$$ None $$1$$ $$0$$ $$0$$ $$-4$$ $$q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{4}+(-4+4\zeta_{6})q^{7}+\cdots$$
1350.2.e.g $$2$$ $$10.780$$ $$\Q(\sqrt{-3})$$ None $$1$$ $$0$$ $$0$$ $$-4$$ $$q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{4}+(-4+4\zeta_{6})q^{7}+\cdots$$
1350.2.e.h $$2$$ $$10.780$$ $$\Q(\sqrt{-3})$$ None $$1$$ $$0$$ $$0$$ $$-2$$ $$q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{4}+(-2+2\zeta_{6})q^{7}+\cdots$$
1350.2.e.i $$2$$ $$10.780$$ $$\Q(\sqrt{-3})$$ None $$1$$ $$0$$ $$0$$ $$1$$ $$q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{4}+(1-\zeta_{6})q^{7}+\cdots$$
1350.2.e.j $$4$$ $$10.780$$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None $$-2$$ $$0$$ $$0$$ $$-4$$ $$q+(-1+\beta _{1})q^{2}-\beta _{1}q^{4}+(-2+2\beta _{1}+\cdots)q^{7}+\cdots$$
1350.2.e.k $$4$$ $$10.780$$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None $$-2$$ $$0$$ $$0$$ $$-4$$ $$q+(-1+\beta _{1})q^{2}-\beta _{1}q^{4}+(-2+2\beta _{1}+\cdots)q^{7}+\cdots$$
1350.2.e.l $$4$$ $$10.780$$ $$\Q(\sqrt{-3}, \sqrt{-11})$$ None $$2$$ $$0$$ $$0$$ $$1$$ $$q+\beta _{1}q^{2}+(-1+\beta _{1})q^{4}+\beta _{3}q^{7}-q^{8}+\cdots$$
1350.2.e.m $$4$$ $$10.780$$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None $$2$$ $$0$$ $$0$$ $$4$$ $$q+(1-\beta _{1})q^{2}-\beta _{1}q^{4}+(2-2\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots$$
1350.2.e.n $$4$$ $$10.780$$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None $$2$$ $$0$$ $$0$$ $$4$$ $$q+(1-\beta _{1})q^{2}-\beta _{1}q^{4}+(2-2\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots$$

Decomposition of $$S_{2}^{\mathrm{old}}(1350, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(1350, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(18, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(54, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(90, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(135, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(225, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(270, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(450, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(675, [\chi])$$$$^{\oplus 2}$$

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 + T + T^{2}$$)($$1 + T + T^{2}$$)($$1 + T + T^{2}$$)($$1 + T + T^{2}$$)($$1 + T + T^{2}$$)($$1 - T + T^{2}$$)($$1 - T + T^{2}$$)($$1 - T + T^{2}$$)($$1 - T + T^{2}$$)($$( 1 + T + T^{2} )^{2}$$)($$( 1 + T + T^{2} )^{2}$$)($$( 1 - T + T^{2} )^{2}$$)($$( 1 - T + T^{2} )^{2}$$)($$( 1 - T + T^{2} )^{2}$$)
$3$ ()()()()()()()()()()()()()()
$5$ ()()()()()()()()()()()()()()
$7$ ($$( 1 - 4 T + 7 T^{2} )( 1 + 5 T + 7 T^{2} )$$)($$( 1 - 4 T + 7 T^{2} )( 1 + 5 T + 7 T^{2} )$$)($$1 - 2 T - 3 T^{2} - 14 T^{3} + 49 T^{4}$$)($$1 - 2 T - 3 T^{2} - 14 T^{3} + 49 T^{4}$$)($$( 1 - 5 T + 7 T^{2} )( 1 + T + 7 T^{2} )$$)($$( 1 - T + 7 T^{2} )( 1 + 5 T + 7 T^{2} )$$)($$( 1 - T + 7 T^{2} )( 1 + 5 T + 7 T^{2} )$$)($$1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4}$$)($$( 1 - 5 T + 7 T^{2} )( 1 + 4 T + 7 T^{2} )$$)($$1 + 4 T + 4 T^{2} - 8 T^{3} - 17 T^{4} - 56 T^{5} + 196 T^{6} + 1372 T^{7} + 2401 T^{8}$$)($$1 + 4 T + 4 T^{2} - 8 T^{3} - 17 T^{4} - 56 T^{5} + 196 T^{6} + 1372 T^{7} + 2401 T^{8}$$)($$1 - T - 5 T^{2} + 8 T^{3} - 20 T^{4} + 56 T^{5} - 245 T^{6} - 343 T^{7} + 2401 T^{8}$$)($$1 - 4 T + 4 T^{2} + 8 T^{3} - 17 T^{4} + 56 T^{5} + 196 T^{6} - 1372 T^{7} + 2401 T^{8}$$)($$1 - 4 T + 4 T^{2} + 8 T^{3} - 17 T^{4} + 56 T^{5} + 196 T^{6} - 1372 T^{7} + 2401 T^{8}$$)
$11$ ($$1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4}$$)($$1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4}$$)($$1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4}$$)($$1 - 11 T^{2} + 121 T^{4}$$)($$1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4}$$)($$1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4}$$)($$1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4}$$)($$1 - 11 T^{2} + 121 T^{4}$$)($$1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4}$$)($$1 + 2 T - 13 T^{2} - 10 T^{3} + 124 T^{4} - 110 T^{5} - 1573 T^{6} + 2662 T^{7} + 14641 T^{8}$$)($$1 + 2 T^{2} - 117 T^{4} + 242 T^{6} + 14641 T^{8}$$)($$1 + 3 T - 7 T^{2} - 18 T^{3} + 36 T^{4} - 198 T^{5} - 847 T^{6} + 3993 T^{7} + 14641 T^{8}$$)($$1 + 2 T - 13 T^{2} - 10 T^{3} + 124 T^{4} - 110 T^{5} - 1573 T^{6} + 2662 T^{7} + 14641 T^{8}$$)($$1 + 2 T^{2} - 117 T^{4} + 242 T^{6} + 14641 T^{8}$$)
$13$ ($$1 + 6 T + 23 T^{2} + 78 T^{3} + 169 T^{4}$$)($$( 1 - 7 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} )$$)($$( 1 - 7 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} )$$)($$1 + 4 T + 3 T^{2} + 52 T^{3} + 169 T^{4}$$)($$1 - 4 T + 3 T^{2} - 52 T^{3} + 169 T^{4}$$)($$1 + 4 T + 3 T^{2} + 52 T^{3} + 169 T^{4}$$)($$1 + 4 T + 3 T^{2} + 52 T^{3} + 169 T^{4}$$)($$1 - 4 T + 3 T^{2} - 52 T^{3} + 169 T^{4}$$)($$1 - 6 T + 23 T^{2} - 78 T^{3} + 169 T^{4}$$)($$1 - 20 T^{2} + 231 T^{4} - 3380 T^{6} + 28561 T^{8}$$)($$1 + 4 T - 8 T^{2} - 8 T^{3} + 199 T^{4} - 104 T^{5} - 1352 T^{6} + 8788 T^{7} + 28561 T^{8}$$)($$1 + 2 T + 10 T^{2} - 64 T^{3} - 185 T^{4} - 832 T^{5} + 1690 T^{6} + 4394 T^{7} + 28561 T^{8}$$)($$1 - 20 T^{2} + 231 T^{4} - 3380 T^{6} + 28561 T^{8}$$)($$1 - 4 T - 8 T^{2} + 8 T^{3} + 199 T^{4} + 104 T^{5} - 1352 T^{6} - 8788 T^{7} + 28561 T^{8}$$)
$17$ ($$( 1 + 2 T + 17 T^{2} )^{2}$$)($$( 1 + 17 T^{2} )^{2}$$)($$( 1 + 3 T + 17 T^{2} )^{2}$$)($$( 1 + 6 T + 17 T^{2} )^{2}$$)($$( 1 - 3 T + 17 T^{2} )^{2}$$)($$( 1 + 3 T + 17 T^{2} )^{2}$$)($$( 1 - 3 T + 17 T^{2} )^{2}$$)($$( 1 - 6 T + 17 T^{2} )^{2}$$)($$( 1 - 2 T + 17 T^{2} )^{2}$$)($$( 1 - 2 T + 11 T^{2} - 34 T^{3} + 289 T^{4} )^{2}$$)($$( 1 + 10 T^{2} + 289 T^{4} )^{2}$$)($$( 1 + 9 T + 46 T^{2} + 153 T^{3} + 289 T^{4} )^{2}$$)($$( 1 + 2 T + 11 T^{2} + 34 T^{3} + 289 T^{4} )^{2}$$)($$( 1 + 10 T^{2} + 289 T^{4} )^{2}$$)
$19$ ($$( 1 - 6 T + 19 T^{2} )^{2}$$)($$( 1 + 4 T + 19 T^{2} )^{2}$$)($$( 1 + T + 19 T^{2} )^{2}$$)($$( 1 + 7 T + 19 T^{2} )^{2}$$)($$( 1 + 4 T + 19 T^{2} )^{2}$$)($$( 1 + 4 T + 19 T^{2} )^{2}$$)($$( 1 - 5 T + 19 T^{2} )^{2}$$)($$( 1 + 7 T + 19 T^{2} )^{2}$$)($$( 1 - 6 T + 19 T^{2} )^{2}$$)($$( 1 + 6 T + 41 T^{2} + 114 T^{3} + 361 T^{4} )^{2}$$)($$( 1 - 10 T + 57 T^{2} - 190 T^{3} + 361 T^{4} )^{2}$$)($$( 1 - T + 30 T^{2} - 19 T^{3} + 361 T^{4} )^{2}$$)($$( 1 + 6 T + 41 T^{2} + 114 T^{3} + 361 T^{4} )^{2}$$)($$( 1 - 10 T + 57 T^{2} - 190 T^{3} + 361 T^{4} )^{2}$$)
$23$ ($$1 + T - 22 T^{2} + 23 T^{3} + 529 T^{4}$$)($$1 + 9 T + 58 T^{2} + 207 T^{3} + 529 T^{4}$$)($$1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4}$$)($$1 - 23 T^{2} + 529 T^{4}$$)($$1 + 6 T + 13 T^{2} + 138 T^{3} + 529 T^{4}$$)($$1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4}$$)($$1 + 6 T + 13 T^{2} + 138 T^{3} + 529 T^{4}$$)($$1 - 23 T^{2} + 529 T^{4}$$)($$1 - T - 22 T^{2} - 23 T^{3} + 529 T^{4}$$)($$1 - 4 T - 10 T^{2} + 80 T^{3} - 221 T^{4} + 1840 T^{5} - 5290 T^{6} - 48668 T^{7} + 279841 T^{8}$$)($$1 - 40 T^{2} + 1071 T^{4} - 21160 T^{6} + 279841 T^{8}$$)($$1 - 3 T - 31 T^{2} + 18 T^{3} + 864 T^{4} + 414 T^{5} - 16399 T^{6} - 36501 T^{7} + 279841 T^{8}$$)($$1 + 4 T - 10 T^{2} - 80 T^{3} - 221 T^{4} - 1840 T^{5} - 5290 T^{6} + 48668 T^{7} + 279841 T^{8}$$)($$1 - 40 T^{2} + 1071 T^{4} - 21160 T^{6} + 279841 T^{8}$$)
$29$ ($$1 - 9 T + 52 T^{2} - 261 T^{3} + 841 T^{4}$$)($$1 - 3 T - 20 T^{2} - 87 T^{3} + 841 T^{4}$$)($$1 - 6 T + 7 T^{2} - 174 T^{3} + 841 T^{4}$$)($$1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4}$$)($$1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4}$$)($$1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4}$$)($$1 - 6 T + 7 T^{2} - 174 T^{3} + 841 T^{4}$$)($$1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4}$$)($$1 - 9 T + 52 T^{2} - 261 T^{3} + 841 T^{4}$$)($$( 1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4} )^{2}$$)($$1 - 52 T^{2} + 1863 T^{4} - 43732 T^{6} + 707281 T^{8}$$)($$1 + 3 T - 43 T^{2} - 18 T^{3} + 1602 T^{4} - 522 T^{5} - 36163 T^{6} + 73167 T^{7} + 707281 T^{8}$$)($$( 1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4} )^{2}$$)($$1 - 52 T^{2} + 1863 T^{4} - 43732 T^{6} + 707281 T^{8}$$)
$31$ ($$1 - 2 T - 27 T^{2} - 62 T^{3} + 961 T^{4}$$)($$( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} )$$)($$( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} )$$)($$1 - 10 T + 69 T^{2} - 310 T^{3} + 961 T^{4}$$)($$1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4}$$)($$1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4}$$)($$1 + 2 T - 27 T^{2} + 62 T^{3} + 961 T^{4}$$)($$1 - 10 T + 69 T^{2} - 310 T^{3} + 961 T^{4}$$)($$1 - 2 T - 27 T^{2} - 62 T^{3} + 961 T^{4}$$)($$1 + 8 T - 8 T^{2} + 80 T^{3} + 2239 T^{4} + 2480 T^{5} - 7688 T^{6} + 238328 T^{7} + 923521 T^{8}$$)($$1 + 4 T - 44 T^{2} - 8 T^{3} + 2143 T^{4} - 248 T^{5} - 42284 T^{6} + 119164 T^{7} + 923521 T^{8}$$)($$1 - 2 T - 26 T^{2} + 64 T^{3} - 185 T^{4} + 1984 T^{5} - 24986 T^{6} - 59582 T^{7} + 923521 T^{8}$$)($$1 + 8 T - 8 T^{2} + 80 T^{3} + 2239 T^{4} + 2480 T^{5} - 7688 T^{6} + 238328 T^{7} + 923521 T^{8}$$)($$1 + 4 T - 44 T^{2} - 8 T^{3} + 2143 T^{4} - 248 T^{5} - 42284 T^{6} + 119164 T^{7} + 923521 T^{8}$$)
$37$ ($$( 1 + 2 T + 37 T^{2} )^{2}$$)($$( 1 + 8 T + 37 T^{2} )^{2}$$)($$( 1 - 4 T + 37 T^{2} )^{2}$$)($$( 1 + 2 T + 37 T^{2} )^{2}$$)($$( 1 - 8 T + 37 T^{2} )^{2}$$)($$( 1 + 8 T + 37 T^{2} )^{2}$$)($$( 1 - 4 T + 37 T^{2} )^{2}$$)($$( 1 - 2 T + 37 T^{2} )^{2}$$)($$( 1 - 2 T + 37 T^{2} )^{2}$$)($$( 1 - 8 T + 37 T^{2} )^{4}$$)($$( 1 + 8 T + 36 T^{2} + 296 T^{3} + 1369 T^{4} )^{2}$$)($$( 1 - 4 T + 37 T^{2} )^{4}$$)($$( 1 + 8 T + 37 T^{2} )^{4}$$)($$( 1 - 8 T + 36 T^{2} - 296 T^{3} + 1369 T^{4} )^{2}$$)
$41$ ($$1 + 11 T + 80 T^{2} + 451 T^{3} + 1681 T^{4}$$)($$1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4}$$)($$1 - 9 T + 40 T^{2} - 369 T^{3} + 1681 T^{4}$$)($$1 - 9 T + 40 T^{2} - 369 T^{3} + 1681 T^{4}$$)($$1 + 6 T - 5 T^{2} + 246 T^{3} + 1681 T^{4}$$)($$1 + 6 T - 5 T^{2} + 246 T^{3} + 1681 T^{4}$$)($$1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4}$$)($$1 - 9 T + 40 T^{2} - 369 T^{3} + 1681 T^{4}$$)($$1 + 11 T + 80 T^{2} + 451 T^{3} + 1681 T^{4}$$)($$( 1 + T - 40 T^{2} + 41 T^{3} + 1681 T^{4} )^{2}$$)($$( 1 - 9 T + 40 T^{2} - 369 T^{3} + 1681 T^{4} )^{2}$$)($$( 1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4} )^{2}$$)($$( 1 + T - 40 T^{2} + 41 T^{3} + 1681 T^{4} )^{2}$$)($$( 1 - 9 T + 40 T^{2} - 369 T^{3} + 1681 T^{4} )^{2}$$)
$43$ ($$1 + 4 T - 27 T^{2} + 172 T^{3} + 1849 T^{4}$$)($$( 1 - 13 T + 43 T^{2} )( 1 + 5 T + 43 T^{2} )$$)($$1 + T - 42 T^{2} + 43 T^{3} + 1849 T^{4}$$)($$1 + T - 42 T^{2} + 43 T^{3} + 1849 T^{4}$$)($$1 - T - 42 T^{2} - 43 T^{3} + 1849 T^{4}$$)($$1 + T - 42 T^{2} + 43 T^{3} + 1849 T^{4}$$)($$1 - 11 T + 78 T^{2} - 473 T^{3} + 1849 T^{4}$$)($$1 - T - 42 T^{2} - 43 T^{3} + 1849 T^{4}$$)($$1 - 4 T - 27 T^{2} - 172 T^{3} + 1849 T^{4}$$)($$1 + 10 T - 5 T^{2} + 190 T^{3} + 4876 T^{4} + 8170 T^{5} - 9245 T^{6} + 795070 T^{7} + 3418801 T^{8}$$)($$1 + 10 T - 5 T^{2} + 190 T^{3} + 4876 T^{4} + 8170 T^{5} - 9245 T^{6} + 795070 T^{7} + 3418801 T^{8}$$)($$1 + 17 T + 139 T^{2} + 1088 T^{3} + 8224 T^{4} + 46784 T^{5} + 257011 T^{6} + 1351619 T^{7} + 3418801 T^{8}$$)($$1 - 10 T - 5 T^{2} - 190 T^{3} + 4876 T^{4} - 8170 T^{5} - 9245 T^{6} - 795070 T^{7} + 3418801 T^{8}$$)($$1 - 10 T - 5 T^{2} - 190 T^{3} + 4876 T^{4} - 8170 T^{5} - 9245 T^{6} - 795070 T^{7} + 3418801 T^{8}$$)
$47$ ($$1 - 7 T + 2 T^{2} - 329 T^{3} + 2209 T^{4}$$)($$1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4}$$)($$1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4}$$)($$1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4}$$)($$1 - 12 T + 97 T^{2} - 564 T^{3} + 2209 T^{4}$$)($$1 + 12 T + 97 T^{2} + 564 T^{3} + 2209 T^{4}$$)($$1 - 47 T^{2} + 2209 T^{4}$$)($$1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4}$$)($$1 + 7 T + 2 T^{2} + 329 T^{3} + 2209 T^{4}$$)($$1 + 4 T - 76 T^{2} - 8 T^{3} + 5503 T^{4} - 376 T^{5} - 167884 T^{6} + 415292 T^{7} + 4879681 T^{8}$$)($$1 + 12 T + 38 T^{2} + 144 T^{3} + 2259 T^{4} + 6768 T^{5} + 83942 T^{6} + 1245876 T^{7} + 4879681 T^{8}$$)($$1 - 9 T - 25 T^{2} - 108 T^{3} + 5220 T^{4} - 5076 T^{5} - 55225 T^{6} - 934407 T^{7} + 4879681 T^{8}$$)($$1 - 4 T - 76 T^{2} + 8 T^{3} + 5503 T^{4} + 376 T^{5} - 167884 T^{6} - 415292 T^{7} + 4879681 T^{8}$$)($$1 - 12 T + 38 T^{2} - 144 T^{3} + 2259 T^{4} - 6768 T^{5} + 83942 T^{6} - 1245876 T^{7} + 4879681 T^{8}$$)
$53$ ($$( 1 + 53 T^{2} )^{2}$$)($$( 1 - 6 T + 53 T^{2} )^{2}$$)($$( 1 - 12 T + 53 T^{2} )^{2}$$)($$( 1 + 12 T + 53 T^{2} )^{2}$$)($$( 1 + 53 T^{2} )^{2}$$)($$( 1 + 53 T^{2} )^{2}$$)($$( 1 - 6 T + 53 T^{2} )^{2}$$)($$( 1 - 12 T + 53 T^{2} )^{2}$$)($$( 1 + 53 T^{2} )^{2}$$)($$( 1 + 12 T + 136 T^{2} + 636 T^{3} + 2809 T^{4} )^{2}$$)($$( 1 - 12 T + 136 T^{2} - 636 T^{3} + 2809 T^{4} )^{2}$$)($$( 1 - 26 T^{2} + 2809 T^{4} )^{2}$$)($$( 1 - 12 T + 136 T^{2} - 636 T^{3} + 2809 T^{4} )^{2}$$)($$( 1 + 12 T + 136 T^{2} + 636 T^{3} + 2809 T^{4} )^{2}$$)
$59$ ($$1 + 4 T - 43 T^{2} + 236 T^{3} + 3481 T^{4}$$)($$1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4}$$)($$1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4}$$)($$1 + 9 T + 22 T^{2} + 531 T^{3} + 3481 T^{4}$$)($$1 + 9 T + 22 T^{2} + 531 T^{3} + 3481 T^{4}$$)($$1 + 9 T + 22 T^{2} + 531 T^{3} + 3481 T^{4}$$)($$1 + 3 T - 50 T^{2} + 177 T^{3} + 3481 T^{4}$$)($$1 + 9 T + 22 T^{2} + 531 T^{3} + 3481 T^{4}$$)($$1 + 4 T - 43 T^{2} + 236 T^{3} + 3481 T^{4}$$)($$1 - 2 T + 35 T^{2} + 298 T^{3} - 2756 T^{4} + 17582 T^{5} + 121835 T^{6} - 410758 T^{7} + 12117361 T^{8}$$)($$1 + 6 T - 85 T^{2} + 18 T^{3} + 9036 T^{4} + 1062 T^{5} - 295885 T^{6} + 1232274 T^{7} + 12117361 T^{8}$$)($$1 - 3 T - 103 T^{2} + 18 T^{3} + 8532 T^{4} + 1062 T^{5} - 358543 T^{6} - 616137 T^{7} + 12117361 T^{8}$$)($$1 - 2 T + 35 T^{2} + 298 T^{3} - 2756 T^{4} + 17582 T^{5} + 121835 T^{6} - 410758 T^{7} + 12117361 T^{8}$$)($$1 + 6 T - 85 T^{2} + 18 T^{3} + 9036 T^{4} + 1062 T^{5} - 295885 T^{6} + 1232274 T^{7} + 12117361 T^{8}$$)
$61$ ($$1 - 7 T - 12 T^{2} - 427 T^{3} + 3721 T^{4}$$)($$( 1 - 14 T + 61 T^{2} )( 1 + T + 61 T^{2} )$$)($$1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4}$$)($$1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4}$$)($$1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4}$$)($$1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4}$$)($$1 - 10 T + 39 T^{2} - 610 T^{3} + 3721 T^{4}$$)($$1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4}$$)($$1 - 7 T - 12 T^{2} - 427 T^{3} + 3721 T^{4}$$)($$1 + 4 T - 104 T^{2} - 8 T^{3} + 9703 T^{4} - 488 T^{5} - 386984 T^{6} + 907924 T^{7} + 13845841 T^{8}$$)($$( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} )^{2}$$)($$1 + T - 47 T^{2} - 74 T^{3} - 1478 T^{4} - 4514 T^{5} - 174887 T^{6} + 226981 T^{7} + 13845841 T^{8}$$)($$1 + 4 T - 104 T^{2} - 8 T^{3} + 9703 T^{4} - 488 T^{5} - 386984 T^{6} + 907924 T^{7} + 13845841 T^{8}$$)($$( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} )^{2}$$)
$67$ ($$( 1 - 5 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} )$$)($$1 + 13 T + 102 T^{2} + 871 T^{3} + 4489 T^{4}$$)($$( 1 - 16 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} )$$)($$1 + 13 T + 102 T^{2} + 871 T^{3} + 4489 T^{4}$$)($$1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4}$$)($$1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4}$$)($$( 1 - 16 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} )$$)($$1 - 13 T + 102 T^{2} - 871 T^{3} + 4489 T^{4}$$)($$( 1 - 16 T + 67 T^{2} )( 1 + 5 T + 67 T^{2} )$$)($$1 + 14 T + 19 T^{2} + 602 T^{3} + 13708 T^{4} + 40334 T^{5} + 85291 T^{6} + 4210682 T^{7} + 20151121 T^{8}$$)($$( 1 - 14 T + 67 T^{2} )^{2}( 1 + 14 T + 129 T^{2} + 938 T^{3} + 4489 T^{4} )$$)($$( 1 + 7 T - 18 T^{2} + 469 T^{3} + 4489 T^{4} )^{2}$$)($$1 - 14 T + 19 T^{2} - 602 T^{3} + 13708 T^{4} - 40334 T^{5} + 85291 T^{6} - 4210682 T^{7} + 20151121 T^{8}$$)($$( 1 + 14 T + 67 T^{2} )^{2}( 1 - 14 T + 129 T^{2} - 938 T^{3} + 4489 T^{4} )$$)
$71$ ($$( 1 - 6 T + 71 T^{2} )^{2}$$)($$( 1 - 6 T + 71 T^{2} )^{2}$$)($$( 1 - 12 T + 71 T^{2} )^{2}$$)($$( 1 + 6 T + 71 T^{2} )^{2}$$)($$( 1 - 6 T + 71 T^{2} )^{2}$$)($$( 1 - 6 T + 71 T^{2} )^{2}$$)($$( 1 + 6 T + 71 T^{2} )^{2}$$)($$( 1 + 6 T + 71 T^{2} )^{2}$$)($$( 1 - 6 T + 71 T^{2} )^{2}$$)($$( 1 + 136 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 + 12 T + 124 T^{2} + 852 T^{3} + 5041 T^{4} )^{2}$$)($$( 1 - 6 T + 71 T^{2} )^{4}$$)($$( 1 + 136 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 + 12 T + 124 T^{2} + 852 T^{3} + 5041 T^{4} )^{2}$$)
$73$ ($$( 1 - 4 T + 73 T^{2} )^{2}$$)($$( 1 - 4 T + 73 T^{2} )^{2}$$)($$( 1 + 11 T + 73 T^{2} )^{2}$$)($$( 1 - T + 73 T^{2} )^{2}$$)($$( 1 - 14 T + 73 T^{2} )^{2}$$)($$( 1 + 14 T + 73 T^{2} )^{2}$$)($$( 1 - 7 T + 73 T^{2} )^{2}$$)($$( 1 + T + 73 T^{2} )^{2}$$)($$( 1 + 4 T + 73 T^{2} )^{2}$$)($$( 1 - 10 T + 75 T^{2} - 730 T^{3} + 5329 T^{4} )^{2}$$)($$( 1 + T + 73 T^{2} )^{4}$$)($$( 1 - 11 T + 102 T^{2} - 803 T^{3} + 5329 T^{4} )^{2}$$)($$( 1 + 10 T + 75 T^{2} + 730 T^{3} + 5329 T^{4} )^{2}$$)($$( 1 - T + 73 T^{2} )^{4}$$)
$79$ ($$1 - 12 T + 65 T^{2} - 948 T^{3} + 6241 T^{4}$$)($$1 - 10 T + 21 T^{2} - 790 T^{3} + 6241 T^{4}$$)($$( 1 - 17 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} )$$)($$1 + 2 T - 75 T^{2} + 158 T^{3} + 6241 T^{4}$$)($$1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4}$$)($$1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4}$$)($$1 + 14 T + 117 T^{2} + 1106 T^{3} + 6241 T^{4}$$)($$1 + 2 T - 75 T^{2} + 158 T^{3} + 6241 T^{4}$$)($$1 - 12 T + 65 T^{2} - 948 T^{3} + 6241 T^{4}$$)($$1 - 104 T^{2} + 4575 T^{4} - 649064 T^{6} + 38950081 T^{8}$$)($$1 + 4 T + 70 T^{2} - 848 T^{3} - 4589 T^{4} - 66992 T^{5} + 436870 T^{6} + 1972156 T^{7} + 38950081 T^{8}$$)($$( 1 + 2 T - 75 T^{2} + 158 T^{3} + 6241 T^{4} )^{2}$$)($$1 - 104 T^{2} + 4575 T^{4} - 649064 T^{6} + 38950081 T^{8}$$)($$1 + 4 T + 70 T^{2} - 848 T^{3} - 4589 T^{4} - 66992 T^{5} + 436870 T^{6} + 1972156 T^{7} + 38950081 T^{8}$$)
$83$ ($$1 - 11 T + 38 T^{2} - 913 T^{3} + 6889 T^{4}$$)($$1 - 9 T - 2 T^{2} - 747 T^{3} + 6889 T^{4}$$)($$1 + 12 T + 61 T^{2} + 996 T^{3} + 6889 T^{4}$$)($$1 + 9 T - 2 T^{2} + 747 T^{3} + 6889 T^{4}$$)($$1 + 9 T - 2 T^{2} + 747 T^{3} + 6889 T^{4}$$)($$1 - 9 T - 2 T^{2} - 747 T^{3} + 6889 T^{4}$$)($$1 + 12 T + 61 T^{2} + 996 T^{3} + 6889 T^{4}$$)($$1 - 9 T - 2 T^{2} - 747 T^{3} + 6889 T^{4}$$)($$1 + 11 T + 38 T^{2} + 913 T^{3} + 6889 T^{4}$$)($$( 1 + 4 T - 67 T^{2} + 332 T^{3} + 6889 T^{4} )^{2}$$)($$1 + 6 T - 133 T^{2} + 18 T^{3} + 18684 T^{4} + 1494 T^{5} - 916237 T^{6} + 3430722 T^{7} + 47458321 T^{8}$$)($$1 + 9 T - 97 T^{2} + 108 T^{3} + 18072 T^{4} + 8964 T^{5} - 668233 T^{6} + 5146083 T^{7} + 47458321 T^{8}$$)($$( 1 - 4 T - 67 T^{2} - 332 T^{3} + 6889 T^{4} )^{2}$$)($$1 - 6 T - 133 T^{2} - 18 T^{3} + 18684 T^{4} - 1494 T^{5} - 916237 T^{6} - 3430722 T^{7} + 47458321 T^{8}$$)
$89$ ($$( 1 + T + 89 T^{2} )^{2}$$)($$( 1 + 9 T + 89 T^{2} )^{2}$$)($$( 1 + 6 T + 89 T^{2} )^{2}$$)($$( 1 + 15 T + 89 T^{2} )^{2}$$)($$( 1 - 9 T + 89 T^{2} )^{2}$$)($$( 1 - 9 T + 89 T^{2} )^{2}$$)($$( 1 + 6 T + 89 T^{2} )^{2}$$)($$( 1 + 15 T + 89 T^{2} )^{2}$$)($$( 1 + T + 89 T^{2} )^{2}$$)($$( 1 + 16 T + 218 T^{2} + 1424 T^{3} + 7921 T^{4} )^{2}$$)($$( 1 - 9 T + 89 T^{2} )^{4}$$)($$( 1 + 15 T + 160 T^{2} + 1335 T^{3} + 7921 T^{4} )^{2}$$)($$( 1 + 16 T + 218 T^{2} + 1424 T^{3} + 7921 T^{4} )^{2}$$)($$( 1 - 9 T + 89 T^{2} )^{4}$$)
$97$ ($$1 + 8 T - 33 T^{2} + 776 T^{3} + 9409 T^{4}$$)($$1 - 2 T - 93 T^{2} - 194 T^{3} + 9409 T^{4}$$)($$( 1 - 19 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} )$$)($$1 - 17 T + 192 T^{2} - 1649 T^{3} + 9409 T^{4}$$)($$1 - 7 T - 48 T^{2} - 679 T^{3} + 9409 T^{4}$$)($$1 + 7 T - 48 T^{2} + 679 T^{3} + 9409 T^{4}$$)($$1 - 11 T + 24 T^{2} - 1067 T^{3} + 9409 T^{4}$$)($$1 + 17 T + 192 T^{2} + 1649 T^{3} + 9409 T^{4}$$)($$1 - 8 T - 33 T^{2} - 776 T^{3} + 9409 T^{4}$$)($$( 1 + 13 T + 72 T^{2} + 1261 T^{3} + 9409 T^{4} )^{2}$$)($$1 - 2 T - 95 T^{2} + 190 T^{3} + 4 T^{4} + 18430 T^{5} - 893855 T^{6} - 1825346 T^{7} + 88529281 T^{8}$$)($$1 + 11 T - 95 T^{2} + 242 T^{3} + 25510 T^{4} + 23474 T^{5} - 893855 T^{6} + 10039403 T^{7} + 88529281 T^{8}$$)($$( 1 - 13 T + 72 T^{2} - 1261 T^{3} + 9409 T^{4} )^{2}$$)($$1 + 2 T - 95 T^{2} - 190 T^{3} + 4 T^{4} - 18430 T^{5} - 893855 T^{6} + 1825346 T^{7} + 88529281 T^{8}$$)