# Properties

 Label 4032.2.b Level 4032 Weight 2 Character orbit b Rep. character $$\chi_{4032}(3583,\cdot)$$ Character field $$\Q$$ Dimension 78 Newform subspaces 18 Sturm bound 1536 Trace bound 25

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## Defining parameters

 Level: $$N$$ $$=$$ $$4032 = 2^{6} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4032.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$28$$ Character field: $$\Q$$ Newform subspaces: $$18$$ Sturm bound: $$1536$$ Trace bound: $$25$$ Distinguishing $$T_p$$: $$5$$, $$11$$, $$19$$

## Dimensions

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

Total New Old
Modular forms 816 82 734
Cusp forms 720 78 642
Eisenstein series 96 4 92

## Trace form

 $$78q + O(q^{10})$$ $$78q - 74q^{25} - 12q^{29} - 20q^{37} - 2q^{49} + 4q^{53} - 16q^{65} + 36q^{77} - 64q^{85} + O(q^{100})$$

## 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.b.a $$2$$ $$32.196$$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$-4$$ $$q+(-2+\zeta_{6})q^{7}+4\zeta_{6}q^{13}-8q^{19}+\cdots$$
4032.2.b.b $$2$$ $$32.196$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$-4$$ $$q-2\zeta_{6}q^{5}+(-2+\zeta_{6})q^{7}+2\zeta_{6}q^{11}+\cdots$$
4032.2.b.c $$2$$ $$32.196$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$-4$$ $$q+(-2-\zeta_{6})q^{7}-2\zeta_{6}q^{11}+4\zeta_{6}q^{17}+\cdots$$
4032.2.b.d $$2$$ $$32.196$$ $$\Q(\sqrt{-6})$$ None $$0$$ $$0$$ $$0$$ $$-2$$ $$q+\beta q^{5}+(-1+\beta )q^{7}-\beta q^{11}-2\beta q^{13}+\cdots$$
4032.2.b.e $$2$$ $$32.196$$ $$\Q(\sqrt{-7})$$ $$\Q(\sqrt{-7})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta q^{7}+2\beta q^{11}+2\beta q^{23}+5q^{25}+\cdots$$
4032.2.b.f $$2$$ $$32.196$$ $$\Q(\sqrt{-6})$$ None $$0$$ $$0$$ $$0$$ $$2$$ $$q+\beta q^{5}+(1-\beta )q^{7}+\beta q^{11}-2\beta q^{13}+\cdots$$
4032.2.b.g $$2$$ $$32.196$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$4$$ $$q+(2+\zeta_{6})q^{7}+2\zeta_{6}q^{11}+4\zeta_{6}q^{17}+\cdots$$
4032.2.b.h $$2$$ $$32.196$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$4$$ $$q-2\zeta_{6}q^{5}+(2-\zeta_{6})q^{7}-2\zeta_{6}q^{11}+\cdots$$
4032.2.b.i $$2$$ $$32.196$$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$4$$ $$q+(2+\zeta_{6})q^{7}-4\zeta_{6}q^{13}+8q^{19}+5q^{25}+\cdots$$
4032.2.b.j $$4$$ $$32.196$$ 4.0.2312.1 None $$0$$ $$0$$ $$0$$ $$-2$$ $$q+(-1+\beta _{1}+\beta _{2}+\beta _{3})q^{5}-\beta _{3}q^{7}+\cdots$$
4032.2.b.k $$4$$ $$32.196$$ $$\Q(\sqrt{-2}, \sqrt{7})$$ $$\Q(\sqrt{-21})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{2}q^{5}-\beta _{3}q^{7}-\beta _{1}q^{11}-\beta _{2}q^{17}+\cdots$$
4032.2.b.l $$4$$ $$32.196$$ $$\Q(\sqrt{-6}, \sqrt{7})$$ $$\Q(\sqrt{-21})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{1}q^{5}-\beta _{3}q^{7}-\beta _{2}q^{11}-3\beta _{1}q^{17}+\cdots$$
4032.2.b.m $$4$$ $$32.196$$ $$\Q(i, \sqrt{7})$$ $$\Q(\sqrt{-7})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{3}q^{7}-\beta _{1}q^{11}+2\beta _{1}q^{23}+5q^{25}+\cdots$$
4032.2.b.n $$4$$ $$32.196$$ 4.0.2312.1 None $$0$$ $$0$$ $$0$$ $$2$$ $$q+(-1+\beta _{1}+\beta _{2}+\beta _{3})q^{5}+\beta _{3}q^{7}+\cdots$$
4032.2.b.o $$8$$ $$32.196$$ 8.0.836829184.2 None $$0$$ $$0$$ $$0$$ $$-4$$ $$q-\beta _{1}q^{5}-\beta _{2}q^{7}+(\beta _{2}+\beta _{4}+\beta _{7})q^{11}+\cdots$$
4032.2.b.p $$8$$ $$32.196$$ $$\Q(\zeta_{16})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{16}^{5}q^{5}+\zeta_{16}^{4}q^{7}+\zeta_{16}q^{11}+\cdots$$
4032.2.b.q $$8$$ $$32.196$$ 8.0.836829184.2 None $$0$$ $$0$$ $$0$$ $$4$$ $$q-\beta _{1}q^{5}+\beta _{2}q^{7}+(-\beta _{2}-\beta _{4}-\beta _{7})q^{11}+\cdots$$
4032.2.b.r $$16$$ $$32.196$$ 16.0.$$\cdots$$.7 None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{2}q^{5}+\beta _{8}q^{7}+(-\beta _{7}-\beta _{11})q^{11}+\cdots$$

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

$$S_{2}^{\mathrm{old}}(4032, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(28, [\chi])$$$$^{\oplus 15}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(84, [\chi])$$$$^{\oplus 10}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(112, [\chi])$$$$^{\oplus 9}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(224, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(252, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(336, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(448, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(672, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1008, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1344, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(2016, [\chi])$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ ($$( 1 - 5 T^{2} )^{2}$$)($$1 + 2 T^{2} + 25 T^{4}$$)($$( 1 - 5 T^{2} )^{2}$$)($$1 - 4 T^{2} + 25 T^{4}$$)($$( 1 - 5 T^{2} )^{2}$$)($$1 - 4 T^{2} + 25 T^{4}$$)($$( 1 - 5 T^{2} )^{2}$$)($$1 + 2 T^{2} + 25 T^{4}$$)($$( 1 - 5 T^{2} )^{2}$$)($$1 - 6 T^{2} + 42 T^{4} - 150 T^{6} + 625 T^{8}$$)($$( 1 + 4 T^{2} + 25 T^{4} )^{2}$$)($$( 1 - 4 T^{2} + 25 T^{4} )^{2}$$)($$( 1 - 5 T^{2} )^{4}$$)($$1 - 6 T^{2} + 42 T^{4} - 150 T^{6} + 625 T^{8}$$)($$1 - 12 T^{2} + 56 T^{4} + 28 T^{6} - 1074 T^{8} + 700 T^{10} + 35000 T^{12} - 187500 T^{14} + 390625 T^{16}$$)($$( 1 - 12 T^{2} + 78 T^{4} - 300 T^{6} + 625 T^{8} )^{2}$$)($$1 - 12 T^{2} + 56 T^{4} + 28 T^{6} - 1074 T^{8} + 700 T^{10} + 35000 T^{12} - 187500 T^{14} + 390625 T^{16}$$)($$( 1 - 6 T^{2} + 42 T^{4} - 150 T^{6} + 625 T^{8} )^{4}$$)
$7$ ($$1 + 4 T + 7 T^{2}$$)($$1 + 4 T + 7 T^{2}$$)($$1 + 4 T + 7 T^{2}$$)($$1 + 2 T + 7 T^{2}$$)($$1 + 7 T^{2}$$)($$1 - 2 T + 7 T^{2}$$)($$1 - 4 T + 7 T^{2}$$)($$1 - 4 T + 7 T^{2}$$)($$1 - 4 T + 7 T^{2}$$)($$1 + 2 T - 2 T^{2} + 14 T^{3} + 49 T^{4}$$)($$( 1 - 7 T^{2} )^{2}$$)($$( 1 - 7 T^{2} )^{2}$$)($$( 1 + 7 T^{2} )^{2}$$)($$1 - 2 T - 2 T^{2} - 14 T^{3} + 49 T^{4}$$)($$1 + 4 T + 8 T^{2} + 20 T^{3} + 46 T^{4} + 140 T^{5} + 392 T^{6} + 1372 T^{7} + 2401 T^{8}$$)($$1 - 4 T^{2} - 26 T^{4} - 196 T^{6} + 2401 T^{8}$$)($$1 - 4 T + 8 T^{2} - 20 T^{3} + 46 T^{4} - 140 T^{5} + 392 T^{6} - 1372 T^{7} + 2401 T^{8}$$)($$( 1 + 8 T^{2} + 46 T^{4} + 392 T^{6} + 2401 T^{8} )^{2}$$)
$11$ ($$( 1 - 11 T^{2} )^{2}$$)($$1 - 10 T^{2} + 121 T^{4}$$)($$1 - 10 T^{2} + 121 T^{4}$$)($$1 - 16 T^{2} + 121 T^{4}$$)($$( 1 - 4 T + 11 T^{2} )( 1 + 4 T + 11 T^{2} )$$)($$1 - 16 T^{2} + 121 T^{4}$$)($$1 - 10 T^{2} + 121 T^{4}$$)($$1 - 10 T^{2} + 121 T^{4}$$)($$( 1 - 11 T^{2} )^{2}$$)($$1 - 34 T^{2} + 514 T^{4} - 4114 T^{6} + 14641 T^{8}$$)($$( 1 - 20 T^{2} + 121 T^{4} )^{2}$$)($$( 1 + 20 T^{2} + 121 T^{4} )^{2}$$)($$( 1 - 6 T^{2} + 121 T^{4} )^{2}$$)($$1 - 34 T^{2} + 514 T^{4} - 4114 T^{6} + 14641 T^{8}$$)($$1 - 20 T^{2} + 296 T^{4} - 2268 T^{6} + 20718 T^{8} - 274428 T^{10} + 4333736 T^{12} - 35431220 T^{14} + 214358881 T^{16}$$)($$( 1 - 18 T^{2} + 121 T^{4} )^{4}$$)($$1 - 20 T^{2} + 296 T^{4} - 2268 T^{6} + 20718 T^{8} - 274428 T^{10} + 4333736 T^{12} - 35431220 T^{14} + 214358881 T^{16}$$)($$( 1 - 18 T^{2} + 170 T^{4} - 2178 T^{6} + 14641 T^{8} )^{4}$$)
$13$ ($$( 1 - 2 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} )$$)($$1 - 14 T^{2} + 169 T^{4}$$)($$( 1 - 13 T^{2} )^{2}$$)($$1 - 2 T^{2} + 169 T^{4}$$)($$( 1 - 13 T^{2} )^{2}$$)($$1 - 2 T^{2} + 169 T^{4}$$)($$( 1 - 13 T^{2} )^{2}$$)($$1 - 14 T^{2} + 169 T^{4}$$)($$( 1 - 2 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} )$$)($$1 - 12 T^{2} + 102 T^{4} - 2028 T^{6} + 28561 T^{8}$$)($$( 1 - 13 T^{2} )^{4}$$)($$( 1 - 13 T^{2} )^{4}$$)($$( 1 - 13 T^{2} )^{4}$$)($$1 - 12 T^{2} + 102 T^{4} - 2028 T^{6} + 28561 T^{8}$$)($$( 1 - 18 T^{2} + 169 T^{4} )^{4}$$)($$( 1 - 12 T^{2} + 366 T^{4} - 2028 T^{6} + 28561 T^{8} )^{2}$$)($$( 1 - 18 T^{2} + 169 T^{4} )^{4}$$)($$( 1 - 12 T^{2} + 102 T^{4} - 2028 T^{6} + 28561 T^{8} )^{4}$$)
$17$ ($$( 1 - 17 T^{2} )^{2}$$)($$( 1 - 17 T^{2} )^{2}$$)($$1 + 14 T^{2} + 289 T^{4}$$)($$1 - 28 T^{2} + 289 T^{4}$$)($$( 1 - 17 T^{2} )^{2}$$)($$1 - 28 T^{2} + 289 T^{4}$$)($$1 + 14 T^{2} + 289 T^{4}$$)($$( 1 - 17 T^{2} )^{2}$$)($$( 1 - 17 T^{2} )^{2}$$)($$1 - 22 T^{2} + 682 T^{4} - 6358 T^{6} + 83521 T^{8}$$)($$( 1 - 20 T^{2} + 289 T^{4} )^{2}$$)($$( 1 + 20 T^{2} + 289 T^{4} )^{2}$$)($$( 1 - 17 T^{2} )^{4}$$)($$1 - 22 T^{2} + 682 T^{4} - 6358 T^{6} + 83521 T^{8}$$)($$1 - 108 T^{2} + 5432 T^{4} - 166628 T^{6} + 3420078 T^{8} - 48155492 T^{10} + 453686072 T^{12} - 2606857452 T^{14} + 6975757441 T^{16}$$)($$( 1 - 4 T + 6 T^{2} - 68 T^{3} + 289 T^{4} )^{2}( 1 + 4 T + 6 T^{2} + 68 T^{3} + 289 T^{4} )^{2}$$)($$1 - 108 T^{2} + 5432 T^{4} - 166628 T^{6} + 3420078 T^{8} - 48155492 T^{10} + 453686072 T^{12} - 2606857452 T^{14} + 6975757441 T^{16}$$)($$( 1 - 22 T^{2} + 682 T^{4} - 6358 T^{6} + 83521 T^{8} )^{4}$$)
$19$ ($$( 1 + 8 T + 19 T^{2} )^{2}$$)($$( 1 + 2 T + 19 T^{2} )^{2}$$)($$( 1 - 4 T + 19 T^{2} )^{2}$$)($$( 1 - 2 T + 19 T^{2} )^{2}$$)($$( 1 + 19 T^{2} )^{2}$$)($$( 1 + 2 T + 19 T^{2} )^{2}$$)($$( 1 + 4 T + 19 T^{2} )^{2}$$)($$( 1 - 2 T + 19 T^{2} )^{2}$$)($$( 1 - 8 T + 19 T^{2} )^{2}$$)($$( 1 - 6 T + 30 T^{2} - 114 T^{3} + 361 T^{4} )^{2}$$)($$( 1 + 10 T^{2} + 361 T^{4} )^{2}$$)($$( 1 + 10 T^{2} + 361 T^{4} )^{2}$$)($$( 1 + 19 T^{2} )^{4}$$)($$( 1 + 6 T + 30 T^{2} + 114 T^{3} + 361 T^{4} )^{2}$$)($$( 1 - 4 T + 32 T^{2} - 132 T^{3} + 558 T^{4} - 2508 T^{5} + 11552 T^{6} - 27436 T^{7} + 130321 T^{8} )^{2}$$)($$( 1 + 36 T^{2} + 1038 T^{4} + 12996 T^{6} + 130321 T^{8} )^{2}$$)($$( 1 + 4 T + 32 T^{2} + 132 T^{3} + 558 T^{4} + 2508 T^{5} + 11552 T^{6} + 27436 T^{7} + 130321 T^{8} )^{2}$$)($$( 1 + 48 T^{2} + 1230 T^{4} + 17328 T^{6} + 130321 T^{8} )^{4}$$)
$23$ ($$( 1 - 23 T^{2} )^{2}$$)($$1 - 34 T^{2} + 529 T^{4}$$)($$1 - 34 T^{2} + 529 T^{4}$$)($$1 + 8 T^{2} + 529 T^{4}$$)($$( 1 - 8 T + 23 T^{2} )( 1 + 8 T + 23 T^{2} )$$)($$1 + 8 T^{2} + 529 T^{4}$$)($$1 - 34 T^{2} + 529 T^{4}$$)($$1 - 34 T^{2} + 529 T^{4}$$)($$( 1 - 23 T^{2} )^{2}$$)($$1 - 82 T^{2} + 2722 T^{4} - 43378 T^{6} + 279841 T^{8}$$)($$( 1 + 4 T^{2} + 529 T^{4} )^{2}$$)($$( 1 - 4 T^{2} + 529 T^{4} )^{2}$$)($$( 1 + 18 T^{2} + 529 T^{4} )^{2}$$)($$1 - 82 T^{2} + 2722 T^{4} - 43378 T^{6} + 279841 T^{8}$$)($$1 - 100 T^{2} + 5000 T^{4} - 174764 T^{6} + 4622926 T^{8} - 92450156 T^{10} + 1399205000 T^{12} - 14803588900 T^{14} + 78310985281 T^{16}$$)($$( 1 - 20 T^{2} + 646 T^{4} - 10580 T^{6} + 279841 T^{8} )^{2}$$)($$1 - 100 T^{2} + 5000 T^{4} - 174764 T^{6} + 4622926 T^{8} - 92450156 T^{10} + 1399205000 T^{12} - 14803588900 T^{14} + 78310985281 T^{16}$$)($$( 1 - 74 T^{2} + 2410 T^{4} - 39146 T^{6} + 279841 T^{8} )^{4}$$)
$29$ ($$( 1 + 29 T^{2} )^{2}$$)($$( 1 - 6 T + 29 T^{2} )^{2}$$)($$( 1 + 6 T + 29 T^{2} )^{2}$$)($$( 1 - 6 T + 29 T^{2} )^{2}$$)($$( 1 + 2 T + 29 T^{2} )^{2}$$)($$( 1 - 6 T + 29 T^{2} )^{2}$$)($$( 1 + 6 T + 29 T^{2} )^{2}$$)($$( 1 - 6 T + 29 T^{2} )^{2}$$)($$( 1 + 29 T^{2} )^{2}$$)($$( 1 + 2 T + 29 T^{2} )^{4}$$)($$( 1 + 29 T^{2} )^{4}$$)($$( 1 + 29 T^{2} )^{4}$$)($$( 1 - 54 T^{2} + 841 T^{4} )^{2}$$)($$( 1 + 2 T + 29 T^{2} )^{4}$$)($$( 1 + 36 T^{2} + 96 T^{3} + 390 T^{4} + 2784 T^{5} + 30276 T^{6} + 707281 T^{8} )^{2}$$)($$( 1 + 4 T + 30 T^{2} + 116 T^{3} + 841 T^{4} )^{4}$$)($$( 1 + 36 T^{2} + 96 T^{3} + 390 T^{4} + 2784 T^{5} + 30276 T^{6} + 707281 T^{8} )^{2}$$)($$( 1 + 44 T^{2} + 1894 T^{4} + 37004 T^{6} + 707281 T^{8} )^{4}$$)
$31$ ($$( 1 - 4 T + 31 T^{2} )^{2}$$)($$( 1 + 8 T + 31 T^{2} )^{2}$$)($$( 1 - 4 T + 31 T^{2} )^{2}$$)($$( 1 - 8 T + 31 T^{2} )^{2}$$)($$( 1 + 31 T^{2} )^{2}$$)($$( 1 + 8 T + 31 T^{2} )^{2}$$)($$( 1 + 4 T + 31 T^{2} )^{2}$$)($$( 1 - 8 T + 31 T^{2} )^{2}$$)($$( 1 + 4 T + 31 T^{2} )^{2}$$)($$( 1 + 31 T^{2} )^{4}$$)($$( 1 - 50 T^{2} + 961 T^{4} )^{2}$$)($$( 1 - 50 T^{2} + 961 T^{4} )^{2}$$)($$( 1 + 31 T^{2} )^{4}$$)($$( 1 + 31 T^{2} )^{4}$$)($$( 1 - 8 T + 100 T^{2} - 616 T^{3} + 4534 T^{4} - 19096 T^{5} + 96100 T^{6} - 238328 T^{7} + 923521 T^{8} )^{2}$$)($$( 1 + 60 T^{2} + 2310 T^{4} + 57660 T^{6} + 923521 T^{8} )^{2}$$)($$( 1 + 8 T + 100 T^{2} + 616 T^{3} + 4534 T^{4} + 19096 T^{5} + 96100 T^{6} + 238328 T^{7} + 923521 T^{8} )^{2}$$)($$( 1 + 84 T^{2} + 3414 T^{4} + 80724 T^{6} + 923521 T^{8} )^{4}$$)
$37$ ($$( 1 + 10 T + 37 T^{2} )^{2}$$)($$( 1 - 2 T + 37 T^{2} )^{2}$$)($$( 1 - 2 T + 37 T^{2} )^{2}$$)($$( 1 + 4 T + 37 T^{2} )^{2}$$)($$( 1 + 6 T + 37 T^{2} )^{2}$$)($$( 1 + 4 T + 37 T^{2} )^{2}$$)($$( 1 - 2 T + 37 T^{2} )^{2}$$)($$( 1 - 2 T + 37 T^{2} )^{2}$$)($$( 1 + 10 T + 37 T^{2} )^{2}$$)($$( 1 - 6 T + 66 T^{2} - 222 T^{3} + 1369 T^{4} )^{2}$$)($$( 1 + 8 T + 37 T^{2} )^{4}$$)($$( 1 - 8 T + 37 T^{2} )^{4}$$)($$( 1 - 6 T + 37 T^{2} )^{4}$$)($$( 1 - 6 T + 66 T^{2} - 222 T^{3} + 1369 T^{4} )^{2}$$)($$( 1 + 4 T + 40 T^{2} - 292 T^{3} - 866 T^{4} - 10804 T^{5} + 54760 T^{6} + 202612 T^{7} + 1874161 T^{8} )^{2}$$)($$( 1 - 4 T + 46 T^{2} - 148 T^{3} + 1369 T^{4} )^{4}$$)($$( 1 + 4 T + 40 T^{2} - 292 T^{3} - 866 T^{4} - 10804 T^{5} + 54760 T^{6} + 202612 T^{7} + 1874161 T^{8} )^{2}$$)($$( 1 + 2 T + 58 T^{2} + 74 T^{3} + 1369 T^{4} )^{8}$$)
$41$ ($$( 1 - 41 T^{2} )^{2}$$)($$1 - 34 T^{2} + 1681 T^{4}$$)($$1 - 34 T^{2} + 1681 T^{4}$$)($$1 - 28 T^{2} + 1681 T^{4}$$)($$( 1 - 41 T^{2} )^{2}$$)($$1 - 28 T^{2} + 1681 T^{4}$$)($$1 - 34 T^{2} + 1681 T^{4}$$)($$1 - 34 T^{2} + 1681 T^{4}$$)($$( 1 - 41 T^{2} )^{2}$$)($$1 - 102 T^{2} + 5130 T^{4} - 171462 T^{6} + 2825761 T^{8}$$)($$( 1 - 68 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 + 68 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 - 41 T^{2} )^{4}$$)($$1 - 102 T^{2} + 5130 T^{4} - 171462 T^{6} + 2825761 T^{8}$$)($$1 - 204 T^{2} + 21176 T^{4} - 1439172 T^{6} + 69360622 T^{8} - 2419248132 T^{10} + 59838314936 T^{12} - 969021265164 T^{14} + 7984925229121 T^{16}$$)($$( 1 - 4 T^{2} + 3238 T^{4} - 6724 T^{6} + 2825761 T^{8} )^{2}$$)($$1 - 204 T^{2} + 21176 T^{4} - 1439172 T^{6} + 69360622 T^{8} - 2419248132 T^{10} + 59838314936 T^{12} - 969021265164 T^{14} + 7984925229121 T^{16}$$)($$( 1 - 118 T^{2} + 6826 T^{4} - 198358 T^{6} + 2825761 T^{8} )^{4}$$)
$43$ ($$( 1 - 8 T + 43 T^{2} )( 1 + 8 T + 43 T^{2} )$$)($$( 1 - 8 T + 43 T^{2} )( 1 + 8 T + 43 T^{2} )$$)($$1 - 74 T^{2} + 1849 T^{4}$$)($$1 - 62 T^{2} + 1849 T^{4}$$)($$( 1 - 12 T + 43 T^{2} )( 1 + 12 T + 43 T^{2} )$$)($$1 - 62 T^{2} + 1849 T^{4}$$)($$1 - 74 T^{2} + 1849 T^{4}$$)($$( 1 - 8 T + 43 T^{2} )( 1 + 8 T + 43 T^{2} )$$)($$( 1 - 8 T + 43 T^{2} )( 1 + 8 T + 43 T^{2} )$$)($$1 - 24 T^{2} + 3774 T^{4} - 44376 T^{6} + 3418801 T^{8}$$)($$( 1 - 43 T^{2} )^{4}$$)($$( 1 - 43 T^{2} )^{4}$$)($$( 1 - 12 T + 43 T^{2} )^{2}( 1 + 12 T + 43 T^{2} )^{2}$$)($$1 - 24 T^{2} + 3774 T^{4} - 44376 T^{6} + 3418801 T^{8}$$)($$1 - 192 T^{2} + 18396 T^{4} - 1162560 T^{6} + 55940774 T^{8} - 2149573440 T^{10} + 62892263196 T^{12} - 1213701705408 T^{14} + 11688200277601 T^{16}$$)($$( 1 - 100 T^{2} + 5686 T^{4} - 184900 T^{6} + 3418801 T^{8} )^{2}$$)($$1 - 192 T^{2} + 18396 T^{4} - 1162560 T^{6} + 55940774 T^{8} - 2149573440 T^{10} + 62892263196 T^{12} - 1213701705408 T^{14} + 11688200277601 T^{16}$$)($$( 1 - 88 T^{2} + 3934 T^{4} - 162712 T^{6} + 3418801 T^{8} )^{4}$$)
$47$ ($$( 1 + 47 T^{2} )^{2}$$)($$( 1 + 47 T^{2} )^{2}$$)($$( 1 + 47 T^{2} )^{2}$$)($$( 1 - 12 T + 47 T^{2} )^{2}$$)($$( 1 + 47 T^{2} )^{2}$$)($$( 1 + 12 T + 47 T^{2} )^{2}$$)($$( 1 + 47 T^{2} )^{2}$$)($$( 1 + 47 T^{2} )^{2}$$)($$( 1 + 47 T^{2} )^{2}$$)($$( 1 + 4 T + 30 T^{2} + 188 T^{3} + 2209 T^{4} )^{2}$$)($$( 1 + 47 T^{2} )^{4}$$)($$( 1 + 47 T^{2} )^{4}$$)($$( 1 + 47 T^{2} )^{4}$$)($$( 1 - 4 T + 30 T^{2} - 188 T^{3} + 2209 T^{4} )^{2}$$)($$( 1 - 8 T + 108 T^{2} - 872 T^{3} + 6758 T^{4} - 40984 T^{5} + 238572 T^{6} - 830584 T^{7} + 4879681 T^{8} )^{2}$$)($$( 1 + 124 T^{2} + 7750 T^{4} + 273916 T^{6} + 4879681 T^{8} )^{2}$$)($$( 1 + 8 T + 108 T^{2} + 872 T^{3} + 6758 T^{4} + 40984 T^{5} + 238572 T^{6} + 830584 T^{7} + 4879681 T^{8} )^{2}$$)($$( 1 + 28 T^{2} + 262 T^{4} + 61852 T^{6} + 4879681 T^{8} )^{4}$$)
$53$ ($$( 1 + 53 T^{2} )^{2}$$)($$( 1 - 6 T + 53 T^{2} )^{2}$$)($$( 1 + 6 T + 53 T^{2} )^{2}$$)($$( 1 + 6 T + 53 T^{2} )^{2}$$)($$( 1 + 10 T + 53 T^{2} )^{2}$$)($$( 1 + 6 T + 53 T^{2} )^{2}$$)($$( 1 + 6 T + 53 T^{2} )^{2}$$)($$( 1 - 6 T + 53 T^{2} )^{2}$$)($$( 1 + 53 T^{2} )^{2}$$)($$( 1 - 8 T + 54 T^{2} - 424 T^{3} + 2809 T^{4} )^{2}$$)($$( 1 + 53 T^{2} )^{4}$$)($$( 1 + 53 T^{2} )^{4}$$)($$( 1 - 6 T^{2} + 2809 T^{4} )^{2}$$)($$( 1 - 8 T + 54 T^{2} - 424 T^{3} + 2809 T^{4} )^{2}$$)($$( 1 - 8 T + 132 T^{2} - 632 T^{3} + 7718 T^{4} - 33496 T^{5} + 370788 T^{6} - 1191016 T^{7} + 7890481 T^{8} )^{2}$$)($$( 1 + 2 T + 53 T^{2} )^{8}$$)($$( 1 - 8 T + 132 T^{2} - 632 T^{3} + 7718 T^{4} - 33496 T^{5} + 370788 T^{6} - 1191016 T^{7} + 7890481 T^{8} )^{2}$$)($$( 1 + 108 T^{2} + 6086 T^{4} + 303372 T^{6} + 7890481 T^{8} )^{4}$$)
$59$ ($$( 1 + 59 T^{2} )^{2}$$)($$( 1 - 6 T + 59 T^{2} )^{2}$$)($$( 1 - 12 T + 59 T^{2} )^{2}$$)($$( 1 - 12 T + 59 T^{2} )^{2}$$)($$( 1 + 59 T^{2} )^{2}$$)($$( 1 + 12 T + 59 T^{2} )^{2}$$)($$( 1 + 12 T + 59 T^{2} )^{2}$$)($$( 1 + 6 T + 59 T^{2} )^{2}$$)($$( 1 + 59 T^{2} )^{2}$$)($$( 1 + 4 T + 59 T^{2} )^{4}$$)($$( 1 + 59 T^{2} )^{4}$$)($$( 1 + 59 T^{2} )^{4}$$)($$( 1 + 59 T^{2} )^{4}$$)($$( 1 - 4 T + 59 T^{2} )^{4}$$)($$( 1 + 8 T + 102 T^{2} + 472 T^{3} + 3481 T^{4} )^{4}$$)($$( 1 + 132 T^{2} + 10926 T^{4} + 459492 T^{6} + 12117361 T^{8} )^{2}$$)($$( 1 - 8 T + 102 T^{2} - 472 T^{3} + 3481 T^{4} )^{4}$$)($$( 1 + 12 T^{2} + 2646 T^{4} + 41772 T^{6} + 12117361 T^{8} )^{4}$$)
$61$ ($$( 1 - 14 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} )$$)($$1 - 110 T^{2} + 3721 T^{4}$$)($$1 + 70 T^{2} + 3721 T^{4}$$)($$( 1 - 61 T^{2} )^{2}$$)($$( 1 - 61 T^{2} )^{2}$$)($$( 1 - 61 T^{2} )^{2}$$)($$1 + 70 T^{2} + 3721 T^{4}$$)($$1 - 110 T^{2} + 3721 T^{4}$$)($$( 1 - 14 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} )$$)($$1 - 132 T^{2} + 10710 T^{4} - 491172 T^{6} + 13845841 T^{8}$$)($$( 1 - 61 T^{2} )^{4}$$)($$( 1 - 61 T^{2} )^{4}$$)($$( 1 - 61 T^{2} )^{4}$$)($$1 - 132 T^{2} + 10710 T^{4} - 491172 T^{6} + 13845841 T^{8}$$)($$1 - 184 T^{2} + 16284 T^{4} - 814280 T^{6} + 38728934 T^{8} - 3029935880 T^{10} + 225465674844 T^{12} - 9479748882424 T^{14} + 191707312997281 T^{16}$$)($$( 1 - 236 T^{2} + 21358 T^{4} - 878156 T^{6} + 13845841 T^{8} )^{2}$$)($$1 - 184 T^{2} + 16284 T^{4} - 814280 T^{6} + 38728934 T^{8} - 3029935880 T^{10} + 225465674844 T^{12} - 9479748882424 T^{14} + 191707312997281 T^{16}$$)($$( 1 + 52 T^{2} + 7846 T^{4} + 193492 T^{6} + 13845841 T^{8} )^{4}$$)
$67$ ($$( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} )$$)($$( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} )$$)($$( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} )$$)($$( 1 - 67 T^{2} )^{2}$$)($$( 1 - 4 T + 67 T^{2} )( 1 + 4 T + 67 T^{2} )$$)($$( 1 - 67 T^{2} )^{2}$$)($$( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} )$$)($$( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} )$$)($$( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} )$$)($$1 - 144 T^{2} + 10830 T^{4} - 646416 T^{6} + 20151121 T^{8}$$)($$( 1 - 67 T^{2} )^{4}$$)($$( 1 - 67 T^{2} )^{4}$$)($$( 1 - 4 T + 67 T^{2} )^{2}( 1 + 4 T + 67 T^{2} )^{2}$$)($$1 - 144 T^{2} + 10830 T^{4} - 646416 T^{6} + 20151121 T^{8}$$)($$1 - 240 T^{2} + 37436 T^{4} - 3815056 T^{6} + 299829030 T^{8} - 17125786384 T^{10} + 754377365756 T^{12} - 21710011720560 T^{14} + 406067677556641 T^{16}$$)($$( 1 - 4 T^{2} - 3818 T^{4} - 17956 T^{6} + 20151121 T^{8} )^{2}$$)($$1 - 240 T^{2} + 37436 T^{4} - 3815056 T^{6} + 299829030 T^{8} - 17125786384 T^{10} + 754377365756 T^{12} - 21710011720560 T^{14} + 406067677556641 T^{16}$$)($$( 1 - 184 T^{2} + 15742 T^{4} - 825976 T^{6} + 20151121 T^{8} )^{4}$$)
$71$ ($$( 1 - 71 T^{2} )^{2}$$)($$1 - 130 T^{2} + 5041 T^{4}$$)($$1 - 34 T^{2} + 5041 T^{4}$$)($$1 + 8 T^{2} + 5041 T^{4}$$)($$( 1 - 16 T + 71 T^{2} )( 1 + 16 T + 71 T^{2} )$$)($$1 + 8 T^{2} + 5041 T^{4}$$)($$1 - 34 T^{2} + 5041 T^{4}$$)($$1 - 130 T^{2} + 5041 T^{4}$$)($$( 1 - 71 T^{2} )^{2}$$)($$1 - 114 T^{2} + 8418 T^{4} - 574674 T^{6} + 25411681 T^{8}$$)($$( 1 + 100 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 - 100 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 + 114 T^{2} + 5041 T^{4} )^{2}$$)($$1 - 114 T^{2} + 8418 T^{4} - 574674 T^{6} + 25411681 T^{8}$$)($$1 - 484 T^{2} + 107912 T^{4} - 14382508 T^{6} + 1250123214 T^{8} - 72502222828 T^{10} + 2742225320072 T^{12} - 62000537417764 T^{14} + 645753531245761 T^{16}$$)($$( 1 - 196 T^{2} + 18534 T^{4} - 988036 T^{6} + 25411681 T^{8} )^{2}$$)($$1 - 484 T^{2} + 107912 T^{4} - 14382508 T^{6} + 1250123214 T^{8} - 72502222828 T^{10} + 2742225320072 T^{12} - 62000537417764 T^{14} + 645753531245761 T^{16}$$)($$( 1 - 106 T^{2} + 9066 T^{4} - 534346 T^{6} + 25411681 T^{8} )^{4}$$)
$73$ ($$( 1 - 10 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} )$$)($$1 - 98 T^{2} + 5329 T^{4}$$)($$( 1 - 10 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} )$$)($$1 + 70 T^{2} + 5329 T^{4}$$)($$( 1 - 73 T^{2} )^{2}$$)($$1 + 70 T^{2} + 5329 T^{4}$$)($$( 1 - 10 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} )$$)($$1 - 98 T^{2} + 5329 T^{4}$$)($$( 1 - 10 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} )$$)($$1 - 236 T^{2} + 24310 T^{4} - 1257644 T^{6} + 28398241 T^{8}$$)($$( 1 - 73 T^{2} )^{4}$$)($$( 1 - 73 T^{2} )^{4}$$)($$( 1 - 73 T^{2} )^{4}$$)($$1 - 236 T^{2} + 24310 T^{4} - 1257644 T^{6} + 28398241 T^{8}$$)($$1 - 408 T^{2} + 80380 T^{4} - 10037032 T^{6} + 869620166 T^{8} - 53487343528 T^{10} + 2282650611580 T^{12} - 61744364325912 T^{14} + 806460091894081 T^{16}$$)($$( 1 - 132 T^{2} + 8742 T^{4} - 703428 T^{6} + 28398241 T^{8} )^{2}$$)($$1 - 408 T^{2} + 80380 T^{4} - 10037032 T^{6} + 869620166 T^{8} - 53487343528 T^{10} + 2282650611580 T^{12} - 61744364325912 T^{14} + 806460091894081 T^{16}$$)($$( 1 - 132 T^{2} + 10662 T^{4} - 703428 T^{6} + 28398241 T^{8} )^{4}$$)
$79$ ($$( 1 - 4 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} )$$)($$1 - 146 T^{2} + 6241 T^{4}$$)($$1 - 50 T^{2} + 6241 T^{4}$$)($$1 - 62 T^{2} + 6241 T^{4}$$)($$( 1 - 8 T + 79 T^{2} )( 1 + 8 T + 79 T^{2} )$$)($$1 - 62 T^{2} + 6241 T^{4}$$)($$1 - 50 T^{2} + 6241 T^{4}$$)($$1 - 146 T^{2} + 6241 T^{4}$$)($$( 1 - 4 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} )$$)($$1 - 288 T^{2} + 33150 T^{4} - 1797408 T^{6} + 38950081 T^{8}$$)($$( 1 - 79 T^{2} )^{4}$$)($$( 1 - 79 T^{2} )^{4}$$)($$( 1 - 8 T + 79 T^{2} )^{2}( 1 + 8 T + 79 T^{2} )^{2}$$)($$1 - 288 T^{2} + 33150 T^{4} - 1797408 T^{6} + 38950081 T^{8}$$)($$1 - 368 T^{2} + 73436 T^{4} - 9593744 T^{6} + 892827078 T^{8} - 59874556304 T^{10} + 2860338148316 T^{12} - 89456183631728 T^{14} + 1517108809906561 T^{16}$$)($$( 1 - 100 T^{2} + 11782 T^{4} - 624100 T^{6} + 38950081 T^{8} )^{2}$$)($$1 - 368 T^{2} + 73436 T^{4} - 9593744 T^{6} + 892827078 T^{8} - 59874556304 T^{10} + 2860338148316 T^{12} - 89456183631728 T^{14} + 1517108809906561 T^{16}$$)($$( 1 - 280 T^{2} + 32014 T^{4} - 1747480 T^{6} + 38950081 T^{8} )^{4}$$)
$83$ ($$( 1 + 83 T^{2} )^{2}$$)($$( 1 + 6 T + 83 T^{2} )^{2}$$)($$( 1 - 12 T + 83 T^{2} )^{2}$$)($$( 1 + 83 T^{2} )^{2}$$)($$( 1 + 83 T^{2} )^{2}$$)($$( 1 + 83 T^{2} )^{2}$$)($$( 1 + 12 T + 83 T^{2} )^{2}$$)($$( 1 - 6 T + 83 T^{2} )^{2}$$)($$( 1 + 83 T^{2} )^{2}$$)($$( 1 - 4 T + 102 T^{2} - 332 T^{3} + 6889 T^{4} )^{2}$$)($$( 1 + 83 T^{2} )^{4}$$)($$( 1 + 83 T^{2} )^{4}$$)($$( 1 + 83 T^{2} )^{4}$$)($$( 1 + 4 T + 102 T^{2} + 332 T^{3} + 6889 T^{4} )^{2}$$)($$( 1 + 8 T + 124 T^{2} + 1480 T^{3} + 7318 T^{4} + 122840 T^{5} + 854236 T^{6} + 4574296 T^{7} + 47458321 T^{8} )^{2}$$)($$( 1 + 196 T^{2} + 19150 T^{4} + 1350244 T^{6} + 47458321 T^{8} )^{2}$$)($$( 1 - 8 T + 124 T^{2} - 1480 T^{3} + 7318 T^{4} - 122840 T^{5} + 854236 T^{6} - 4574296 T^{7} + 47458321 T^{8} )^{2}$$)($$( 1 + 172 T^{2} + 16822 T^{4} + 1184908 T^{6} + 47458321 T^{8} )^{4}$$)
$89$ ($$( 1 - 89 T^{2} )^{2}$$)($$1 - 130 T^{2} + 7921 T^{4}$$)($$1 - 130 T^{2} + 7921 T^{4}$$)($$1 - 28 T^{2} + 7921 T^{4}$$)($$( 1 - 89 T^{2} )^{2}$$)($$1 - 28 T^{2} + 7921 T^{4}$$)($$1 - 130 T^{2} + 7921 T^{4}$$)($$1 - 130 T^{2} + 7921 T^{4}$$)($$( 1 - 89 T^{2} )^{2}$$)($$1 - 294 T^{2} + 36618 T^{4} - 2328774 T^{6} + 62742241 T^{8}$$)($$( 1 + 172 T^{2} + 7921 T^{4} )^{2}$$)($$( 1 - 172 T^{2} + 7921 T^{4} )^{2}$$)($$( 1 - 89 T^{2} )^{4}$$)($$1 - 294 T^{2} + 36618 T^{4} - 2328774 T^{6} + 62742241 T^{8}$$)($$1 + 52 T^{2} - 328 T^{4} + 280764 T^{6} + 120770670 T^{8} + 2223931644 T^{10} - 20579455048 T^{12} + 25843027129972 T^{14} + 3936588805702081 T^{16}$$)($$( 1 - 324 T^{2} + 41958 T^{4} - 2566404 T^{6} + 62742241 T^{8} )^{2}$$)($$1 + 52 T^{2} - 328 T^{4} + 280764 T^{6} + 120770670 T^{8} + 2223931644 T^{10} - 20579455048 T^{12} + 25843027129972 T^{14} + 3936588805702081 T^{16}$$)($$( 1 - 294 T^{2} + 36618 T^{4} - 2328774 T^{6} + 62742241 T^{8} )^{4}$$)
$97$ ($$( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} )$$)($$( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} )$$)($$( 1 - 97 T^{2} )^{2}$$)($$1 - 170 T^{2} + 9409 T^{4}$$)($$( 1 - 97 T^{2} )^{2}$$)($$1 - 170 T^{2} + 9409 T^{4}$$)($$( 1 - 97 T^{2} )^{2}$$)($$( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} )$$)($$( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} )$$)($$1 - 204 T^{2} + 28950 T^{4} - 1919436 T^{6} + 88529281 T^{8}$$)($$( 1 - 97 T^{2} )^{4}$$)($$( 1 - 97 T^{2} )^{4}$$)($$( 1 - 97 T^{2} )^{4}$$)($$1 - 204 T^{2} + 28950 T^{4} - 1919436 T^{6} + 88529281 T^{8}$$)($$1 - 344 T^{2} + 58172 T^{4} - 6404840 T^{6} + 615495942 T^{8} - 60263139560 T^{10} + 5149925334332 T^{12} - 286542369695576 T^{14} + 7837433594376961 T^{16}$$)($$( 1 - 68 T^{2} + 19462 T^{4} - 639812 T^{6} + 88529281 T^{8} )^{2}$$)($$1 - 344 T^{2} + 58172 T^{4} - 6404840 T^{6} + 615495942 T^{8} - 60263139560 T^{10} + 5149925334332 T^{12} - 286542369695576 T^{14} + 7837433594376961 T^{16}$$)($$( 1 - 4 T - 74 T^{2} - 388 T^{3} + 9409 T^{4} )^{4}( 1 + 4 T - 74 T^{2} + 388 T^{3} + 9409 T^{4} )^{4}$$)
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