# Properties

 Label 3024.2.k Level $3024$ Weight $2$ Character orbit 3024.k Rep. character $\chi_{3024}(1889,\cdot)$ Character field $\Q$ Dimension $64$ Newform subspaces $12$ Sturm bound $1152$ Trace bound $25$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$3024 = 2^{4} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3024.k (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$21$$ Character field: $$\Q$$ Newform subspaces: $$12$$ Sturm bound: $$1152$$ Trace bound: $$25$$ Distinguishing $$T_p$$: $$5$$, $$11$$, $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(3024, [\chi])$$.

Total New Old
Modular forms 612 64 548
Cusp forms 540 64 476
Eisenstein series 72 0 72

## Trace form

 $$64q + 2q^{7} + O(q^{10})$$ $$64q + 2q^{7} + 56q^{25} - 32q^{43} + 28q^{67} - 60q^{79} + 8q^{85} - 42q^{91} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(3024, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
3024.2.k.a $$2$$ $$24.147$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-6$$ $$-4$$ $$q-3q^{5}+(-2-\zeta_{6})q^{7}+3\zeta_{6}q^{11}+\cdots$$
3024.2.k.b $$2$$ $$24.147$$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$-1$$ $$q+(-1+\zeta_{6})q^{7}+(-1+2\zeta_{6})q^{13}+\cdots$$
3024.2.k.c $$2$$ $$24.147$$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$5$$ $$q+(2+\zeta_{6})q^{7}+(1-2\zeta_{6})q^{13}+(-5+\cdots)q^{19}+\cdots$$
3024.2.k.d $$2$$ $$24.147$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$6$$ $$-4$$ $$q+3q^{5}+(-2+\zeta_{6})q^{7}+3\zeta_{6}q^{11}+\cdots$$
3024.2.k.e $$4$$ $$24.147$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$-8$$ $$q-\zeta_{12}^{2}q^{5}+(-2+\zeta_{12})q^{7}+\zeta_{12}^{3}q^{11}+\cdots$$
3024.2.k.f $$4$$ $$24.147$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$-8$$ $$q-2\zeta_{12}^{2}q^{5}+(-2+\zeta_{12})q^{7}+2\zeta_{12}^{3}q^{11}+\cdots$$
3024.2.k.g $$4$$ $$24.147$$ $$\Q(\sqrt{-2}, \sqrt{3})$$ None $$0$$ $$0$$ $$0$$ $$-4$$ $$q-\beta _{2}q^{5}+(-1-\beta _{3})q^{7}-\beta _{1}q^{11}+\cdots$$
3024.2.k.h $$4$$ $$24.147$$ $$\Q(\sqrt{-2}, \sqrt{3})$$ None $$0$$ $$0$$ $$0$$ $$4$$ $$q-\beta _{2}q^{5}+(1+\beta _{1})q^{7}+\beta _{3}q^{11}+\beta _{1}q^{13}+\cdots$$
3024.2.k.i $$4$$ $$24.147$$ $$\Q(\sqrt{-3}, \sqrt{-5})$$ None $$0$$ $$0$$ $$0$$ $$8$$ $$q+\beta _{3}q^{5}+(2-\beta _{2})q^{7}-\beta _{1}q^{11}-2\beta _{2}q^{13}+\cdots$$
3024.2.k.j $$4$$ $$24.147$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$10$$ $$q-\zeta_{12}^{3}q^{5}+(3-\zeta_{12})q^{7}+(2\zeta_{12}^{2}+\cdots)q^{11}+\cdots$$
3024.2.k.k $$16$$ $$24.147$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$2$$ $$q-\beta _{9}q^{5}+\beta _{1}q^{7}+\beta _{10}q^{11}-\beta _{5}q^{13}+\cdots$$
3024.2.k.l $$16$$ $$24.147$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$2$$ $$q+\beta _{1}q^{5}+\beta _{12}q^{7}-\beta _{10}q^{11}+(\beta _{4}+\cdots)q^{13}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(3024, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(3024, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(42, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(63, [\chi])$$$$^{\oplus 10}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(84, [\chi])$$$$^{\oplus 9}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(168, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(189, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(252, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(336, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(378, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(504, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(756, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1008, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1512, [\chi])$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ ($$( 1 + 3 T + 5 T^{2} )^{2}$$)($$( 1 + 5 T^{2} )^{2}$$)($$( 1 + 5 T^{2} )^{2}$$)($$( 1 - 3 T + 5 T^{2} )^{2}$$)($$( 1 + 7 T^{2} + 25 T^{4} )^{2}$$)($$( 1 - 2 T^{2} + 25 T^{4} )^{2}$$)($$( 1 + 7 T^{2} + 25 T^{4} )^{2}$$)($$( 1 + 7 T^{2} + 25 T^{4} )^{2}$$)($$( 1 - 5 T^{2} + 25 T^{4} )^{2}$$)($$( 1 + 7 T^{2} + 25 T^{4} )^{2}$$)($$( 1 + 23 T^{2} + 273 T^{4} + 2194 T^{6} + 12806 T^{8} + 54850 T^{10} + 170625 T^{12} + 359375 T^{14} + 390625 T^{16} )^{2}$$)($$( 1 + 11 T^{2} + 45 T^{4} + 34 T^{6} - 214 T^{8} + 850 T^{10} + 28125 T^{12} + 171875 T^{14} + 390625 T^{16} )^{2}$$)
$7$ ($$1 + 4 T + 7 T^{2}$$)($$1 + T + 7 T^{2}$$)($$1 - 5 T + 7 T^{2}$$)($$1 + 4 T + 7 T^{2}$$)($$( 1 + 4 T + 7 T^{2} )^{2}$$)($$( 1 + 4 T + 7 T^{2} )^{2}$$)($$( 1 + 2 T + 7 T^{2} )^{2}$$)($$( 1 - 2 T + 7 T^{2} )^{2}$$)($$( 1 - 4 T + 7 T^{2} )^{2}$$)($$( 1 - 5 T + 7 T^{2} )^{2}$$)($$( 1 - T - 5 T^{3} - 34 T^{4} - 35 T^{5} - 343 T^{7} + 2401 T^{8} )^{2}$$)($$( 1 - T + 7 T^{3} + 14 T^{4} + 49 T^{5} - 343 T^{7} + 2401 T^{8} )^{2}$$)
$11$ ($$1 + 5 T^{2} + 121 T^{4}$$)($$( 1 - 11 T^{2} )^{2}$$)($$( 1 - 11 T^{2} )^{2}$$)($$1 + 5 T^{2} + 121 T^{4}$$)($$( 1 - 13 T^{2} + 121 T^{4} )^{2}$$)($$( 1 + 14 T^{2} + 121 T^{4} )^{2}$$)($$( 1 - 20 T^{2} + 121 T^{4} )^{2}$$)($$( 1 - 4 T^{2} + 121 T^{4} )^{2}$$)($$( 1 - 17 T^{2} + 121 T^{4} )^{2}$$)($$( 1 - 13 T^{2} + 121 T^{4} )^{2}$$)($$( 1 - 37 T^{2} + 801 T^{4} - 12446 T^{6} + 149966 T^{8} - 1505966 T^{10} + 11727441 T^{12} - 65547757 T^{14} + 214358881 T^{16} )^{2}$$)($$( 1 - 55 T^{2} + 1590 T^{4} - 29597 T^{6} + 386186 T^{8} - 3581237 T^{10} + 23279190 T^{12} - 97435855 T^{14} + 214358881 T^{16} )^{2}$$)
$13$ ($$1 - 14 T^{2} + 169 T^{4}$$)($$( 1 - 5 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} )$$)($$( 1 - 7 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )$$)($$1 - 14 T^{2} + 169 T^{4}$$)($$( 1 - 14 T^{2} + 169 T^{4} )^{2}$$)($$( 1 - 7 T + 13 T^{2} )^{2}( 1 + 7 T + 13 T^{2} )^{2}$$)($$( 1 - 20 T^{2} + 169 T^{4} )^{2}$$)($$( 1 - 20 T^{2} + 169 T^{4} )^{2}$$)($$( 1 - 14 T^{2} + 169 T^{4} )^{2}$$)($$( 1 - 2 T + 13 T^{2} )^{2}( 1 + 2 T + 13 T^{2} )^{2}$$)($$( 1 - 49 T^{2} + 1398 T^{4} - 28055 T^{6} + 418226 T^{8} - 4741295 T^{10} + 39928278 T^{12} - 236513641 T^{14} + 815730721 T^{16} )^{2}$$)($$( 1 - 49 T^{2} + 1362 T^{4} - 26411 T^{6} + 392258 T^{8} - 4463459 T^{10} + 38900082 T^{12} - 236513641 T^{14} + 815730721 T^{16} )^{2}$$)
$17$ ($$( 1 - 6 T + 17 T^{2} )^{2}$$)($$( 1 + 17 T^{2} )^{2}$$)($$( 1 + 17 T^{2} )^{2}$$)($$( 1 + 6 T + 17 T^{2} )^{2}$$)($$( 1 - 14 T^{2} + 289 T^{4} )^{2}$$)($$( 1 + 31 T^{2} + 289 T^{4} )^{2}$$)($$( 1 + 7 T^{2} + 289 T^{4} )^{2}$$)($$( 1 + 31 T^{2} + 289 T^{4} )^{2}$$)($$( 1 + 17 T^{2} )^{4}$$)($$( 1 - 14 T^{2} + 289 T^{4} )^{2}$$)($$( 1 + 72 T^{2} + 2652 T^{4} + 65784 T^{6} + 1248326 T^{8} + 19011576 T^{10} + 221497692 T^{12} + 1737904968 T^{14} + 6975757441 T^{16} )^{2}$$)($$( 1 + 87 T^{2} + 3537 T^{4} + 92514 T^{6} + 1788926 T^{8} + 26736546 T^{10} + 295413777 T^{12} + 2099968503 T^{14} + 6975757441 T^{16} )^{2}$$)
$19$ ($$1 - 35 T^{2} + 361 T^{4}$$)($$( 1 - 7 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} )$$)($$( 1 - T + 19 T^{2} )( 1 + T + 19 T^{2} )$$)($$1 - 35 T^{2} + 361 T^{4}$$)($$( 1 - 35 T^{2} + 361 T^{4} )^{2}$$)($$( 1 + 10 T^{2} + 361 T^{4} )^{2}$$)($$( 1 + 16 T^{2} + 361 T^{4} )^{2}$$)($$( 1 - 32 T^{2} + 361 T^{4} )^{2}$$)($$( 1 - 7 T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2}$$)($$( 1 - 8 T + 19 T^{2} )^{2}( 1 + 8 T + 19 T^{2} )^{2}$$)($$( 1 - 100 T^{2} + 4914 T^{4} - 156608 T^{6} + 3515195 T^{8} - 56535488 T^{10} + 640397394 T^{12} - 4704588100 T^{14} + 16983563041 T^{16} )^{2}$$)($$( 1 - 112 T^{2} + 5952 T^{4} - 197456 T^{6} + 4482014 T^{8} - 71281616 T^{10} + 775670592 T^{12} - 5269138672 T^{14} + 16983563041 T^{16} )^{2}$$)
$23$ ($$1 - 19 T^{2} + 529 T^{4}$$)($$( 1 - 23 T^{2} )^{2}$$)($$( 1 - 23 T^{2} )^{2}$$)($$1 - 19 T^{2} + 529 T^{4}$$)($$( 1 - 37 T^{2} + 529 T^{4} )^{2}$$)($$( 1 - 37 T^{2} + 529 T^{4} )^{2}$$)($$( 1 - 38 T^{2} + 529 T^{4} )^{2}$$)($$( 1 + 26 T^{2} + 529 T^{4} )^{2}$$)($$( 1 - 41 T^{2} + 529 T^{4} )^{2}$$)($$( 1 - 10 T^{2} + 529 T^{4} )^{2}$$)($$( 1 - 21 T^{2} + 697 T^{4} - 11718 T^{6} + 312438 T^{8} - 6198822 T^{10} + 195049177 T^{12} - 3108753669 T^{14} + 78310985281 T^{16} )^{2}$$)($$( 1 - 99 T^{2} + 5182 T^{4} - 187533 T^{6} + 5005890 T^{8} - 99204957 T^{10} + 1450136062 T^{12} - 14655553011 T^{14} + 78310985281 T^{16} )^{2}$$)
$29$ ($$1 + 50 T^{2} + 841 T^{4}$$)($$( 1 - 29 T^{2} )^{2}$$)($$( 1 - 29 T^{2} )^{2}$$)($$1 + 50 T^{2} + 841 T^{4}$$)($$( 1 - 22 T^{2} + 841 T^{4} )^{2}$$)($$( 1 - 49 T^{2} + 841 T^{4} )^{2}$$)($$( 1 - 8 T^{2} + 841 T^{4} )^{2}$$)($$( 1 - 40 T^{2} + 841 T^{4} )^{2}$$)($$( 1 - 38 T^{2} + 841 T^{4} )^{2}$$)($$( 1 - 22 T^{2} + 841 T^{4} )^{2}$$)($$( 1 - 172 T^{2} + 14052 T^{4} - 714740 T^{6} + 24798806 T^{8} - 601096340 T^{10} + 9938712612 T^{12} - 102309611212 T^{14} + 500246412961 T^{16} )^{2}$$)($$( 1 - 103 T^{2} + 5226 T^{4} - 168029 T^{6} + 4755218 T^{8} - 141312389 T^{10} + 3696250506 T^{12} - 61266802063 T^{14} + 500246412961 T^{16} )^{2}$$)
$31$ ($$1 - 35 T^{2} + 961 T^{4}$$)($$( 1 - 4 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} )$$)($$( 1 - 4 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} )$$)($$1 - 35 T^{2} + 961 T^{4}$$)($$( 1 - 35 T^{2} + 961 T^{4} )^{2}$$)($$( 1 - 35 T^{2} + 961 T^{4} )^{2}$$)($$( 1 - 56 T^{2} + 961 T^{4} )^{2}$$)($$( 1 - 8 T^{2} + 961 T^{4} )^{2}$$)($$( 1 - 11 T + 31 T^{2} )^{2}( 1 + 11 T + 31 T^{2} )^{2}$$)($$( 1 - 35 T^{2} + 961 T^{4} )^{2}$$)($$( 1 - 187 T^{2} + 16353 T^{4} - 883634 T^{6} + 32653478 T^{8} - 849172274 T^{10} + 15102338913 T^{12} - 165963188347 T^{14} + 852891037441 T^{16} )^{2}$$)($$( 1 - 94 T^{2} + 5349 T^{4} - 230570 T^{6} + 7832744 T^{8} - 221577770 T^{10} + 4939913829 T^{12} - 83425346014 T^{14} + 852891037441 T^{16} )^{2}$$)
$37$ ($$( 1 - T + 37 T^{2} )^{2}$$)($$( 1 + 11 T + 37 T^{2} )^{2}$$)($$( 1 - T + 37 T^{2} )^{2}$$)($$( 1 - T + 37 T^{2} )^{2}$$)($$( 1 - 7 T + 37 T^{2} )^{4}$$)($$( 1 + 2 T + 37 T^{2} )^{4}$$)($$( 1 - 5 T + 37 T^{2} )^{4}$$)($$( 1 - T + 37 T^{2} )^{4}$$)($$( 1 + T + 37 T^{2} )^{4}$$)($$( 1 + 2 T + 37 T^{2} )^{4}$$)($$( 1 + 2 T + 52 T^{2} - 100 T^{3} + 949 T^{4} - 3700 T^{5} + 71188 T^{6} + 101306 T^{7} + 1874161 T^{8} )^{4}$$)($$( 1 + 2 T + 49 T^{2} + 410 T^{3} + 1084 T^{4} + 15170 T^{5} + 67081 T^{6} + 101306 T^{7} + 1874161 T^{8} )^{4}$$)
$41$ ($$( 1 - 3 T + 41 T^{2} )^{2}$$)($$( 1 + 41 T^{2} )^{2}$$)($$( 1 + 41 T^{2} )^{2}$$)($$( 1 + 3 T + 41 T^{2} )^{2}$$)($$( 1 - 65 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 + 34 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 + 7 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 + 79 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 + 67 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 + 70 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 + 135 T^{2} + 11289 T^{4} + 646218 T^{6} + 29821310 T^{8} + 1086292458 T^{10} + 31900015929 T^{12} + 641264072535 T^{14} + 7984925229121 T^{16} )^{2}$$)($$( 1 + 102 T^{2} + 10017 T^{4} + 548706 T^{6} + 28065644 T^{8} + 922374786 T^{10} + 28305647937 T^{12} + 484510632582 T^{14} + 7984925229121 T^{16} )^{2}$$)
$43$ ($$( 1 + 10 T + 43 T^{2} )^{2}$$)($$( 1 - 8 T + 43 T^{2} )^{2}$$)($$( 1 - 8 T + 43 T^{2} )^{2}$$)($$( 1 + 10 T + 43 T^{2} )^{2}$$)($$( 1 - 2 T + 43 T^{2} )^{4}$$)($$( 1 - 11 T + 43 T^{2} )^{4}$$)($$( 1 + 5 T + 43 T^{2} )^{4}$$)($$( 1 + 7 T + 43 T^{2} )^{4}$$)($$( 1 + 2 T + 43 T^{2} )^{4}$$)($$( 1 - 2 T + 43 T^{2} )^{4}$$)($$( 1 + 2 T + 40 T^{2} - 118 T^{3} - 194 T^{4} - 5074 T^{5} + 73960 T^{6} + 159014 T^{7} + 3418801 T^{8} )^{4}$$)($$( 1 + 5 T + 109 T^{2} + 248 T^{3} + 5122 T^{4} + 10664 T^{5} + 201541 T^{6} + 397535 T^{7} + 3418801 T^{8} )^{4}$$)
$47$ ($$( 1 + 6 T + 47 T^{2} )^{2}$$)($$( 1 + 47 T^{2} )^{2}$$)($$( 1 + 47 T^{2} )^{2}$$)($$( 1 - 6 T + 47 T^{2} )^{2}$$)($$( 1 + 82 T^{2} + 2209 T^{4} )^{2}$$)($$( 1 + 46 T^{2} + 2209 T^{4} )^{2}$$)($$( 1 + 19 T^{2} + 2209 T^{4} )^{2}$$)($$( 1 - 53 T^{2} + 2209 T^{4} )^{2}$$)($$( 1 + 34 T^{2} + 2209 T^{4} )^{2}$$)($$( 1 + 82 T^{2} + 2209 T^{4} )^{2}$$)($$( 1 + 20 T^{2} + 4068 T^{4} + 4972 T^{6} + 7535798 T^{8} + 10983148 T^{10} + 19850542308 T^{12} + 215584306580 T^{14} + 23811286661761 T^{16} )^{2}$$)($$( 1 + 278 T^{2} + 35193 T^{4} + 2749546 T^{6} + 150940940 T^{8} + 6073747114 T^{10} + 171730613433 T^{12} + 2996621861462 T^{14} + 23811286661761 T^{16} )^{2}$$)
$53$ ($$( 1 - 53 T^{2} )^{2}$$)($$( 1 - 53 T^{2} )^{2}$$)($$( 1 - 53 T^{2} )^{2}$$)($$( 1 - 53 T^{2} )^{2}$$)($$( 1 + 38 T^{2} + 2809 T^{4} )^{2}$$)($$( 1 - 97 T^{2} + 2809 T^{4} )^{2}$$)($$( 1 + 22 T^{2} + 2809 T^{4} )^{2}$$)($$( 1 - 53 T^{2} )^{4}$$)($$( 1 - 26 T^{2} + 2809 T^{4} )^{2}$$)($$( 1 - 97 T^{2} + 2809 T^{4} )^{2}$$)($$( 1 - 184 T^{2} + 17436 T^{4} - 1193288 T^{6} + 68587430 T^{8} - 3351945992 T^{10} + 137578426716 T^{12} - 4078242447736 T^{14} + 62259690411361 T^{16} )^{2}$$)($$( 1 - 142 T^{2} + 13833 T^{4} - 1028534 T^{6} + 58011620 T^{8} - 2889152006 T^{10} + 109149023673 T^{12} - 3147339280318 T^{14} + 62259690411361 T^{16} )^{2}$$)
$59$ ($$( 1 - 6 T + 59 T^{2} )^{2}$$)($$( 1 + 59 T^{2} )^{2}$$)($$( 1 + 59 T^{2} )^{2}$$)($$( 1 + 6 T + 59 T^{2} )^{2}$$)($$( 1 + 106 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 + 43 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 + 43 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 + 43 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 + 58 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 + 106 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 + 148 T^{2} + 10276 T^{4} + 457612 T^{6} + 23078422 T^{8} + 1592947372 T^{10} + 124518001636 T^{12} + 6242718978868 T^{14} + 146830437604321 T^{16} )^{2}$$)($$( 1 + 283 T^{2} + 41785 T^{4} + 4061626 T^{6} + 280646974 T^{8} + 14138520106 T^{10} + 506323929385 T^{12} + 11937091020403 T^{14} + 146830437604321 T^{16} )^{2}$$)
$61$ ($$1 + 70 T^{2} + 3721 T^{4}$$)($$( 1 - T + 61 T^{2} )( 1 + T + 61 T^{2} )$$)($$( 1 - 13 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} )$$)($$1 + 70 T^{2} + 3721 T^{4}$$)($$( 1 - 14 T + 61 T^{2} )^{2}( 1 + 14 T + 61 T^{2} )^{2}$$)($$( 1 + 70 T^{2} + 3721 T^{4} )^{2}$$)($$( 1 - 116 T^{2} + 3721 T^{4} )^{2}$$)($$( 1 - 116 T^{2} + 3721 T^{4} )^{2}$$)($$( 1 - 14 T + 61 T^{2} )^{2}( 1 + 14 T + 61 T^{2} )^{2}$$)($$( 1 - 14 T + 61 T^{2} )^{2}( 1 + 14 T + 61 T^{2} )^{2}$$)($$( 1 - 317 T^{2} + 50098 T^{4} - 5135915 T^{6} + 370298362 T^{8} - 19110739715 T^{10} + 693648942418 T^{12} - 16331958672437 T^{14} + 191707312997281 T^{16} )^{2}$$)($$( 1 - 236 T^{2} + 32440 T^{4} - 3088964 T^{6} + 216205582 T^{8} - 11494035044 T^{10} + 449159082040 T^{12} - 12158808349196 T^{14} + 191707312997281 T^{16} )^{2}$$)
$67$ ($$( 1 + 2 T + 67 T^{2} )^{2}$$)($$( 1 + 5 T + 67 T^{2} )^{2}$$)($$( 1 + 11 T + 67 T^{2} )^{2}$$)($$( 1 + 2 T + 67 T^{2} )^{2}$$)($$( 1 + 2 T + 67 T^{2} )^{4}$$)($$( 1 - 7 T + 67 T^{2} )^{4}$$)($$( 1 + 2 T + 67 T^{2} )^{4}$$)($$( 1 - 10 T + 67 T^{2} )^{4}$$)($$( 1 - 10 T + 67 T^{2} )^{4}$$)($$( 1 + 2 T + 67 T^{2} )^{4}$$)($$( 1 - 7 T + 114 T^{2} - 35 T^{3} + 3554 T^{4} - 2345 T^{5} + 511746 T^{6} - 2105341 T^{7} + 20151121 T^{8} )^{4}$$)($$( 1 + 11 T + 198 T^{2} + 2071 T^{3} + 17498 T^{4} + 138757 T^{5} + 888822 T^{6} + 3308393 T^{7} + 20151121 T^{8} )^{4}$$)
$71$ ($$1 - 115 T^{2} + 5041 T^{4}$$)($$( 1 - 71 T^{2} )^{2}$$)($$( 1 - 71 T^{2} )^{2}$$)($$1 - 115 T^{2} + 5041 T^{4}$$)($$( 1 - 133 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 - 133 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 + 58 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 - 70 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 - 17 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 + 2 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 - 341 T^{2} + 63193 T^{4} - 7537622 T^{6} + 634982182 T^{8} - 37997152502 T^{10} + 1605840357433 T^{12} - 43682196817061 T^{14} + 645753531245761 T^{16} )^{2}$$)($$( 1 - 167 T^{2} + 18034 T^{4} - 1223801 T^{6} + 87222970 T^{8} - 6169180841 T^{10} + 458274255154 T^{12} - 21392747414807 T^{14} + 645753531245761 T^{16} )^{2}$$)
$73$ ($$1 - 134 T^{2} + 5329 T^{4}$$)($$( 1 - 7 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} )$$)($$( 1 - 17 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} )$$)($$1 - 134 T^{2} + 5329 T^{4}$$)($$( 1 - 134 T^{2} + 5329 T^{4} )^{2}$$)($$( 1 - 98 T^{2} + 5329 T^{4} )^{2}$$)($$( 1 - 73 T^{2} )^{4}$$)($$( 1 - 14 T + 73 T^{2} )^{2}( 1 + 14 T + 73 T^{2} )^{2}$$)($$( 1 - 38 T^{2} + 5329 T^{4} )^{2}$$)($$( 1 + T^{2} + 5329 T^{4} )^{2}$$)($$( 1 - 321 T^{2} + 57726 T^{4} - 6821535 T^{6} + 583944098 T^{8} - 36351960015 T^{10} + 1639316859966 T^{12} - 48578286638769 T^{14} + 806460091894081 T^{16} )^{2}$$)($$( 1 - 117 T^{2} + 15114 T^{4} - 1400907 T^{6} + 111599882 T^{8} - 7465433403 T^{10} + 429211014474 T^{12} - 17706104475813 T^{14} + 806460091894081 T^{16} )^{2}$$)
$79$ ($$( 1 + 14 T + 79 T^{2} )^{2}$$)($$( 1 + 17 T + 79 T^{2} )^{2}$$)($$( 1 - 13 T + 79 T^{2} )^{2}$$)($$( 1 + 14 T + 79 T^{2} )^{2}$$)($$( 1 - 10 T + 79 T^{2} )^{4}$$)($$( 1 + 8 T + 79 T^{2} )^{4}$$)($$( 1 - 13 T + 79 T^{2} )^{4}$$)($$( 1 + 5 T + 79 T^{2} )^{4}$$)($$( 1 + 2 T + 79 T^{2} )^{4}$$)($$( 1 + 8 T + 79 T^{2} )^{4}$$)($$( 1 - 11 T + 246 T^{2} - 2515 T^{3} + 26570 T^{4} - 198685 T^{5} + 1535286 T^{6} - 5423429 T^{7} + 38950081 T^{8} )^{4}$$)($$( 1 + 10 T + 261 T^{2} + 1850 T^{3} + 28244 T^{4} + 146150 T^{5} + 1628901 T^{6} + 4930390 T^{7} + 38950081 T^{8} )^{4}$$)
$83$ ($$( 1 - 6 T + 83 T^{2} )^{2}$$)($$( 1 + 83 T^{2} )^{2}$$)($$( 1 + 83 T^{2} )^{2}$$)($$( 1 + 6 T + 83 T^{2} )^{2}$$)($$( 1 - 134 T^{2} + 6889 T^{4} )^{2}$$)($$( 1 + 154 T^{2} + 6889 T^{4} )^{2}$$)($$( 1 + 163 T^{2} + 6889 T^{4} )^{2}$$)($$( 1 + 19 T^{2} + 6889 T^{4} )^{2}$$)($$( 1 + 106 T^{2} + 6889 T^{4} )^{2}$$)($$( 1 + 163 T^{2} + 6889 T^{4} )^{2}$$)($$( 1 + 132 T^{2} + 12996 T^{4} + 1459932 T^{6} + 170705174 T^{8} + 10057471548 T^{10} + 616768339716 T^{12} + 43156129284708 T^{14} + 2252292232139041 T^{16} )^{2}$$)($$( 1 + 339 T^{2} + 56745 T^{4} + 6361962 T^{6} + 567506462 T^{8} + 43827556218 T^{10} + 2693022425145 T^{12} + 110832786572091 T^{14} + 2252292232139041 T^{16} )^{2}$$)
$89$ ($$( 1 + 9 T + 89 T^{2} )^{2}$$)($$( 1 + 89 T^{2} )^{2}$$)($$( 1 + 89 T^{2} )^{2}$$)($$( 1 - 9 T + 89 T^{2} )^{2}$$)($$( 1 + 151 T^{2} + 7921 T^{4} )^{2}$$)($$( 1 + 151 T^{2} + 7921 T^{4} )^{2}$$)($$( 1 + 70 T^{2} + 7921 T^{4} )^{2}$$)($$( 1 + 70 T^{2} + 7921 T^{4} )^{2}$$)($$( 1 + 43 T^{2} + 7921 T^{4} )^{2}$$)($$( 1 + 70 T^{2} + 7921 T^{4} )^{2}$$)($$( 1 + 351 T^{2} + 57993 T^{4} + 6040074 T^{6} + 534549758 T^{8} + 47843426154 T^{10} + 3638610782313 T^{12} + 174440433127311 T^{14} + 3936588805702081 T^{16} )^{2}$$)($$( 1 + 321 T^{2} + 55134 T^{4} + 6909375 T^{6} + 691071074 T^{8} + 54729159375 T^{10} + 3459230715294 T^{12} + 159530994398481 T^{14} + 3936588805702081 T^{16} )^{2}$$)
$97$ ($$1 - 146 T^{2} + 9409 T^{4}$$)($$( 1 - 19 T + 97 T^{2} )( 1 + 19 T + 97 T^{2} )$$)($$( 1 - 5 T + 97 T^{2} )( 1 + 5 T + 97 T^{2} )$$)($$1 - 146 T^{2} + 9409 T^{4}$$)($$( 1 - 14 T + 97 T^{2} )^{2}( 1 + 14 T + 97 T^{2} )^{2}$$)($$( 1 - 146 T^{2} + 9409 T^{4} )^{2}$$)($$( 1 + 100 T^{2} + 9409 T^{4} )^{2}$$)($$( 1 - 188 T^{2} + 9409 T^{4} )^{2}$$)($$( 1 - 14 T + 97 T^{2} )^{2}( 1 + 14 T + 97 T^{2} )^{2}$$)($$( 1 - 47 T^{2} + 9409 T^{4} )^{2}$$)($$( 1 - 285 T^{2} + 64986 T^{4} - 9080499 T^{6} + 1059084650 T^{8} - 85438415091 T^{10} + 5753163855066 T^{12} - 237397021404765 T^{14} + 7837433594376961 T^{16} )^{2}$$)($$( 1 - 537 T^{2} + 140634 T^{4} - 23455755 T^{6} + 2707648082 T^{8} - 220695198795 T^{10} + 12450226904154 T^{12} - 447305966646873 T^{14} + 7837433594376961 T^{16} )^{2}$$)