# Properties

 Label 1575.2.d Level 1575 Weight 2 Character orbit d Rep. character $$\chi_{1575}(1324,\cdot)$$ Character field $$\Q$$ Dimension 46 Newform subspaces 12 Sturm bound 480 Trace bound 11

# Related objects

## Defining parameters

 Level: $$N$$ = $$1575 = 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 1575.d (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$5$$ Character field: $$\Q$$ Newform subspaces: $$12$$ Sturm bound: $$480$$ Trace bound: $$11$$ Distinguishing $$T_p$$: $$2$$, $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(1575, [\chi])$$.

Total New Old
Modular forms 264 46 218
Cusp forms 216 46 170
Eisenstein series 48 0 48

## Trace form

 $$46q - 56q^{4} + O(q^{10})$$ $$46q - 56q^{4} - 4q^{11} + 4q^{14} + 60q^{16} - 8q^{19} + 64q^{26} - 12q^{29} + 44q^{31} - 20q^{34} + 40q^{41} - 62q^{44} + 14q^{46} - 46q^{49} - 6q^{56} - 4q^{59} + 24q^{61} - 42q^{64} - 24q^{71} + 38q^{74} - 20q^{76} - 12q^{79} - 62q^{86} - 8q^{89} - 4q^{91} - 4q^{94} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(1575, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
1575.2.d.a $$2$$ $$12.576$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}+q^{4}-iq^{7}+3iq^{8}-4q^{11}+\cdots$$
1575.2.d.b $$2$$ $$12.576$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}+q^{4}-iq^{7}+3iq^{8}-6iq^{13}+\cdots$$
1575.2.d.c $$2$$ $$12.576$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+2q^{4}-iq^{7}+3q^{11}+5iq^{13}+4q^{16}+\cdots$$
1575.2.d.d $$4$$ $$12.576$$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{2}q^{2}-3q^{4}+\beta _{1}q^{7}+\beta _{2}q^{8}+(-2+\cdots)q^{11}+\cdots$$
1575.2.d.e $$4$$ $$12.576$$ $$\Q(i, \sqrt{17})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+(-3+\beta _{3})q^{4}+\beta _{2}q^{7}+(-\beta _{1}+\cdots)q^{8}+\cdots$$
1575.2.d.f $$4$$ $$12.576$$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}-\beta _{3})q^{2}+3\beta _{2}q^{4}-\beta _{3}q^{7}+(-4\beta _{1}+\cdots)q^{8}+\cdots$$
1575.2.d.g $$4$$ $$12.576$$ $$\Q(i, \sqrt{13})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+(-2+\beta _{3})q^{4}+\beta _{2}q^{7}+3\beta _{2}q^{8}+\cdots$$
1575.2.d.h $$4$$ $$12.576$$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\zeta_{8}+\zeta_{8}^{2})q^{2}+(-1-2\zeta_{8}^{3})q^{4}+\cdots$$
1575.2.d.i $$4$$ $$12.576$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{12}^{2}q^{2}-q^{4}-\zeta_{12}q^{7}+\zeta_{12}^{2}q^{8}+\cdots$$
1575.2.d.j $$4$$ $$12.576$$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\zeta_{8}+\zeta_{8}^{2})q^{2}+(-1-2\zeta_{8}^{3})q^{4}+\cdots$$
1575.2.d.k $$4$$ $$12.576$$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+\beta _{3}q^{7}+(2\beta _{1}+\cdots)q^{8}+\cdots$$
1575.2.d.l $$8$$ $$12.576$$ 8.0.$$\cdots$$.2 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{2}q^{2}+(-2+\beta _{6})q^{4}-\beta _{4}q^{7}+(-2\beta _{2}+\cdots)q^{8}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(1575, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(1575, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(35, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(75, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(105, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(175, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(225, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(315, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(525, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 - 3 T^{2} + 4 T^{4}$$)($$1 - 3 T^{2} + 4 T^{4}$$)($$( 1 - 2 T^{2} )^{2}$$)($$( 1 + T^{2} + 4 T^{4} )^{2}$$)($$1 + T^{2} + 4 T^{4} + 4 T^{6} + 16 T^{8}$$)($$1 - T^{2} - 3 T^{4} - 4 T^{6} + 16 T^{8}$$)($$1 - T^{2} + 5 T^{4} - 4 T^{6} + 16 T^{8}$$)($$1 - 2 T^{2} + T^{4} - 8 T^{6} + 16 T^{8}$$)($$( 1 - T^{2} + 4 T^{4} )^{2}$$)($$1 - 2 T^{2} + T^{4} - 8 T^{6} + 16 T^{8}$$)($$1 - 5 T^{2} + 13 T^{4} - 20 T^{6} + 16 T^{8}$$)($$( 1 + T^{2} + T^{4} + 4 T^{6} + 16 T^{8} )^{2}$$)
$3$ 1
$5$ 1
$7$ ($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{4}$$)
$11$ ($$( 1 + 4 T + 11 T^{2} )^{2}$$)($$( 1 + 11 T^{2} )^{2}$$)($$( 1 - 3 T + 11 T^{2} )^{2}$$)($$( 1 + 4 T + 6 T^{2} + 44 T^{3} + 121 T^{4} )^{2}$$)($$( 1 + T + 18 T^{2} + 11 T^{3} + 121 T^{4} )^{2}$$)($$( 1 - 2 T + 3 T^{2} - 22 T^{3} + 121 T^{4} )^{2}$$)($$( 1 - 3 T + 11 T^{2} )^{4}$$)($$( 1 + 4 T + 18 T^{2} + 44 T^{3} + 121 T^{4} )^{2}$$)($$( 1 + 10 T^{2} + 121 T^{4} )^{2}$$)($$( 1 - 4 T + 18 T^{2} - 44 T^{3} + 121 T^{4} )^{2}$$)($$( 1 + 4 T + 21 T^{2} + 44 T^{3} + 121 T^{4} )^{2}$$)($$( 1 + 2 T^{2} + 127 T^{4} + 242 T^{6} + 14641 T^{8} )^{2}$$)
$13$ ($$1 - 22 T^{2} + 169 T^{4}$$)($$( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )$$)($$1 - T^{2} + 169 T^{4}$$)($$( 1 - 6 T^{2} + 169 T^{4} )^{2}$$)($$1 - 31 T^{2} + 472 T^{4} - 5239 T^{6} + 28561 T^{8}$$)($$1 - 24 T^{2} + 302 T^{4} - 4056 T^{6} + 28561 T^{8}$$)($$1 - 24 T^{2} + 430 T^{4} - 4056 T^{6} + 28561 T^{8}$$)($$1 - 28 T^{2} + 406 T^{4} - 4732 T^{6} + 28561 T^{8}$$)($$( 1 - 22 T^{2} + 169 T^{4} )^{2}$$)($$1 - 28 T^{2} + 406 T^{4} - 4732 T^{6} + 28561 T^{8}$$)($$1 - 40 T^{2} + 718 T^{4} - 6760 T^{6} + 28561 T^{8}$$)($$( 1 + 8 T^{2} + 238 T^{4} + 1352 T^{6} + 28561 T^{8} )^{2}$$)
$17$ ($$1 + 2 T^{2} + 289 T^{4}$$)($$( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} )$$)($$1 - 25 T^{2} + 289 T^{4}$$)($$( 1 - 8 T + 17 T^{2} )^{2}( 1 + 8 T + 17 T^{2} )^{2}$$)($$1 - 47 T^{2} + 1024 T^{4} - 13583 T^{6} + 83521 T^{8}$$)($$1 + 24 T^{2} + 542 T^{4} + 6936 T^{6} + 83521 T^{8}$$)($$1 - 40 T^{2} + 926 T^{4} - 11560 T^{6} + 83521 T^{8}$$)($$1 + 4 T^{2} + 70 T^{4} + 1156 T^{6} + 83521 T^{8}$$)($$( 1 - 22 T^{2} + 289 T^{4} )^{2}$$)($$1 + 4 T^{2} + 70 T^{4} + 1156 T^{6} + 83521 T^{8}$$)($$1 - 20 T^{2} + 358 T^{4} - 5780 T^{6} + 83521 T^{8}$$)($$( 1 - 32 T^{2} + 718 T^{4} - 9248 T^{6} + 83521 T^{8} )^{2}$$)
$19$ ($$( 1 + 4 T + 19 T^{2} )^{2}$$)($$( 1 - 8 T + 19 T^{2} )^{2}$$)($$( 1 + 2 T + 19 T^{2} )^{2}$$)($$( 1 + 4 T + 22 T^{2} + 76 T^{3} + 361 T^{4} )^{2}$$)($$( 1 - 6 T + 30 T^{2} - 114 T^{3} + 361 T^{4} )^{2}$$)($$( 1 - 2 T + 34 T^{2} - 38 T^{3} + 361 T^{4} )^{2}$$)($$( 1 + 6 T + 34 T^{2} + 114 T^{3} + 361 T^{4} )^{2}$$)($$( 1 + 30 T^{2} + 361 T^{4} )^{2}$$)($$( 1 - 4 T + 19 T^{2} )^{4}$$)($$( 1 + 30 T^{2} + 361 T^{4} )^{2}$$)($$( 1 + 18 T^{2} + 361 T^{4} )^{2}$$)($$( 1 + 6 T + 18 T^{2} + 114 T^{3} + 361 T^{4} )^{4}$$)
$23$ ($$( 1 - 23 T^{2} )^{2}$$)($$1 + 18 T^{2} + 529 T^{4}$$)($$1 - 10 T^{2} + 529 T^{4}$$)($$( 1 - 30 T^{2} + 529 T^{4} )^{2}$$)($$1 - 56 T^{2} + 1774 T^{4} - 29624 T^{6} + 279841 T^{8}$$)($$( 1 - 21 T^{2} + 529 T^{4} )^{2}$$)($$1 + 14 T^{2} + 899 T^{4} + 7406 T^{6} + 279841 T^{8}$$)($$1 - 20 T^{2} + 646 T^{4} - 10580 T^{6} + 279841 T^{8}$$)($$( 1 - 34 T^{2} + 529 T^{4} )^{2}$$)($$1 - 20 T^{2} + 646 T^{4} - 10580 T^{6} + 279841 T^{8}$$)($$1 - 50 T^{2} + 1363 T^{4} - 26450 T^{6} + 279841 T^{8}$$)($$( 1 - 50 T^{2} + 1567 T^{4} - 26450 T^{6} + 279841 T^{8} )^{2}$$)
$29$ ($$( 1 + 2 T + 29 T^{2} )^{2}$$)($$( 1 + 2 T + 29 T^{2} )^{2}$$)($$( 1 - 3 T + 29 T^{2} )^{2}$$)($$( 1 + 2 T + 29 T^{2} )^{4}$$)($$( 1 - T + 20 T^{2} - 29 T^{3} + 841 T^{4} )^{2}$$)($$( 1 + 4 T + 17 T^{2} + 116 T^{3} + 841 T^{4} )^{2}$$)($$( 1 + 8 T + 61 T^{2} + 232 T^{3} + 841 T^{4} )^{2}$$)($$( 1 - 8 T + 29 T^{2} )^{4}$$)($$( 1 + 29 T^{2} )^{4}$$)($$( 1 + 8 T + 29 T^{2} )^{4}$$)($$( 1 - 5 T + 29 T^{2} )^{4}$$)($$( 1 + 42 T^{2} + 1079 T^{4} + 35322 T^{6} + 707281 T^{8} )^{2}$$)
$31$ ($$( 1 + 31 T^{2} )^{2}$$)($$( 1 - 4 T + 31 T^{2} )^{2}$$)($$( 1 + 4 T + 31 T^{2} )^{2}$$)($$( 1 - 12 T + 78 T^{2} - 372 T^{3} + 961 T^{4} )^{2}$$)($$( 1 + 31 T^{2} )^{4}$$)($$( 1 + 42 T^{2} + 961 T^{4} )^{2}$$)($$( 1 + 10 T^{2} + 961 T^{4} )^{2}$$)($$( 1 - 10 T^{2} + 961 T^{4} )^{2}$$)($$( 1 + 4 T + 31 T^{2} )^{4}$$)($$( 1 - 10 T^{2} + 961 T^{4} )^{2}$$)($$( 1 + 6 T + 26 T^{2} + 186 T^{3} + 961 T^{4} )^{2}$$)($$( 1 - 6 T + 31 T^{2} )^{8}$$)
$37$ ($$1 - 38 T^{2} + 1369 T^{4}$$)($$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$)($$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$)($$1 + 20 T^{2} + 1558 T^{4} + 27380 T^{6} + 1874161 T^{8}$$)($$( 1 - 38 T^{2} + 1369 T^{4} )^{2}$$)($$1 - 106 T^{2} + 5467 T^{4} - 145114 T^{6} + 1874161 T^{8}$$)($$1 - 26 T^{2} + 1035 T^{4} - 35594 T^{6} + 1874161 T^{8}$$)($$( 1 - 38 T^{2} + 1369 T^{4} )^{2}$$)($$( 1 - 12 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2}$$)($$( 1 - 38 T^{2} + 1369 T^{4} )^{2}$$)($$( 1 - 65 T^{2} + 1369 T^{4} )^{2}$$)($$( 1 - 65 T^{2} + 1369 T^{4} )^{4}$$)
$41$ ($$( 1 + 2 T + 41 T^{2} )^{2}$$)($$( 1 - 6 T + 41 T^{2} )^{2}$$)($$( 1 - 12 T + 41 T^{2} )^{2}$$)($$( 1 - 2 T + 41 T^{2} )^{4}$$)($$( 1 + 2 T + 66 T^{2} + 82 T^{3} + 1681 T^{4} )^{2}$$)($$( 1 - 8 T + 41 T^{2} )^{4}$$)($$( 1 + 41 T^{2} )^{4}$$)($$( 1 - 4 T + 54 T^{2} - 164 T^{3} + 1681 T^{4} )^{2}$$)($$( 1 - 26 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 + 4 T + 54 T^{2} + 164 T^{3} + 1681 T^{4} )^{2}$$)($$( 1 + 14 T + 126 T^{2} + 574 T^{3} + 1681 T^{4} )^{2}$$)($$( 1 + 20 T^{2} + 1606 T^{4} + 33620 T^{6} + 2825761 T^{8} )^{2}$$)
$43$ ($$1 - 70 T^{2} + 1849 T^{4}$$)($$1 - 70 T^{2} + 1849 T^{4}$$)($$1 + 14 T^{2} + 1849 T^{4}$$)($$( 1 - 6 T^{2} + 1849 T^{4} )^{2}$$)($$1 - 88 T^{2} + 3934 T^{4} - 162712 T^{6} + 3418801 T^{8}$$)($$1 - 90 T^{2} + 5003 T^{4} - 166410 T^{6} + 3418801 T^{8}$$)($$1 - 74 T^{2} + 3195 T^{4} - 136826 T^{6} + 3418801 T^{8}$$)($$1 - 76 T^{2} + 3094 T^{4} - 140524 T^{6} + 3418801 T^{8}$$)($$( 1 - 70 T^{2} + 1849 T^{4} )^{2}$$)($$1 - 76 T^{2} + 3094 T^{4} - 140524 T^{6} + 3418801 T^{8}$$)($$1 - 130 T^{2} + 7603 T^{4} - 240370 T^{6} + 3418801 T^{8}$$)($$( 1 - 82 T^{2} + 3523 T^{4} - 151618 T^{6} + 3418801 T^{8} )^{2}$$)
$47$ ($$( 1 - 47 T^{2} )^{2}$$)($$1 - 30 T^{2} + 2209 T^{4}$$)($$1 - 13 T^{2} + 2209 T^{4}$$)($$1 + 4 T^{2} - 698 T^{4} + 8836 T^{6} + 4879681 T^{8}$$)($$1 - 99 T^{2} + 5912 T^{4} - 218691 T^{6} + 4879681 T^{8}$$)($$1 - 128 T^{2} + 8014 T^{4} - 282752 T^{6} + 4879681 T^{8}$$)($$1 - 64 T^{2} + 2894 T^{4} - 141376 T^{6} + 4879681 T^{8}$$)($$( 1 - 78 T^{2} + 2209 T^{4} )^{2}$$)($$( 1 - 46 T^{2} + 2209 T^{4} )^{2}$$)($$( 1 - 78 T^{2} + 2209 T^{4} )^{2}$$)($$( 1 - 90 T^{2} + 2209 T^{4} )^{2}$$)($$( 1 + 24 T^{2} + 3518 T^{4} + 53016 T^{6} + 4879681 T^{8} )^{2}$$)
$53$ ($$1 - 70 T^{2} + 2809 T^{4}$$)($$1 - 6 T^{2} + 2809 T^{4}$$)($$1 + 38 T^{2} + 2809 T^{4}$$)($$1 - 44 T^{2} + 982 T^{4} - 123596 T^{6} + 7890481 T^{8}$$)($$1 - 176 T^{2} + 13294 T^{4} - 494384 T^{6} + 7890481 T^{8}$$)($$1 - 140 T^{2} + 9238 T^{4} - 393260 T^{6} + 7890481 T^{8}$$)($$1 - 76 T^{2} + 3734 T^{4} - 213484 T^{6} + 7890481 T^{8}$$)($$1 - 68 T^{2} + 4726 T^{4} - 191012 T^{6} + 7890481 T^{8}$$)($$( 1 - 58 T^{2} + 2809 T^{4} )^{2}$$)($$1 - 68 T^{2} + 4726 T^{4} - 191012 T^{6} + 7890481 T^{8}$$)($$1 - 140 T^{2} + 9238 T^{4} - 393260 T^{6} + 7890481 T^{8}$$)($$( 1 - 68 T^{2} + 4918 T^{4} - 191012 T^{6} + 7890481 T^{8} )^{2}$$)
$59$ ($$( 1 - 12 T + 59 T^{2} )^{2}$$)($$( 1 - 4 T + 59 T^{2} )^{2}$$)($$( 1 + 59 T^{2} )^{2}$$)($$( 1 + 38 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 + 4 T + 59 T^{2} )^{4}$$)($$( 1 + 6 T + 122 T^{2} + 354 T^{3} + 3481 T^{4} )^{2}$$)($$( 1 + 14 T + 154 T^{2} + 826 T^{3} + 3481 T^{4} )^{2}$$)($$( 1 + 4 T + 59 T^{2} )^{4}$$)($$( 1 + 70 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 - 4 T + 59 T^{2} )^{4}$$)($$( 1 - 10 T + 98 T^{2} - 590 T^{3} + 3481 T^{4} )^{2}$$)($$( 1 + 24 T^{2} + 6062 T^{4} + 83544 T^{6} + 12117361 T^{8} )^{2}$$)
$61$ ($$( 1 + 2 T + 61 T^{2} )^{2}$$)($$( 1 + 2 T + 61 T^{2} )^{2}$$)($$( 1 - 8 T + 61 T^{2} )^{2}$$)($$( 1 + 2 T + 61 T^{2} )^{4}$$)($$( 1 - 6 T - 22 T^{2} - 366 T^{3} + 3721 T^{4} )^{2}$$)($$( 1 + 4 T - 54 T^{2} + 244 T^{3} + 3721 T^{4} )^{2}$$)($$( 1 - 12 T + 106 T^{2} - 732 T^{3} + 3721 T^{4} )^{2}$$)($$( 1 - 6 T + 61 T^{2} )^{4}$$)($$( 1 + 10 T + 61 T^{2} )^{4}$$)($$( 1 - 6 T + 61 T^{2} )^{4}$$)($$( 1 + 6 T + 86 T^{2} + 366 T^{3} + 3721 T^{4} )^{2}$$)($$( 1 + 61 T^{2} )^{8}$$)
$67$ ($$1 - 118 T^{2} + 4489 T^{4}$$)($$1 - 118 T^{2} + 4489 T^{4}$$)($$1 - 118 T^{2} + 4489 T^{4}$$)($$( 1 - 118 T^{2} + 4489 T^{4} )^{2}$$)($$1 - 124 T^{2} + 11734 T^{4} - 556636 T^{6} + 20151121 T^{8}$$)($$1 - 146 T^{2} + 11427 T^{4} - 655394 T^{6} + 20151121 T^{8}$$)($$1 + 46 T^{2} + 2019 T^{4} + 206494 T^{6} + 20151121 T^{8}$$)($$1 + 20 T^{2} + 886 T^{4} + 89780 T^{6} + 20151121 T^{8}$$)($$( 1 - 118 T^{2} + 4489 T^{4} )^{2}$$)($$1 + 20 T^{2} + 886 T^{4} + 89780 T^{6} + 20151121 T^{8}$$)($$1 - 250 T^{2} + 24523 T^{4} - 1122250 T^{6} + 20151121 T^{8}$$)($$( 1 - 178 T^{2} + 15043 T^{4} - 799042 T^{6} + 20151121 T^{8} )^{2}$$)
$71$ ($$( 1 + 71 T^{2} )^{2}$$)($$( 1 - 12 T + 71 T^{2} )^{2}$$)($$( 1 + 71 T^{2} )^{2}$$)($$( 1 + 20 T + 222 T^{2} + 1420 T^{3} + 5041 T^{4} )^{2}$$)($$( 1 + 8 T + 71 T^{2} )^{4}$$)($$( 1 - 10 T + 147 T^{2} - 710 T^{3} + 5041 T^{4} )^{2}$$)($$( 1 - 3 T + 71 T^{2} )^{4}$$)($$( 1 + 4 T + 74 T^{2} + 284 T^{3} + 5041 T^{4} )^{2}$$)($$( 1 + 34 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 - 4 T + 74 T^{2} - 284 T^{3} + 5041 T^{4} )^{2}$$)($$( 1 + 4 T + 101 T^{2} + 284 T^{3} + 5041 T^{4} )^{2}$$)($$( 1 + 90 T^{2} + 2711 T^{4} + 453690 T^{6} + 25411681 T^{8} )^{2}$$)
$73$ ($$( 1 - 16 T + 73 T^{2} )( 1 + 16 T + 73 T^{2} )$$)($$1 - 142 T^{2} + 5329 T^{4}$$)($$1 - 142 T^{2} + 5329 T^{4}$$)($$1 - 124 T^{2} + 9382 T^{4} - 660796 T^{6} + 28398241 T^{8}$$)($$1 - 124 T^{2} + 10150 T^{4} - 660796 T^{6} + 28398241 T^{8}$$)($$1 - 280 T^{2} + 30238 T^{4} - 1492120 T^{6} + 28398241 T^{8}$$)($$1 - 248 T^{2} + 25566 T^{4} - 1321592 T^{6} + 28398241 T^{8}$$)($$1 + 116 T^{2} + 10822 T^{4} + 618164 T^{6} + 28398241 T^{8}$$)($$( 1 + 50 T^{2} + 5329 T^{4} )^{2}$$)($$1 + 116 T^{2} + 10822 T^{4} + 618164 T^{6} + 28398241 T^{8}$$)($$1 - 40 T^{2} + 8638 T^{4} - 213160 T^{6} + 28398241 T^{8}$$)($$( 1 - 232 T^{2} + 23998 T^{4} - 1236328 T^{6} + 28398241 T^{8} )^{2}$$)
$79$ ($$( 1 - 16 T + 79 T^{2} )^{2}$$)($$( 1 + 8 T + 79 T^{2} )^{2}$$)($$( 1 - T + 79 T^{2} )^{2}$$)($$( 1 + 8 T + 94 T^{2} + 632 T^{3} + 6241 T^{4} )^{2}$$)($$( 1 - 9 T + 174 T^{2} - 711 T^{3} + 6241 T^{4} )^{2}$$)($$( 1 + 8 T + 169 T^{2} + 632 T^{3} + 6241 T^{4} )^{2}$$)($$( 1 - 8 T + 57 T^{2} - 632 T^{3} + 6241 T^{4} )^{2}$$)($$( 1 + 126 T^{2} + 6241 T^{4} )^{2}$$)($$( 1 + 8 T + 79 T^{2} )^{4}$$)($$( 1 + 126 T^{2} + 6241 T^{4} )^{2}$$)($$( 1 + 33 T^{2} + 6241 T^{4} )^{2}$$)($$( 1 + 129 T^{2} + 6241 T^{4} )^{4}$$)
$83$ ($$1 - 22 T^{2} + 6889 T^{4}$$)($$1 - 150 T^{2} + 6889 T^{4}$$)($$1 - 22 T^{2} + 6889 T^{4}$$)($$1 - 44 T^{2} - 6218 T^{4} - 303116 T^{6} + 47458321 T^{8}$$)($$( 1 - 150 T^{2} + 6889 T^{4} )^{2}$$)($$1 - 164 T^{2} + 15382 T^{4} - 1129796 T^{6} + 47458321 T^{8}$$)($$1 - 196 T^{2} + 20054 T^{4} - 1350244 T^{6} + 47458321 T^{8}$$)($$( 1 - 102 T^{2} + 6889 T^{4} )^{2}$$)($$( 1 - 83 T^{2} )^{4}$$)($$( 1 - 102 T^{2} + 6889 T^{4} )^{2}$$)($$1 - 240 T^{2} + 27998 T^{4} - 1653360 T^{6} + 47458321 T^{8}$$)($$( 1 - 36 T^{2} - 2602 T^{4} - 248004 T^{6} + 47458321 T^{8} )^{2}$$)
$89$ ($$( 1 + 14 T + 89 T^{2} )^{2}$$)($$( 1 + 6 T + 89 T^{2} )^{2}$$)($$( 1 + 12 T + 89 T^{2} )^{2}$$)($$( 1 + 2 T + 89 T^{2} )^{4}$$)($$( 1 - 6 T + 170 T^{2} - 534 T^{3} + 7921 T^{4} )^{2}$$)($$( 1 - 2 T + 134 T^{2} - 178 T^{3} + 7921 T^{4} )^{2}$$)($$( 1 + 6 T + 70 T^{2} + 534 T^{3} + 7921 T^{4} )^{2}$$)($$( 1 - 12 T + 86 T^{2} - 1068 T^{3} + 7921 T^{4} )^{2}$$)($$( 1 + 166 T^{2} + 7921 T^{4} )^{2}$$)($$( 1 + 12 T + 86 T^{2} + 1068 T^{3} + 7921 T^{4} )^{2}$$)($$( 1 - 30 T + 398 T^{2} - 2670 T^{3} + 7921 T^{4} )^{2}$$)($$( 1 + 160 T^{2} + 12846 T^{4} + 1267360 T^{6} + 62742241 T^{8} )^{2}$$)
$97$ ($$( 1 - 8 T + 97 T^{2} )( 1 + 8 T + 97 T^{2} )$$)($$( 1 - 8 T + 97 T^{2} )( 1 + 8 T + 97 T^{2} )$$)($$1 - 193 T^{2} + 9409 T^{4}$$)($$1 - 316 T^{2} + 42502 T^{4} - 2973244 T^{6} + 88529281 T^{8}$$)($$1 - 135 T^{2} + 14768 T^{4} - 1270215 T^{6} + 88529281 T^{8}$$)($$1 - 220 T^{2} + 25798 T^{4} - 2069980 T^{6} + 88529281 T^{8}$$)($$1 - 156 T^{2} + 11590 T^{4} - 1467804 T^{6} + 88529281 T^{8}$$)($$1 - 300 T^{2} + 40166 T^{4} - 2822700 T^{6} + 88529281 T^{8}$$)($$( 1 + 2 T^{2} + 9409 T^{4} )^{2}$$)($$1 - 300 T^{2} + 40166 T^{4} - 2822700 T^{6} + 88529281 T^{8}$$)($$1 - 360 T^{2} + 51038 T^{4} - 3387240 T^{6} + 88529281 T^{8}$$)($$( 1 - 78 T^{2} + 9409 T^{4} )^{4}$$)