# Properties

 Label 504.2.s Level 504 Weight 2 Character orbit s Rep. character $$\chi_{504}(289,\cdot)$$ Character field $$\Q(\zeta_{3})$$ Dimension 20 Newform subspaces 9 Sturm bound 192 Trace bound 7

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## Defining parameters

 Level: $$N$$ = $$504 = 2^{3} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 504.s (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$7$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$9$$ Sturm bound: $$192$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$5$$, $$11$$, $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(504, [\chi])$$.

Total New Old
Modular forms 224 20 204
Cusp forms 160 20 140
Eisenstein series 64 0 64

## Trace form

 $$20q - 2q^{5} + O(q^{10})$$ $$20q - 2q^{5} - 2q^{11} - 6q^{17} - 10q^{19} + 2q^{23} - 20q^{25} + 16q^{29} + 6q^{31} + 30q^{35} - 2q^{37} + 8q^{43} + 6q^{47} + 36q^{49} + 18q^{53} + 4q^{55} + 18q^{59} + 10q^{61} + 20q^{65} + 2q^{67} - 48q^{71} + 22q^{73} - 50q^{77} - 14q^{79} - 56q^{83} - 60q^{85} - 34q^{89} - 44q^{91} - 38q^{95} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(504, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
504.2.s.a $$2$$ $$4.024$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-4$$ $$5$$ $$q+(-4+4\zeta_{6})q^{5}+(3-\zeta_{6})q^{7}-3q^{13}+\cdots$$
504.2.s.b $$2$$ $$4.024$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-1$$ $$-5$$ $$q+(-1+\zeta_{6})q^{5}+(-2-\zeta_{6})q^{7}+5\zeta_{6}q^{11}+\cdots$$
504.2.s.c $$2$$ $$4.024$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-1$$ $$-4$$ $$q+(-1+\zeta_{6})q^{5}+(-3+2\zeta_{6})q^{7}+3\zeta_{6}q^{11}+\cdots$$
504.2.s.d $$2$$ $$4.024$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-1$$ $$1$$ $$q+(-1+\zeta_{6})q^{5}+(2-3\zeta_{6})q^{7}+3\zeta_{6}q^{11}+\cdots$$
504.2.s.e $$2$$ $$4.024$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-1$$ $$4$$ $$q+(-1+\zeta_{6})q^{5}+(1+2\zeta_{6})q^{7}-\zeta_{6}q^{11}+\cdots$$
504.2.s.f $$2$$ $$4.024$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$1$$ $$-5$$ $$q+(1-\zeta_{6})q^{5}+(-2-\zeta_{6})q^{7}-5\zeta_{6}q^{11}+\cdots$$
504.2.s.g $$2$$ $$4.024$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$2$$ $$5$$ $$q+(2-2\zeta_{6})q^{5}+(3-\zeta_{6})q^{7}-6\zeta_{6}q^{11}+\cdots$$
504.2.s.h $$2$$ $$4.024$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$4$$ $$5$$ $$q+(4-4\zeta_{6})q^{5}+(3-\zeta_{6})q^{7}-3q^{13}+\cdots$$
504.2.s.i $$4$$ $$4.024$$ $$\Q(\sqrt{-3}, \sqrt{-19})$$ None $$0$$ $$0$$ $$-1$$ $$-6$$ $$q+(-1+2\beta _{1}-\beta _{3})q^{5}+(-1-\beta _{1}+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(504, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(504, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(21, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(28, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(42, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(56, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(63, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(84, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(126, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(168, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(252, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ ($$1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4}$$)($$1 + T - 4 T^{2} + 5 T^{3} + 25 T^{4}$$)($$1 + T - 4 T^{2} + 5 T^{3} + 25 T^{4}$$)($$1 + T - 4 T^{2} + 5 T^{3} + 25 T^{4}$$)($$1 + T - 4 T^{2} + 5 T^{3} + 25 T^{4}$$)($$1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4}$$)($$1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4}$$)($$1 - 4 T + 11 T^{2} - 20 T^{3} + 25 T^{4}$$)($$( 1 + T + 5 T^{2} )^{2}( 1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4} )$$)
$7$ ($$1 - 5 T + 7 T^{2}$$)($$1 + 5 T + 7 T^{2}$$)($$1 + 4 T + 7 T^{2}$$)($$1 - T + 7 T^{2}$$)($$1 - 4 T + 7 T^{2}$$)($$1 + 5 T + 7 T^{2}$$)($$1 - 5 T + 7 T^{2}$$)($$1 - 5 T + 7 T^{2}$$)($$( 1 + 3 T + 7 T^{2} )^{2}$$)
$11$ ($$1 - 11 T^{2} + 121 T^{4}$$)($$1 - 5 T + 14 T^{2} - 55 T^{3} + 121 T^{4}$$)($$1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4}$$)($$1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4}$$)($$1 + T - 10 T^{2} + 11 T^{3} + 121 T^{4}$$)($$1 + 5 T + 14 T^{2} + 55 T^{3} + 121 T^{4}$$)($$1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4}$$)($$1 - 11 T^{2} + 121 T^{4}$$)($$1 + T - 7 T^{2} - 14 T^{3} - 68 T^{4} - 154 T^{5} - 847 T^{6} + 1331 T^{7} + 14641 T^{8}$$)
$13$ ($$( 1 + 3 T + 13 T^{2} )^{2}$$)($$( 1 - 2 T + 13 T^{2} )^{2}$$)($$( 1 + 6 T + 13 T^{2} )^{2}$$)($$( 1 - 4 T + 13 T^{2} )^{2}$$)($$( 1 - 2 T + 13 T^{2} )^{2}$$)($$( 1 - 2 T + 13 T^{2} )^{2}$$)($$( 1 + 3 T + 13 T^{2} )^{2}$$)($$( 1 + 3 T + 13 T^{2} )^{2}$$)($$( 1 - 5 T + 18 T^{2} - 65 T^{3} + 169 T^{4} )^{2}$$)
$17$ ($$1 - 4 T - T^{2} - 68 T^{3} + 289 T^{4}$$)($$1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4}$$)($$1 + 5 T + 8 T^{2} + 85 T^{3} + 289 T^{4}$$)($$1 - 17 T^{2} + 289 T^{4}$$)($$1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4}$$)($$1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4}$$)($$1 - 4 T - T^{2} - 68 T^{3} + 289 T^{4}$$)($$1 + 4 T - T^{2} + 68 T^{3} + 289 T^{4}$$)($$( 1 + 4 T - T^{2} + 68 T^{3} + 289 T^{4} )^{2}$$)
$19$ ($$( 1 - T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$)($$1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4}$$)($$( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$)($$1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4}$$)($$1 + 5 T + 6 T^{2} + 95 T^{3} + 361 T^{4}$$)($$1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4}$$)($$1 - 5 T + 6 T^{2} - 95 T^{3} + 361 T^{4}$$)($$( 1 - T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$)($$1 - 5 T - 5 T^{2} + 40 T^{3} + 64 T^{4} + 760 T^{5} - 1805 T^{6} - 34295 T^{7} + 130321 T^{8}$$)
$23$ ($$1 + 4 T - 7 T^{2} + 92 T^{3} + 529 T^{4}$$)($$1 + 6 T + 13 T^{2} + 138 T^{3} + 529 T^{4}$$)($$1 + 7 T + 26 T^{2} + 161 T^{3} + 529 T^{4}$$)($$1 - 8 T + 41 T^{2} - 184 T^{3} + 529 T^{4}$$)($$1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4}$$)($$1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4}$$)($$1 + 4 T - 7 T^{2} + 92 T^{3} + 529 T^{4}$$)($$1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4}$$)($$( 1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4} )^{2}$$)
$29$ ($$( 1 + 8 T + 29 T^{2} )^{2}$$)($$( 1 - 3 T + 29 T^{2} )^{2}$$)($$( 1 + 2 T + 29 T^{2} )^{2}$$)($$( 1 - 3 T + 29 T^{2} )^{2}$$)($$( 1 - 6 T + 29 T^{2} )^{2}$$)($$( 1 + 3 T + 29 T^{2} )^{2}$$)($$( 1 - 4 T + 29 T^{2} )^{2}$$)($$( 1 - 8 T + 29 T^{2} )^{2}$$)($$( 1 + 3 T + 46 T^{2} + 87 T^{3} + 841 T^{4} )^{2}$$)
$31$ ($$1 - 5 T - 6 T^{2} - 155 T^{3} + 961 T^{4}$$)($$1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4}$$)($$1 - 5 T - 6 T^{2} - 155 T^{3} + 961 T^{4}$$)($$1 - 5 T - 6 T^{2} - 155 T^{3} + 961 T^{4}$$)($$1 - T - 30 T^{2} - 31 T^{3} + 961 T^{4}$$)($$1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4}$$)($$( 1 - 4 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} )$$)($$1 - 5 T - 6 T^{2} - 155 T^{3} + 961 T^{4}$$)($$( 1 - T - 30 T^{2} - 31 T^{3} + 961 T^{4} )^{2}$$)
$37$ ($$1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4}$$)($$1 - 2 T - 33 T^{2} - 74 T^{3} + 1369 T^{4}$$)($$1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4}$$)($$1 + 8 T + 27 T^{2} + 296 T^{3} + 1369 T^{4}$$)($$1 - 5 T - 12 T^{2} - 185 T^{3} + 1369 T^{4}$$)($$1 - 2 T - 33 T^{2} - 74 T^{3} + 1369 T^{4}$$)($$1 - 9 T + 44 T^{2} - 333 T^{3} + 1369 T^{4}$$)($$1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4}$$)($$1 + 3 T - 53 T^{2} - 36 T^{3} + 2142 T^{4} - 1332 T^{5} - 72557 T^{6} + 151959 T^{7} + 1874161 T^{8}$$)
$41$ ($$( 1 - 8 T + 41 T^{2} )^{2}$$)($$( 1 + 8 T + 41 T^{2} )^{2}$$)($$( 1 - 2 T + 41 T^{2} )^{2}$$)($$( 1 + 8 T + 41 T^{2} )^{2}$$)($$( 1 - 10 T + 41 T^{2} )^{2}$$)($$( 1 - 8 T + 41 T^{2} )^{2}$$)($$( 1 - 2 T + 41 T^{2} )^{2}$$)($$( 1 + 8 T + 41 T^{2} )^{2}$$)($$( 1 + 6 T + 34 T^{2} + 246 T^{3} + 1681 T^{4} )^{2}$$)
$43$ ($$( 1 - 11 T + 43 T^{2} )^{2}$$)($$( 1 + 4 T + 43 T^{2} )^{2}$$)($$( 1 + 4 T + 43 T^{2} )^{2}$$)($$( 1 - 6 T + 43 T^{2} )^{2}$$)($$( 1 + 4 T + 43 T^{2} )^{2}$$)($$( 1 + 4 T + 43 T^{2} )^{2}$$)($$( 1 + T + 43 T^{2} )^{2}$$)($$( 1 - 11 T + 43 T^{2} )^{2}$$)($$( 1 + 7 T + 84 T^{2} + 301 T^{3} + 1849 T^{4} )^{2}$$)
$47$ ($$1 + 4 T - 31 T^{2} + 188 T^{3} + 2209 T^{4}$$)($$1 - 4 T - 31 T^{2} - 188 T^{3} + 2209 T^{4}$$)($$1 - 5 T - 22 T^{2} - 235 T^{3} + 2209 T^{4}$$)($$1 - 10 T + 53 T^{2} - 470 T^{3} + 2209 T^{4}$$)($$1 - T - 46 T^{2} - 47 T^{3} + 2209 T^{4}$$)($$1 + 4 T - 31 T^{2} + 188 T^{3} + 2209 T^{4}$$)($$1 - 2 T - 43 T^{2} - 94 T^{3} + 2209 T^{4}$$)($$1 - 4 T - 31 T^{2} - 188 T^{3} + 2209 T^{4}$$)($$( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} )^{2}$$)
$53$ ($$1 + 4 T - 37 T^{2} + 212 T^{3} + 2809 T^{4}$$)($$1 - 9 T + 28 T^{2} - 477 T^{3} + 2809 T^{4}$$)($$1 + T - 52 T^{2} + 53 T^{3} + 2809 T^{4}$$)($$1 - 9 T + 28 T^{2} - 477 T^{3} + 2809 T^{4}$$)($$1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4}$$)($$1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4}$$)($$1 - 8 T + 11 T^{2} - 424 T^{3} + 2809 T^{4}$$)($$1 - 4 T - 37 T^{2} - 212 T^{3} + 2809 T^{4}$$)($$1 - 11 T - T^{2} - 176 T^{3} + 5662 T^{4} - 9328 T^{5} - 2809 T^{6} - 1637647 T^{7} + 7890481 T^{8}$$)
$59$ ($$1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4}$$)($$1 + 3 T - 50 T^{2} + 177 T^{3} + 3481 T^{4}$$)($$1 - 15 T + 166 T^{2} - 885 T^{3} + 3481 T^{4}$$)($$1 + 5 T - 34 T^{2} + 295 T^{3} + 3481 T^{4}$$)($$1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4}$$)($$1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4}$$)($$1 - 59 T^{2} + 3481 T^{4}$$)($$1 - 12 T + 85 T^{2} - 708 T^{3} + 3481 T^{4}$$)($$1 - 5 T - 85 T^{2} + 40 T^{3} + 7144 T^{4} + 2360 T^{5} - 295885 T^{6} - 1026895 T^{7} + 12117361 T^{8}$$)
$61$ ($$1 - 2 T - 57 T^{2} - 122 T^{3} + 3721 T^{4}$$)($$1 - 12 T + 83 T^{2} - 732 T^{3} + 3721 T^{4}$$)($$1 - 5 T - 36 T^{2} - 305 T^{3} + 3721 T^{4}$$)($$1 - 10 T + 39 T^{2} - 610 T^{3} + 3721 T^{4}$$)($$1 + 3 T - 52 T^{2} + 183 T^{3} + 3721 T^{4}$$)($$1 - 12 T + 83 T^{2} - 732 T^{3} + 3721 T^{4}$$)($$1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4}$$)($$1 - 2 T - 57 T^{2} - 122 T^{3} + 3721 T^{4}$$)($$( 1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4} )^{2}$$)
$67$ ($$1 - 3 T - 58 T^{2} - 201 T^{3} + 4489 T^{4}$$)($$1 + 2 T - 63 T^{2} + 134 T^{3} + 4489 T^{4}$$)($$1 - 9 T + 14 T^{2} - 603 T^{3} + 4489 T^{4}$$)($$1 + 6 T - 31 T^{2} + 402 T^{3} + 4489 T^{4}$$)($$( 1 - 5 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} )$$)($$1 + 2 T - 63 T^{2} + 134 T^{3} + 4489 T^{4}$$)($$1 - 15 T + 158 T^{2} - 1005 T^{3} + 4489 T^{4}$$)($$1 - 3 T - 58 T^{2} - 201 T^{3} + 4489 T^{4}$$)($$1 + 7 T - 83 T^{2} - 14 T^{3} + 9652 T^{4} - 938 T^{5} - 372587 T^{6} + 2105341 T^{7} + 20151121 T^{8}$$)
$71$ ($$( 1 - 12 T + 71 T^{2} )^{2}$$)($$( 1 - 8 T + 71 T^{2} )^{2}$$)($$( 1 + 71 T^{2} )^{2}$$)($$( 1 + 10 T + 71 T^{2} )^{2}$$)($$( 1 + 16 T + 71 T^{2} )^{2}$$)($$( 1 + 8 T + 71 T^{2} )^{2}$$)($$( 1 - 6 T + 71 T^{2} )^{2}$$)($$( 1 + 12 T + 71 T^{2} )^{2}$$)($$( 1 + 2 T + 71 T^{2} )^{4}$$)
$73$ ($$1 + T - 72 T^{2} + 73 T^{3} + 5329 T^{4}$$)($$1 - 14 T + 123 T^{2} - 1022 T^{3} + 5329 T^{4}$$)($$( 1 - 10 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} )$$)($$1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4}$$)($$( 1 - 10 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} )$$)($$1 - 14 T + 123 T^{2} - 1022 T^{3} + 5329 T^{4}$$)($$1 - 11 T + 48 T^{2} - 803 T^{3} + 5329 T^{4}$$)($$1 + T - 72 T^{2} + 73 T^{3} + 5329 T^{4}$$)($$1 - T - 131 T^{2} + 14 T^{3} + 12022 T^{4} + 1022 T^{5} - 698099 T^{6} - 389017 T^{7} + 28398241 T^{8}$$)
$79$ ($$1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4}$$)($$1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4}$$)($$1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4}$$)($$1 + 11 T + 42 T^{2} + 869 T^{3} + 6241 T^{4}$$)($$1 - 11 T + 42 T^{2} - 869 T^{3} + 6241 T^{4}$$)($$1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4}$$)($$1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4}$$)($$1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4}$$)($$1 + 8 T - 53 T^{2} - 328 T^{3} + 2392 T^{4} - 25912 T^{5} - 330773 T^{6} + 3944312 T^{7} + 38950081 T^{8}$$)
$83$ ($$( 1 - 12 T + 83 T^{2} )^{2}$$)($$( 1 + 17 T + 83 T^{2} )^{2}$$)($$( 1 + 12 T + 83 T^{2} )^{2}$$)($$( 1 + 7 T + 83 T^{2} )^{2}$$)($$( 1 - 4 T + 83 T^{2} )^{2}$$)($$( 1 - 17 T + 83 T^{2} )^{2}$$)($$( 1 + 6 T + 83 T^{2} )^{2}$$)($$( 1 + 12 T + 83 T^{2} )^{2}$$)($$( 1 + 7 T + 164 T^{2} + 581 T^{3} + 6889 T^{4} )^{2}$$)
$89$ ($$1 + 8 T - 25 T^{2} + 712 T^{3} + 7921 T^{4}$$)($$1 - 18 T + 235 T^{2} - 1602 T^{3} + 7921 T^{4}$$)($$1 - 7 T - 40 T^{2} - 623 T^{3} + 7921 T^{4}$$)($$1 + 18 T + 235 T^{2} + 1602 T^{3} + 7921 T^{4}$$)($$1 + 9 T - 8 T^{2} + 801 T^{3} + 7921 T^{4}$$)($$1 + 18 T + 235 T^{2} + 1602 T^{3} + 7921 T^{4}$$)($$1 + 8 T - 25 T^{2} + 712 T^{3} + 7921 T^{4}$$)($$1 - 8 T - 25 T^{2} - 712 T^{3} + 7921 T^{4}$$)($$1 + 6 T - 94 T^{2} - 288 T^{3} + 5775 T^{4} - 25632 T^{5} - 744574 T^{6} + 4229814 T^{7} + 62742241 T^{8}$$)
$97$ ($$( 1 + 2 T + 97 T^{2} )^{2}$$)($$( 1 - 3 T + 97 T^{2} )^{2}$$)($$( 1 + 2 T + 97 T^{2} )^{2}$$)($$( 1 + 17 T + 97 T^{2} )^{2}$$)($$( 1 - 6 T + 97 T^{2} )^{2}$$)($$( 1 - 3 T + 97 T^{2} )^{2}$$)($$( 1 + 14 T + 97 T^{2} )^{2}$$)($$( 1 + 2 T + 97 T^{2} )^{2}$$)($$( 1 - 25 T + 336 T^{2} - 2425 T^{3} + 9409 T^{4} )^{2}$$)
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