# Properties

 Label 420.2.l Level $420$ Weight $2$ Character orbit 420.l Rep. character $\chi_{420}(239,\cdot)$ Character field $\Q$ Dimension $72$ Newform subspaces $8$ Sturm bound $192$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$420 = 2^{2} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 420.l (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$60$$ Character field: $$\Q$$ Newform subspaces: $$8$$ Sturm bound: $$192$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$11$$, $$17$$, $$43$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(420, [\chi])$$.

Total New Old
Modular forms 104 72 32
Cusp forms 88 72 16
Eisenstein series 16 0 16

## Trace form

 $$72q + 4q^{4} + O(q^{10})$$ $$72q + 4q^{4} + 8q^{10} - 12q^{16} - 44q^{24} - 24q^{30} - 32q^{34} + 4q^{36} - 8q^{40} - 24q^{45} - 24q^{46} + 72q^{49} + 68q^{54} + 52q^{60} - 64q^{61} + 52q^{64} - 76q^{66} - 48q^{69} + 12q^{70} - 24q^{76} - 16q^{81} + 24q^{85} - 56q^{90} - 152q^{94} + 60q^{96} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(420, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
420.2.l.a $$4$$ $$3.354$$ $$\Q(\zeta_{8})$$ None $$0$$ $$-4$$ $$0$$ $$4$$ $$q-\zeta_{8}^{2}q^{2}+(-1+\zeta_{8}^{2})q^{3}-2q^{4}+\cdots$$
420.2.l.b $$4$$ $$3.354$$ $$\Q(\zeta_{8})$$ None $$0$$ $$4$$ $$0$$ $$-4$$ $$q+\zeta_{8}^{2}q^{2}+(1-\zeta_{8}^{2})q^{3}-2q^{4}+(\zeta_{8}^{2}+\cdots)q^{5}+\cdots$$
420.2.l.c $$8$$ $$3.354$$ 8.0.386672896.3 None $$0$$ $$-2$$ $$-8$$ $$-8$$ $$q-\beta _{3}q^{2}+\beta _{2}q^{3}+(1-\beta _{4}-\beta _{6})q^{4}+\cdots$$
420.2.l.d $$8$$ $$3.354$$ 8.0.386672896.3 None $$0$$ $$-2$$ $$8$$ $$-8$$ $$q+\beta _{3}q^{2}-\beta _{6}q^{3}+(1-\beta _{4}-\beta _{6})q^{4}+\cdots$$
420.2.l.e $$8$$ $$3.354$$ 8.0.386672896.3 None $$0$$ $$2$$ $$-8$$ $$8$$ $$q-\beta _{3}q^{2}+\beta _{6}q^{3}+(1-\beta _{4}-\beta _{6})q^{4}+\cdots$$
420.2.l.f $$8$$ $$3.354$$ 8.0.386672896.3 None $$0$$ $$2$$ $$8$$ $$8$$ $$q+\beta _{3}q^{2}-\beta _{2}q^{3}+(1-\beta _{4}-\beta _{6})q^{4}+\cdots$$
420.2.l.g $$16$$ $$3.354$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$-16$$ $$q+\beta _{3}q^{2}+\beta _{4}q^{3}+\beta _{11}q^{4}+\beta _{5}q^{5}+\cdots$$
420.2.l.h $$16$$ $$3.354$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$16$$ $$q+\beta _{3}q^{2}+\beta _{6}q^{3}+\beta _{11}q^{4}+(-\beta _{4}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(420, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(420, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(60, [\chi])$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$( 1 + 2 T^{2} )^{2}$$)($$( 1 + 2 T^{2} )^{2}$$)($$1 - T^{2} - 2 T^{3} + 2 T^{4} - 4 T^{5} - 4 T^{6} + 16 T^{8}$$)($$1 - T^{2} + 2 T^{3} + 2 T^{4} + 4 T^{5} - 4 T^{6} + 16 T^{8}$$)($$1 - T^{2} - 2 T^{3} + 2 T^{4} - 4 T^{5} - 4 T^{6} + 16 T^{8}$$)($$1 - T^{2} + 2 T^{3} + 2 T^{4} + 4 T^{5} - 4 T^{6} + 16 T^{8}$$)($$1 - 3 T^{2} + 7 T^{4} - 13 T^{6} + 28 T^{8} - 52 T^{10} + 112 T^{12} - 192 T^{14} + 256 T^{16}$$)($$1 - 3 T^{2} + 7 T^{4} - 13 T^{6} + 28 T^{8} - 52 T^{10} + 112 T^{12} - 192 T^{14} + 256 T^{16}$$)
$3$ ($$( 1 + 2 T + 3 T^{2} )^{2}$$)($$( 1 - 2 T + 3 T^{2} )^{2}$$)($$1 + 2 T + T^{2} + 2 T^{3} + 4 T^{4} + 6 T^{5} + 9 T^{6} + 54 T^{7} + 81 T^{8}$$)($$1 + 2 T + T^{2} + 2 T^{3} + 4 T^{4} + 6 T^{5} + 9 T^{6} + 54 T^{7} + 81 T^{8}$$)($$1 - 2 T + T^{2} - 2 T^{3} + 4 T^{4} - 6 T^{5} + 9 T^{6} - 54 T^{7} + 81 T^{8}$$)($$1 - 2 T + T^{2} - 2 T^{3} + 4 T^{4} - 6 T^{5} + 9 T^{6} - 54 T^{7} + 81 T^{8}$$)($$( 1 - 4 T^{3} - 2 T^{4} - 12 T^{5} + 81 T^{8} )^{2}$$)($$( 1 + 4 T^{3} - 2 T^{4} + 12 T^{5} + 81 T^{8} )^{2}$$)
$5$ ($$1 - 8 T^{2} + 25 T^{4}$$)($$1 - 8 T^{2} + 25 T^{4}$$)($$( 1 + 2 T + 5 T^{2} )^{4}$$)($$( 1 - 2 T + 5 T^{2} )^{4}$$)($$( 1 + 2 T + 5 T^{2} )^{4}$$)($$( 1 - 2 T + 5 T^{2} )^{4}$$)($$1 - 16 T^{2} + 156 T^{4} - 1152 T^{6} + 6534 T^{8} - 28800 T^{10} + 97500 T^{12} - 250000 T^{14} + 390625 T^{16}$$)($$1 - 16 T^{2} + 156 T^{4} - 1152 T^{6} + 6534 T^{8} - 28800 T^{10} + 97500 T^{12} - 250000 T^{14} + 390625 T^{16}$$)
$7$ ($$( 1 - T )^{4}$$)($$( 1 + T )^{4}$$)($$( 1 + T )^{8}$$)($$( 1 + T )^{8}$$)($$( 1 - T )^{8}$$)($$( 1 - T )^{8}$$)($$( 1 + T )^{16}$$)($$( 1 - T )^{16}$$)
$11$ ($$( 1 + 4 T^{2} + 121 T^{4} )^{2}$$)($$( 1 + 4 T^{2} + 121 T^{4} )^{2}$$)($$( 1 - 2 T + 33 T^{2} - 50 T^{3} + 500 T^{4} - 550 T^{5} + 3993 T^{6} - 2662 T^{7} + 14641 T^{8} )^{2}$$)($$( 1 + 2 T + 33 T^{2} + 50 T^{3} + 500 T^{4} + 550 T^{5} + 3993 T^{6} + 2662 T^{7} + 14641 T^{8} )^{2}$$)($$( 1 + 2 T + 33 T^{2} + 50 T^{3} + 500 T^{4} + 550 T^{5} + 3993 T^{6} + 2662 T^{7} + 14641 T^{8} )^{2}$$)($$( 1 - 2 T + 33 T^{2} - 50 T^{3} + 500 T^{4} - 550 T^{5} + 3993 T^{6} - 2662 T^{7} + 14641 T^{8} )^{2}$$)($$( 1 + 22 T^{2} + 432 T^{4} + 4986 T^{6} + 65534 T^{8} + 603306 T^{10} + 6324912 T^{12} + 38974342 T^{14} + 214358881 T^{16} )^{2}$$)($$( 1 + 22 T^{2} + 432 T^{4} + 4986 T^{6} + 65534 T^{8} + 603306 T^{10} + 6324912 T^{12} + 38974342 T^{14} + 214358881 T^{16} )^{2}$$)
$13$ ($$( 1 - 4 T + 13 T^{2} )^{2}( 1 + 4 T + 13 T^{2} )^{2}$$)($$( 1 - 4 T + 13 T^{2} )^{2}( 1 + 4 T + 13 T^{2} )^{2}$$)($$1 - 34 T^{2} + 593 T^{4} - 8610 T^{6} + 122100 T^{8} - 1455090 T^{10} + 16936673 T^{12} - 164111506 T^{14} + 815730721 T^{16}$$)($$1 - 34 T^{2} + 593 T^{4} - 8610 T^{6} + 122100 T^{8} - 1455090 T^{10} + 16936673 T^{12} - 164111506 T^{14} + 815730721 T^{16}$$)($$1 - 34 T^{2} + 593 T^{4} - 8610 T^{6} + 122100 T^{8} - 1455090 T^{10} + 16936673 T^{12} - 164111506 T^{14} + 815730721 T^{16}$$)($$1 - 34 T^{2} + 593 T^{4} - 8610 T^{6} + 122100 T^{8} - 1455090 T^{10} + 16936673 T^{12} - 164111506 T^{14} + 815730721 T^{16}$$)($$( 1 - 66 T^{2} + 2096 T^{4} - 42782 T^{6} + 637630 T^{8} - 7230158 T^{10} + 59863856 T^{12} - 318569394 T^{14} + 815730721 T^{16} )^{2}$$)($$( 1 - 66 T^{2} + 2096 T^{4} - 42782 T^{6} + 637630 T^{8} - 7230158 T^{10} + 59863856 T^{12} - 318569394 T^{14} + 815730721 T^{16} )^{2}$$)
$17$ ($$( 1 + 16 T^{2} + 289 T^{4} )^{2}$$)($$( 1 + 16 T^{2} + 289 T^{4} )^{2}$$)($$( 1 - 2 T + 33 T^{2} - 50 T^{3} + 644 T^{4} - 850 T^{5} + 9537 T^{6} - 9826 T^{7} + 83521 T^{8} )^{2}$$)($$( 1 + 2 T + 33 T^{2} + 50 T^{3} + 644 T^{4} + 850 T^{5} + 9537 T^{6} + 9826 T^{7} + 83521 T^{8} )^{2}$$)($$( 1 - 2 T + 33 T^{2} - 50 T^{3} + 644 T^{4} - 850 T^{5} + 9537 T^{6} - 9826 T^{7} + 83521 T^{8} )^{2}$$)($$( 1 + 2 T + 33 T^{2} + 50 T^{3} + 644 T^{4} + 850 T^{5} + 9537 T^{6} + 9826 T^{7} + 83521 T^{8} )^{2}$$)($$( 1 + 60 T^{2} + 2308 T^{4} + 59972 T^{6} + 1182838 T^{8} + 17331908 T^{10} + 192766468 T^{12} + 1448254140 T^{14} + 6975757441 T^{16} )^{2}$$)($$( 1 + 60 T^{2} + 2308 T^{4} + 59972 T^{6} + 1182838 T^{8} + 17331908 T^{10} + 192766468 T^{12} + 1448254140 T^{14} + 6975757441 T^{16} )^{2}$$)
$19$ ($$( 1 - 2 T^{2} + 361 T^{4} )^{2}$$)($$( 1 - 2 T^{2} + 361 T^{4} )^{2}$$)($$1 - 64 T^{2} + 1180 T^{4} + 10560 T^{6} - 634906 T^{8} + 3812160 T^{10} + 153778780 T^{12} - 3010936384 T^{14} + 16983563041 T^{16}$$)($$1 - 64 T^{2} + 1180 T^{4} + 10560 T^{6} - 634906 T^{8} + 3812160 T^{10} + 153778780 T^{12} - 3010936384 T^{14} + 16983563041 T^{16}$$)($$1 - 64 T^{2} + 1180 T^{4} + 10560 T^{6} - 634906 T^{8} + 3812160 T^{10} + 153778780 T^{12} - 3010936384 T^{14} + 16983563041 T^{16}$$)($$1 - 64 T^{2} + 1180 T^{4} + 10560 T^{6} - 634906 T^{8} + 3812160 T^{10} + 153778780 T^{12} - 3010936384 T^{14} + 16983563041 T^{16}$$)($$( 1 - 114 T^{2} + 6104 T^{4} - 202286 T^{6} + 4583854 T^{8} - 73025246 T^{10} + 795479384 T^{12} - 5363230434 T^{14} + 16983563041 T^{16} )^{2}$$)($$( 1 - 114 T^{2} + 6104 T^{4} - 202286 T^{6} + 4583854 T^{8} - 73025246 T^{10} + 795479384 T^{12} - 5363230434 T^{14} + 16983563041 T^{16} )^{2}$$)
$23$ ($$( 1 - 44 T^{2} + 529 T^{4} )^{2}$$)($$( 1 - 44 T^{2} + 529 T^{4} )^{2}$$)($$1 - 88 T^{2} + 4220 T^{4} - 145000 T^{6} + 3793350 T^{8} - 76705000 T^{10} + 1180929020 T^{12} - 13027158232 T^{14} + 78310985281 T^{16}$$)($$1 - 88 T^{2} + 4220 T^{4} - 145000 T^{6} + 3793350 T^{8} - 76705000 T^{10} + 1180929020 T^{12} - 13027158232 T^{14} + 78310985281 T^{16}$$)($$1 - 88 T^{2} + 4220 T^{4} - 145000 T^{6} + 3793350 T^{8} - 76705000 T^{10} + 1180929020 T^{12} - 13027158232 T^{14} + 78310985281 T^{16}$$)($$1 - 88 T^{2} + 4220 T^{4} - 145000 T^{6} + 3793350 T^{8} - 76705000 T^{10} + 1180929020 T^{12} - 13027158232 T^{14} + 78310985281 T^{16}$$)($$( 1 - 72 T^{2} + 3292 T^{4} - 107896 T^{6} + 2808070 T^{8} - 57076984 T^{10} + 921236572 T^{12} - 10658584008 T^{14} + 78310985281 T^{16} )^{2}$$)($$( 1 - 72 T^{2} + 3292 T^{4} - 107896 T^{6} + 2808070 T^{8} - 57076984 T^{10} + 921236572 T^{12} - 10658584008 T^{14} + 78310985281 T^{16} )^{2}$$)
$29$ ($$( 1 - 50 T^{2} + 841 T^{4} )^{2}$$)($$( 1 - 50 T^{2} + 841 T^{4} )^{2}$$)($$1 - 162 T^{2} + 12689 T^{4} - 630306 T^{6} + 21708724 T^{8} - 530087346 T^{10} + 8974688609 T^{12} - 96361378002 T^{14} + 500246412961 T^{16}$$)($$1 - 162 T^{2} + 12689 T^{4} - 630306 T^{6} + 21708724 T^{8} - 530087346 T^{10} + 8974688609 T^{12} - 96361378002 T^{14} + 500246412961 T^{16}$$)($$1 - 162 T^{2} + 12689 T^{4} - 630306 T^{6} + 21708724 T^{8} - 530087346 T^{10} + 8974688609 T^{12} - 96361378002 T^{14} + 500246412961 T^{16}$$)($$1 - 162 T^{2} + 12689 T^{4} - 630306 T^{6} + 21708724 T^{8} - 530087346 T^{10} + 8974688609 T^{12} - 96361378002 T^{14} + 500246412961 T^{16}$$)($$( 1 - 68 T^{2} + 3892 T^{4} - 150428 T^{6} + 5129398 T^{8} - 126509948 T^{10} + 2752737652 T^{12} - 40447985828 T^{14} + 500246412961 T^{16} )^{2}$$)($$( 1 - 68 T^{2} + 3892 T^{4} - 150428 T^{6} + 5129398 T^{8} - 126509948 T^{10} + 2752737652 T^{12} - 40447985828 T^{14} + 500246412961 T^{16} )^{2}$$)
$31$ ($$( 1 - 31 T^{2} )^{4}$$)($$( 1 - 31 T^{2} )^{4}$$)($$1 - 56 T^{2} + 2684 T^{4} - 115208 T^{6} + 3486342 T^{8} - 110714888 T^{10} + 2478730364 T^{12} - 49700206136 T^{14} + 852891037441 T^{16}$$)($$1 - 56 T^{2} + 2684 T^{4} - 115208 T^{6} + 3486342 T^{8} - 110714888 T^{10} + 2478730364 T^{12} - 49700206136 T^{14} + 852891037441 T^{16}$$)($$1 - 56 T^{2} + 2684 T^{4} - 115208 T^{6} + 3486342 T^{8} - 110714888 T^{10} + 2478730364 T^{12} - 49700206136 T^{14} + 852891037441 T^{16}$$)($$1 - 56 T^{2} + 2684 T^{4} - 115208 T^{6} + 3486342 T^{8} - 110714888 T^{10} + 2478730364 T^{12} - 49700206136 T^{14} + 852891037441 T^{16}$$)($$( 1 - 158 T^{2} + 12408 T^{4} - 637698 T^{6} + 23269838 T^{8} - 612827778 T^{10} + 11459048568 T^{12} - 140225581598 T^{14} + 852891037441 T^{16} )^{2}$$)($$( 1 - 158 T^{2} + 12408 T^{4} - 637698 T^{6} + 23269838 T^{8} - 612827778 T^{10} + 11459048568 T^{12} - 140225581598 T^{14} + 852891037441 T^{16} )^{2}$$)
$37$ ($$( 1 - 38 T^{2} + 1369 T^{4} )^{2}$$)($$( 1 - 38 T^{2} + 1369 T^{4} )^{2}$$)($$1 - 160 T^{2} + 12124 T^{4} - 608352 T^{6} + 24287270 T^{8} - 832833888 T^{10} + 22722327964 T^{12} - 410516225440 T^{14} + 3512479453921 T^{16}$$)($$1 - 160 T^{2} + 12124 T^{4} - 608352 T^{6} + 24287270 T^{8} - 832833888 T^{10} + 22722327964 T^{12} - 410516225440 T^{14} + 3512479453921 T^{16}$$)($$1 - 160 T^{2} + 12124 T^{4} - 608352 T^{6} + 24287270 T^{8} - 832833888 T^{10} + 22722327964 T^{12} - 410516225440 T^{14} + 3512479453921 T^{16}$$)($$1 - 160 T^{2} + 12124 T^{4} - 608352 T^{6} + 24287270 T^{8} - 832833888 T^{10} + 22722327964 T^{12} - 410516225440 T^{14} + 3512479453921 T^{16}$$)($$( 1 - 148 T^{2} + 12948 T^{4} - 755340 T^{6} + 32573238 T^{8} - 1034060460 T^{10} + 24266636628 T^{12} - 379727508532 T^{14} + 3512479453921 T^{16} )^{2}$$)($$( 1 - 148 T^{2} + 12948 T^{4} - 755340 T^{6} + 32573238 T^{8} - 1034060460 T^{10} + 24266636628 T^{12} - 379727508532 T^{14} + 3512479453921 T^{16} )^{2}$$)
$41$ ($$( 1 - 80 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 - 80 T^{2} + 1681 T^{4} )^{2}$$)($$1 - 224 T^{2} + 24956 T^{4} - 1767712 T^{6} + 86340934 T^{8} - 2971523872 T^{10} + 70519691516 T^{12} - 1064023349984 T^{14} + 7984925229121 T^{16}$$)($$1 - 224 T^{2} + 24956 T^{4} - 1767712 T^{6} + 86340934 T^{8} - 2971523872 T^{10} + 70519691516 T^{12} - 1064023349984 T^{14} + 7984925229121 T^{16}$$)($$1 - 224 T^{2} + 24956 T^{4} - 1767712 T^{6} + 86340934 T^{8} - 2971523872 T^{10} + 70519691516 T^{12} - 1064023349984 T^{14} + 7984925229121 T^{16}$$)($$1 - 224 T^{2} + 24956 T^{4} - 1767712 T^{6} + 86340934 T^{8} - 2971523872 T^{10} + 70519691516 T^{12} - 1064023349984 T^{14} + 7984925229121 T^{16}$$)($$( 1 - 84 T^{2} + 6292 T^{4} - 269932 T^{6} + 12822550 T^{8} - 453755692 T^{10} + 17779688212 T^{12} - 399008756244 T^{14} + 7984925229121 T^{16} )^{2}$$)($$( 1 - 84 T^{2} + 6292 T^{4} - 269932 T^{6} + 12822550 T^{8} - 453755692 T^{10} + 17779688212 T^{12} - 399008756244 T^{14} + 7984925229121 T^{16} )^{2}$$)
$43$ ($$( 1 - 8 T + 43 T^{2} )^{4}$$)($$( 1 + 8 T + 43 T^{2} )^{4}$$)($$( 1 - 4 T + 116 T^{2} - 324 T^{3} + 6150 T^{4} - 13932 T^{5} + 214484 T^{6} - 318028 T^{7} + 3418801 T^{8} )^{2}$$)($$( 1 - 4 T + 116 T^{2} - 324 T^{3} + 6150 T^{4} - 13932 T^{5} + 214484 T^{6} - 318028 T^{7} + 3418801 T^{8} )^{2}$$)($$( 1 + 4 T + 116 T^{2} + 324 T^{3} + 6150 T^{4} + 13932 T^{5} + 214484 T^{6} + 318028 T^{7} + 3418801 T^{8} )^{2}$$)($$( 1 + 4 T + 116 T^{2} + 324 T^{3} + 6150 T^{4} + 13932 T^{5} + 214484 T^{6} + 318028 T^{7} + 3418801 T^{8} )^{2}$$)($$( 1 - 4 T + 76 T^{2} - 260 T^{3} + 4454 T^{4} - 11180 T^{5} + 140524 T^{6} - 318028 T^{7} + 3418801 T^{8} )^{4}$$)($$( 1 + 4 T + 76 T^{2} + 260 T^{3} + 4454 T^{4} + 11180 T^{5} + 140524 T^{6} + 318028 T^{7} + 3418801 T^{8} )^{4}$$)
$47$ ($$( 1 - 86 T^{2} + 2209 T^{4} )^{2}$$)($$( 1 - 86 T^{2} + 2209 T^{4} )^{2}$$)($$1 - 354 T^{2} + 55793 T^{4} - 5112098 T^{6} + 297791076 T^{8} - 11292624482 T^{10} + 272252042033 T^{12} - 3815842226466 T^{14} + 23811286661761 T^{16}$$)($$1 - 354 T^{2} + 55793 T^{4} - 5112098 T^{6} + 297791076 T^{8} - 11292624482 T^{10} + 272252042033 T^{12} - 3815842226466 T^{14} + 23811286661761 T^{16}$$)($$1 - 354 T^{2} + 55793 T^{4} - 5112098 T^{6} + 297791076 T^{8} - 11292624482 T^{10} + 272252042033 T^{12} - 3815842226466 T^{14} + 23811286661761 T^{16}$$)($$1 - 354 T^{2} + 55793 T^{4} - 5112098 T^{6} + 297791076 T^{8} - 11292624482 T^{10} + 272252042033 T^{12} - 3815842226466 T^{14} + 23811286661761 T^{16}$$)($$( 1 - 152 T^{2} + 16476 T^{4} - 1161512 T^{6} + 64108614 T^{8} - 2565780008 T^{10} + 80397624156 T^{12} - 1638440730008 T^{14} + 23811286661761 T^{16} )^{2}$$)($$( 1 - 152 T^{2} + 16476 T^{4} - 1161512 T^{6} + 64108614 T^{8} - 2565780008 T^{10} + 80397624156 T^{12} - 1638440730008 T^{14} + 23811286661761 T^{16} )^{2}$$)
$53$ ($$( 1 + 34 T^{2} + 2809 T^{4} )^{2}$$)($$( 1 + 34 T^{2} + 2809 T^{4} )^{2}$$)($$( 1 + 12 T^{2} + 256 T^{3} + 1046 T^{4} + 13568 T^{5} + 33708 T^{6} + 7890481 T^{8} )^{2}$$)($$( 1 + 12 T^{2} - 256 T^{3} + 1046 T^{4} - 13568 T^{5} + 33708 T^{6} + 7890481 T^{8} )^{2}$$)($$( 1 + 12 T^{2} + 256 T^{3} + 1046 T^{4} + 13568 T^{5} + 33708 T^{6} + 7890481 T^{8} )^{2}$$)($$( 1 + 12 T^{2} - 256 T^{3} + 1046 T^{4} - 13568 T^{5} + 33708 T^{6} + 7890481 T^{8} )^{2}$$)($$( 1 + 278 T^{2} + 37272 T^{4} + 3219258 T^{6} + 198845358 T^{8} + 9042895722 T^{10} + 294094007832 T^{12} + 6161692393862 T^{14} + 62259690411361 T^{16} )^{2}$$)($$( 1 + 278 T^{2} + 37272 T^{4} + 3219258 T^{6} + 198845358 T^{8} + 9042895722 T^{10} + 294094007832 T^{12} + 6161692393862 T^{14} + 62259690411361 T^{16} )^{2}$$)
$59$ ($$( 1 + 59 T^{2} )^{4}$$)($$( 1 + 59 T^{2} )^{4}$$)($$( 1 - 4 T + 96 T^{2} + 444 T^{3} + 2126 T^{4} + 26196 T^{5} + 334176 T^{6} - 821516 T^{7} + 12117361 T^{8} )^{2}$$)($$( 1 + 4 T + 96 T^{2} - 444 T^{3} + 2126 T^{4} - 26196 T^{5} + 334176 T^{6} + 821516 T^{7} + 12117361 T^{8} )^{2}$$)($$( 1 + 4 T + 96 T^{2} - 444 T^{3} + 2126 T^{4} - 26196 T^{5} + 334176 T^{6} + 821516 T^{7} + 12117361 T^{8} )^{2}$$)($$( 1 - 4 T + 96 T^{2} + 444 T^{3} + 2126 T^{4} + 26196 T^{5} + 334176 T^{6} - 821516 T^{7} + 12117361 T^{8} )^{2}$$)($$( 1 + 304 T^{2} + 44556 T^{4} + 4247040 T^{6} + 291601382 T^{8} + 14783946240 T^{10} + 539901136716 T^{12} + 12822882226864 T^{14} + 146830437604321 T^{16} )^{2}$$)($$( 1 + 304 T^{2} + 44556 T^{4} + 4247040 T^{6} + 291601382 T^{8} + 14783946240 T^{10} + 539901136716 T^{12} + 12822882226864 T^{14} + 146830437604321 T^{16} )^{2}$$)
$61$ ($$( 1 + 10 T + 61 T^{2} )^{4}$$)($$( 1 + 10 T + 61 T^{2} )^{4}$$)($$( 1 + 8 T + 96 T^{2} + 232 T^{3} + 2958 T^{4} + 14152 T^{5} + 357216 T^{6} + 1815848 T^{7} + 13845841 T^{8} )^{2}$$)($$( 1 + 8 T + 96 T^{2} + 232 T^{3} + 2958 T^{4} + 14152 T^{5} + 357216 T^{6} + 1815848 T^{7} + 13845841 T^{8} )^{2}$$)($$( 1 + 8 T + 96 T^{2} + 232 T^{3} + 2958 T^{4} + 14152 T^{5} + 357216 T^{6} + 1815848 T^{7} + 13845841 T^{8} )^{2}$$)($$( 1 + 8 T + 96 T^{2} + 232 T^{3} + 2958 T^{4} + 14152 T^{5} + 357216 T^{6} + 1815848 T^{7} + 13845841 T^{8} )^{2}$$)($$( 1 - 10 T + 236 T^{2} - 1522 T^{3} + 20638 T^{4} - 92842 T^{5} + 878156 T^{6} - 2269810 T^{7} + 13845841 T^{8} )^{4}$$)($$( 1 - 10 T + 236 T^{2} - 1522 T^{3} + 20638 T^{4} - 92842 T^{5} + 878156 T^{6} - 2269810 T^{7} + 13845841 T^{8} )^{4}$$)
$67$ ($$( 1 + 4 T + 67 T^{2} )^{4}$$)($$( 1 - 4 T + 67 T^{2} )^{4}$$)($$( 1 - 12 T + 144 T^{2} - 300 T^{3} + 3470 T^{4} - 20100 T^{5} + 646416 T^{6} - 3609156 T^{7} + 20151121 T^{8} )^{2}$$)($$( 1 - 12 T + 144 T^{2} - 300 T^{3} + 3470 T^{4} - 20100 T^{5} + 646416 T^{6} - 3609156 T^{7} + 20151121 T^{8} )^{2}$$)($$( 1 + 12 T + 144 T^{2} + 300 T^{3} + 3470 T^{4} + 20100 T^{5} + 646416 T^{6} + 3609156 T^{7} + 20151121 T^{8} )^{2}$$)($$( 1 + 12 T + 144 T^{2} + 300 T^{3} + 3470 T^{4} + 20100 T^{5} + 646416 T^{6} + 3609156 T^{7} + 20151121 T^{8} )^{2}$$)($$( 1 + 14 T + 232 T^{2} + 2310 T^{3} + 21246 T^{4} + 154770 T^{5} + 1041448 T^{6} + 4210682 T^{7} + 20151121 T^{8} )^{4}$$)($$( 1 - 14 T + 232 T^{2} - 2310 T^{3} + 21246 T^{4} - 154770 T^{5} + 1041448 T^{6} - 4210682 T^{7} + 20151121 T^{8} )^{4}$$)
$71$ ($$( 1 - 20 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 - 20 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 + 8 T + 71 T^{2} )^{8}$$)($$( 1 - 8 T + 71 T^{2} )^{8}$$)($$( 1 - 8 T + 71 T^{2} )^{8}$$)($$( 1 + 8 T + 71 T^{2} )^{8}$$)($$( 1 + 226 T^{2} + 18304 T^{4} + 119582 T^{6} - 52128898 T^{8} + 602812862 T^{10} + 465135409024 T^{12} + 28950664166146 T^{14} + 645753531245761 T^{16} )^{2}$$)($$( 1 + 226 T^{2} + 18304 T^{4} + 119582 T^{6} - 52128898 T^{8} + 602812862 T^{10} + 465135409024 T^{12} + 28950664166146 T^{14} + 645753531245761 T^{16} )^{2}$$)
$73$ ($$( 1 - 16 T + 73 T^{2} )^{2}( 1 + 16 T + 73 T^{2} )^{2}$$)($$( 1 - 16 T + 73 T^{2} )^{2}( 1 + 16 T + 73 T^{2} )^{2}$$)($$( 1 - 82 T^{2} + 5329 T^{4} )^{4}$$)($$( 1 - 82 T^{2} + 5329 T^{4} )^{4}$$)($$( 1 - 82 T^{2} + 5329 T^{4} )^{4}$$)($$( 1 - 82 T^{2} + 5329 T^{4} )^{4}$$)($$( 1 - 234 T^{2} + 39488 T^{4} - 4281494 T^{6} + 369066238 T^{8} - 22816081526 T^{10} + 1121389740608 T^{12} - 35412208951626 T^{14} + 806460091894081 T^{16} )^{2}$$)($$( 1 - 234 T^{2} + 39488 T^{4} - 4281494 T^{6} + 369066238 T^{8} - 22816081526 T^{10} + 1121389740608 T^{12} - 35412208951626 T^{14} + 806460091894081 T^{16} )^{2}$$)
$79$ ($$( 1 - 79 T^{2} )^{4}$$)($$( 1 - 79 T^{2} )^{4}$$)($$1 - 130 T^{2} + 3089 T^{4} - 667586 T^{6} + 117965092 T^{8} - 4166404226 T^{10} + 120316800209 T^{12} - 31601369217730 T^{14} + 1517108809906561 T^{16}$$)($$1 - 130 T^{2} + 3089 T^{4} - 667586 T^{6} + 117965092 T^{8} - 4166404226 T^{10} + 120316800209 T^{12} - 31601369217730 T^{14} + 1517108809906561 T^{16}$$)($$1 - 130 T^{2} + 3089 T^{4} - 667586 T^{6} + 117965092 T^{8} - 4166404226 T^{10} + 120316800209 T^{12} - 31601369217730 T^{14} + 1517108809906561 T^{16}$$)($$1 - 130 T^{2} + 3089 T^{4} - 667586 T^{6} + 117965092 T^{8} - 4166404226 T^{10} + 120316800209 T^{12} - 31601369217730 T^{14} + 1517108809906561 T^{16}$$)($$( 1 - 300 T^{2} + 46388 T^{4} - 4850900 T^{6} + 410340886 T^{8} - 30274466900 T^{10} + 1806816357428 T^{12} - 72926236656300 T^{14} + 1517108809906561 T^{16} )^{2}$$)($$( 1 - 300 T^{2} + 46388 T^{4} - 4850900 T^{6} + 410340886 T^{8} - 30274466900 T^{10} + 1806816357428 T^{12} - 72926236656300 T^{14} + 1517108809906561 T^{16} )^{2}$$)
$83$ ($$( 1 - 18 T + 83 T^{2} )^{2}( 1 + 18 T + 83 T^{2} )^{2}$$)($$( 1 - 18 T + 83 T^{2} )^{2}( 1 + 18 T + 83 T^{2} )^{2}$$)($$1 - 232 T^{2} + 25980 T^{4} - 1355288 T^{6} + 66491558 T^{8} - 9336579032 T^{10} + 1232967179580 T^{12} - 75850166621608 T^{14} + 2252292232139041 T^{16}$$)($$1 - 232 T^{2} + 25980 T^{4} - 1355288 T^{6} + 66491558 T^{8} - 9336579032 T^{10} + 1232967179580 T^{12} - 75850166621608 T^{14} + 2252292232139041 T^{16}$$)($$1 - 232 T^{2} + 25980 T^{4} - 1355288 T^{6} + 66491558 T^{8} - 9336579032 T^{10} + 1232967179580 T^{12} - 75850166621608 T^{14} + 2252292232139041 T^{16}$$)($$1 - 232 T^{2} + 25980 T^{4} - 1355288 T^{6} + 66491558 T^{8} - 9336579032 T^{10} + 1232967179580 T^{12} - 75850166621608 T^{14} + 2252292232139041 T^{16}$$)($$( 1 - 284 T^{2} + 25348 T^{4} + 201580 T^{6} - 152756234 T^{8} + 1388684620 T^{10} + 1202973520708 T^{12} - 92851066036796 T^{14} + 2252292232139041 T^{16} )^{2}$$)($$( 1 - 284 T^{2} + 25348 T^{4} + 201580 T^{6} - 152756234 T^{8} + 1388684620 T^{10} + 1202973520708 T^{12} - 92851066036796 T^{14} + 2252292232139041 T^{16} )^{2}$$)
$89$ ($$( 1 - 128 T^{2} + 7921 T^{4} )^{2}$$)($$( 1 - 128 T^{2} + 7921 T^{4} )^{2}$$)($$1 - 360 T^{2} + 59996 T^{4} - 6261336 T^{6} + 552707974 T^{8} - 49596042456 T^{10} + 3764283491036 T^{12} - 178913264745960 T^{14} + 3936588805702081 T^{16}$$)($$1 - 360 T^{2} + 59996 T^{4} - 6261336 T^{6} + 552707974 T^{8} - 49596042456 T^{10} + 3764283491036 T^{12} - 178913264745960 T^{14} + 3936588805702081 T^{16}$$)($$1 - 360 T^{2} + 59996 T^{4} - 6261336 T^{6} + 552707974 T^{8} - 49596042456 T^{10} + 3764283491036 T^{12} - 178913264745960 T^{14} + 3936588805702081 T^{16}$$)($$1 - 360 T^{2} + 59996 T^{4} - 6261336 T^{6} + 552707974 T^{8} - 49596042456 T^{10} + 3764283491036 T^{12} - 178913264745960 T^{14} + 3936588805702081 T^{16}$$)($$( 1 - 184 T^{2} + 18396 T^{4} - 1862280 T^{6} + 181662342 T^{8} - 14751119880 T^{10} + 1154206265436 T^{12} - 91444557536824 T^{14} + 3936588805702081 T^{16} )^{2}$$)($$( 1 - 184 T^{2} + 18396 T^{4} - 1862280 T^{6} + 181662342 T^{8} - 14751119880 T^{10} + 1154206265436 T^{12} - 91444557536824 T^{14} + 3936588805702081 T^{16} )^{2}$$)
$97$ ($$( 1 - 158 T^{2} + 9409 T^{4} )^{2}$$)($$( 1 - 158 T^{2} + 9409 T^{4} )^{2}$$)($$1 - 82 T^{2} - 11855 T^{4} - 200178 T^{6} + 251318948 T^{8} - 1883474802 T^{10} - 1049514626255 T^{12} - 68303704404178 T^{14} + 7837433594376961 T^{16}$$)($$1 - 82 T^{2} - 11855 T^{4} - 200178 T^{6} + 251318948 T^{8} - 1883474802 T^{10} - 1049514626255 T^{12} - 68303704404178 T^{14} + 7837433594376961 T^{16}$$)($$1 - 82 T^{2} - 11855 T^{4} - 200178 T^{6} + 251318948 T^{8} - 1883474802 T^{10} - 1049514626255 T^{12} - 68303704404178 T^{14} + 7837433594376961 T^{16}$$)($$1 - 82 T^{2} - 11855 T^{4} - 200178 T^{6} + 251318948 T^{8} - 1883474802 T^{10} - 1049514626255 T^{12} - 68303704404178 T^{14} + 7837433594376961 T^{16}$$)($$( 1 - 266 T^{2} + 44320 T^{4} - 5389846 T^{6} + 579408254 T^{8} - 50713061014 T^{10} + 3923617733920 T^{12} - 221570553311114 T^{14} + 7837433594376961 T^{16} )^{2}$$)($$( 1 - 266 T^{2} + 44320 T^{4} - 5389846 T^{6} + 579408254 T^{8} - 50713061014 T^{10} + 3923617733920 T^{12} - 221570553311114 T^{14} + 7837433594376961 T^{16} )^{2}$$)