# Properties

 Label 2304.2.d Level 2304 Weight 2 Character orbit d Rep. character $$\chi_{2304}(1153,\cdot)$$ Character field $$\Q$$ Dimension 38 Newform subspaces 19 Sturm bound 768 Trace bound 23

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$2304 = 2^{8} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2304.d (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$8$$ Character field: $$\Q$$ Newform subspaces: $$19$$ Sturm bound: $$768$$ Trace bound: $$23$$ Distinguishing $$T_p$$: $$5$$, $$7$$, $$11$$, $$17$$, $$23$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(2304, [\chi])$$.

Total New Old
Modular forms 432 42 390
Cusp forms 336 38 298
Eisenstein series 96 4 92

## Trace form

 $$38q + O(q^{10})$$ $$38q + 4q^{17} - 26q^{25} - 4q^{41} - 10q^{49} + 40q^{65} + 20q^{73} - 20q^{89} - 36q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(2304, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
2304.2.d.a $$2$$ $$18.398$$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$-8$$ $$q-4q^{7}-iq^{13}+4iq^{19}+5q^{25}+\cdots$$
2304.2.d.b $$2$$ $$18.398$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$-8$$ $$q+iq^{5}-4q^{7}+iq^{11}+iq^{13}+2q^{17}+\cdots$$
2304.2.d.c $$2$$ $$18.398$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$-8$$ $$q+iq^{5}-4q^{7}-2iq^{11}-iq^{13}+6q^{17}+\cdots$$
2304.2.d.d $$2$$ $$18.398$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$-4$$ $$q-2q^{7}+2iq^{11}-3iq^{13}-6q^{17}+\cdots$$
2304.2.d.e $$2$$ $$18.398$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$-4$$ $$q+iq^{5}-2q^{7}+2iq^{11}+iq^{13}-4q^{17}+\cdots$$
2304.2.d.f $$2$$ $$18.398$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$-4$$ $$q+2iq^{5}-2q^{7}-2iq^{11}-iq^{13}+\cdots$$
2304.2.d.g $$2$$ $$18.398$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$-4$$ $$q+iq^{5}-2q^{7}+2iq^{11}-iq^{13}+4q^{17}+\cdots$$
2304.2.d.h $$2$$ $$18.398$$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+2iq^{5}-3iq^{13}-8q^{17}-11q^{25}+\cdots$$
2304.2.d.i $$2$$ $$18.398$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{5}-2iq^{11}+iq^{13}-2q^{17}+\cdots$$
2304.2.d.j $$2$$ $$18.398$$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{5}-3iq^{13}-2q^{17}+q^{25}-5iq^{29}+\cdots$$
2304.2.d.k $$2$$ $$18.398$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{5}+2iq^{11}+iq^{13}-2q^{17}+\cdots$$
2304.2.d.l $$2$$ $$18.398$$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+2iq^{5}+3iq^{13}+8q^{17}-11q^{25}+\cdots$$
2304.2.d.m $$2$$ $$18.398$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$4$$ $$q+2q^{7}-2iq^{11}-3iq^{13}-6q^{17}+\cdots$$
2304.2.d.n $$2$$ $$18.398$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$4$$ $$q+iq^{5}+2q^{7}-2iq^{11}+iq^{13}-4q^{17}+\cdots$$
2304.2.d.o $$2$$ $$18.398$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$4$$ $$q+2iq^{5}+2q^{7}+2iq^{11}-iq^{13}+\cdots$$
2304.2.d.p $$2$$ $$18.398$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$4$$ $$q+iq^{5}+2q^{7}-2iq^{11}-iq^{13}+4q^{17}+\cdots$$
2304.2.d.q $$2$$ $$18.398$$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$8$$ $$q+4q^{7}-iq^{13}-4iq^{19}+5q^{25}+\cdots$$
2304.2.d.r $$2$$ $$18.398$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$8$$ $$q+iq^{5}+4q^{7}-iq^{11}+iq^{13}+2q^{17}+\cdots$$
2304.2.d.s $$2$$ $$18.398$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$8$$ $$q+iq^{5}+4q^{7}+2iq^{11}-iq^{13}+6q^{17}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(2304, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(2304, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(24, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(64, [\chi])$$$$^{\oplus 9}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(72, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(96, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(128, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(192, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(256, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(288, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(384, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(576, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(768, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1152, [\chi])$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ ($$( 1 - 5 T^{2} )^{2}$$)($$( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$)($$( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$)($$( 1 - 5 T^{2} )^{2}$$)($$( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$)($$( 1 - 2 T + 5 T^{2} )( 1 + 2 T + 5 T^{2} )$$)($$( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$)($$( 1 - 2 T + 5 T^{2} )( 1 + 2 T + 5 T^{2} )$$)($$( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$)($$( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$)($$( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$)($$( 1 - 2 T + 5 T^{2} )( 1 + 2 T + 5 T^{2} )$$)($$( 1 - 5 T^{2} )^{2}$$)($$( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$)($$( 1 - 2 T + 5 T^{2} )( 1 + 2 T + 5 T^{2} )$$)($$( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$)($$( 1 - 5 T^{2} )^{2}$$)($$( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$)($$( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$)
$7$ ($$( 1 + 4 T + 7 T^{2} )^{2}$$)($$( 1 + 4 T + 7 T^{2} )^{2}$$)($$( 1 + 4 T + 7 T^{2} )^{2}$$)($$( 1 + 2 T + 7 T^{2} )^{2}$$)($$( 1 + 2 T + 7 T^{2} )^{2}$$)($$( 1 + 2 T + 7 T^{2} )^{2}$$)($$( 1 + 2 T + 7 T^{2} )^{2}$$)($$( 1 + 7 T^{2} )^{2}$$)($$( 1 + 7 T^{2} )^{2}$$)($$( 1 + 7 T^{2} )^{2}$$)($$( 1 + 7 T^{2} )^{2}$$)($$( 1 + 7 T^{2} )^{2}$$)($$( 1 - 2 T + 7 T^{2} )^{2}$$)($$( 1 - 2 T + 7 T^{2} )^{2}$$)($$( 1 - 2 T + 7 T^{2} )^{2}$$)($$( 1 - 2 T + 7 T^{2} )^{2}$$)($$( 1 - 4 T + 7 T^{2} )^{2}$$)($$( 1 - 4 T + 7 T^{2} )^{2}$$)($$( 1 - 4 T + 7 T^{2} )^{2}$$)
$11$ ($$( 1 - 11 T^{2} )^{2}$$)($$1 - 18 T^{2} + 121 T^{4}$$)($$1 - 6 T^{2} + 121 T^{4}$$)($$1 - 6 T^{2} + 121 T^{4}$$)($$1 - 6 T^{2} + 121 T^{4}$$)($$1 - 6 T^{2} + 121 T^{4}$$)($$1 - 6 T^{2} + 121 T^{4}$$)($$( 1 - 11 T^{2} )^{2}$$)($$1 - 6 T^{2} + 121 T^{4}$$)($$( 1 - 11 T^{2} )^{2}$$)($$1 - 6 T^{2} + 121 T^{4}$$)($$( 1 - 11 T^{2} )^{2}$$)($$1 - 6 T^{2} + 121 T^{4}$$)($$1 - 6 T^{2} + 121 T^{4}$$)($$1 - 6 T^{2} + 121 T^{4}$$)($$1 - 6 T^{2} + 121 T^{4}$$)($$( 1 - 11 T^{2} )^{2}$$)($$1 - 18 T^{2} + 121 T^{4}$$)($$1 - 6 T^{2} + 121 T^{4}$$)
$13$ ($$1 - 22 T^{2} + 169 T^{4}$$)($$1 - 22 T^{2} + 169 T^{4}$$)($$1 - 22 T^{2} + 169 T^{4}$$)($$( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )$$)($$1 - 22 T^{2} + 169 T^{4}$$)($$1 - 22 T^{2} + 169 T^{4}$$)($$1 - 22 T^{2} + 169 T^{4}$$)($$( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )$$)($$1 - 22 T^{2} + 169 T^{4}$$)($$( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )$$)($$1 - 22 T^{2} + 169 T^{4}$$)($$( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )$$)($$( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )$$)($$1 - 22 T^{2} + 169 T^{4}$$)($$1 - 22 T^{2} + 169 T^{4}$$)($$1 - 22 T^{2} + 169 T^{4}$$)($$1 - 22 T^{2} + 169 T^{4}$$)($$1 - 22 T^{2} + 169 T^{4}$$)($$1 - 22 T^{2} + 169 T^{4}$$)
$17$ ($$( 1 + 17 T^{2} )^{2}$$)($$( 1 - 2 T + 17 T^{2} )^{2}$$)($$( 1 - 6 T + 17 T^{2} )^{2}$$)($$( 1 + 6 T + 17 T^{2} )^{2}$$)($$( 1 + 4 T + 17 T^{2} )^{2}$$)($$( 1 - 2 T + 17 T^{2} )^{2}$$)($$( 1 - 4 T + 17 T^{2} )^{2}$$)($$( 1 + 8 T + 17 T^{2} )^{2}$$)($$( 1 + 2 T + 17 T^{2} )^{2}$$)($$( 1 + 2 T + 17 T^{2} )^{2}$$)($$( 1 + 2 T + 17 T^{2} )^{2}$$)($$( 1 - 8 T + 17 T^{2} )^{2}$$)($$( 1 + 6 T + 17 T^{2} )^{2}$$)($$( 1 + 4 T + 17 T^{2} )^{2}$$)($$( 1 - 2 T + 17 T^{2} )^{2}$$)($$( 1 - 4 T + 17 T^{2} )^{2}$$)($$( 1 + 17 T^{2} )^{2}$$)($$( 1 - 2 T + 17 T^{2} )^{2}$$)($$( 1 - 6 T + 17 T^{2} )^{2}$$)
$19$ ($$1 + 26 T^{2} + 361 T^{4}$$)($$1 - 34 T^{2} + 361 T^{4}$$)($$1 - 22 T^{2} + 361 T^{4}$$)($$( 1 - 19 T^{2} )^{2}$$)($$1 - 22 T^{2} + 361 T^{4}$$)($$1 + 26 T^{2} + 361 T^{4}$$)($$1 - 22 T^{2} + 361 T^{4}$$)($$( 1 - 19 T^{2} )^{2}$$)($$1 - 22 T^{2} + 361 T^{4}$$)($$( 1 - 19 T^{2} )^{2}$$)($$1 - 22 T^{2} + 361 T^{4}$$)($$( 1 - 19 T^{2} )^{2}$$)($$( 1 - 19 T^{2} )^{2}$$)($$1 - 22 T^{2} + 361 T^{4}$$)($$1 + 26 T^{2} + 361 T^{4}$$)($$1 - 22 T^{2} + 361 T^{4}$$)($$1 + 26 T^{2} + 361 T^{4}$$)($$1 - 34 T^{2} + 361 T^{4}$$)($$1 - 22 T^{2} + 361 T^{4}$$)
$23$ ($$( 1 + 23 T^{2} )^{2}$$)($$( 1 + 4 T + 23 T^{2} )^{2}$$)($$( 1 + 23 T^{2} )^{2}$$)($$( 1 - 4 T + 23 T^{2} )^{2}$$)($$( 1 + 8 T + 23 T^{2} )^{2}$$)($$( 1 - 4 T + 23 T^{2} )^{2}$$)($$( 1 - 8 T + 23 T^{2} )^{2}$$)($$( 1 + 23 T^{2} )^{2}$$)($$( 1 + 8 T + 23 T^{2} )^{2}$$)($$( 1 + 23 T^{2} )^{2}$$)($$( 1 - 8 T + 23 T^{2} )^{2}$$)($$( 1 + 23 T^{2} )^{2}$$)($$( 1 + 4 T + 23 T^{2} )^{2}$$)($$( 1 - 8 T + 23 T^{2} )^{2}$$)($$( 1 + 4 T + 23 T^{2} )^{2}$$)($$( 1 + 8 T + 23 T^{2} )^{2}$$)($$( 1 + 23 T^{2} )^{2}$$)($$( 1 - 4 T + 23 T^{2} )^{2}$$)($$( 1 + 23 T^{2} )^{2}$$)
$29$ ($$( 1 - 29 T^{2} )^{2}$$)($$1 - 22 T^{2} + 841 T^{4}$$)($$1 - 54 T^{2} + 841 T^{4}$$)($$( 1 - 10 T + 29 T^{2} )( 1 + 10 T + 29 T^{2} )$$)($$1 - 22 T^{2} + 841 T^{4}$$)($$( 1 - 29 T^{2} )^{2}$$)($$1 - 22 T^{2} + 841 T^{4}$$)($$( 1 - 10 T + 29 T^{2} )( 1 + 10 T + 29 T^{2} )$$)($$1 - 22 T^{2} + 841 T^{4}$$)($$( 1 - 4 T + 29 T^{2} )( 1 + 4 T + 29 T^{2} )$$)($$1 - 22 T^{2} + 841 T^{4}$$)($$( 1 - 10 T + 29 T^{2} )( 1 + 10 T + 29 T^{2} )$$)($$( 1 - 10 T + 29 T^{2} )( 1 + 10 T + 29 T^{2} )$$)($$1 - 22 T^{2} + 841 T^{4}$$)($$( 1 - 29 T^{2} )^{2}$$)($$1 - 22 T^{2} + 841 T^{4}$$)($$( 1 - 29 T^{2} )^{2}$$)($$1 - 22 T^{2} + 841 T^{4}$$)($$1 - 54 T^{2} + 841 T^{4}$$)
$31$ ($$( 1 - 4 T + 31 T^{2} )^{2}$$)($$( 1 + 31 T^{2} )^{2}$$)($$( 1 + 4 T + 31 T^{2} )^{2}$$)($$( 1 - 10 T + 31 T^{2} )^{2}$$)($$( 1 + 6 T + 31 T^{2} )^{2}$$)($$( 1 + 6 T + 31 T^{2} )^{2}$$)($$( 1 + 6 T + 31 T^{2} )^{2}$$)($$( 1 + 31 T^{2} )^{2}$$)($$( 1 - 8 T + 31 T^{2} )^{2}$$)($$( 1 + 31 T^{2} )^{2}$$)($$( 1 + 8 T + 31 T^{2} )^{2}$$)($$( 1 + 31 T^{2} )^{2}$$)($$( 1 + 10 T + 31 T^{2} )^{2}$$)($$( 1 - 6 T + 31 T^{2} )^{2}$$)($$( 1 - 6 T + 31 T^{2} )^{2}$$)($$( 1 - 6 T + 31 T^{2} )^{2}$$)($$( 1 + 4 T + 31 T^{2} )^{2}$$)($$( 1 + 31 T^{2} )^{2}$$)($$( 1 - 4 T + 31 T^{2} )^{2}$$)
$37$ ($$1 + 26 T^{2} + 1369 T^{4}$$)($$1 + 26 T^{2} + 1369 T^{4}$$)($$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$)($$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$)($$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$)($$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$)($$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$)($$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$)($$1 - 38 T^{2} + 1369 T^{4}$$)($$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$)($$1 - 38 T^{2} + 1369 T^{4}$$)($$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$)($$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$)($$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$)($$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$)($$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$)($$1 + 26 T^{2} + 1369 T^{4}$$)($$1 + 26 T^{2} + 1369 T^{4}$$)($$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$)
$41$ ($$( 1 + 41 T^{2} )^{2}$$)($$( 1 + 6 T + 41 T^{2} )^{2}$$)($$( 1 - 2 T + 41 T^{2} )^{2}$$)($$( 1 + 2 T + 41 T^{2} )^{2}$$)($$( 1 - 12 T + 41 T^{2} )^{2}$$)($$( 1 - 6 T + 41 T^{2} )^{2}$$)($$( 1 + 12 T + 41 T^{2} )^{2}$$)($$( 1 + 8 T + 41 T^{2} )^{2}$$)($$( 1 + 6 T + 41 T^{2} )^{2}$$)($$( 1 - 10 T + 41 T^{2} )^{2}$$)($$( 1 + 6 T + 41 T^{2} )^{2}$$)($$( 1 - 8 T + 41 T^{2} )^{2}$$)($$( 1 + 2 T + 41 T^{2} )^{2}$$)($$( 1 - 12 T + 41 T^{2} )^{2}$$)($$( 1 - 6 T + 41 T^{2} )^{2}$$)($$( 1 + 12 T + 41 T^{2} )^{2}$$)($$( 1 + 41 T^{2} )^{2}$$)($$( 1 + 6 T + 41 T^{2} )^{2}$$)($$( 1 - 2 T + 41 T^{2} )^{2}$$)
$43$ ($$1 - 22 T^{2} + 1849 T^{4}$$)($$1 - 50 T^{2} + 1849 T^{4}$$)($$1 - 70 T^{2} + 1849 T^{4}$$)($$1 - 22 T^{2} + 1849 T^{4}$$)($$1 + 58 T^{2} + 1849 T^{4}$$)($$( 1 - 43 T^{2} )^{2}$$)($$1 + 58 T^{2} + 1849 T^{4}$$)($$( 1 - 43 T^{2} )^{2}$$)($$1 - 70 T^{2} + 1849 T^{4}$$)($$( 1 - 43 T^{2} )^{2}$$)($$1 - 70 T^{2} + 1849 T^{4}$$)($$( 1 - 43 T^{2} )^{2}$$)($$1 - 22 T^{2} + 1849 T^{4}$$)($$1 + 58 T^{2} + 1849 T^{4}$$)($$( 1 - 43 T^{2} )^{2}$$)($$1 + 58 T^{2} + 1849 T^{4}$$)($$1 - 22 T^{2} + 1849 T^{4}$$)($$1 - 50 T^{2} + 1849 T^{4}$$)($$1 - 70 T^{2} + 1849 T^{4}$$)
$47$ ($$( 1 + 47 T^{2} )^{2}$$)($$( 1 + 8 T + 47 T^{2} )^{2}$$)($$( 1 - 8 T + 47 T^{2} )^{2}$$)($$( 1 - 12 T + 47 T^{2} )^{2}$$)($$( 1 - 8 T + 47 T^{2} )^{2}$$)($$( 1 + 4 T + 47 T^{2} )^{2}$$)($$( 1 + 8 T + 47 T^{2} )^{2}$$)($$( 1 + 47 T^{2} )^{2}$$)($$( 1 + 47 T^{2} )^{2}$$)($$( 1 + 47 T^{2} )^{2}$$)($$( 1 + 47 T^{2} )^{2}$$)($$( 1 + 47 T^{2} )^{2}$$)($$( 1 + 12 T + 47 T^{2} )^{2}$$)($$( 1 + 8 T + 47 T^{2} )^{2}$$)($$( 1 - 4 T + 47 T^{2} )^{2}$$)($$( 1 - 8 T + 47 T^{2} )^{2}$$)($$( 1 + 47 T^{2} )^{2}$$)($$( 1 - 8 T + 47 T^{2} )^{2}$$)($$( 1 + 8 T + 47 T^{2} )^{2}$$)
$53$ ($$( 1 - 53 T^{2} )^{2}$$)($$1 - 70 T^{2} + 2809 T^{4}$$)($$1 - 6 T^{2} + 2809 T^{4}$$)($$1 + 38 T^{2} + 2809 T^{4}$$)($$1 - 70 T^{2} + 2809 T^{4}$$)($$( 1 - 53 T^{2} )^{2}$$)($$1 - 70 T^{2} + 2809 T^{4}$$)($$( 1 - 14 T + 53 T^{2} )( 1 + 14 T + 53 T^{2} )$$)($$1 - 102 T^{2} + 2809 T^{4}$$)($$( 1 - 4 T + 53 T^{2} )( 1 + 4 T + 53 T^{2} )$$)($$1 - 102 T^{2} + 2809 T^{4}$$)($$( 1 - 14 T + 53 T^{2} )( 1 + 14 T + 53 T^{2} )$$)($$1 + 38 T^{2} + 2809 T^{4}$$)($$1 - 70 T^{2} + 2809 T^{4}$$)($$( 1 - 53 T^{2} )^{2}$$)($$1 - 70 T^{2} + 2809 T^{4}$$)($$( 1 - 53 T^{2} )^{2}$$)($$1 - 70 T^{2} + 2809 T^{4}$$)($$1 - 6 T^{2} + 2809 T^{4}$$)
$59$ ($$( 1 - 59 T^{2} )^{2}$$)($$1 + 78 T^{2} + 3481 T^{4}$$)($$1 - 102 T^{2} + 3481 T^{4}$$)($$1 - 102 T^{2} + 3481 T^{4}$$)($$1 - 54 T^{2} + 3481 T^{4}$$)($$1 - 102 T^{2} + 3481 T^{4}$$)($$1 - 54 T^{2} + 3481 T^{4}$$)($$( 1 - 59 T^{2} )^{2}$$)($$1 - 102 T^{2} + 3481 T^{4}$$)($$( 1 - 59 T^{2} )^{2}$$)($$1 - 102 T^{2} + 3481 T^{4}$$)($$( 1 - 59 T^{2} )^{2}$$)($$1 - 102 T^{2} + 3481 T^{4}$$)($$1 - 54 T^{2} + 3481 T^{4}$$)($$1 - 102 T^{2} + 3481 T^{4}$$)($$1 - 54 T^{2} + 3481 T^{4}$$)($$( 1 - 59 T^{2} )^{2}$$)($$1 + 78 T^{2} + 3481 T^{4}$$)($$1 - 102 T^{2} + 3481 T^{4}$$)
$61$ ($$1 + 74 T^{2} + 3721 T^{4}$$)($$1 - 118 T^{2} + 3721 T^{4}$$)($$1 - 86 T^{2} + 3721 T^{4}$$)($$1 - 118 T^{2} + 3721 T^{4}$$)($$( 1 - 12 T + 61 T^{2} )( 1 + 12 T + 61 T^{2} )$$)($$1 + 74 T^{2} + 3721 T^{4}$$)($$( 1 - 12 T + 61 T^{2} )( 1 + 12 T + 61 T^{2} )$$)($$( 1 - 12 T + 61 T^{2} )( 1 + 12 T + 61 T^{2} )$$)($$1 - 118 T^{2} + 3721 T^{4}$$)($$( 1 - 12 T + 61 T^{2} )( 1 + 12 T + 61 T^{2} )$$)($$1 - 118 T^{2} + 3721 T^{4}$$)($$( 1 - 12 T + 61 T^{2} )( 1 + 12 T + 61 T^{2} )$$)($$1 - 118 T^{2} + 3721 T^{4}$$)($$( 1 - 12 T + 61 T^{2} )( 1 + 12 T + 61 T^{2} )$$)($$1 + 74 T^{2} + 3721 T^{4}$$)($$( 1 - 12 T + 61 T^{2} )( 1 + 12 T + 61 T^{2} )$$)($$1 + 74 T^{2} + 3721 T^{4}$$)($$1 - 118 T^{2} + 3721 T^{4}$$)($$1 - 86 T^{2} + 3721 T^{4}$$)
$67$ ($$1 + 122 T^{2} + 4489 T^{4}$$)($$1 - 34 T^{2} + 4489 T^{4}$$)($$1 - 118 T^{2} + 4489 T^{4}$$)($$1 - 118 T^{2} + 4489 T^{4}$$)($$1 - 70 T^{2} + 4489 T^{4}$$)($$1 - 118 T^{2} + 4489 T^{4}$$)($$1 - 70 T^{2} + 4489 T^{4}$$)($$( 1 - 67 T^{2} )^{2}$$)($$1 - 118 T^{2} + 4489 T^{4}$$)($$( 1 - 67 T^{2} )^{2}$$)($$1 - 118 T^{2} + 4489 T^{4}$$)($$( 1 - 67 T^{2} )^{2}$$)($$1 - 118 T^{2} + 4489 T^{4}$$)($$1 - 70 T^{2} + 4489 T^{4}$$)($$1 - 118 T^{2} + 4489 T^{4}$$)($$1 - 70 T^{2} + 4489 T^{4}$$)($$1 + 122 T^{2} + 4489 T^{4}$$)($$1 - 34 T^{2} + 4489 T^{4}$$)($$1 - 118 T^{2} + 4489 T^{4}$$)
$71$ ($$( 1 + 71 T^{2} )^{2}$$)($$( 1 + 12 T + 71 T^{2} )^{2}$$)($$( 1 - 16 T + 71 T^{2} )^{2}$$)($$( 1 + 4 T + 71 T^{2} )^{2}$$)($$( 1 + 71 T^{2} )^{2}$$)($$( 1 - 12 T + 71 T^{2} )^{2}$$)($$( 1 + 71 T^{2} )^{2}$$)($$( 1 + 71 T^{2} )^{2}$$)($$( 1 - 8 T + 71 T^{2} )^{2}$$)($$( 1 + 71 T^{2} )^{2}$$)($$( 1 + 8 T + 71 T^{2} )^{2}$$)($$( 1 + 71 T^{2} )^{2}$$)($$( 1 - 4 T + 71 T^{2} )^{2}$$)($$( 1 + 71 T^{2} )^{2}$$)($$( 1 + 12 T + 71 T^{2} )^{2}$$)($$( 1 + 71 T^{2} )^{2}$$)($$( 1 + 71 T^{2} )^{2}$$)($$( 1 - 12 T + 71 T^{2} )^{2}$$)($$( 1 + 16 T + 71 T^{2} )^{2}$$)
$73$ ($$( 1 - 10 T + 73 T^{2} )^{2}$$)($$( 1 + 14 T + 73 T^{2} )^{2}$$)($$( 1 - 6 T + 73 T^{2} )^{2}$$)($$( 1 - 10 T + 73 T^{2} )^{2}$$)($$( 1 + 2 T + 73 T^{2} )^{2}$$)($$( 1 - 10 T + 73 T^{2} )^{2}$$)($$( 1 + 2 T + 73 T^{2} )^{2}$$)($$( 1 + 6 T + 73 T^{2} )^{2}$$)($$( 1 + 10 T + 73 T^{2} )^{2}$$)($$( 1 - 6 T + 73 T^{2} )^{2}$$)($$( 1 + 10 T + 73 T^{2} )^{2}$$)($$( 1 + 6 T + 73 T^{2} )^{2}$$)($$( 1 - 10 T + 73 T^{2} )^{2}$$)($$( 1 + 2 T + 73 T^{2} )^{2}$$)($$( 1 - 10 T + 73 T^{2} )^{2}$$)($$( 1 + 2 T + 73 T^{2} )^{2}$$)($$( 1 - 10 T + 73 T^{2} )^{2}$$)($$( 1 + 14 T + 73 T^{2} )^{2}$$)($$( 1 - 6 T + 73 T^{2} )^{2}$$)
$79$ ($$( 1 - 4 T + 79 T^{2} )^{2}$$)($$( 1 - 8 T + 79 T^{2} )^{2}$$)($$( 1 + 4 T + 79 T^{2} )^{2}$$)($$( 1 + 6 T + 79 T^{2} )^{2}$$)($$( 1 + 14 T + 79 T^{2} )^{2}$$)($$( 1 - 10 T + 79 T^{2} )^{2}$$)($$( 1 + 14 T + 79 T^{2} )^{2}$$)($$( 1 + 79 T^{2} )^{2}$$)($$( 1 + 8 T + 79 T^{2} )^{2}$$)($$( 1 + 79 T^{2} )^{2}$$)($$( 1 - 8 T + 79 T^{2} )^{2}$$)($$( 1 + 79 T^{2} )^{2}$$)($$( 1 - 6 T + 79 T^{2} )^{2}$$)($$( 1 - 14 T + 79 T^{2} )^{2}$$)($$( 1 + 10 T + 79 T^{2} )^{2}$$)($$( 1 - 14 T + 79 T^{2} )^{2}$$)($$( 1 + 4 T + 79 T^{2} )^{2}$$)($$( 1 + 8 T + 79 T^{2} )^{2}$$)($$( 1 - 4 T + 79 T^{2} )^{2}$$)
$83$ ($$( 1 - 83 T^{2} )^{2}$$)($$1 - 130 T^{2} + 6889 T^{4}$$)($$1 - 22 T^{2} + 6889 T^{4}$$)($$1 - 22 T^{2} + 6889 T^{4}$$)($$1 - 22 T^{2} + 6889 T^{4}$$)($$1 - 22 T^{2} + 6889 T^{4}$$)($$1 - 22 T^{2} + 6889 T^{4}$$)($$( 1 - 83 T^{2} )^{2}$$)($$1 - 150 T^{2} + 6889 T^{4}$$)($$( 1 - 83 T^{2} )^{2}$$)($$1 - 150 T^{2} + 6889 T^{4}$$)($$( 1 - 83 T^{2} )^{2}$$)($$1 - 22 T^{2} + 6889 T^{4}$$)($$1 - 22 T^{2} + 6889 T^{4}$$)($$1 - 22 T^{2} + 6889 T^{4}$$)($$1 - 22 T^{2} + 6889 T^{4}$$)($$( 1 - 83 T^{2} )^{2}$$)($$1 - 130 T^{2} + 6889 T^{4}$$)($$1 - 22 T^{2} + 6889 T^{4}$$)
$89$ ($$( 1 + 89 T^{2} )^{2}$$)($$( 1 + 2 T + 89 T^{2} )^{2}$$)($$( 1 - 10 T + 89 T^{2} )^{2}$$)($$( 1 - 2 T + 89 T^{2} )^{2}$$)($$( 1 + 8 T + 89 T^{2} )^{2}$$)($$( 1 + 14 T + 89 T^{2} )^{2}$$)($$( 1 - 8 T + 89 T^{2} )^{2}$$)($$( 1 - 16 T + 89 T^{2} )^{2}$$)($$( 1 + 6 T + 89 T^{2} )^{2}$$)($$( 1 - 10 T + 89 T^{2} )^{2}$$)($$( 1 + 6 T + 89 T^{2} )^{2}$$)($$( 1 + 16 T + 89 T^{2} )^{2}$$)($$( 1 - 2 T + 89 T^{2} )^{2}$$)($$( 1 + 8 T + 89 T^{2} )^{2}$$)($$( 1 + 14 T + 89 T^{2} )^{2}$$)($$( 1 - 8 T + 89 T^{2} )^{2}$$)($$( 1 + 89 T^{2} )^{2}$$)($$( 1 + 2 T + 89 T^{2} )^{2}$$)($$( 1 - 10 T + 89 T^{2} )^{2}$$)
$97$ ($$( 1 - 14 T + 97 T^{2} )^{2}$$)($$( 1 + 2 T + 97 T^{2} )^{2}$$)($$( 1 + 14 T + 97 T^{2} )^{2}$$)($$( 1 + 6 T + 97 T^{2} )^{2}$$)($$( 1 + 2 T + 97 T^{2} )^{2}$$)($$( 1 - 10 T + 97 T^{2} )^{2}$$)($$( 1 + 2 T + 97 T^{2} )^{2}$$)($$( 1 + 18 T + 97 T^{2} )^{2}$$)($$( 1 - 2 T + 97 T^{2} )^{2}$$)($$( 1 - 18 T + 97 T^{2} )^{2}$$)($$( 1 - 2 T + 97 T^{2} )^{2}$$)($$( 1 + 18 T + 97 T^{2} )^{2}$$)($$( 1 + 6 T + 97 T^{2} )^{2}$$)($$( 1 + 2 T + 97 T^{2} )^{2}$$)($$( 1 - 10 T + 97 T^{2} )^{2}$$)($$( 1 + 2 T + 97 T^{2} )^{2}$$)($$( 1 - 14 T + 97 T^{2} )^{2}$$)($$( 1 + 2 T + 97 T^{2} )^{2}$$)($$( 1 + 14 T + 97 T^{2} )^{2}$$)