# Properties

 Label 3150.2.g Level 3150 Weight 2 Character orbit g Rep. character $$\chi_{3150}(2899,\cdot)$$ Character field $$\Q$$ Dimension 44 Newform subspaces 22 Sturm bound 1440 Trace bound 26

# Related objects

## Defining parameters

 Level: $$N$$ = $$3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 3150.g (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$5$$ Character field: $$\Q$$ Newform subspaces: $$22$$ Sturm bound: $$1440$$ Trace bound: $$26$$ Distinguishing $$T_p$$: $$11$$, $$13$$, $$17$$, $$19$$, $$29$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(3150, [\chi])$$.

Total New Old
Modular forms 768 44 724
Cusp forms 672 44 628
Eisenstein series 96 0 96

## Trace form

 $$44q - 44q^{4} + O(q^{10})$$ $$44q - 44q^{4} - 4q^{11} - 4q^{14} + 44q^{16} - 16q^{19} - 12q^{26} + 16q^{29} - 40q^{31} + 28q^{34} - 44q^{41} + 4q^{44} - 16q^{46} - 44q^{49} + 4q^{56} + 20q^{59} + 36q^{61} - 44q^{64} + 24q^{71} + 16q^{74} + 16q^{76} - 56q^{79} + 56q^{86} - 36q^{89} + 20q^{91} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(3150, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
3150.2.g.a $$2$$ $$25.153$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-q^{4}-iq^{7}+iq^{8}-6q^{11}+\cdots$$
3150.2.g.b $$2$$ $$25.153$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-q^{4}-iq^{7}+iq^{8}-4q^{11}+\cdots$$
3150.2.g.c $$2$$ $$25.153$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-q^{4}+iq^{7}-iq^{8}-4q^{11}+\cdots$$
3150.2.g.d $$2$$ $$25.153$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-q^{4}+iq^{7}-iq^{8}-4q^{11}+\cdots$$
3150.2.g.e $$2$$ $$25.153$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-q^{4}-iq^{7}-iq^{8}-4q^{11}+\cdots$$
3150.2.g.f $$2$$ $$25.153$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-q^{4}-iq^{7}-iq^{8}-3q^{11}+\cdots$$
3150.2.g.g $$2$$ $$25.153$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-q^{4}+iq^{7}-iq^{8}-2q^{11}+\cdots$$
3150.2.g.h $$2$$ $$25.153$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-q^{4}+iq^{7}+iq^{8}-2q^{11}+\cdots$$
3150.2.g.i $$2$$ $$25.153$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-q^{4}-iq^{7}+iq^{8}+2iq^{13}+\cdots$$
3150.2.g.j $$2$$ $$25.153$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-q^{4}-iq^{7}+iq^{8}-4iq^{13}+\cdots$$
3150.2.g.k $$2$$ $$25.153$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-q^{4}-iq^{7}+iq^{8}-iq^{13}+\cdots$$
3150.2.g.l $$2$$ $$25.153$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-q^{4}+iq^{7}-iq^{8}-2iq^{13}+\cdots$$
3150.2.g.m $$2$$ $$25.153$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-q^{4}-iq^{7}-iq^{8}+2iq^{13}+\cdots$$
3150.2.g.n $$2$$ $$25.153$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-q^{4}-iq^{7}-iq^{8}-iq^{13}+\cdots$$
3150.2.g.o $$2$$ $$25.153$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-q^{4}-iq^{7}-iq^{8}+2iq^{13}+\cdots$$
3150.2.g.p $$2$$ $$25.153$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-q^{4}+iq^{7}+iq^{8}+2q^{11}+\cdots$$
3150.2.g.q $$2$$ $$25.153$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-q^{4}+iq^{7}-iq^{8}+4q^{11}+\cdots$$
3150.2.g.r $$2$$ $$25.153$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-q^{4}-iq^{7}+iq^{8}+4q^{11}+\cdots$$
3150.2.g.s $$2$$ $$25.153$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-q^{4}+iq^{7}+iq^{8}+4q^{11}+\cdots$$
3150.2.g.t $$2$$ $$25.153$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-q^{4}-iq^{7}-iq^{8}+4q^{11}+\cdots$$
3150.2.g.u $$2$$ $$25.153$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-q^{4}+iq^{7}+iq^{8}+4q^{11}+\cdots$$
3150.2.g.v $$2$$ $$25.153$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-q^{4}-iq^{7}+iq^{8}+5q^{11}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(3150, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(3150, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(30, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(35, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(50, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(70, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(75, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(90, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(105, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(150, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(175, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(210, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(225, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(315, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(350, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(450, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(525, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(630, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1050, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1575, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)
$3$ ()()()()()()()()()()()()()()()()()()()()()()
$5$ ()()()()()()()()()()()()()()()()()()()()()()
$7$ ($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)
$11$ ($$( 1 + 6 T + 11 T^{2} )^{2}$$)($$( 1 + 4 T + 11 T^{2} )^{2}$$)($$( 1 + 4 T + 11 T^{2} )^{2}$$)($$( 1 + 4 T + 11 T^{2} )^{2}$$)($$( 1 + 4 T + 11 T^{2} )^{2}$$)($$( 1 + 3 T + 11 T^{2} )^{2}$$)($$( 1 + 2 T + 11 T^{2} )^{2}$$)($$( 1 + 2 T + 11 T^{2} )^{2}$$)($$( 1 + 11 T^{2} )^{2}$$)($$( 1 + 11 T^{2} )^{2}$$)($$( 1 + 11 T^{2} )^{2}$$)($$( 1 + 11 T^{2} )^{2}$$)($$( 1 + 11 T^{2} )^{2}$$)($$( 1 + 11 T^{2} )^{2}$$)($$( 1 + 11 T^{2} )^{2}$$)($$( 1 - 2 T + 11 T^{2} )^{2}$$)($$( 1 - 4 T + 11 T^{2} )^{2}$$)($$( 1 - 4 T + 11 T^{2} )^{2}$$)($$( 1 - 4 T + 11 T^{2} )^{2}$$)($$( 1 - 4 T + 11 T^{2} )^{2}$$)($$( 1 - 4 T + 11 T^{2} )^{2}$$)($$( 1 - 5 T + 11 T^{2} )^{2}$$)
$13$ ($$1 - 25 T^{2} + 169 T^{4}$$)($$( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )$$)($$( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )$$)($$1 - 17 T^{2} + 169 T^{4}$$)($$1 - 22 T^{2} + 169 T^{4}$$)($$1 - 22 T^{2} + 169 T^{4}$$)($$1 - 25 T^{2} + 169 T^{4}$$)($$1 - 25 T^{2} + 169 T^{4}$$)($$1 - 22 T^{2} + 169 T^{4}$$)($$( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} )$$)($$1 - 25 T^{2} + 169 T^{4}$$)($$1 - 22 T^{2} + 169 T^{4}$$)($$1 - 22 T^{2} + 169 T^{4}$$)($$1 - 25 T^{2} + 169 T^{4}$$)($$1 - 22 T^{2} + 169 T^{4}$$)($$1 + 23 T^{2} + 169 T^{4}$$)($$1 - 22 T^{2} + 169 T^{4}$$)($$( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )$$)($$( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )$$)($$1 - 22 T^{2} + 169 T^{4}$$)($$1 - 17 T^{2} + 169 T^{4}$$)($$( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )$$)
$17$ ($$1 - 25 T^{2} + 289 T^{4}$$)($$1 - 18 T^{2} + 289 T^{4}$$)($$( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} )$$)($$1 + 15 T^{2} + 289 T^{4}$$)($$( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} )$$)($$1 - 25 T^{2} + 289 T^{4}$$)($$1 - 33 T^{2} + 289 T^{4}$$)($$1 - 25 T^{2} + 289 T^{4}$$)($$1 + 2 T^{2} + 289 T^{4}$$)($$1 + 2 T^{2} + 289 T^{4}$$)($$1 - 25 T^{2} + 289 T^{4}$$)($$( 1 - 17 T^{2} )^{2}$$)($$( 1 - 17 T^{2} )^{2}$$)($$1 - 25 T^{2} + 289 T^{4}$$)($$1 + 2 T^{2} + 289 T^{4}$$)($$1 + 15 T^{2} + 289 T^{4}$$)($$( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} )$$)($$( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} )$$)($$1 - 18 T^{2} + 289 T^{4}$$)($$1 + 2 T^{2} + 289 T^{4}$$)($$1 + 15 T^{2} + 289 T^{4}$$)($$1 - 33 T^{2} + 289 T^{4}$$)
$19$ ($$( 1 - 4 T + 19 T^{2} )^{2}$$)($$( 1 + 6 T + 19 T^{2} )^{2}$$)($$( 1 + 19 T^{2} )^{2}$$)($$( 1 - 6 T + 19 T^{2} )^{2}$$)($$( 1 - 4 T + 19 T^{2} )^{2}$$)($$( 1 - 7 T + 19 T^{2} )^{2}$$)($$( 1 + 4 T + 19 T^{2} )^{2}$$)($$( 1 + 19 T^{2} )^{2}$$)($$( 1 + 8 T + 19 T^{2} )^{2}$$)($$( 1 + 2 T + 19 T^{2} )^{2}$$)($$( 1 + 2 T + 19 T^{2} )^{2}$$)($$( 1 + 2 T + 19 T^{2} )^{2}$$)($$( 1 + 2 T + 19 T^{2} )^{2}$$)($$( 1 + 2 T + 19 T^{2} )^{2}$$)($$( 1 - 4 T + 19 T^{2} )^{2}$$)($$( 1 + 8 T + 19 T^{2} )^{2}$$)($$( 1 + 4 T + 19 T^{2} )^{2}$$)($$( 1 - 4 T + 19 T^{2} )^{2}$$)($$( 1 + 6 T + 19 T^{2} )^{2}$$)($$( 1 + 19 T^{2} )^{2}$$)($$( 1 - 6 T + 19 T^{2} )^{2}$$)($$( 1 - 3 T + 19 T^{2} )^{2}$$)
$23$ ($$1 - 37 T^{2} + 529 T^{4}$$)($$( 1 - 23 T^{2} )^{2}$$)($$( 1 - 23 T^{2} )^{2}$$)($$1 + 35 T^{2} + 529 T^{4}$$)($$1 + 18 T^{2} + 529 T^{4}$$)($$( 1 - 23 T^{2} )^{2}$$)($$1 + 3 T^{2} + 529 T^{4}$$)($$1 - 45 T^{2} + 529 T^{4}$$)($$( 1 - 23 T^{2} )^{2}$$)($$( 1 - 23 T^{2} )^{2}$$)($$1 - 37 T^{2} + 529 T^{4}$$)($$( 1 - 23 T^{2} )^{2}$$)($$( 1 - 23 T^{2} )^{2}$$)($$1 - 37 T^{2} + 529 T^{4}$$)($$( 1 - 23 T^{2} )^{2}$$)($$1 - 21 T^{2} + 529 T^{4}$$)($$1 + 18 T^{2} + 529 T^{4}$$)($$1 + 18 T^{2} + 529 T^{4}$$)($$( 1 - 23 T^{2} )^{2}$$)($$1 + 18 T^{2} + 529 T^{4}$$)($$1 + 35 T^{2} + 529 T^{4}$$)($$( 1 - 23 T^{2} )^{2}$$)
$29$ ($$( 1 - 3 T + 29 T^{2} )^{2}$$)($$( 1 - 6 T + 29 T^{2} )^{2}$$)($$( 1 - 6 T + 29 T^{2} )^{2}$$)($$( 1 - 3 T + 29 T^{2} )^{2}$$)($$( 1 - 6 T + 29 T^{2} )^{2}$$)($$( 1 + 6 T + 29 T^{2} )^{2}$$)($$( 1 - T + 29 T^{2} )^{2}$$)($$( 1 + 5 T + 29 T^{2} )^{2}$$)($$( 1 - 6 T + 29 T^{2} )^{2}$$)($$( 1 + 6 T + 29 T^{2} )^{2}$$)($$( 1 + 3 T + 29 T^{2} )^{2}$$)($$( 1 - 6 T + 29 T^{2} )^{2}$$)($$( 1 + 6 T + 29 T^{2} )^{2}$$)($$( 1 - 3 T + 29 T^{2} )^{2}$$)($$( 1 + 6 T + 29 T^{2} )^{2}$$)($$( 1 - 9 T + 29 T^{2} )^{2}$$)($$( 1 + 2 T + 29 T^{2} )^{2}$$)($$( 1 + 2 T + 29 T^{2} )^{2}$$)($$( 1 + 6 T + 29 T^{2} )^{2}$$)($$( 1 - 10 T + 29 T^{2} )^{2}$$)($$( 1 + 3 T + 29 T^{2} )^{2}$$)($$( 1 + 6 T + 29 T^{2} )^{2}$$)
$31$ ($$( 1 - 5 T + 31 T^{2} )^{2}$$)($$( 1 + 4 T + 31 T^{2} )^{2}$$)($$( 1 - 8 T + 31 T^{2} )^{2}$$)($$( 1 + 7 T + 31 T^{2} )^{2}$$)($$( 1 + 8 T + 31 T^{2} )^{2}$$)($$( 1 + 4 T + 31 T^{2} )^{2}$$)($$( 1 - 3 T + 31 T^{2} )^{2}$$)($$( 1 - 7 T + 31 T^{2} )^{2}$$)($$( 1 + 4 T + 31 T^{2} )^{2}$$)($$( 1 + 4 T + 31 T^{2} )^{2}$$)($$( 1 + T + 31 T^{2} )^{2}$$)($$( 1 - 8 T + 31 T^{2} )^{2}$$)($$( 1 - 8 T + 31 T^{2} )^{2}$$)($$( 1 + T + 31 T^{2} )^{2}$$)($$( 1 + 4 T + 31 T^{2} )^{2}$$)($$( 1 - T + 31 T^{2} )^{2}$$)($$( 1 + 31 T^{2} )^{2}$$)($$( 1 + 31 T^{2} )^{2}$$)($$( 1 + 4 T + 31 T^{2} )^{2}$$)($$( 1 + 8 T + 31 T^{2} )^{2}$$)($$( 1 + 7 T + 31 T^{2} )^{2}$$)($$( 1 + 4 T + 31 T^{2} )^{2}$$)
$37$ ($$1 + 26 T^{2} + 1369 T^{4}$$)($$1 - 10 T^{2} + 1369 T^{4}$$)($$1 + 26 T^{2} + 1369 T^{4}$$)($$1 + 26 T^{2} + 1369 T^{4}$$)($$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$)($$1 - 10 T^{2} + 1369 T^{4}$$)($$1 - 38 T^{2} + 1369 T^{4}$$)($$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$)($$1 + 26 T^{2} + 1369 T^{4}$$)($$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$)($$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$)($$1 - 58 T^{2} + 1369 T^{4}$$)($$1 - 58 T^{2} + 1369 T^{4}$$)($$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$)($$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$)($$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$)($$1 - 38 T^{2} + 1369 T^{4}$$)($$1 + 26 T^{2} + 1369 T^{4}$$)($$1 - 10 T^{2} + 1369 T^{4}$$)($$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$)($$1 + 26 T^{2} + 1369 T^{4}$$)($$1 - 10 T^{2} + 1369 T^{4}$$)
$41$ ($$( 1 + 9 T + 41 T^{2} )^{2}$$)($$( 1 - 10 T + 41 T^{2} )^{2}$$)($$( 1 + 2 T + 41 T^{2} )^{2}$$)($$( 1 - T + 41 T^{2} )^{2}$$)($$( 1 + 2 T + 41 T^{2} )^{2}$$)($$( 1 - 9 T + 41 T^{2} )^{2}$$)($$( 1 - 3 T + 41 T^{2} )^{2}$$)($$( 1 + 7 T + 41 T^{2} )^{2}$$)($$( 1 - 6 T + 41 T^{2} )^{2}$$)($$( 1 + 6 T + 41 T^{2} )^{2}$$)($$( 1 + 3 T + 41 T^{2} )^{2}$$)($$( 1 + 6 T + 41 T^{2} )^{2}$$)($$( 1 - 6 T + 41 T^{2} )^{2}$$)($$( 1 - 3 T + 41 T^{2} )^{2}$$)($$( 1 + 6 T + 41 T^{2} )^{2}$$)($$( 1 + 11 T + 41 T^{2} )^{2}$$)($$( 1 - 6 T + 41 T^{2} )^{2}$$)($$( 1 - 6 T + 41 T^{2} )^{2}$$)($$( 1 + 10 T + 41 T^{2} )^{2}$$)($$( 1 - 2 T + 41 T^{2} )^{2}$$)($$( 1 + T + 41 T^{2} )^{2}$$)($$( 1 + 11 T + 41 T^{2} )^{2}$$)
$43$ ($$1 - 85 T^{2} + 1849 T^{4}$$)($$1 - 82 T^{2} + 1849 T^{4}$$)($$1 - 70 T^{2} + 1849 T^{4}$$)($$1 + 83 T^{2} + 1849 T^{4}$$)($$1 + 58 T^{2} + 1849 T^{4}$$)($$1 - 22 T^{2} + 1849 T^{4}$$)($$1 - 85 T^{2} + 1849 T^{4}$$)($$1 + 35 T^{2} + 1849 T^{4}$$)($$1 - 70 T^{2} + 1849 T^{4}$$)($$1 - 22 T^{2} + 1849 T^{4}$$)($$1 - 37 T^{2} + 1849 T^{4}$$)($$1 - 82 T^{2} + 1849 T^{4}$$)($$1 - 82 T^{2} + 1849 T^{4}$$)($$1 - 37 T^{2} + 1849 T^{4}$$)($$1 - 22 T^{2} + 1849 T^{4}$$)($$1 - 77 T^{2} + 1849 T^{4}$$)($$1 - 70 T^{2} + 1849 T^{4}$$)($$1 - 70 T^{2} + 1849 T^{4}$$)($$1 - 82 T^{2} + 1849 T^{4}$$)($$1 - 22 T^{2} + 1849 T^{4}$$)($$1 + 83 T^{2} + 1849 T^{4}$$)($$1 - 22 T^{2} + 1849 T^{4}$$)
$47$ ($$( 1 - 47 T^{2} )^{2}$$)($$1 + 6 T^{2} + 2209 T^{4}$$)($$1 - 30 T^{2} + 2209 T^{4}$$)($$1 - 90 T^{2} + 2209 T^{4}$$)($$1 - 30 T^{2} + 2209 T^{4}$$)($$1 - 58 T^{2} + 2209 T^{4}$$)($$1 + 50 T^{2} + 2209 T^{4}$$)($$1 - 30 T^{2} + 2209 T^{4}$$)($$( 1 - 47 T^{2} )^{2}$$)($$1 + 50 T^{2} + 2209 T^{4}$$)($$1 - 58 T^{2} + 2209 T^{4}$$)($$1 - 58 T^{2} + 2209 T^{4}$$)($$1 - 58 T^{2} + 2209 T^{4}$$)($$1 - 58 T^{2} + 2209 T^{4}$$)($$1 + 50 T^{2} + 2209 T^{4}$$)($$1 - 78 T^{2} + 2209 T^{4}$$)($$( 1 - 47 T^{2} )^{2}$$)($$( 1 - 47 T^{2} )^{2}$$)($$1 + 6 T^{2} + 2209 T^{4}$$)($$1 - 78 T^{2} + 2209 T^{4}$$)($$1 - 90 T^{2} + 2209 T^{4}$$)($$1 - 90 T^{2} + 2209 T^{4}$$)
$53$ ($$1 - 25 T^{2} + 2809 T^{4}$$)($$( 1 - 4 T + 53 T^{2} )( 1 + 4 T + 53 T^{2} )$$)($$1 - 102 T^{2} + 2809 T^{4}$$)($$1 - 105 T^{2} + 2809 T^{4}$$)($$1 - 70 T^{2} + 2809 T^{4}$$)($$1 + 38 T^{2} + 2809 T^{4}$$)($$1 + 15 T^{2} + 2809 T^{4}$$)($$1 - 105 T^{2} + 2809 T^{4}$$)($$1 - 70 T^{2} + 2809 T^{4}$$)($$1 - 70 T^{2} + 2809 T^{4}$$)($$1 - 25 T^{2} + 2809 T^{4}$$)($$1 - 70 T^{2} + 2809 T^{4}$$)($$1 - 70 T^{2} + 2809 T^{4}$$)($$1 - 25 T^{2} + 2809 T^{4}$$)($$1 - 70 T^{2} + 2809 T^{4}$$)($$1 - 97 T^{2} + 2809 T^{4}$$)($$1 - 6 T^{2} + 2809 T^{4}$$)($$1 - 70 T^{2} + 2809 T^{4}$$)($$( 1 - 4 T + 53 T^{2} )( 1 + 4 T + 53 T^{2} )$$)($$1 - 6 T^{2} + 2809 T^{4}$$)($$1 - 105 T^{2} + 2809 T^{4}$$)($$( 1 - 14 T + 53 T^{2} )( 1 + 14 T + 53 T^{2} )$$)
$59$ ($$( 1 - 9 T + 59 T^{2} )^{2}$$)($$( 1 - 4 T + 59 T^{2} )^{2}$$)($$( 1 + 8 T + 59 T^{2} )^{2}$$)($$( 1 + 11 T + 59 T^{2} )^{2}$$)($$( 1 - 4 T + 59 T^{2} )^{2}$$)($$( 1 - 12 T + 59 T^{2} )^{2}$$)($$( 1 + 3 T + 59 T^{2} )^{2}$$)($$( 1 + 5 T + 59 T^{2} )^{2}$$)($$( 1 + 12 T + 59 T^{2} )^{2}$$)($$( 1 + 6 T + 59 T^{2} )^{2}$$)($$( 1 + 3 T + 59 T^{2} )^{2}$$)($$( 1 - 12 T + 59 T^{2} )^{2}$$)($$( 1 + 12 T + 59 T^{2} )^{2}$$)($$( 1 - 3 T + 59 T^{2} )^{2}$$)($$( 1 + 12 T + 59 T^{2} )^{2}$$)($$( 1 - 7 T + 59 T^{2} )^{2}$$)($$( 1 - 12 T + 59 T^{2} )^{2}$$)($$( 1 - 4 T + 59 T^{2} )^{2}$$)($$( 1 + 4 T + 59 T^{2} )^{2}$$)($$( 1 - 4 T + 59 T^{2} )^{2}$$)($$( 1 - 11 T + 59 T^{2} )^{2}$$)($$( 1 - 4 T + 59 T^{2} )^{2}$$)
$61$ ($$( 1 - 11 T + 61 T^{2} )^{2}$$)($$( 1 + 8 T + 61 T^{2} )^{2}$$)($$( 1 + 14 T + 61 T^{2} )^{2}$$)($$( 1 - 13 T + 61 T^{2} )^{2}$$)($$( 1 + 2 T + 61 T^{2} )^{2}$$)($$( 1 + 10 T + 61 T^{2} )^{2}$$)($$( 1 - 5 T + 61 T^{2} )^{2}$$)($$( 1 + 3 T + 61 T^{2} )^{2}$$)($$( 1 + 10 T + 61 T^{2} )^{2}$$)($$( 1 - 8 T + 61 T^{2} )^{2}$$)($$( 1 + T + 61 T^{2} )^{2}$$)($$( 1 - 8 T + 61 T^{2} )^{2}$$)($$( 1 - 8 T + 61 T^{2} )^{2}$$)($$( 1 + T + 61 T^{2} )^{2}$$)($$( 1 - 2 T + 61 T^{2} )^{2}$$)($$( 1 + 5 T + 61 T^{2} )^{2}$$)($$( 1 - 14 T + 61 T^{2} )^{2}$$)($$( 1 - 6 T + 61 T^{2} )^{2}$$)($$( 1 + 8 T + 61 T^{2} )^{2}$$)($$( 1 + 6 T + 61 T^{2} )^{2}$$)($$( 1 - 13 T + 61 T^{2} )^{2}$$)($$( 1 + 2 T + 61 T^{2} )^{2}$$)
$67$ ($$1 - 118 T^{2} + 4489 T^{4}$$)($$1 - 98 T^{2} + 4489 T^{4}$$)($$1 + 10 T^{2} + 4489 T^{4}$$)($$( 1 - 67 T^{2} )^{2}$$)($$1 + 10 T^{2} + 4489 T^{4}$$)($$1 - 85 T^{2} + 4489 T^{4}$$)($$1 + 10 T^{2} + 4489 T^{4}$$)($$1 + 10 T^{2} + 4489 T^{4}$$)($$1 - 118 T^{2} + 4489 T^{4}$$)($$1 - 118 T^{2} + 4489 T^{4}$$)($$1 - 70 T^{2} + 4489 T^{4}$$)($$1 - 130 T^{2} + 4489 T^{4}$$)($$1 - 130 T^{2} + 4489 T^{4}$$)($$1 - 70 T^{2} + 4489 T^{4}$$)($$1 - 70 T^{2} + 4489 T^{4}$$)($$1 + 10 T^{2} + 4489 T^{4}$$)($$1 + 10 T^{2} + 4489 T^{4}$$)($$1 - 118 T^{2} + 4489 T^{4}$$)($$1 - 98 T^{2} + 4489 T^{4}$$)($$( 1 - 67 T^{2} )^{2}$$)($$( 1 - 67 T^{2} )^{2}$$)($$1 - 53 T^{2} + 4489 T^{4}$$)
$71$ ($$( 1 - 12 T + 71 T^{2} )^{2}$$)($$( 1 + 2 T + 71 T^{2} )^{2}$$)($$( 1 - 16 T + 71 T^{2} )^{2}$$)($$( 1 + 8 T + 71 T^{2} )^{2}$$)($$( 1 + 8 T + 71 T^{2} )^{2}$$)($$( 1 + 6 T + 71 T^{2} )^{2}$$)($$( 1 + 4 T + 71 T^{2} )^{2}$$)($$( 1 + 12 T + 71 T^{2} )^{2}$$)($$( 1 + 12 T + 71 T^{2} )^{2}$$)($$( 1 + 71 T^{2} )^{2}$$)($$( 1 + 71 T^{2} )^{2}$$)($$( 1 + 6 T + 71 T^{2} )^{2}$$)($$( 1 - 6 T + 71 T^{2} )^{2}$$)($$( 1 + 71 T^{2} )^{2}$$)($$( 1 + 71 T^{2} )^{2}$$)($$( 1 - 4 T + 71 T^{2} )^{2}$$)($$( 1 - 8 T + 71 T^{2} )^{2}$$)($$( 1 + 8 T + 71 T^{2} )^{2}$$)($$( 1 - 2 T + 71 T^{2} )^{2}$$)($$( 1 - 12 T + 71 T^{2} )^{2}$$)($$( 1 - 8 T + 71 T^{2} )^{2}$$)($$( 1 - 10 T + 71 T^{2} )^{2}$$)
$73$ ($$1 - 46 T^{2} + 5329 T^{4}$$)($$1 - 46 T^{2} + 5329 T^{4}$$)($$1 - 142 T^{2} + 5329 T^{4}$$)($$1 - 82 T^{2} + 5329 T^{4}$$)($$1 + 50 T^{2} + 5329 T^{4}$$)($$1 - 121 T^{2} + 5329 T^{4}$$)($$1 + 50 T^{2} + 5329 T^{4}$$)($$( 1 - 16 T + 73 T^{2} )( 1 + 16 T + 73 T^{2} )$$)($$1 - 46 T^{2} + 5329 T^{4}$$)($$1 - 142 T^{2} + 5329 T^{4}$$)($$1 - 130 T^{2} + 5329 T^{4}$$)($$1 - 142 T^{2} + 5329 T^{4}$$)($$1 - 142 T^{2} + 5329 T^{4}$$)($$1 - 130 T^{2} + 5329 T^{4}$$)($$1 + 50 T^{2} + 5329 T^{4}$$)($$1 - 46 T^{2} + 5329 T^{4}$$)($$1 - 46 T^{2} + 5329 T^{4}$$)($$1 - 46 T^{2} + 5329 T^{4}$$)($$1 - 46 T^{2} + 5329 T^{4}$$)($$( 1 - 16 T + 73 T^{2} )( 1 + 16 T + 73 T^{2} )$$)($$1 - 82 T^{2} + 5329 T^{4}$$)($$1 - 97 T^{2} + 5329 T^{4}$$)
$79$ ($$( 1 - 10 T + 79 T^{2} )^{2}$$)($$( 1 + 16 T + 79 T^{2} )^{2}$$)($$( 1 - 8 T + 79 T^{2} )^{2}$$)($$( 1 + 4 T + 79 T^{2} )^{2}$$)($$( 1 + 79 T^{2} )^{2}$$)($$( 1 + 14 T + 79 T^{2} )^{2}$$)($$( 1 - 2 T + 79 T^{2} )^{2}$$)($$( 1 + 10 T + 79 T^{2} )^{2}$$)($$( 1 + 8 T + 79 T^{2} )^{2}$$)($$( 1 + 8 T + 79 T^{2} )^{2}$$)($$( 1 + 8 T + 79 T^{2} )^{2}$$)($$( 1 - 16 T + 79 T^{2} )^{2}$$)($$( 1 - 16 T + 79 T^{2} )^{2}$$)($$( 1 + 8 T + 79 T^{2} )^{2}$$)($$( 1 - 16 T + 79 T^{2} )^{2}$$)($$( 1 - 6 T + 79 T^{2} )^{2}$$)($$( 1 + 16 T + 79 T^{2} )^{2}$$)($$( 1 + 79 T^{2} )^{2}$$)($$( 1 + 16 T + 79 T^{2} )^{2}$$)($$( 1 - 8 T + 79 T^{2} )^{2}$$)($$( 1 + 4 T + 79 T^{2} )^{2}$$)($$( 1 - 2 T + 79 T^{2} )^{2}$$)
$83$ ($$1 - 85 T^{2} + 6889 T^{4}$$)($$1 - 102 T^{2} + 6889 T^{4}$$)($$1 - 102 T^{2} + 6889 T^{4}$$)($$1 - 117 T^{2} + 6889 T^{4}$$)($$1 - 22 T^{2} + 6889 T^{4}$$)($$1 - 85 T^{2} + 6889 T^{4}$$)($$1 - 157 T^{2} + 6889 T^{4}$$)($$1 - 45 T^{2} + 6889 T^{4}$$)($$1 - 22 T^{2} + 6889 T^{4}$$)($$1 - 130 T^{2} + 6889 T^{4}$$)($$1 + 59 T^{2} + 6889 T^{4}$$)($$( 1 - 83 T^{2} )^{2}$$)($$( 1 - 83 T^{2} )^{2}$$)($$1 + 59 T^{2} + 6889 T^{4}$$)($$1 - 22 T^{2} + 6889 T^{4}$$)($$1 - 85 T^{2} + 6889 T^{4}$$)($$1 - 22 T^{2} + 6889 T^{4}$$)($$1 - 150 T^{2} + 6889 T^{4}$$)($$1 - 102 T^{2} + 6889 T^{4}$$)($$1 - 150 T^{2} + 6889 T^{4}$$)($$1 - 117 T^{2} + 6889 T^{4}$$)($$1 - 45 T^{2} + 6889 T^{4}$$)
$89$ ($$( 1 + 6 T + 89 T^{2} )^{2}$$)($$( 1 + 2 T + 89 T^{2} )^{2}$$)($$( 1 - 10 T + 89 T^{2} )^{2}$$)($$( 1 + 14 T + 89 T^{2} )^{2}$$)($$( 1 - 2 T + 89 T^{2} )^{2}$$)($$( 1 + 15 T + 89 T^{2} )^{2}$$)($$( 1 - 10 T + 89 T^{2} )^{2}$$)($$( 1 + 10 T + 89 T^{2} )^{2}$$)($$( 1 + 6 T + 89 T^{2} )^{2}$$)($$( 1 + 6 T + 89 T^{2} )^{2}$$)($$( 1 + 6 T + 89 T^{2} )^{2}$$)($$( 1 - 6 T + 89 T^{2} )^{2}$$)($$( 1 + 6 T + 89 T^{2} )^{2}$$)($$( 1 - 6 T + 89 T^{2} )^{2}$$)($$( 1 - 6 T + 89 T^{2} )^{2}$$)($$( 1 + 10 T + 89 T^{2} )^{2}$$)($$( 1 - 10 T + 89 T^{2} )^{2}$$)($$( 1 + 6 T + 89 T^{2} )^{2}$$)($$( 1 - 2 T + 89 T^{2} )^{2}$$)($$( 1 - 14 T + 89 T^{2} )^{2}$$)($$( 1 - 14 T + 89 T^{2} )^{2}$$)($$( 1 + 11 T + 89 T^{2} )^{2}$$)
$97$ ($$1 + 2 T^{2} + 9409 T^{4}$$)($$1 - 190 T^{2} + 9409 T^{4}$$)($$1 - 190 T^{2} + 9409 T^{4}$$)($$( 1 - 18 T + 97 T^{2} )( 1 + 18 T + 97 T^{2} )$$)($$1 - 94 T^{2} + 9409 T^{4}$$)($$1 - 94 T^{2} + 9409 T^{4}$$)($$1 - 94 T^{2} + 9409 T^{4}$$)($$1 - 190 T^{2} + 9409 T^{4}$$)($$1 - 94 T^{2} + 9409 T^{4}$$)($$1 - 94 T^{2} + 9409 T^{4}$$)($$( 1 - 18 T + 97 T^{2} )( 1 + 18 T + 97 T^{2} )$$)($$1 - 94 T^{2} + 9409 T^{4}$$)($$1 - 94 T^{2} + 9409 T^{4}$$)($$( 1 - 18 T + 97 T^{2} )( 1 + 18 T + 97 T^{2} )$$)($$1 + 2 T^{2} + 9409 T^{4}$$)($$1 - 94 T^{2} + 9409 T^{4}$$)($$1 - 190 T^{2} + 9409 T^{4}$$)($$1 + 2 T^{2} + 9409 T^{4}$$)($$1 - 190 T^{2} + 9409 T^{4}$$)($$1 - 190 T^{2} + 9409 T^{4}$$)($$( 1 - 18 T + 97 T^{2} )( 1 + 18 T + 97 T^{2} )$$)($$1 - 94 T^{2} + 9409 T^{4}$$)