# Properties

 Label 2100.2.d Level 2100 Weight 2 Character orbit d Rep. character $$\chi_{2100}(1301,\cdot)$$ Character field $$\Q$$ Dimension 50 Newform subspaces 12 Sturm bound 960 Trace bound 43

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2100.d (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$21$$ Character field: $$\Q$$ Newform subspaces: $$12$$ Sturm bound: $$960$$ Trace bound: $$43$$ Distinguishing $$T_p$$: $$11$$, $$13$$, $$17$$, $$37$$, $$41$$, $$43$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(2100, [\chi])$$.

Total New Old
Modular forms 516 50 466
Cusp forms 444 50 394
Eisenstein series 72 0 72

## Trace form

 $$50q + 6q^{9} + O(q^{10})$$ $$50q + 6q^{9} + 8q^{21} + 12q^{37} + 16q^{43} - 4q^{49} - 20q^{51} - 20q^{63} + 16q^{79} - 10q^{81} - 22q^{91} + 56q^{93} - 28q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(2100, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
2100.2.d.a $$2$$ $$16.769$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-3$$ $$0$$ $$-4$$ $$q+(-1-\zeta_{6})q^{3}+(-3+2\zeta_{6})q^{7}+3\zeta_{6}q^{9}+\cdots$$
2100.2.d.b $$2$$ $$16.769$$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$-4$$ $$q-\zeta_{6}q^{3}+(-2-\zeta_{6})q^{7}-3q^{9}+4\zeta_{6}q^{13}+\cdots$$
2100.2.d.c $$2$$ $$16.769$$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$-1$$ $$q+(-1+2\zeta_{6})q^{3}+(1-3\zeta_{6})q^{7}-3q^{9}+\cdots$$
2100.2.d.d $$2$$ $$16.769$$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$1$$ $$q+(1-2\zeta_{6})q^{3}+(-1+3\zeta_{6})q^{7}-3q^{9}+\cdots$$
2100.2.d.e $$2$$ $$16.769$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$3$$ $$0$$ $$-4$$ $$q+(1+\zeta_{6})q^{3}+(-1-2\zeta_{6})q^{7}+3\zeta_{6}q^{9}+\cdots$$
2100.2.d.f $$4$$ $$16.769$$ $$\Q(\sqrt{-2}, \sqrt{5})$$ None $$0$$ $$-4$$ $$0$$ $$0$$ $$q+(-1-\beta _{1})q^{3}+(\beta _{1}+\beta _{3})q^{7}+(-1+\cdots)q^{9}+\cdots$$
2100.2.d.g $$4$$ $$16.769$$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$-2$$ $$0$$ $$6$$ $$q+(-1+\beta _{3})q^{3}+(1+\beta _{2}+\beta _{3})q^{7}+\cdots$$
2100.2.d.h $$4$$ $$16.769$$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$2$$ $$0$$ $$6$$ $$q+(1-\beta _{3})q^{3}+(1-\beta _{1}-\beta _{2})q^{7}+(1+\cdots)q^{9}+\cdots$$
2100.2.d.i $$4$$ $$16.769$$ $$\Q(\sqrt{-2}, \sqrt{5})$$ None $$0$$ $$4$$ $$0$$ $$0$$ $$q+(1+\beta _{1})q^{3}+(-\beta _{1}+\beta _{3})q^{7}+(-1+\cdots)q^{9}+\cdots$$
2100.2.d.j $$8$$ $$16.769$$ 8.0.121550625.1 $$\Q(\sqrt{-35})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{3}q^{3}+(-\beta _{2}+\beta _{3})q^{7}-\beta _{1}q^{9}+\cdots$$
2100.2.d.k $$8$$ $$16.769$$ 8.0.$$\cdots$$.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}+\beta _{3}-\beta _{6})q^{3}+(\beta _{3}+\beta _{5})q^{7}+\cdots$$
2100.2.d.l $$8$$ $$16.769$$ 8.0.$$\cdots$$.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}+\beta _{3}-\beta _{6})q^{3}+(\beta _{3}-\beta _{5})q^{7}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(2100, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(2100, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(42, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(84, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(105, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(210, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(420, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(525, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1050, [\chi])$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ ($$1 + 3 T + 3 T^{2}$$)($$1 + 3 T^{2}$$)($$1 + 3 T^{2}$$)($$1 + 3 T^{2}$$)($$1 - 3 T + 3 T^{2}$$)($$( 1 + 2 T + 3 T^{2} )^{2}$$)($$1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4}$$)($$1 - 2 T + 2 T^{2} - 6 T^{3} + 9 T^{4}$$)($$( 1 - 2 T + 3 T^{2} )^{2}$$)($$1 - T^{2} - 8 T^{4} - 9 T^{6} + 81 T^{8}$$)($$1 - 6 T^{2} + 22 T^{4} - 54 T^{6} + 81 T^{8}$$)($$1 - 6 T^{2} + 22 T^{4} - 54 T^{6} + 81 T^{8}$$)
$5$ 1
$7$ ($$1 + 4 T + 7 T^{2}$$)($$1 + 4 T + 7 T^{2}$$)($$1 + T + 7 T^{2}$$)($$1 - T + 7 T^{2}$$)($$1 + 4 T + 7 T^{2}$$)($$1 - 6 T^{2} + 49 T^{4}$$)($$1 - 6 T + 18 T^{2} - 42 T^{3} + 49 T^{4}$$)($$1 - 6 T + 18 T^{2} - 42 T^{3} + 49 T^{4}$$)($$1 - 6 T^{2} + 49 T^{4}$$)($$( 1 + 7 T^{2} )^{4}$$)($$( 1 - 6 T^{2} + 49 T^{4} )^{2}$$)($$( 1 - 6 T^{2} + 49 T^{4} )^{2}$$)
$11$ ($$1 + 5 T^{2} + 121 T^{4}$$)($$( 1 - 11 T^{2} )^{2}$$)($$( 1 - 11 T^{2} )^{2}$$)($$( 1 - 11 T^{2} )^{2}$$)($$1 + 5 T^{2} + 121 T^{4}$$)($$( 1 - 6 T + 11 T^{2} )^{2}( 1 + 6 T + 11 T^{2} )^{2}$$)($$1 - 16 T^{2} + 126 T^{4} - 1936 T^{6} + 14641 T^{8}$$)($$1 - 16 T^{2} + 126 T^{4} - 1936 T^{6} + 14641 T^{8}$$)($$( 1 - 6 T + 11 T^{2} )^{2}( 1 + 6 T + 11 T^{2} )^{2}$$)($$( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} )^{2}( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} )^{2}$$)($$( 1 - 28 T^{2} + 393 T^{4} - 3388 T^{6} + 14641 T^{8} )^{2}$$)($$( 1 - 28 T^{2} + 393 T^{4} - 3388 T^{6} + 14641 T^{8} )^{2}$$)
$13$ ($$( 1 - 7 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )$$)($$( 1 - 2 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} )$$)($$( 1 - 7 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )$$)($$( 1 - 7 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )$$)($$( 1 - 7 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )$$)($$( 1 - 18 T^{2} + 169 T^{4} )^{2}$$)($$1 - 40 T^{2} + 718 T^{4} - 6760 T^{6} + 28561 T^{8}$$)($$1 - 40 T^{2} + 718 T^{4} - 6760 T^{6} + 28561 T^{8}$$)($$( 1 - 18 T^{2} + 169 T^{4} )^{2}$$)($$( 1 + 19 T^{2} + 192 T^{4} + 3211 T^{6} + 28561 T^{8} )^{2}$$)($$( 1 - 16 T^{2} + 82 T^{4} - 2704 T^{6} + 28561 T^{8} )^{2}$$)($$( 1 - 16 T^{2} + 82 T^{4} - 2704 T^{6} + 28561 T^{8} )^{2}$$)
$17$ ($$( 1 - 3 T + 17 T^{2} )^{2}$$)($$( 1 + 17 T^{2} )^{2}$$)($$( 1 + 17 T^{2} )^{2}$$)($$( 1 + 17 T^{2} )^{2}$$)($$( 1 + 3 T + 17 T^{2} )^{2}$$)($$( 1 + 4 T + 17 T^{2} )^{4}$$)($$( 1 + 14 T^{2} + 289 T^{4} )^{2}$$)($$( 1 + 14 T^{2} + 289 T^{4} )^{2}$$)($$( 1 - 4 T + 17 T^{2} )^{4}$$)($$( 1 - 29 T^{2} + 552 T^{4} - 8381 T^{6} + 83521 T^{8} )^{2}$$)($$( 1 + 46 T^{2} + 1062 T^{4} + 13294 T^{6} + 83521 T^{8} )^{2}$$)($$( 1 + 46 T^{2} + 1062 T^{4} + 13294 T^{6} + 83521 T^{8} )^{2}$$)
$19$ ($$( 1 - 8 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$)($$( 1 - 8 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$)($$( 1 - 7 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} )$$)($$( 1 - 7 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} )$$)($$( 1 - 8 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$)($$( 1 - 6 T + 19 T^{2} )^{2}( 1 + 6 T + 19 T^{2} )^{2}$$)($$1 - 16 T^{2} + 286 T^{4} - 5776 T^{6} + 130321 T^{8}$$)($$1 - 16 T^{2} + 286 T^{4} - 5776 T^{6} + 130321 T^{8}$$)($$( 1 - 6 T + 19 T^{2} )^{2}( 1 + 6 T + 19 T^{2} )^{2}$$)($$( 1 - 19 T^{2} )^{8}$$)($$( 1 - 46 T^{2} + 1126 T^{4} - 16606 T^{6} + 130321 T^{8} )^{2}$$)($$( 1 - 46 T^{2} + 1126 T^{4} - 16606 T^{6} + 130321 T^{8} )^{2}$$)
$23$ ($$( 1 - 23 T^{2} )^{2}$$)($$( 1 - 23 T^{2} )^{2}$$)($$( 1 - 23 T^{2} )^{2}$$)($$( 1 - 23 T^{2} )^{2}$$)($$( 1 - 23 T^{2} )^{2}$$)($$( 1 - 6 T^{2} + 529 T^{4} )^{2}$$)($$( 1 - 2 T + 2 T^{2} - 46 T^{3} + 529 T^{4} )( 1 + 2 T + 2 T^{2} + 46 T^{3} + 529 T^{4} )$$)($$( 1 - 2 T + 2 T^{2} - 46 T^{3} + 529 T^{4} )( 1 + 2 T + 2 T^{2} + 46 T^{3} + 529 T^{4} )$$)($$( 1 - 6 T^{2} + 529 T^{4} )^{2}$$)($$( 1 - 23 T^{2} )^{8}$$)($$( 1 - 12 T^{2} - 31 T^{4} - 6348 T^{6} + 279841 T^{8} )^{2}$$)($$( 1 - 12 T^{2} - 31 T^{4} - 6348 T^{6} + 279841 T^{8} )^{2}$$)
$29$ ($$1 - 31 T^{2} + 841 T^{4}$$)($$( 1 - 29 T^{2} )^{2}$$)($$( 1 - 29 T^{2} )^{2}$$)($$( 1 - 29 T^{2} )^{2}$$)($$1 - 31 T^{2} + 841 T^{4}$$)($$( 1 - 26 T^{2} + 841 T^{4} )^{2}$$)($$( 1 - 10 T + 29 T^{2} )^{2}( 1 + 10 T + 29 T^{2} )^{2}$$)($$( 1 - 10 T + 29 T^{2} )^{2}( 1 + 10 T + 29 T^{2} )^{2}$$)($$( 1 - 26 T^{2} + 841 T^{4} )^{2}$$)($$( 1 - 9 T + 52 T^{2} - 261 T^{3} + 841 T^{4} )^{2}( 1 + 9 T + 52 T^{2} + 261 T^{3} + 841 T^{4} )^{2}$$)($$( 1 - 12 T^{2} + 1313 T^{4} - 10092 T^{6} + 707281 T^{8} )^{2}$$)($$( 1 - 12 T^{2} + 1313 T^{4} - 10092 T^{6} + 707281 T^{8} )^{2}$$)
$31$ ($$( 1 - 4 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} )$$)($$( 1 - 4 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} )$$)($$( 1 - 11 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} )$$)($$( 1 - 11 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} )$$)($$( 1 - 4 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} )$$)($$( 1 - 22 T^{2} + 961 T^{4} )^{2}$$)($$1 - 112 T^{2} + 5038 T^{4} - 107632 T^{6} + 923521 T^{8}$$)($$1 - 112 T^{2} + 5038 T^{4} - 107632 T^{6} + 923521 T^{8}$$)($$( 1 - 22 T^{2} + 961 T^{4} )^{2}$$)($$( 1 - 31 T^{2} )^{8}$$)($$( 1 - 94 T^{2} + 4006 T^{4} - 90334 T^{6} + 923521 T^{8} )^{2}$$)($$( 1 - 94 T^{2} + 4006 T^{4} - 90334 T^{6} + 923521 T^{8} )^{2}$$)
$37$ ($$( 1 - 8 T + 37 T^{2} )^{2}$$)($$( 1 + 10 T + 37 T^{2} )^{2}$$)($$( 1 + 10 T + 37 T^{2} )^{2}$$)($$( 1 - 10 T + 37 T^{2} )^{2}$$)($$( 1 - 8 T + 37 T^{2} )^{2}$$)($$( 1 - 6 T^{2} + 1369 T^{4} )^{2}$$)($$( 1 + 54 T^{2} + 1369 T^{4} )^{2}$$)($$( 1 + 54 T^{2} + 1369 T^{4} )^{2}$$)($$( 1 - 6 T^{2} + 1369 T^{4} )^{2}$$)($$( 1 + 37 T^{2} )^{8}$$)($$( 1 + 69 T^{2} + 1369 T^{4} )^{4}$$)($$( 1 + 69 T^{2} + 1369 T^{4} )^{4}$$)
$41$ ($$( 1 + 6 T + 41 T^{2} )^{2}$$)($$( 1 + 41 T^{2} )^{2}$$)($$( 1 + 41 T^{2} )^{2}$$)($$( 1 + 41 T^{2} )^{2}$$)($$( 1 - 6 T + 41 T^{2} )^{2}$$)($$( 1 + 62 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 + 16 T + 126 T^{2} + 656 T^{3} + 1681 T^{4} )^{2}$$)($$( 1 - 16 T + 126 T^{2} - 656 T^{3} + 1681 T^{4} )^{2}$$)($$( 1 + 62 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 + 41 T^{2} )^{8}$$)($$( 1 + 74 T^{2} + 4606 T^{4} + 124394 T^{6} + 2825761 T^{8} )^{2}$$)($$( 1 + 74 T^{2} + 4606 T^{4} + 124394 T^{6} + 2825761 T^{8} )^{2}$$)
$43$ ($$( 1 + 10 T + 43 T^{2} )^{2}$$)($$( 1 - 8 T + 43 T^{2} )^{2}$$)($$( 1 + 13 T + 43 T^{2} )^{2}$$)($$( 1 - 13 T + 43 T^{2} )^{2}$$)($$( 1 + 10 T + 43 T^{2} )^{2}$$)($$( 1 + 66 T^{2} + 1849 T^{4} )^{2}$$)($$( 1 - 10 T + 106 T^{2} - 430 T^{3} + 1849 T^{4} )^{2}$$)($$( 1 - 10 T + 106 T^{2} - 430 T^{3} + 1849 T^{4} )^{2}$$)($$( 1 + 66 T^{2} + 1849 T^{4} )^{2}$$)($$( 1 + 43 T^{2} )^{8}$$)($$( 1 + 10 T + 91 T^{2} + 430 T^{3} + 1849 T^{4} )^{4}$$)($$( 1 - 10 T + 91 T^{2} - 430 T^{3} + 1849 T^{4} )^{4}$$)
$47$ ($$( 1 - 3 T + 47 T^{2} )^{2}$$)($$( 1 + 47 T^{2} )^{2}$$)($$( 1 + 47 T^{2} )^{2}$$)($$( 1 + 47 T^{2} )^{2}$$)($$( 1 + 3 T + 47 T^{2} )^{2}$$)($$( 1 + 6 T + 47 T^{2} )^{4}$$)($$( 1 - 2 T + 90 T^{2} - 94 T^{3} + 2209 T^{4} )^{2}$$)($$( 1 + 2 T + 90 T^{2} + 94 T^{3} + 2209 T^{4} )^{2}$$)($$( 1 - 6 T + 47 T^{2} )^{4}$$)($$( 1 + 31 T^{2} - 1248 T^{4} + 68479 T^{6} + 4879681 T^{8} )^{2}$$)($$( 1 - 34 T^{2} + 1582 T^{4} - 75106 T^{6} + 4879681 T^{8} )^{2}$$)($$( 1 - 34 T^{2} + 1582 T^{4} - 75106 T^{6} + 4879681 T^{8} )^{2}$$)
$53$ ($$1 + 2 T^{2} + 2809 T^{4}$$)($$( 1 - 53 T^{2} )^{2}$$)($$( 1 - 53 T^{2} )^{2}$$)($$( 1 - 53 T^{2} )^{2}$$)($$1 + 2 T^{2} + 2809 T^{4}$$)($$( 1 - 66 T^{2} + 2809 T^{4} )^{2}$$)($$1 - 104 T^{2} + 7342 T^{4} - 292136 T^{6} + 7890481 T^{8}$$)($$1 - 104 T^{2} + 7342 T^{4} - 292136 T^{6} + 7890481 T^{8}$$)($$( 1 - 66 T^{2} + 2809 T^{4} )^{2}$$)($$( 1 - 53 T^{2} )^{8}$$)($$( 1 - 32 T^{2} + 5374 T^{4} - 89888 T^{6} + 7890481 T^{8} )^{2}$$)($$( 1 - 32 T^{2} + 5374 T^{4} - 89888 T^{6} + 7890481 T^{8} )^{2}$$)
$59$ ($$( 1 + 6 T + 59 T^{2} )^{2}$$)($$( 1 + 59 T^{2} )^{2}$$)($$( 1 + 59 T^{2} )^{2}$$)($$( 1 + 59 T^{2} )^{2}$$)($$( 1 - 6 T + 59 T^{2} )^{2}$$)($$( 1 + 38 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 + 38 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 + 38 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 + 38 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 + 59 T^{2} )^{8}$$)($$( 1 + 76 T^{2} + 3906 T^{4} + 264556 T^{6} + 12117361 T^{8} )^{2}$$)($$( 1 + 76 T^{2} + 3906 T^{4} + 264556 T^{6} + 12117361 T^{8} )^{2}$$)
$61$ ($$( 1 - 14 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} )$$)($$( 1 - 14 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} )$$)($$( 1 - T + 61 T^{2} )( 1 + T + 61 T^{2} )$$)($$( 1 - T + 61 T^{2} )( 1 + T + 61 T^{2} )$$)($$( 1 - 14 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} )$$)($$( 1 - 61 T^{2} )^{4}$$)($$1 - 52 T^{2} + 2998 T^{4} - 193492 T^{6} + 13845841 T^{8}$$)($$1 - 52 T^{2} + 2998 T^{4} - 193492 T^{6} + 13845841 T^{8}$$)($$( 1 - 61 T^{2} )^{4}$$)($$( 1 - 61 T^{2} )^{8}$$)($$( 1 - 94 T^{2} + 6526 T^{4} - 349774 T^{6} + 13845841 T^{8} )^{2}$$)($$( 1 - 94 T^{2} + 6526 T^{4} - 349774 T^{6} + 13845841 T^{8} )^{2}$$)
$67$ ($$( 1 + 2 T + 67 T^{2} )^{2}$$)($$( 1 - 16 T + 67 T^{2} )^{2}$$)($$( 1 + 11 T + 67 T^{2} )^{2}$$)($$( 1 - 11 T + 67 T^{2} )^{2}$$)($$( 1 + 2 T + 67 T^{2} )^{2}$$)($$( 1 + 114 T^{2} + 4489 T^{4} )^{2}$$)($$( 1 + 6 T + 98 T^{2} + 402 T^{3} + 4489 T^{4} )^{2}$$)($$( 1 + 6 T + 98 T^{2} + 402 T^{3} + 4489 T^{4} )^{2}$$)($$( 1 + 114 T^{2} + 4489 T^{4} )^{2}$$)($$( 1 + 67 T^{2} )^{8}$$)($$( 1 + 89 T^{2} + 4489 T^{4} )^{4}$$)($$( 1 + 89 T^{2} + 4489 T^{4} )^{4}$$)
$71$ ($$1 - 34 T^{2} + 5041 T^{4}$$)($$( 1 - 71 T^{2} )^{2}$$)($$( 1 - 71 T^{2} )^{2}$$)($$( 1 - 71 T^{2} )^{2}$$)($$1 - 34 T^{2} + 5041 T^{4}$$)($$( 1 + 58 T^{2} + 5041 T^{4} )^{2}$$)($$1 - 32 T^{2} + 9838 T^{4} - 161312 T^{6} + 25411681 T^{8}$$)($$1 - 32 T^{2} + 9838 T^{4} - 161312 T^{6} + 25411681 T^{8}$$)($$( 1 + 58 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 - 12 T + 71 T^{2} )^{4}( 1 + 12 T + 71 T^{2} )^{4}$$)($$( 1 - 164 T^{2} + 13681 T^{4} - 826724 T^{6} + 25411681 T^{8} )^{2}$$)($$( 1 - 164 T^{2} + 13681 T^{4} - 826724 T^{6} + 25411681 T^{8} )^{2}$$)
$73$ ($$1 - 98 T^{2} + 5329 T^{4}$$)($$( 1 - 10 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} )$$)($$( 1 - 10 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} )$$)($$( 1 - 10 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} )$$)($$1 - 98 T^{2} + 5329 T^{4}$$)($$( 1 - 74 T^{2} + 5329 T^{4} )^{2}$$)($$1 - 120 T^{2} + 12638 T^{4} - 639480 T^{6} + 28398241 T^{8}$$)($$1 - 120 T^{2} + 12638 T^{4} - 639480 T^{6} + 28398241 T^{8}$$)($$( 1 - 74 T^{2} + 5329 T^{4} )^{2}$$)($$( 1 - 34 T^{2} + 5329 T^{4} )^{4}$$)($$( 1 - 144 T^{2} + 5329 T^{4} )^{4}$$)($$( 1 - 144 T^{2} + 5329 T^{4} )^{4}$$)
$79$ ($$( 1 + 13 T + 79 T^{2} )^{2}$$)($$( 1 + 4 T + 79 T^{2} )^{2}$$)($$( 1 + 4 T + 79 T^{2} )^{2}$$)($$( 1 + 4 T + 79 T^{2} )^{2}$$)($$( 1 + 13 T + 79 T^{2} )^{2}$$)($$( 1 - 12 T + 79 T^{2} )^{4}$$)($$( 1 - 12 T + 174 T^{2} - 948 T^{3} + 6241 T^{4} )^{2}$$)($$( 1 - 12 T + 174 T^{2} - 948 T^{3} + 6241 T^{4} )^{2}$$)($$( 1 - 12 T + 79 T^{2} )^{4}$$)($$( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} )^{4}$$)($$( 1 + 6 T + 147 T^{2} + 474 T^{3} + 6241 T^{4} )^{4}$$)($$( 1 + 6 T + 147 T^{2} + 474 T^{3} + 6241 T^{4} )^{4}$$)
$83$ ($$( 1 + 12 T + 83 T^{2} )^{2}$$)($$( 1 + 83 T^{2} )^{2}$$)($$( 1 + 83 T^{2} )^{2}$$)($$( 1 + 83 T^{2} )^{2}$$)($$( 1 - 12 T + 83 T^{2} )^{2}$$)($$( 1 - 10 T + 83 T^{2} )^{4}$$)($$( 1 - 10 T + 186 T^{2} - 830 T^{3} + 6889 T^{4} )^{2}$$)($$( 1 + 10 T + 186 T^{2} + 830 T^{3} + 6889 T^{4} )^{2}$$)($$( 1 + 10 T + 83 T^{2} )^{4}$$)($$( 1 + 86 T^{2} + 6889 T^{4} )^{4}$$)($$( 1 + 242 T^{2} + 28294 T^{4} + 1667138 T^{6} + 47458321 T^{8} )^{2}$$)($$( 1 + 242 T^{2} + 28294 T^{4} + 1667138 T^{6} + 47458321 T^{8} )^{2}$$)
$89$ ($$( 1 + 89 T^{2} )^{2}$$)($$( 1 + 89 T^{2} )^{2}$$)($$( 1 + 89 T^{2} )^{2}$$)($$( 1 + 89 T^{2} )^{2}$$)($$( 1 + 89 T^{2} )^{2}$$)($$( 1 + 158 T^{2} + 7921 T^{4} )^{2}$$)($$( 1 + 16 T + 222 T^{2} + 1424 T^{3} + 7921 T^{4} )^{2}$$)($$( 1 - 16 T + 222 T^{2} - 1424 T^{3} + 7921 T^{4} )^{2}$$)($$( 1 + 158 T^{2} + 7921 T^{4} )^{2}$$)($$( 1 + 89 T^{2} )^{8}$$)($$( 1 + 116 T^{2} + 6706 T^{4} + 918836 T^{6} + 62742241 T^{8} )^{2}$$)($$( 1 + 116 T^{2} + 6706 T^{4} + 918836 T^{6} + 62742241 T^{8} )^{2}$$)
$97$ ($$1 - 191 T^{2} + 9409 T^{4}$$)($$( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} )$$)($$( 1 - 19 T + 97 T^{2} )( 1 + 19 T + 97 T^{2} )$$)($$( 1 - 19 T + 97 T^{2} )( 1 + 19 T + 97 T^{2} )$$)($$1 - 191 T^{2} + 9409 T^{4}$$)($$( 1 - 122 T^{2} + 9409 T^{4} )^{2}$$)($$1 - 120 T^{2} + 17918 T^{4} - 1129080 T^{6} + 88529281 T^{8}$$)($$1 - 120 T^{2} + 17918 T^{4} - 1129080 T^{6} + 88529281 T^{8}$$)($$( 1 - 122 T^{2} + 9409 T^{4} )^{2}$$)($$( 1 - 149 T^{2} + 12792 T^{4} - 1401941 T^{6} + 88529281 T^{8} )^{2}$$)($$( 1 - 294 T^{2} + 38222 T^{4} - 2766246 T^{6} + 88529281 T^{8} )^{2}$$)($$( 1 - 294 T^{2} + 38222 T^{4} - 2766246 T^{6} + 88529281 T^{8} )^{2}$$)