# Properties

 Label 525.2.b Level 525 Weight 2 Character orbit b Rep. character $$\chi_{525}(251,\cdot)$$ Character field $$\Q$$ Dimension 44 Newform subspaces 10 Sturm bound 160 Trace bound 7

# Related objects

## Defining parameters

 Level: $$N$$ = $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 525.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$21$$ Character field: $$\Q$$ Newform subspaces: $$10$$ Sturm bound: $$160$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$2$$, $$11$$, $$17$$, $$37$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(525, [\chi])$$.

Total New Old
Modular forms 92 56 36
Cusp forms 68 44 24
Eisenstein series 24 12 12

## Trace form

 $$44q - 32q^{4} + 8q^{7} + O(q^{10})$$ $$44q - 32q^{4} + 8q^{7} + 8q^{16} + 4q^{18} - 18q^{21} - 8q^{22} - 16q^{28} + 16q^{37} - 36q^{42} + 40q^{43} - 24q^{46} - 34q^{49} + 12q^{51} + 12q^{57} - 8q^{58} + 44q^{63} + 64q^{64} - 24q^{67} - 64q^{72} - 84q^{78} - 104q^{79} + 108q^{84} - 16q^{88} + 6q^{91} - 12q^{93} - 36q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(525, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
525.2.b.a $$2$$ $$4.192$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$-4$$ $$q-\zeta_{6}q^{2}-\zeta_{6}q^{3}-q^{4}-3q^{6}+(-2+\cdots)q^{7}+\cdots$$
525.2.b.b $$2$$ $$4.192$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$-4$$ $$q-\zeta_{6}q^{2}+\zeta_{6}q^{3}-q^{4}+3q^{6}+(-2+\cdots)q^{7}+\cdots$$
525.2.b.c $$2$$ $$4.192$$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$-5$$ $$q+(1-2\zeta_{6})q^{3}+2q^{4}+(-2-\zeta_{6})q^{7}+\cdots$$
525.2.b.d $$2$$ $$4.192$$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$5$$ $$q+(1-2\zeta_{6})q^{3}+2q^{4}+(3-\zeta_{6})q^{7}+\cdots$$
525.2.b.e $$4$$ $$4.192$$ $$\Q(\sqrt{-3}, \sqrt{-11})$$ None $$0$$ $$-1$$ $$0$$ $$8$$ $$q+(-\beta _{1}-\beta _{2}+\beta _{3})q^{2}+\beta _{3}q^{3}+(-1+\cdots)q^{4}+\cdots$$
525.2.b.f $$4$$ $$4.192$$ $$\Q(\sqrt{5}, \sqrt{-7})$$ $$\Q(\sqrt{-35})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{3}q^{3}+2q^{4}+(\beta _{1}-\beta _{3})q^{7}+(-1+\cdots)q^{9}+\cdots$$
525.2.b.g $$4$$ $$4.192$$ $$\Q(\sqrt{-3}, \sqrt{-11})$$ None $$0$$ $$1$$ $$0$$ $$8$$ $$q+(-\beta _{1}-\beta _{2}+\beta _{3})q^{2}-\beta _{3}q^{3}+(-1+\cdots)q^{4}+\cdots$$
525.2.b.h $$8$$ $$4.192$$ 8.0.$$\cdots$$.7 None $$0$$ $$0$$ $$0$$ $$-8$$ $$q+\beta _{2}q^{2}-\beta _{7}q^{3}+(-2+\beta _{4})q^{4}+\beta _{3}q^{6}+\cdots$$
525.2.b.i $$8$$ $$4.192$$ 8.0.$$\cdots$$.7 None $$0$$ $$0$$ $$0$$ $$8$$ $$q+\beta _{3}q^{2}+\beta _{1}q^{3}+(-2+\beta _{4})q^{4}-\beta _{5}q^{6}+\cdots$$
525.2.b.j $$8$$ $$4.192$$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{24}q^{2}-\zeta_{24}^{3}q^{3}-q^{4}+(\zeta_{24}^{2}+\cdots)q^{6}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(525, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(525, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(105, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 - T^{2} + 4 T^{4}$$)($$1 - T^{2} + 4 T^{4}$$)($$( 1 - 2 T^{2} )^{2}$$)($$( 1 - 2 T^{2} )^{2}$$)($$1 - T^{2} - 4 T^{6} + 16 T^{8}$$)($$( 1 - 2 T^{2} )^{4}$$)($$1 - T^{2} - 4 T^{6} + 16 T^{8}$$)($$( 1 - T^{2} + 3 T^{4} - 4 T^{6} + 16 T^{8} )^{2}$$)($$( 1 - T^{2} + 3 T^{4} - 4 T^{6} + 16 T^{8} )^{2}$$)($$( 1 - T^{2} + 4 T^{4} )^{4}$$)
$3$ ($$1 + 3 T^{2}$$)($$1 + 3 T^{2}$$)($$1 + 3 T^{2}$$)($$1 + 3 T^{2}$$)($$1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4}$$)($$1 + T^{2} + 9 T^{4}$$)($$1 - T - 2 T^{2} - 3 T^{3} + 9 T^{4}$$)($$1 - 2 T^{2} - 2 T^{4} - 18 T^{6} + 81 T^{8}$$)($$1 - 2 T^{2} - 2 T^{4} - 18 T^{6} + 81 T^{8}$$)($$( 1 - 2 T^{2} + 9 T^{4} )^{2}$$)
$5$ 1
$7$ ($$1 + 4 T + 7 T^{2}$$)($$1 + 4 T + 7 T^{2}$$)($$1 + 5 T + 7 T^{2}$$)($$1 - 5 T + 7 T^{2}$$)($$( 1 - 4 T + 7 T^{2} )^{2}$$)($$( 1 + 7 T^{2} )^{2}$$)($$( 1 - 4 T + 7 T^{2} )^{2}$$)($$( 1 + 2 T + 7 T^{2} )^{4}$$)($$( 1 - 2 T + 7 T^{2} )^{4}$$)($$( 1 - 10 T^{2} + 49 T^{4} )^{2}$$)
$11$ ($$1 - 10 T^{2} + 121 T^{4}$$)($$1 - 10 T^{2} + 121 T^{4}$$)($$( 1 - 11 T^{2} )^{2}$$)($$( 1 - 11 T^{2} )^{2}$$)($$1 - 37 T^{2} + 576 T^{4} - 4477 T^{6} + 14641 T^{8}$$)($$( 1 - 3 T + 11 T^{2} )^{2}( 1 + 3 T + 11 T^{2} )^{2}$$)($$1 - 37 T^{2} + 576 T^{4} - 4477 T^{6} + 14641 T^{8}$$)($$( 1 - 16 T^{2} + 285 T^{4} - 1936 T^{6} + 14641 T^{8} )^{2}$$)($$( 1 - 16 T^{2} + 285 T^{4} - 1936 T^{6} + 14641 T^{8} )^{2}$$)($$( 1 - 6 T + 11 T^{2} )^{4}( 1 + 6 T + 11 T^{2} )^{4}$$)
$13$ ($$( 1 - 13 T^{2} )^{2}$$)($$( 1 - 13 T^{2} )^{2}$$)($$( 1 - 5 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} )$$)($$( 1 - 5 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} )$$)($$1 - T^{2} + 264 T^{4} - 169 T^{6} + 28561 T^{8}$$)($$( 1 - 19 T^{2} + 169 T^{4} )^{2}$$)($$1 - T^{2} + 264 T^{4} - 169 T^{6} + 28561 T^{8}$$)($$( 1 - 20 T^{2} + 169 T^{4} )^{4}$$)($$( 1 - 20 T^{2} + 169 T^{4} )^{4}$$)($$( 1 - 6 T + 13 T^{2} )^{4}( 1 + 6 T + 13 T^{2} )^{4}$$)
$17$ ($$( 1 + 6 T + 17 T^{2} )^{2}$$)($$( 1 - 6 T + 17 T^{2} )^{2}$$)($$( 1 + 17 T^{2} )^{2}$$)($$( 1 + 17 T^{2} )^{2}$$)($$( 1 + 3 T + 28 T^{2} + 51 T^{3} + 289 T^{4} )^{2}$$)($$( 1 + 29 T^{2} + 289 T^{4} )^{2}$$)($$( 1 - 3 T + 28 T^{2} - 51 T^{3} + 289 T^{4} )^{2}$$)($$( 1 + 26 T^{2} + 558 T^{4} + 7514 T^{6} + 83521 T^{8} )^{2}$$)($$( 1 + 26 T^{2} + 558 T^{4} + 7514 T^{6} + 83521 T^{8} )^{2}$$)($$( 1 + 26 T^{2} + 289 T^{4} )^{4}$$)
$19$ ($$( 1 - 8 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$)($$( 1 - 8 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$)($$( 1 - T + 19 T^{2} )( 1 + T + 19 T^{2} )$$)($$( 1 - T + 19 T^{2} )( 1 + T + 19 T^{2} )$$)($$( 1 - 8 T + 19 T^{2} )^{2}( 1 + 8 T + 19 T^{2} )^{2}$$)($$( 1 - 19 T^{2} )^{4}$$)($$( 1 - 8 T + 19 T^{2} )^{2}( 1 + 8 T + 19 T^{2} )^{2}$$)($$( 1 - 10 T^{2} + 558 T^{4} - 3610 T^{6} + 130321 T^{8} )^{2}$$)($$( 1 - 10 T^{2} + 558 T^{4} - 3610 T^{6} + 130321 T^{8} )^{2}$$)($$( 1 - 19 T^{2} )^{8}$$)
$23$ ($$1 - 34 T^{2} + 529 T^{4}$$)($$1 - 34 T^{2} + 529 T^{4}$$)($$( 1 - 23 T^{2} )^{2}$$)($$( 1 - 23 T^{2} )^{2}$$)($$1 - 16 T^{2} - 66 T^{4} - 8464 T^{6} + 279841 T^{8}$$)($$( 1 - 23 T^{2} )^{4}$$)($$1 - 16 T^{2} - 66 T^{4} - 8464 T^{6} + 279841 T^{8}$$)($$( 1 - 64 T^{2} + 1893 T^{4} - 33856 T^{6} + 279841 T^{8} )^{2}$$)($$( 1 - 64 T^{2} + 1893 T^{4} - 33856 T^{6} + 279841 T^{8} )^{2}$$)($$( 1 - 34 T^{2} + 529 T^{4} )^{4}$$)
$29$ ($$1 - 10 T^{2} + 841 T^{4}$$)($$1 - 10 T^{2} + 841 T^{4}$$)($$( 1 - 29 T^{2} )^{2}$$)($$( 1 - 29 T^{2} )^{2}$$)($$1 - 97 T^{2} + 3960 T^{4} - 81577 T^{6} + 707281 T^{8}$$)($$( 1 - 9 T + 29 T^{2} )^{2}( 1 + 9 T + 29 T^{2} )^{2}$$)($$1 - 97 T^{2} + 3960 T^{4} - 81577 T^{6} + 707281 T^{8}$$)($$( 1 - 88 T^{2} + 3429 T^{4} - 74008 T^{6} + 707281 T^{8} )^{2}$$)($$( 1 - 88 T^{2} + 3429 T^{4} - 74008 T^{6} + 707281 T^{8} )^{2}$$)($$( 1 - 26 T^{2} + 841 T^{4} )^{4}$$)
$31$ ($$1 - 50 T^{2} + 961 T^{4}$$)($$1 - 50 T^{2} + 961 T^{4}$$)($$( 1 - 7 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} )$$)($$( 1 - 7 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} )$$)($$( 1 - 50 T^{2} + 961 T^{4} )^{2}$$)($$( 1 - 31 T^{2} )^{4}$$)($$( 1 - 50 T^{2} + 961 T^{4} )^{2}$$)($$( 1 - 58 T^{2} + 2574 T^{4} - 55738 T^{6} + 923521 T^{8} )^{2}$$)($$( 1 - 58 T^{2} + 2574 T^{4} - 55738 T^{6} + 923521 T^{8} )^{2}$$)($$( 1 + 34 T^{2} + 961 T^{4} )^{4}$$)
$37$ ($$( 1 - 2 T + 37 T^{2} )^{2}$$)($$( 1 - 2 T + 37 T^{2} )^{2}$$)($$( 1 - 10 T + 37 T^{2} )^{2}$$)($$( 1 + 10 T + 37 T^{2} )^{2}$$)($$( 1 - 2 T + 42 T^{2} - 74 T^{3} + 1369 T^{4} )^{2}$$)($$( 1 + 37 T^{2} )^{4}$$)($$( 1 - 2 T + 42 T^{2} - 74 T^{3} + 1369 T^{4} )^{2}$$)($$( 1 - 8 T + 69 T^{2} - 296 T^{3} + 1369 T^{4} )^{4}$$)($$( 1 + 8 T + 69 T^{2} + 296 T^{3} + 1369 T^{4} )^{4}$$)($$( 1 + 37 T^{2} )^{8}$$)
$41$ ($$( 1 - 6 T + 41 T^{2} )^{2}$$)($$( 1 + 6 T + 41 T^{2} )^{2}$$)($$( 1 + 41 T^{2} )^{2}$$)($$( 1 + 41 T^{2} )^{2}$$)($$( 1 + 6 T + 41 T^{2} )^{4}$$)($$( 1 + 41 T^{2} )^{4}$$)($$( 1 - 6 T + 41 T^{2} )^{4}$$)($$( 1 + 38 T^{2} + 2022 T^{4} + 63878 T^{6} + 2825761 T^{8} )^{2}$$)($$( 1 + 38 T^{2} + 2022 T^{4} + 63878 T^{6} + 2825761 T^{8} )^{2}$$)($$( 1 + 70 T^{2} + 1681 T^{4} )^{4}$$)
$43$ ($$( 1 - 8 T + 43 T^{2} )^{2}$$)($$( 1 - 8 T + 43 T^{2} )^{2}$$)($$( 1 + 5 T + 43 T^{2} )^{2}$$)($$( 1 - 5 T + 43 T^{2} )^{2}$$)($$( 1 - 2 T + 54 T^{2} - 86 T^{3} + 1849 T^{4} )^{2}$$)($$( 1 + 43 T^{2} )^{4}$$)($$( 1 - 2 T + 54 T^{2} - 86 T^{3} + 1849 T^{4} )^{2}$$)($$( 1 + 4 T + 69 T^{2} + 172 T^{3} + 1849 T^{4} )^{4}$$)($$( 1 - 4 T + 69 T^{2} - 172 T^{3} + 1849 T^{4} )^{4}$$)($$( 1 + 62 T^{2} + 1849 T^{4} )^{4}$$)
$47$ ($$( 1 - 12 T + 47 T^{2} )^{2}$$)($$( 1 + 12 T + 47 T^{2} )^{2}$$)($$( 1 + 47 T^{2} )^{2}$$)($$( 1 + 47 T^{2} )^{2}$$)($$( 1 - 9 T + 106 T^{2} - 423 T^{3} + 2209 T^{4} )^{2}$$)($$( 1 - 31 T^{2} + 2209 T^{4} )^{2}$$)($$( 1 + 9 T + 106 T^{2} + 423 T^{3} + 2209 T^{4} )^{2}$$)($$( 1 + 146 T^{2} + 9558 T^{4} + 322514 T^{6} + 4879681 T^{8} )^{2}$$)($$( 1 + 146 T^{2} + 9558 T^{4} + 322514 T^{6} + 4879681 T^{8} )^{2}$$)($$( 1 + 86 T^{2} + 2209 T^{4} )^{4}$$)
$53$ ($$( 1 - 53 T^{2} )^{2}$$)($$( 1 - 53 T^{2} )^{2}$$)($$( 1 - 53 T^{2} )^{2}$$)($$( 1 - 53 T^{2} )^{2}$$)($$1 - 136 T^{2} + 9054 T^{4} - 382024 T^{6} + 7890481 T^{8}$$)($$( 1 - 53 T^{2} )^{4}$$)($$1 - 136 T^{2} + 9054 T^{4} - 382024 T^{6} + 7890481 T^{8}$$)($$( 1 - 16 T^{2} - 1122 T^{4} - 44944 T^{6} + 7890481 T^{8} )^{2}$$)($$( 1 - 16 T^{2} - 1122 T^{4} - 44944 T^{6} + 7890481 T^{8} )^{2}$$)($$( 1 - 53 T^{2} )^{8}$$)
$59$ ($$( 1 + 12 T + 59 T^{2} )^{2}$$)($$( 1 - 12 T + 59 T^{2} )^{2}$$)($$( 1 + 59 T^{2} )^{2}$$)($$( 1 + 59 T^{2} )^{2}$$)($$( 1 + 6 T + 94 T^{2} + 354 T^{3} + 3481 T^{4} )^{2}$$)($$( 1 + 59 T^{2} )^{4}$$)($$( 1 - 6 T + 94 T^{2} - 354 T^{3} + 3481 T^{4} )^{2}$$)($$( 1 + 68 T^{2} + 7362 T^{4} + 236708 T^{6} + 12117361 T^{8} )^{2}$$)($$( 1 + 68 T^{2} + 7362 T^{4} + 236708 T^{6} + 12117361 T^{8} )^{2}$$)($$( 1 + 70 T^{2} + 3481 T^{4} )^{4}$$)
$61$ ($$( 1 - 14 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} )$$)($$( 1 - 14 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} )$$)($$( 1 - 13 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} )$$)($$( 1 - 13 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} )$$)($$( 1 - 14 T + 61 T^{2} )^{2}( 1 + 14 T + 61 T^{2} )^{2}$$)($$( 1 - 61 T^{2} )^{4}$$)($$( 1 - 14 T + 61 T^{2} )^{2}( 1 + 14 T + 61 T^{2} )^{2}$$)($$( 1 - 178 T^{2} + 15174 T^{4} - 662338 T^{6} + 13845841 T^{8} )^{2}$$)($$( 1 - 178 T^{2} + 15174 T^{4} - 662338 T^{6} + 13845841 T^{8} )^{2}$$)($$( 1 - 26 T^{2} + 3721 T^{4} )^{4}$$)
$67$ ($$( 1 + 8 T + 67 T^{2} )^{2}$$)($$( 1 + 8 T + 67 T^{2} )^{2}$$)($$( 1 - 5 T + 67 T^{2} )^{2}$$)($$( 1 + 5 T + 67 T^{2} )^{2}$$)($$( 1 - 2 T + 102 T^{2} - 134 T^{3} + 4489 T^{4} )^{2}$$)($$( 1 + 67 T^{2} )^{4}$$)($$( 1 - 2 T + 102 T^{2} - 134 T^{3} + 4489 T^{4} )^{2}$$)($$( 1 + 10 T + 75 T^{2} + 670 T^{3} + 4489 T^{4} )^{4}$$)($$( 1 - 10 T + 75 T^{2} - 670 T^{3} + 4489 T^{4} )^{4}$$)($$( 1 + 110 T^{2} + 4489 T^{4} )^{4}$$)
$71$ ($$1 - 130 T^{2} + 5041 T^{4}$$)($$1 - 130 T^{2} + 5041 T^{4}$$)($$( 1 - 71 T^{2} )^{2}$$)($$( 1 - 71 T^{2} )^{2}$$)($$1 - 100 T^{2} + 4134 T^{4} - 504100 T^{6} + 25411681 T^{8}$$)($$( 1 - 12 T + 71 T^{2} )^{2}( 1 + 12 T + 71 T^{2} )^{2}$$)($$1 - 100 T^{2} + 4134 T^{4} - 504100 T^{6} + 25411681 T^{8}$$)($$( 1 - 256 T^{2} + 26445 T^{4} - 1290496 T^{6} + 25411681 T^{8} )^{2}$$)($$( 1 - 256 T^{2} + 26445 T^{4} - 1290496 T^{6} + 25411681 T^{8} )^{2}$$)($$( 1 - 134 T^{2} + 5041 T^{4} )^{4}$$)
$73$ ($$1 - 98 T^{2} + 5329 T^{4}$$)($$1 - 98 T^{2} + 5329 T^{4}$$)($$( 1 - 10 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} )$$)($$( 1 - 10 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} )$$)($$( 1 - 98 T^{2} + 5329 T^{4} )^{2}$$)($$( 1 - 34 T^{2} + 5329 T^{4} )^{2}$$)($$( 1 - 98 T^{2} + 5329 T^{4} )^{2}$$)($$( 1 + 8 T^{2} - 1422 T^{4} + 42632 T^{6} + 28398241 T^{8} )^{2}$$)($$( 1 + 8 T^{2} - 1422 T^{4} + 42632 T^{6} + 28398241 T^{8} )^{2}$$)($$( 1 - 82 T^{2} + 5329 T^{4} )^{4}$$)
$79$ ($$( 1 - 8 T + 79 T^{2} )^{2}$$)($$( 1 - 8 T + 79 T^{2} )^{2}$$)($$( 1 + 4 T + 79 T^{2} )^{2}$$)($$( 1 + 4 T + 79 T^{2} )^{2}$$)($$( 1 - T + 150 T^{2} - 79 T^{3} + 6241 T^{4} )^{2}$$)($$( 1 - T + 79 T^{2} )^{4}$$)($$( 1 - T + 150 T^{2} - 79 T^{3} + 6241 T^{4} )^{2}$$)($$( 1 + 8 T + 153 T^{2} + 632 T^{3} + 6241 T^{4} )^{4}$$)($$( 1 + 8 T + 153 T^{2} + 632 T^{3} + 6241 T^{4} )^{4}$$)($$( 1 + 8 T + 79 T^{2} )^{8}$$)
$83$ ($$( 1 + 83 T^{2} )^{2}$$)($$( 1 + 83 T^{2} )^{2}$$)($$( 1 + 83 T^{2} )^{2}$$)($$( 1 + 83 T^{2} )^{2}$$)($$( 1 - 12 T + 70 T^{2} - 996 T^{3} + 6889 T^{4} )^{2}$$)($$( 1 + 86 T^{2} + 6889 T^{4} )^{2}$$)($$( 1 + 12 T + 70 T^{2} + 996 T^{3} + 6889 T^{4} )^{2}$$)($$( 1 + 38 T^{2} + 4878 T^{4} + 261782 T^{6} + 47458321 T^{8} )^{2}$$)($$( 1 + 38 T^{2} + 4878 T^{4} + 261782 T^{6} + 47458321 T^{8} )^{2}$$)($$( 1 + 158 T^{2} + 6889 T^{4} )^{4}$$)
$89$ ($$( 1 - 6 T + 89 T^{2} )^{2}$$)($$( 1 + 6 T + 89 T^{2} )^{2}$$)($$( 1 + 89 T^{2} )^{2}$$)($$( 1 + 89 T^{2} )^{2}$$)($$( 1 - 18 T + 226 T^{2} - 1602 T^{3} + 7921 T^{4} )^{2}$$)($$( 1 + 89 T^{2} )^{4}$$)($$( 1 + 18 T + 226 T^{2} + 1602 T^{3} + 7921 T^{4} )^{2}$$)($$( 1 + 188 T^{2} + 23922 T^{4} + 1489148 T^{6} + 62742241 T^{8} )^{2}$$)($$( 1 + 188 T^{2} + 23922 T^{4} + 1489148 T^{6} + 62742241 T^{8} )^{2}$$)($$( 1 + 70 T^{2} + 7921 T^{4} )^{4}$$)
$97$ ($$1 - 146 T^{2} + 9409 T^{4}$$)($$1 - 146 T^{2} + 9409 T^{4}$$)($$( 1 - 5 T + 97 T^{2} )( 1 + 5 T + 97 T^{2} )$$)($$( 1 - 5 T + 97 T^{2} )( 1 + 5 T + 97 T^{2} )$$)($$1 - 265 T^{2} + 32736 T^{4} - 2493385 T^{6} + 88529281 T^{8}$$)($$( 1 + 149 T^{2} + 9409 T^{4} )^{2}$$)($$1 - 265 T^{2} + 32736 T^{4} - 2493385 T^{6} + 88529281 T^{8}$$)($$( 1 - 250 T^{2} + 29718 T^{4} - 2352250 T^{6} + 88529281 T^{8} )^{2}$$)($$( 1 - 250 T^{2} + 29718 T^{4} - 2352250 T^{6} + 88529281 T^{8} )^{2}$$)($$( 1 - 18 T + 97 T^{2} )^{4}( 1 + 18 T + 97 T^{2} )^{4}$$)