# Properties

 Label 4032.2.c Level 4032 Weight 2 Character orbit c Rep. character $$\chi_{4032}(2017,\cdot)$$ Character field $$\Q$$ Dimension 60 Newform subspaces 18 Sturm bound 1536 Trace bound 25

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$4032 = 2^{6} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4032.c (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$8$$ Character field: $$\Q$$ Newform subspaces: $$18$$ Sturm bound: $$1536$$ Trace bound: $$25$$ Distinguishing $$T_p$$: $$5$$, $$11$$, $$13$$, $$17$$, $$23$$, $$31$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(4032, [\chi])$$.

Total New Old
Modular forms 816 60 756
Cusp forms 720 60 660
Eisenstein series 96 0 96

## Trace form

 $$60q + O(q^{10})$$ $$60q + 24q^{17} - 84q^{25} - 72q^{41} + 60q^{49} + 24q^{73} + 24q^{89} + 24q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(4032, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
4032.2.c.a $$2$$ $$32.196$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$-2$$ $$q+iq^{5}-q^{7}+2iq^{11}-iq^{13}-2q^{17}+\cdots$$
4032.2.c.b $$2$$ $$32.196$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$-2$$ $$q+2iq^{5}-q^{7}+iq^{11}-2iq^{13}-2q^{17}+\cdots$$
4032.2.c.c $$2$$ $$32.196$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$-2$$ $$q+iq^{5}-q^{7}-iq^{11}-iq^{13}-2q^{17}+\cdots$$
4032.2.c.d $$2$$ $$32.196$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$-2$$ $$q+iq^{5}-q^{7}+2iq^{11}+iq^{13}+2q^{17}+\cdots$$
4032.2.c.e $$2$$ $$32.196$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$-2$$ $$q+iq^{5}-q^{7}-iq^{11}+3iq^{13}+6q^{17}+\cdots$$
4032.2.c.f $$2$$ $$32.196$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$2$$ $$q+2iq^{5}+q^{7}-iq^{11}-2iq^{13}-2q^{17}+\cdots$$
4032.2.c.g $$2$$ $$32.196$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$2$$ $$q+iq^{5}+q^{7}+iq^{11}-iq^{13}-2q^{17}+\cdots$$
4032.2.c.h $$2$$ $$32.196$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$2$$ $$q+iq^{5}+q^{7}-2iq^{11}-iq^{13}-2q^{17}+\cdots$$
4032.2.c.i $$2$$ $$32.196$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$2$$ $$q+iq^{5}+q^{7}-2iq^{11}+iq^{13}+2q^{17}+\cdots$$
4032.2.c.j $$2$$ $$32.196$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$2$$ $$q+iq^{5}+q^{7}+iq^{11}+3iq^{13}+6q^{17}+\cdots$$
4032.2.c.k $$4$$ $$32.196$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$-4$$ $$q-\zeta_{12}q^{5}-q^{7}-2\zeta_{12}q^{11}+(-2\zeta_{12}+\cdots)q^{13}+\cdots$$
4032.2.c.l $$4$$ $$32.196$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$-4$$ $$q-\zeta_{12}^{3}q^{5}-q^{7}+(-\zeta_{12}^{2}-\zeta_{12}^{3})q^{11}+\cdots$$
4032.2.c.m $$4$$ $$32.196$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$-4$$ $$q-\zeta_{12}^{3}q^{5}-q^{7}+(\zeta_{12}^{2}+3\zeta_{12}^{3})q^{11}+\cdots$$
4032.2.c.n $$4$$ $$32.196$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$4$$ $$q-\zeta_{12}q^{5}+q^{7}+2\zeta_{12}q^{11}+(-2\zeta_{12}+\cdots)q^{13}+\cdots$$
4032.2.c.o $$4$$ $$32.196$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$4$$ $$q-\zeta_{12}^{3}q^{5}+q^{7}+(\zeta_{12}^{2}+\zeta_{12}^{3})q^{11}+\cdots$$
4032.2.c.p $$4$$ $$32.196$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$4$$ $$q-\zeta_{12}^{3}q^{5}+q^{7}+(-\zeta_{12}^{2}-3\zeta_{12}^{3})q^{11}+\cdots$$
4032.2.c.q $$8$$ $$32.196$$ 8.0.897122304.10 None $$0$$ $$0$$ $$0$$ $$-8$$ $$q-\beta _{5}q^{5}-q^{7}+\beta _{5}q^{11}+(-\beta _{4}-\beta _{6}+\cdots)q^{13}+\cdots$$
4032.2.c.r $$8$$ $$32.196$$ 8.0.897122304.10 None $$0$$ $$0$$ $$0$$ $$8$$ $$q-\beta _{5}q^{5}+q^{7}-\beta _{5}q^{11}+(-\beta _{4}-\beta _{6}+\cdots)q^{13}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(4032, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(4032, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(24, [\chi])$$$$^{\oplus 16}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(56, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(64, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(72, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(96, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(168, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(192, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(224, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(288, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(448, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(504, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(576, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(672, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1344, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(2016, [\chi])$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ ($$( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$)($$( 1 - 2 T + 5 T^{2} )( 1 + 2 T + 5 T^{2} )$$)($$( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$)($$( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$)($$( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$)($$( 1 - 2 T + 5 T^{2} )( 1 + 2 T + 5 T^{2} )$$)($$( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$)($$( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$)($$( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$)($$( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$)($$1 - 12 T^{2} + 74 T^{4} - 300 T^{6} + 625 T^{8}$$)($$1 - 12 T^{2} + 74 T^{4} - 300 T^{6} + 625 T^{8}$$)($$1 - 12 T^{2} + 74 T^{4} - 300 T^{6} + 625 T^{8}$$)($$1 - 12 T^{2} + 74 T^{4} - 300 T^{6} + 625 T^{8}$$)($$1 - 12 T^{2} + 74 T^{4} - 300 T^{6} + 625 T^{8}$$)($$1 - 12 T^{2} + 74 T^{4} - 300 T^{6} + 625 T^{8}$$)($$( 1 + 2 T^{4} + 625 T^{8} )^{2}$$)($$( 1 + 2 T^{4} + 625 T^{8} )^{2}$$)
$7$ ($$( 1 + T )^{2}$$)($$( 1 + T )^{2}$$)($$( 1 + T )^{2}$$)($$( 1 + T )^{2}$$)($$( 1 + T )^{2}$$)($$( 1 - T )^{2}$$)($$( 1 - T )^{2}$$)($$( 1 - T )^{2}$$)($$( 1 - T )^{2}$$)($$( 1 - T )^{2}$$)($$( 1 + T )^{4}$$)($$( 1 + T )^{4}$$)($$( 1 + T )^{4}$$)($$( 1 - T )^{4}$$)($$( 1 - T )^{4}$$)($$( 1 - T )^{4}$$)($$( 1 + T )^{8}$$)($$( 1 - T )^{8}$$)
$11$ ($$1 - 6 T^{2} + 121 T^{4}$$)($$1 - 18 T^{2} + 121 T^{4}$$)($$1 - 18 T^{2} + 121 T^{4}$$)($$1 - 6 T^{2} + 121 T^{4}$$)($$1 - 18 T^{2} + 121 T^{4}$$)($$1 - 18 T^{2} + 121 T^{4}$$)($$1 - 18 T^{2} + 121 T^{4}$$)($$1 - 6 T^{2} + 121 T^{4}$$)($$1 - 6 T^{2} + 121 T^{4}$$)($$1 - 18 T^{2} + 121 T^{4}$$)($$1 - 12 T^{2} + 86 T^{4} - 1452 T^{6} + 14641 T^{8}$$)($$1 - 36 T^{2} + 554 T^{4} - 4356 T^{6} + 14641 T^{8}$$)($$1 + 12 T^{2} + 170 T^{4} + 1452 T^{6} + 14641 T^{8}$$)($$1 - 12 T^{2} + 86 T^{4} - 1452 T^{6} + 14641 T^{8}$$)($$1 - 36 T^{2} + 554 T^{4} - 4356 T^{6} + 14641 T^{8}$$)($$1 + 12 T^{2} + 170 T^{4} + 1452 T^{6} + 14641 T^{8}$$)($$( 1 - 24 T^{2} + 338 T^{4} - 2904 T^{6} + 14641 T^{8} )^{2}$$)($$( 1 - 24 T^{2} + 338 T^{4} - 2904 T^{6} + 14641 T^{8} )^{2}$$)
$13$ ($$1 - 22 T^{2} + 169 T^{4}$$)($$( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} )$$)($$1 - 22 T^{2} + 169 T^{4}$$)($$1 - 22 T^{2} + 169 T^{4}$$)($$( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )$$)($$( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} )$$)($$1 - 22 T^{2} + 169 T^{4}$$)($$1 - 22 T^{2} + 169 T^{4}$$)($$1 - 22 T^{2} + 169 T^{4}$$)($$( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )$$)($$1 + 4 T^{2} + 42 T^{4} + 676 T^{6} + 28561 T^{8}$$)($$( 1 - 6 T + 13 T^{2} )^{2}( 1 + 6 T + 13 T^{2} )^{2}$$)($$( 1 - 13 T^{2} )^{4}$$)($$1 + 4 T^{2} + 42 T^{4} + 676 T^{6} + 28561 T^{8}$$)($$( 1 - 6 T + 13 T^{2} )^{2}( 1 + 6 T + 13 T^{2} )^{2}$$)($$( 1 - 13 T^{2} )^{4}$$)($$( 1 - 20 T^{2} + 246 T^{4} - 3380 T^{6} + 28561 T^{8} )^{2}$$)($$( 1 - 20 T^{2} + 246 T^{4} - 3380 T^{6} + 28561 T^{8} )^{2}$$)
$17$ ($$( 1 + 2 T + 17 T^{2} )^{2}$$)($$( 1 + 2 T + 17 T^{2} )^{2}$$)($$( 1 + 2 T + 17 T^{2} )^{2}$$)($$( 1 - 2 T + 17 T^{2} )^{2}$$)($$( 1 - 6 T + 17 T^{2} )^{2}$$)($$( 1 + 2 T + 17 T^{2} )^{2}$$)($$( 1 + 2 T + 17 T^{2} )^{2}$$)($$( 1 + 2 T + 17 T^{2} )^{2}$$)($$( 1 - 2 T + 17 T^{2} )^{2}$$)($$( 1 - 6 T + 17 T^{2} )^{2}$$)($$( 1 + 2 T + 17 T^{2} )^{4}$$)($$( 1 - 2 T + 8 T^{2} - 34 T^{3} + 289 T^{4} )^{2}$$)($$( 1 - 6 T + 40 T^{2} - 102 T^{3} + 289 T^{4} )^{2}$$)($$( 1 + 2 T + 17 T^{2} )^{4}$$)($$( 1 - 2 T + 8 T^{2} - 34 T^{3} + 289 T^{4} )^{2}$$)($$( 1 - 6 T + 40 T^{2} - 102 T^{3} + 289 T^{4} )^{2}$$)($$( 1 + 24 T^{2} + 290 T^{4} + 6936 T^{6} + 83521 T^{8} )^{2}$$)($$( 1 + 24 T^{2} + 290 T^{4} + 6936 T^{6} + 83521 T^{8} )^{2}$$)
$19$ ($$( 1 - 19 T^{2} )^{2}$$)($$1 - 2 T^{2} + 361 T^{4}$$)($$( 1 - 19 T^{2} )^{2}$$)($$( 1 - 19 T^{2} )^{2}$$)($$1 + 26 T^{2} + 361 T^{4}$$)($$1 - 2 T^{2} + 361 T^{4}$$)($$( 1 - 19 T^{2} )^{2}$$)($$( 1 - 19 T^{2} )^{2}$$)($$( 1 - 19 T^{2} )^{2}$$)($$1 + 26 T^{2} + 361 T^{4}$$)($$1 - 52 T^{2} + 1290 T^{4} - 18772 T^{6} + 130321 T^{8}$$)($$( 1 - 8 T + 19 T^{2} )^{2}( 1 + 8 T + 19 T^{2} )^{2}$$)($$1 - 20 T^{2} + 54 T^{4} - 7220 T^{6} + 130321 T^{8}$$)($$1 - 52 T^{2} + 1290 T^{4} - 18772 T^{6} + 130321 T^{8}$$)($$( 1 - 8 T + 19 T^{2} )^{2}( 1 + 8 T + 19 T^{2} )^{2}$$)($$1 - 20 T^{2} + 54 T^{4} - 7220 T^{6} + 130321 T^{8}$$)($$( 1 - 8 T + 19 T^{2} )^{4}( 1 + 8 T + 19 T^{2} )^{4}$$)($$( 1 - 8 T + 19 T^{2} )^{4}( 1 + 8 T + 19 T^{2} )^{4}$$)
$23$ ($$( 1 + 6 T + 23 T^{2} )^{2}$$)($$( 1 + 23 T^{2} )^{2}$$)($$( 1 + 23 T^{2} )^{2}$$)($$( 1 - 6 T + 23 T^{2} )^{2}$$)($$( 1 - 8 T + 23 T^{2} )^{2}$$)($$( 1 + 23 T^{2} )^{2}$$)($$( 1 + 23 T^{2} )^{2}$$)($$( 1 - 6 T + 23 T^{2} )^{2}$$)($$( 1 + 6 T + 23 T^{2} )^{2}$$)($$( 1 + 8 T + 23 T^{2} )^{2}$$)($$( 1 + 34 T^{2} + 529 T^{4} )^{2}$$)($$( 1 - 6 T + 52 T^{2} - 138 T^{3} + 529 T^{4} )^{2}$$)($$( 1 + 14 T + 92 T^{2} + 322 T^{3} + 529 T^{4} )^{2}$$)($$( 1 + 34 T^{2} + 529 T^{4} )^{2}$$)($$( 1 + 6 T + 52 T^{2} + 138 T^{3} + 529 T^{4} )^{2}$$)($$( 1 - 14 T + 92 T^{2} - 322 T^{3} + 529 T^{4} )^{2}$$)($$( 1 + 32 T^{2} + 882 T^{4} + 16928 T^{6} + 279841 T^{8} )^{2}$$)($$( 1 + 32 T^{2} + 882 T^{4} + 16928 T^{6} + 279841 T^{8} )^{2}$$)
$29$ ($$1 - 54 T^{2} + 841 T^{4}$$)($$1 + 6 T^{2} + 841 T^{4}$$)($$1 + 6 T^{2} + 841 T^{4}$$)($$1 - 54 T^{2} + 841 T^{4}$$)($$( 1 - 29 T^{2} )^{2}$$)($$1 + 6 T^{2} + 841 T^{4}$$)($$1 + 6 T^{2} + 841 T^{4}$$)($$1 - 54 T^{2} + 841 T^{4}$$)($$1 - 54 T^{2} + 841 T^{4}$$)($$( 1 - 29 T^{2} )^{2}$$)($$1 - 84 T^{2} + 3254 T^{4} - 70644 T^{6} + 707281 T^{8}$$)($$1 - 60 T^{2} + 1814 T^{4} - 50460 T^{6} + 707281 T^{8}$$)($$( 1 - 46 T^{2} + 841 T^{4} )^{2}$$)($$1 - 84 T^{2} + 3254 T^{4} - 70644 T^{6} + 707281 T^{8}$$)($$1 - 60 T^{2} + 1814 T^{4} - 50460 T^{6} + 707281 T^{8}$$)($$( 1 - 46 T^{2} + 841 T^{4} )^{2}$$)($$( 1 - 4 T + 14 T^{2} - 116 T^{3} + 841 T^{4} )^{2}( 1 + 4 T + 14 T^{2} + 116 T^{3} + 841 T^{4} )^{2}$$)($$( 1 - 4 T + 14 T^{2} - 116 T^{3} + 841 T^{4} )^{2}( 1 + 4 T + 14 T^{2} + 116 T^{3} + 841 T^{4} )^{2}$$)
$31$ ($$( 1 + 4 T + 31 T^{2} )^{2}$$)($$( 1 - 8 T + 31 T^{2} )^{2}$$)($$( 1 - 8 T + 31 T^{2} )^{2}$$)($$( 1 + 4 T + 31 T^{2} )^{2}$$)($$( 1 + 8 T + 31 T^{2} )^{2}$$)($$( 1 + 8 T + 31 T^{2} )^{2}$$)($$( 1 + 8 T + 31 T^{2} )^{2}$$)($$( 1 - 4 T + 31 T^{2} )^{2}$$)($$( 1 - 4 T + 31 T^{2} )^{2}$$)($$( 1 - 8 T + 31 T^{2} )^{2}$$)($$( 1 + 4 T + 31 T^{2} )^{4}$$)($$( 1 - 2 T + 31 T^{2} )^{4}$$)($$( 1 + 4 T + 18 T^{2} + 124 T^{3} + 961 T^{4} )^{2}$$)($$( 1 - 4 T + 31 T^{2} )^{4}$$)($$( 1 + 2 T + 31 T^{2} )^{4}$$)($$( 1 - 4 T + 18 T^{2} - 124 T^{3} + 961 T^{4} )^{2}$$)($$( 1 - 2 T + 31 T^{2} )^{8}$$)($$( 1 + 2 T + 31 T^{2} )^{8}$$)
$37$ ($$1 - 58 T^{2} + 1369 T^{4}$$)($$1 - 10 T^{2} + 1369 T^{4}$$)($$1 - 58 T^{2} + 1369 T^{4}$$)($$1 - 58 T^{2} + 1369 T^{4}$$)($$1 - 58 T^{2} + 1369 T^{4}$$)($$1 - 10 T^{2} + 1369 T^{4}$$)($$1 - 58 T^{2} + 1369 T^{4}$$)($$1 - 58 T^{2} + 1369 T^{4}$$)($$1 - 58 T^{2} + 1369 T^{4}$$)($$1 - 58 T^{2} + 1369 T^{4}$$)($$1 - 116 T^{2} + 5910 T^{4} - 158804 T^{6} + 1874161 T^{8}$$)($$( 1 - 12 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2}$$)($$( 1 + 26 T^{2} + 1369 T^{4} )^{2}$$)($$1 - 116 T^{2} + 5910 T^{4} - 158804 T^{6} + 1874161 T^{8}$$)($$( 1 - 12 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2}$$)($$( 1 + 26 T^{2} + 1369 T^{4} )^{2}$$)($$( 1 + 4 T^{2} - 330 T^{4} + 5476 T^{6} + 1874161 T^{8} )^{2}$$)($$( 1 + 4 T^{2} - 330 T^{4} + 5476 T^{6} + 1874161 T^{8} )^{2}$$)
$41$ ($$( 1 + 10 T + 41 T^{2} )^{2}$$)($$( 1 + 10 T + 41 T^{2} )^{2}$$)($$( 1 - 2 T + 41 T^{2} )^{2}$$)($$( 1 - 10 T + 41 T^{2} )^{2}$$)($$( 1 + 6 T + 41 T^{2} )^{2}$$)($$( 1 + 10 T + 41 T^{2} )^{2}$$)($$( 1 - 2 T + 41 T^{2} )^{2}$$)($$( 1 + 10 T + 41 T^{2} )^{2}$$)($$( 1 - 10 T + 41 T^{2} )^{2}$$)($$( 1 + 6 T + 41 T^{2} )^{2}$$)($$( 1 - 2 T + 41 T^{2} )^{4}$$)($$( 1 + 2 T + 8 T^{2} + 82 T^{3} + 1681 T^{4} )^{2}$$)($$( 1 + 6 T + 88 T^{2} + 246 T^{3} + 1681 T^{4} )^{2}$$)($$( 1 - 2 T + 41 T^{2} )^{4}$$)($$( 1 + 2 T + 8 T^{2} + 82 T^{3} + 1681 T^{4} )^{2}$$)($$( 1 + 6 T + 88 T^{2} + 246 T^{3} + 1681 T^{4} )^{2}$$)($$( 1 + 120 T^{2} + 6530 T^{4} + 201720 T^{6} + 2825761 T^{8} )^{2}$$)($$( 1 + 120 T^{2} + 6530 T^{4} + 201720 T^{6} + 2825761 T^{8} )^{2}$$)
$43$ ($$1 - 82 T^{2} + 1849 T^{4}$$)($$1 - 82 T^{2} + 1849 T^{4}$$)($$1 + 14 T^{2} + 1849 T^{4}$$)($$1 - 82 T^{2} + 1849 T^{4}$$)($$1 - 50 T^{2} + 1849 T^{4}$$)($$1 - 82 T^{2} + 1849 T^{4}$$)($$1 + 14 T^{2} + 1849 T^{4}$$)($$1 - 82 T^{2} + 1849 T^{4}$$)($$1 - 82 T^{2} + 1849 T^{4}$$)($$1 - 50 T^{2} + 1849 T^{4}$$)($$1 - 140 T^{2} + 8406 T^{4} - 258860 T^{6} + 3418801 T^{8}$$)($$1 - 68 T^{2} + 4086 T^{4} - 125732 T^{6} + 3418801 T^{8}$$)($$1 - 4 T^{2} - 3210 T^{4} - 7396 T^{6} + 3418801 T^{8}$$)($$1 - 140 T^{2} + 8406 T^{4} - 258860 T^{6} + 3418801 T^{8}$$)($$1 - 68 T^{2} + 4086 T^{4} - 125732 T^{6} + 3418801 T^{8}$$)($$1 - 4 T^{2} - 3210 T^{4} - 7396 T^{6} + 3418801 T^{8}$$)($$( 1 - 68 T^{2} + 4086 T^{4} - 125732 T^{6} + 3418801 T^{8} )^{2}$$)($$( 1 - 68 T^{2} + 4086 T^{4} - 125732 T^{6} + 3418801 T^{8} )^{2}$$)
$47$ ($$( 1 - 8 T + 47 T^{2} )^{2}$$)($$( 1 - 8 T + 47 T^{2} )^{2}$$)($$( 1 + 4 T + 47 T^{2} )^{2}$$)($$( 1 + 8 T + 47 T^{2} )^{2}$$)($$( 1 - 12 T + 47 T^{2} )^{2}$$)($$( 1 + 8 T + 47 T^{2} )^{2}$$)($$( 1 - 4 T + 47 T^{2} )^{2}$$)($$( 1 + 8 T + 47 T^{2} )^{2}$$)($$( 1 - 8 T + 47 T^{2} )^{2}$$)($$( 1 + 12 T + 47 T^{2} )^{2}$$)($$( 1 + 8 T + 62 T^{2} + 376 T^{3} + 2209 T^{4} )^{2}$$)($$( 1 + 4 T + 47 T^{2} )^{4}$$)($$( 1 + 46 T^{2} + 2209 T^{4} )^{2}$$)($$( 1 - 8 T + 62 T^{2} - 376 T^{3} + 2209 T^{4} )^{2}$$)($$( 1 - 4 T + 47 T^{2} )^{4}$$)($$( 1 + 46 T^{2} + 2209 T^{4} )^{2}$$)($$( 1 + 60 T^{2} + 4550 T^{4} + 132540 T^{6} + 4879681 T^{8} )^{2}$$)($$( 1 + 60 T^{2} + 4550 T^{4} + 132540 T^{6} + 4879681 T^{8} )^{2}$$)
$53$ ($$1 - 70 T^{2} + 2809 T^{4}$$)($$( 1 - 53 T^{2} )^{2}$$)($$1 + 38 T^{2} + 2809 T^{4}$$)($$1 - 70 T^{2} + 2809 T^{4}$$)($$1 + 38 T^{2} + 2809 T^{4}$$)($$( 1 - 53 T^{2} )^{2}$$)($$1 + 38 T^{2} + 2809 T^{4}$$)($$1 - 70 T^{2} + 2809 T^{4}$$)($$1 - 70 T^{2} + 2809 T^{4}$$)($$1 + 38 T^{2} + 2809 T^{4}$$)($$( 1 + 38 T^{2} + 2809 T^{4} )^{2}$$)($$( 1 - 94 T^{2} + 2809 T^{4} )^{2}$$)($$( 1 - 94 T^{2} + 2809 T^{4} )^{2}$$)($$( 1 + 38 T^{2} + 2809 T^{4} )^{2}$$)($$( 1 - 94 T^{2} + 2809 T^{4} )^{2}$$)($$( 1 - 94 T^{2} + 2809 T^{4} )^{2}$$)($$( 1 - 53 T^{2} )^{8}$$)($$( 1 - 53 T^{2} )^{8}$$)
$59$ ($$1 - 102 T^{2} + 3481 T^{4}$$)($$1 - 18 T^{2} + 3481 T^{4}$$)($$1 - 102 T^{2} + 3481 T^{4}$$)($$1 - 102 T^{2} + 3481 T^{4}$$)($$1 - 102 T^{2} + 3481 T^{4}$$)($$1 - 18 T^{2} + 3481 T^{4}$$)($$1 - 102 T^{2} + 3481 T^{4}$$)($$1 - 102 T^{2} + 3481 T^{4}$$)($$1 - 102 T^{2} + 3481 T^{4}$$)($$1 - 102 T^{2} + 3481 T^{4}$$)($$1 - 84 T^{2} + 8426 T^{4} - 292404 T^{6} + 12117361 T^{8}$$)($$( 1 - 54 T^{2} + 3481 T^{4} )^{2}$$)($$1 - 108 T^{2} + 6806 T^{4} - 375948 T^{6} + 12117361 T^{8}$$)($$1 - 84 T^{2} + 8426 T^{4} - 292404 T^{6} + 12117361 T^{8}$$)($$( 1 - 54 T^{2} + 3481 T^{4} )^{2}$$)($$1 - 108 T^{2} + 6806 T^{4} - 375948 T^{6} + 12117361 T^{8}$$)($$( 1 - 108 T^{2} + 9110 T^{4} - 375948 T^{6} + 12117361 T^{8} )^{2}$$)($$( 1 - 108 T^{2} + 9110 T^{4} - 375948 T^{6} + 12117361 T^{8} )^{2}$$)
$61$ ($$( 1 - 12 T + 61 T^{2} )( 1 + 12 T + 61 T^{2} )$$)($$1 - 106 T^{2} + 3721 T^{4}$$)($$1 - 118 T^{2} + 3721 T^{4}$$)($$( 1 - 12 T + 61 T^{2} )( 1 + 12 T + 61 T^{2} )$$)($$( 1 - 12 T + 61 T^{2} )( 1 + 12 T + 61 T^{2} )$$)($$1 - 106 T^{2} + 3721 T^{4}$$)($$1 - 118 T^{2} + 3721 T^{4}$$)($$( 1 - 12 T + 61 T^{2} )( 1 + 12 T + 61 T^{2} )$$)($$( 1 - 12 T + 61 T^{2} )( 1 + 12 T + 61 T^{2} )$$)($$( 1 - 12 T + 61 T^{2} )( 1 + 12 T + 61 T^{2} )$$)($$1 + 52 T^{2} + 7530 T^{4} + 193492 T^{6} + 13845841 T^{8}$$)($$1 - 20 T^{2} + 5814 T^{4} - 74420 T^{6} + 13845841 T^{8}$$)($$1 - 212 T^{2} + 18486 T^{4} - 788852 T^{6} + 13845841 T^{8}$$)($$1 + 52 T^{2} + 7530 T^{4} + 193492 T^{6} + 13845841 T^{8}$$)($$1 - 20 T^{2} + 5814 T^{4} - 74420 T^{6} + 13845841 T^{8}$$)($$1 - 212 T^{2} + 18486 T^{4} - 788852 T^{6} + 13845841 T^{8}$$)($$( 1 - 212 T^{2} + 18486 T^{4} - 788852 T^{6} + 13845841 T^{8} )^{2}$$)($$( 1 - 212 T^{2} + 18486 T^{4} - 788852 T^{6} + 13845841 T^{8} )^{2}$$)
$67$ ($$1 - 130 T^{2} + 4489 T^{4}$$)($$1 - 130 T^{2} + 4489 T^{4}$$)($$1 - 130 T^{2} + 4489 T^{4}$$)($$1 - 130 T^{2} + 4489 T^{4}$$)($$1 - 130 T^{2} + 4489 T^{4}$$)($$1 - 130 T^{2} + 4489 T^{4}$$)($$1 - 130 T^{2} + 4489 T^{4}$$)($$1 - 130 T^{2} + 4489 T^{4}$$)($$1 - 130 T^{2} + 4489 T^{4}$$)($$1 - 130 T^{2} + 4489 T^{4}$$)($$( 1 - 70 T^{2} + 4489 T^{4} )^{2}$$)($$1 - 20 T^{2} + 2166 T^{4} - 89780 T^{6} + 20151121 T^{8}$$)($$1 - 212 T^{2} + 19446 T^{4} - 951668 T^{6} + 20151121 T^{8}$$)($$( 1 - 70 T^{2} + 4489 T^{4} )^{2}$$)($$1 - 20 T^{2} + 2166 T^{4} - 89780 T^{6} + 20151121 T^{8}$$)($$1 - 212 T^{2} + 19446 T^{4} - 951668 T^{6} + 20151121 T^{8}$$)($$( 1 - 34 T^{2} + 4489 T^{4} )^{4}$$)($$( 1 - 34 T^{2} + 4489 T^{4} )^{4}$$)
$71$ ($$( 1 - 2 T + 71 T^{2} )^{2}$$)($$( 1 - 8 T + 71 T^{2} )^{2}$$)($$( 1 - 8 T + 71 T^{2} )^{2}$$)($$( 1 + 2 T + 71 T^{2} )^{2}$$)($$( 1 + 71 T^{2} )^{2}$$)($$( 1 + 8 T + 71 T^{2} )^{2}$$)($$( 1 + 8 T + 71 T^{2} )^{2}$$)($$( 1 + 2 T + 71 T^{2} )^{2}$$)($$( 1 - 2 T + 71 T^{2} )^{2}$$)($$( 1 + 71 T^{2} )^{2}$$)($$( 1 + 8 T + 110 T^{2} + 568 T^{3} + 5041 T^{4} )^{2}$$)($$( 1 - 10 T + 92 T^{2} - 710 T^{3} + 5041 T^{4} )^{2}$$)($$( 1 + 18 T + 196 T^{2} + 1278 T^{3} + 5041 T^{4} )^{2}$$)($$( 1 - 8 T + 110 T^{2} - 568 T^{3} + 5041 T^{4} )^{2}$$)($$( 1 + 10 T + 92 T^{2} + 710 T^{3} + 5041 T^{4} )^{2}$$)($$( 1 - 18 T + 196 T^{2} - 1278 T^{3} + 5041 T^{4} )^{2}$$)($$( 1 + 96 T^{2} + 12338 T^{4} + 483936 T^{6} + 25411681 T^{8} )^{2}$$)($$( 1 + 96 T^{2} + 12338 T^{4} + 483936 T^{6} + 25411681 T^{8} )^{2}$$)
$73$ ($$( 1 + 6 T + 73 T^{2} )^{2}$$)($$( 1 + 6 T + 73 T^{2} )^{2}$$)($$( 1 - 6 T + 73 T^{2} )^{2}$$)($$( 1 + 6 T + 73 T^{2} )^{2}$$)($$( 1 - 6 T + 73 T^{2} )^{2}$$)($$( 1 + 6 T + 73 T^{2} )^{2}$$)($$( 1 - 6 T + 73 T^{2} )^{2}$$)($$( 1 + 6 T + 73 T^{2} )^{2}$$)($$( 1 + 6 T + 73 T^{2} )^{2}$$)($$( 1 - 6 T + 73 T^{2} )^{2}$$)($$( 1 - 12 T + 134 T^{2} - 876 T^{3} + 5329 T^{4} )^{2}$$)($$( 1 + 24 T + 278 T^{2} + 1752 T^{3} + 5329 T^{4} )^{2}$$)($$( 1 + 38 T^{2} + 5329 T^{4} )^{2}$$)($$( 1 - 12 T + 134 T^{2} - 876 T^{3} + 5329 T^{4} )^{2}$$)($$( 1 + 24 T + 278 T^{2} + 1752 T^{3} + 5329 T^{4} )^{2}$$)($$( 1 + 38 T^{2} + 5329 T^{4} )^{2}$$)($$( 1 - 12 T + 134 T^{2} - 876 T^{3} + 5329 T^{4} )^{4}$$)($$( 1 - 12 T + 134 T^{2} - 876 T^{3} + 5329 T^{4} )^{4}$$)
$79$ ($$( 1 + 8 T + 79 T^{2} )^{2}$$)($$( 1 + 8 T + 79 T^{2} )^{2}$$)($$( 1 - 16 T + 79 T^{2} )^{2}$$)($$( 1 + 8 T + 79 T^{2} )^{2}$$)($$( 1 + 79 T^{2} )^{2}$$)($$( 1 - 8 T + 79 T^{2} )^{2}$$)($$( 1 + 16 T + 79 T^{2} )^{2}$$)($$( 1 - 8 T + 79 T^{2} )^{2}$$)($$( 1 - 8 T + 79 T^{2} )^{2}$$)($$( 1 + 79 T^{2} )^{2}$$)($$( 1 - 8 T + 126 T^{2} - 632 T^{3} + 6241 T^{4} )^{2}$$)($$( 1 + 4 T + 54 T^{2} + 316 T^{3} + 6241 T^{4} )^{2}$$)($$( 1 + 12 T + 182 T^{2} + 948 T^{3} + 6241 T^{4} )^{2}$$)($$( 1 + 8 T + 126 T^{2} + 632 T^{3} + 6241 T^{4} )^{2}$$)($$( 1 - 4 T + 54 T^{2} - 316 T^{3} + 6241 T^{4} )^{2}$$)($$( 1 - 12 T + 182 T^{2} - 948 T^{3} + 6241 T^{4} )^{2}$$)($$( 1 - 8 T + 126 T^{2} - 632 T^{3} + 6241 T^{4} )^{4}$$)($$( 1 + 8 T + 126 T^{2} + 632 T^{3} + 6241 T^{4} )^{4}$$)
$83$ ($$1 - 22 T^{2} + 6889 T^{4}$$)($$1 - 130 T^{2} + 6889 T^{4}$$)($$( 1 - 83 T^{2} )^{2}$$)($$1 - 22 T^{2} + 6889 T^{4}$$)($$1 - 102 T^{2} + 6889 T^{4}$$)($$1 - 130 T^{2} + 6889 T^{4}$$)($$( 1 - 83 T^{2} )^{2}$$)($$1 - 22 T^{2} + 6889 T^{4}$$)($$1 - 22 T^{2} + 6889 T^{4}$$)($$1 - 102 T^{2} + 6889 T^{4}$$)($$1 - 164 T^{2} + 17802 T^{4} - 1129796 T^{6} + 47458321 T^{8}$$)($$1 - 236 T^{2} + 25974 T^{4} - 1625804 T^{6} + 47458321 T^{8}$$)($$1 - 108 T^{2} + 14966 T^{4} - 744012 T^{6} + 47458321 T^{8}$$)($$1 - 164 T^{2} + 17802 T^{4} - 1129796 T^{6} + 47458321 T^{8}$$)($$1 - 236 T^{2} + 25974 T^{4} - 1625804 T^{6} + 47458321 T^{8}$$)($$1 - 108 T^{2} + 14966 T^{4} - 744012 T^{6} + 47458321 T^{8}$$)($$( 1 - 92 T^{2} + 8982 T^{4} - 633788 T^{6} + 47458321 T^{8} )^{2}$$)($$( 1 - 92 T^{2} + 8982 T^{4} - 633788 T^{6} + 47458321 T^{8} )^{2}$$)
$89$ ($$( 1 - 2 T + 89 T^{2} )^{2}$$)($$( 1 + 10 T + 89 T^{2} )^{2}$$)($$( 1 - 14 T + 89 T^{2} )^{2}$$)($$( 1 + 2 T + 89 T^{2} )^{2}$$)($$( 1 - 6 T + 89 T^{2} )^{2}$$)($$( 1 + 10 T + 89 T^{2} )^{2}$$)($$( 1 - 14 T + 89 T^{2} )^{2}$$)($$( 1 - 2 T + 89 T^{2} )^{2}$$)($$( 1 + 2 T + 89 T^{2} )^{2}$$)($$( 1 - 6 T + 89 T^{2} )^{2}$$)($$( 1 - 4 T - 10 T^{2} - 356 T^{3} + 7921 T^{4} )^{2}$$)($$( 1 + 14 T + 200 T^{2} + 1246 T^{3} + 7921 T^{4} )^{2}$$)($$( 1 - 6 T + 184 T^{2} - 534 T^{3} + 7921 T^{4} )^{2}$$)($$( 1 - 4 T - 10 T^{2} - 356 T^{3} + 7921 T^{4} )^{2}$$)($$( 1 + 14 T + 200 T^{2} + 1246 T^{3} + 7921 T^{4} )^{2}$$)($$( 1 - 6 T + 184 T^{2} - 534 T^{3} + 7921 T^{4} )^{2}$$)($$( 1 + 120 T^{2} + 5570 T^{4} + 950520 T^{6} + 62742241 T^{8} )^{2}$$)($$( 1 + 120 T^{2} + 5570 T^{4} + 950520 T^{6} + 62742241 T^{8} )^{2}$$)
$97$ ($$( 1 - 2 T + 97 T^{2} )^{2}$$)($$( 1 - 2 T + 97 T^{2} )^{2}$$)($$( 1 + 10 T + 97 T^{2} )^{2}$$)($$( 1 - 2 T + 97 T^{2} )^{2}$$)($$( 1 + 10 T + 97 T^{2} )^{2}$$)($$( 1 - 2 T + 97 T^{2} )^{2}$$)($$( 1 + 10 T + 97 T^{2} )^{2}$$)($$( 1 - 2 T + 97 T^{2} )^{2}$$)($$( 1 - 2 T + 97 T^{2} )^{2}$$)($$( 1 + 10 T + 97 T^{2} )^{2}$$)($$( 1 - 4 T + 150 T^{2} - 388 T^{3} + 9409 T^{4} )^{2}$$)($$( 1 - 16 T + 150 T^{2} - 1552 T^{3} + 9409 T^{4} )^{2}$$)($$( 1 + 8 T + 102 T^{2} + 776 T^{3} + 9409 T^{4} )^{2}$$)($$( 1 - 4 T + 150 T^{2} - 388 T^{3} + 9409 T^{4} )^{2}$$)($$( 1 - 16 T + 150 T^{2} - 1552 T^{3} + 9409 T^{4} )^{2}$$)($$( 1 + 8 T + 102 T^{2} + 776 T^{3} + 9409 T^{4} )^{2}$$)($$( 1 - 4 T + 150 T^{2} - 388 T^{3} + 9409 T^{4} )^{4}$$)($$( 1 - 4 T + 150 T^{2} - 388 T^{3} + 9409 T^{4} )^{4}$$)