# Properties

 Label 847.2.b Level 847 Weight 2 Character orbit b Rep. character $$\chi_{847}(846,\cdot)$$ Character field $$\Q$$ Dimension 64 Newform subspaces 6 Sturm bound 176 Trace bound 4

# Related objects

## Defining parameters

 Level: $$N$$ = $$847 = 7 \cdot 11^{2}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 847.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$77$$ Character field: $$\Q$$ Newform subspaces: $$6$$ Sturm bound: $$176$$ Trace bound: $$4$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(847, [\chi])$$.

Total New Old
Modular forms 100 80 20
Cusp forms 76 64 12
Eisenstein series 24 16 8

## Trace form

 $$64q - 50q^{4} - 48q^{9} + O(q^{10})$$ $$64q - 50q^{4} - 48q^{9} - 2q^{14} - 20q^{15} + 26q^{16} + 8q^{23} - 40q^{25} + 70q^{36} + 24q^{37} + 16q^{42} + 28q^{49} - 24q^{53} - 82q^{56} - 52q^{58} - 68q^{60} + 86q^{64} - 68q^{67} + 28q^{70} - 20q^{71} + 76q^{78} + 8q^{81} + 4q^{86} + 76q^{91} - 20q^{92} + 20q^{93} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(847, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
847.2.b.a $$8$$ $$6.763$$ 8.0.$$\cdots$$.11 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{2}+\beta _{4})q^{2}-\beta _{3}q^{3}+(-2+\beta _{6}+\cdots)q^{4}+\cdots$$
847.2.b.b $$8$$ $$6.763$$ 8.0.37515625.1 $$\Q(\sqrt{-7})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+(-1+\beta _{2})q^{4}+(-\beta _{1}+\beta _{4}+\cdots)q^{7}+\cdots$$
847.2.b.c $$8$$ $$6.763$$ 8.0.4956160000.2 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{3}+\beta _{5})q^{2}+\beta _{6}q^{3}+(-1-\beta _{4}+\cdots)q^{4}+\cdots$$
847.2.b.d $$8$$ $$6.763$$ 8.0.$$\cdots$$.2 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{6}q^{2}-\beta _{1}q^{4}+\beta _{7}q^{5}+(-\beta _{3}-\beta _{5}+\cdots)q^{7}+\cdots$$
847.2.b.e $$16$$ $$6.763$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(-\beta _{3}+\beta _{10})q^{2}+(-\beta _{5}+\beta _{14}+\beta _{15})q^{3}+\cdots$$
847.2.b.f $$16$$ $$6.763$$ $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}+\beta _{5})q^{2}-\beta _{14}q^{3}+(1-\beta _{6}-\beta _{8}+\cdots)q^{4}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(847, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(847, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(77, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$( 1 + T^{4} + 16 T^{8} )^{2}$$)($$( 1 - T - T^{2} + 3 T^{3} - T^{4} + 6 T^{5} - 4 T^{6} - 8 T^{7} + 16 T^{8} )( 1 + T - T^{2} - 3 T^{3} - T^{4} - 6 T^{5} - 4 T^{6} + 8 T^{7} + 16 T^{8} )$$)($$( 1 - 2 T^{2} + 4 T^{4} - 8 T^{6} + 16 T^{8} )^{2}$$)($$( 1 - 4 T^{2} + 9 T^{4} - 16 T^{6} + 16 T^{8} )^{2}$$)($$( 1 - 4 T^{2} + 15 T^{4} - 44 T^{6} + 89 T^{8} - 176 T^{10} + 240 T^{12} - 256 T^{14} + 256 T^{16} )^{2}$$)($$( 1 - 4 T + 4 T^{2} + 7 T^{3} - 21 T^{4} + 14 T^{5} + 16 T^{6} - 32 T^{7} + 16 T^{8} )^{2}( 1 + 4 T + 4 T^{2} - 7 T^{3} - 21 T^{4} - 14 T^{5} + 16 T^{6} + 32 T^{7} + 16 T^{8} )^{2}$$)
$3$ ($$( 1 - 2 T^{2} + 12 T^{4} - 18 T^{6} + 81 T^{8} )^{2}$$)($$( 1 - 3 T^{2} )^{8}$$)($$( 1 - 4 T^{2} + 17 T^{4} - 36 T^{6} + 81 T^{8} )^{2}$$)($$( 1 - 3 T^{2} )^{8}$$)($$( 1 - 2 T^{2} + 17 T^{4} + 14 T^{6} + 85 T^{8} + 126 T^{10} + 1377 T^{12} - 1458 T^{14} + 6561 T^{16} )^{2}$$)($$( 1 - 4 T^{2} + 27 T^{4} - 82 T^{6} + 355 T^{8} - 738 T^{10} + 2187 T^{12} - 2916 T^{14} + 6561 T^{16} )^{2}$$)
$5$ ($$( 1 + 22 T^{4} + 625 T^{8} )^{2}$$)($$( 1 - 5 T^{2} )^{8}$$)($$( 1 - 12 T^{2} + 81 T^{4} - 300 T^{6} + 625 T^{8} )^{2}$$)($$( 1 - 3 T + 5 T^{2} )^{4}( 1 + 3 T + 5 T^{2} )^{4}$$)($$( 1 - 30 T^{2} + 426 T^{4} - 3756 T^{6} + 22511 T^{8} - 93900 T^{10} + 266250 T^{12} - 468750 T^{14} + 390625 T^{16} )^{2}$$)($$( 1 - 10 T^{2} + 90 T^{4} - 635 T^{6} + 3395 T^{8} - 15875 T^{10} + 56250 T^{12} - 156250 T^{14} + 390625 T^{16} )^{2}$$)
$7$ ($$1 + 4 T^{2} - 10 T^{4} + 196 T^{6} + 2401 T^{8}$$)($$( 1 + 7 T^{2} )^{4}$$)($$1 + 2 T^{2} - 26 T^{4} + 98 T^{6} + 2401 T^{8}$$)($$( 1 - 8 T^{2} + 49 T^{4} )^{2}$$)($$1 - 8 T^{2} + 108 T^{4} - 1144 T^{6} + 6374 T^{8} - 56056 T^{10} + 259308 T^{12} - 941192 T^{14} + 5764801 T^{16}$$)($$1 - 24 T^{2} + 252 T^{4} - 1512 T^{6} + 8390 T^{8} - 74088 T^{10} + 605052 T^{12} - 2823576 T^{14} + 5764801 T^{16}$$)
$11$ ()()()()()()
$13$ ($$( 1 + 46 T^{2} + 860 T^{4} + 7774 T^{6} + 28561 T^{8} )^{2}$$)($$( 1 + 13 T^{2} )^{8}$$)($$( 1 - 4 T^{2} + 262 T^{4} - 676 T^{6} + 28561 T^{8} )^{2}$$)($$( 1 + 8 T^{2} - 9 T^{4} + 1352 T^{6} + 28561 T^{8} )^{2}$$)($$( 1 + 52 T^{2} + 1644 T^{4} + 33884 T^{6} + 518582 T^{8} + 5726396 T^{10} + 46954284 T^{12} + 250994068 T^{14} + 815730721 T^{16} )^{2}$$)($$( 1 + 84 T^{2} + 3312 T^{4} + 79227 T^{6} + 1252715 T^{8} + 13389363 T^{10} + 94594032 T^{12} + 405451956 T^{14} + 815730721 T^{16} )^{2}$$)
$17$ ($$( 1 + 58 T^{2} + 1412 T^{4} + 16762 T^{6} + 83521 T^{8} )^{2}$$)($$( 1 + 17 T^{2} )^{8}$$)($$( 1 - 12 T^{2} + 114 T^{4} - 3468 T^{6} + 83521 T^{8} )^{2}$$)($$( 1 + 24 T^{2} + 359 T^{4} + 6936 T^{6} + 83521 T^{8} )^{2}$$)($$( 1 + 42 T^{2} + 1233 T^{4} + 28986 T^{6} + 533141 T^{8} + 8376954 T^{10} + 102981393 T^{12} + 1013777898 T^{14} + 6975757441 T^{16} )^{2}$$)($$( 1 + 81 T^{2} + 3507 T^{4} + 99528 T^{6} + 1993145 T^{8} + 28763592 T^{10} + 292908147 T^{12} + 1955143089 T^{14} + 6975757441 T^{16} )^{2}$$)
$19$ ($$( 1 + 28 T^{2} + 470 T^{4} + 10108 T^{6} + 130321 T^{8} )^{2}$$)($$( 1 + 19 T^{2} )^{8}$$)($$( 1 + 62 T^{2} + 1678 T^{4} + 22382 T^{6} + 130321 T^{8} )^{2}$$)($$( 1 + 16 T^{2} + 361 T^{4} )^{4}$$)($$( 1 + 58 T^{2} + 2274 T^{4} + 60020 T^{6} + 1295735 T^{8} + 21667220 T^{10} + 296349954 T^{12} + 2728661098 T^{14} + 16983563041 T^{16} )^{2}$$)($$( 1 + 57 T^{2} + 1173 T^{4} + 10794 T^{6} + 84845 T^{8} + 3896634 T^{10} + 152866533 T^{12} + 2681615217 T^{14} + 16983563041 T^{16} )^{2}$$)
$23$ ($$( 1 + 23 T^{2} )^{8}$$)($$( 1 - 8 T + 41 T^{2} - 144 T^{3} + 209 T^{4} - 3312 T^{5} + 21689 T^{6} - 97336 T^{7} + 279841 T^{8} )^{2}$$)($$( 1 + 2 T + 27 T^{2} + 46 T^{3} + 529 T^{4} )^{4}$$)($$( 1 + 6 T + 52 T^{2} + 138 T^{3} + 529 T^{4} )^{4}$$)($$( 1 - 8 T + 67 T^{2} - 376 T^{3} + 2337 T^{4} - 8648 T^{5} + 35443 T^{6} - 97336 T^{7} + 279841 T^{8} )^{4}$$)($$( 1 + T + 35 T^{2} + 23 T^{3} + 529 T^{4} )^{8}$$)
$29$ ($$( 1 - 24 T^{2} + 34 T^{4} - 20184 T^{6} + 707281 T^{8} )^{2}$$)($$( 1 - 2 T - 25 T^{2} + 108 T^{3} + 509 T^{4} + 3132 T^{5} - 21025 T^{6} - 48778 T^{7} + 707281 T^{8} )( 1 + 2 T - 25 T^{2} - 108 T^{3} + 509 T^{4} - 3132 T^{5} - 21025 T^{6} + 48778 T^{7} + 707281 T^{8} )$$)($$( 1 - 62 T^{2} + 2238 T^{4} - 52142 T^{6} + 707281 T^{8} )^{2}$$)($$( 1 - 64 T^{2} + 2559 T^{4} - 53824 T^{6} + 707281 T^{8} )^{2}$$)($$( 1 - 148 T^{2} + 11011 T^{4} - 529684 T^{6} + 18048601 T^{8} - 445464244 T^{10} + 7787871091 T^{12} - 88033851508 T^{14} + 500246412961 T^{16} )^{2}$$)($$( 1 - 209 T^{2} + 19740 T^{4} - 1097431 T^{6} + 39229319 T^{8} - 922939471 T^{10} + 13961726940 T^{12} - 124318074089 T^{14} + 500246412961 T^{16} )^{2}$$)
$31$ ($$( 1 + 26 T^{2} + 2084 T^{4} + 24986 T^{6} + 923521 T^{8} )^{2}$$)($$( 1 - 31 T^{2} )^{8}$$)($$( 1 - 92 T^{2} + 3793 T^{4} - 88412 T^{6} + 923521 T^{8} )^{2}$$)($$( 1 - 36 T^{2} + 794 T^{4} - 34596 T^{6} + 923521 T^{8} )^{2}$$)($$( 1 - 154 T^{2} + 12138 T^{4} - 623876 T^{6} + 22716143 T^{8} - 599544836 T^{10} + 11209697898 T^{12} - 136675566874 T^{14} + 852891037441 T^{16} )^{2}$$)($$( 1 - 183 T^{2} + 15988 T^{4} - 870381 T^{6} + 32291445 T^{8} - 836436141 T^{10} + 14765253748 T^{12} - 162413173623 T^{14} + 852891037441 T^{16} )^{2}$$)
$37$ ($$( 1 - 10 T + 92 T^{2} - 370 T^{3} + 1369 T^{4} )^{4}$$)($$( 1 - 6 T - T^{2} + 228 T^{3} - 1331 T^{4} + 8436 T^{5} - 1369 T^{6} - 303918 T^{7} + 1874161 T^{8} )^{2}$$)($$( 1 + 3 T + 37 T^{2} )^{8}$$)($$( 1 + 10 T + 87 T^{2} + 370 T^{3} + 1369 T^{4} )^{4}$$)($$( 1 - 18 T + 238 T^{2} - 2106 T^{3} + 14910 T^{4} - 77922 T^{5} + 325822 T^{6} - 911754 T^{7} + 1874161 T^{8} )^{4}$$)($$( 1 + 9 T + 114 T^{2} + 753 T^{3} + 6389 T^{4} + 27861 T^{5} + 156066 T^{6} + 455877 T^{7} + 1874161 T^{8} )^{4}$$)
$41$ ($$( 1 + 94 T^{2} + 5228 T^{4} + 158014 T^{6} + 2825761 T^{8} )^{2}$$)($$( 1 + 41 T^{2} )^{8}$$)($$( 1 + 150 T^{2} + 8982 T^{4} + 252150 T^{6} + 2825761 T^{8} )^{2}$$)($$( 1 + 120 T^{2} + 6599 T^{4} + 201720 T^{6} + 2825761 T^{8} )^{2}$$)($$( 1 + 198 T^{2} + 19146 T^{4} + 1201164 T^{6} + 55935071 T^{8} + 2019156684 T^{10} + 54102020106 T^{12} + 940520639718 T^{14} + 7984925229121 T^{16} )^{2}$$)($$( 1 + 68 T^{2} + 2763 T^{4} + 134806 T^{6} + 5734715 T^{8} + 226608886 T^{10} + 7807577643 T^{12} + 323007088388 T^{14} + 7984925229121 T^{16} )^{2}$$)
$43$ ($$( 1 - 72 T^{2} + 1849 T^{4} )^{4}$$)($$( 1 - 12 T + 101 T^{2} - 696 T^{3} + 4009 T^{4} - 29928 T^{5} + 186749 T^{6} - 954084 T^{7} + 3418801 T^{8} )( 1 + 12 T + 101 T^{2} + 696 T^{3} + 4009 T^{4} + 29928 T^{5} + 186749 T^{6} + 954084 T^{7} + 3418801 T^{8} )$$)($$( 1 + 2 T^{2} + 2854 T^{4} + 3698 T^{6} + 3418801 T^{8} )^{2}$$)($$( 1 - 60 T^{2} + 4166 T^{4} - 110940 T^{6} + 3418801 T^{8} )^{2}$$)($$( 1 - 236 T^{2} + 27603 T^{4} - 2048308 T^{6} + 104925905 T^{8} - 3787321492 T^{10} + 94369164003 T^{12} - 1491841679564 T^{14} + 11688200277601 T^{16} )^{2}$$)($$( 1 - 13 T + 43 T^{2} )^{8}( 1 + 13 T + 43 T^{2} )^{8}$$)
$47$ ($$( 1 - 150 T^{2} + 9700 T^{4} - 331350 T^{6} + 4879681 T^{8} )^{2}$$)($$( 1 - 47 T^{2} )^{8}$$)($$( 1 - 48 T^{2} + 4494 T^{4} - 106032 T^{6} + 4879681 T^{8} )^{2}$$)($$( 1 - 100 T^{2} + 5466 T^{4} - 220900 T^{6} + 4879681 T^{8} )^{2}$$)($$( 1 - 72 T^{2} + 4821 T^{4} - 132198 T^{6} + 5984465 T^{8} - 292025382 T^{10} + 23524942101 T^{12} - 776103503688 T^{14} + 23811286661761 T^{16} )^{2}$$)($$( 1 - 226 T^{2} + 24862 T^{4} - 1818763 T^{6} + 98315935 T^{8} - 4017647467 T^{10} + 121318629022 T^{12} - 2436102664354 T^{14} + 23811286661761 T^{16} )^{2}$$)
$53$ ($$( 1 - 6 T + 52 T^{2} - 318 T^{3} + 2809 T^{4} )^{4}$$)($$( 1 + 10 T + 47 T^{2} - 60 T^{3} - 3091 T^{4} - 3180 T^{5} + 132023 T^{6} + 1488770 T^{7} + 7890481 T^{8} )^{2}$$)($$( 1 + 2 T + 53 T^{2} )^{8}$$)($$( 1 + 6 T + 67 T^{2} + 318 T^{3} + 2809 T^{4} )^{4}$$)($$( 1 + 2 T + 65 T^{2} + 170 T^{3} + 4561 T^{4} + 9010 T^{5} + 182585 T^{6} + 297754 T^{7} + 7890481 T^{8} )^{4}$$)($$( 1 - 5 T + 202 T^{2} - 775 T^{3} + 15789 T^{4} - 41075 T^{5} + 567418 T^{6} - 744385 T^{7} + 7890481 T^{8} )^{4}$$)
$59$ ($$( 1 - 54 T^{2} + 7348 T^{4} - 187974 T^{6} + 12117361 T^{8} )^{2}$$)($$( 1 - 59 T^{2} )^{8}$$)($$( 1 - 108 T^{2} + 9753 T^{4} - 375948 T^{6} + 12117361 T^{8} )^{2}$$)($$( 1 - 74 T^{2} + 3481 T^{4} )^{4}$$)($$( 1 - 312 T^{2} + 49077 T^{4} - 4945830 T^{6} + 346685561 T^{8} - 17216434230 T^{10} + 594683725797 T^{12} - 13160326495992 T^{14} + 146830437604321 T^{16} )^{2}$$)($$( 1 - 247 T^{2} + 32808 T^{4} - 3008579 T^{6} + 204923225 T^{8} - 10472863499 T^{10} + 397546379688 T^{12} - 10418591809327 T^{14} + 146830437604321 T^{16} )^{2}$$)
$61$ ($$( 1 + 122 T^{2} + 7460 T^{4} + 453962 T^{6} + 13845841 T^{8} )^{2}$$)($$( 1 + 61 T^{2} )^{8}$$)($$( 1 + 230 T^{2} + 20662 T^{4} + 855830 T^{6} + 13845841 T^{8} )^{2}$$)($$( 1 + 56 T^{2} + 3721 T^{4} )^{4}$$)($$( 1 + 160 T^{2} + 14349 T^{4} + 678590 T^{6} + 34880249 T^{8} + 2525033390 T^{10} + 198673972509 T^{12} + 8243259897760 T^{14} + 191707312997281 T^{16} )^{2}$$)($$( 1 + 163 T^{2} + 18018 T^{4} + 1294271 T^{6} + 86965925 T^{8} + 4815982391 T^{10} + 249474363138 T^{12} + 8397821020843 T^{14} + 191707312997281 T^{16} )^{2}$$)
$67$ ($$( 1 + 16 T + 170 T^{2} + 1072 T^{3} + 4489 T^{4} )^{4}$$)($$( 1 - 4 T - 51 T^{2} + 472 T^{3} + 1529 T^{4} + 31624 T^{5} - 228939 T^{6} - 1203052 T^{7} + 20151121 T^{8} )^{2}$$)($$( 1 + 6 T + 63 T^{2} + 402 T^{3} + 4489 T^{4} )^{4}$$)($$( 1 + 2 T + 60 T^{2} + 134 T^{3} + 4489 T^{4} )^{4}$$)($$( 1 - 6 T + 261 T^{2} - 1110 T^{3} + 25853 T^{4} - 74370 T^{5} + 1171629 T^{6} - 1804578 T^{7} + 20151121 T^{8} )^{4}$$)($$( 1 + T + 159 T^{2} - 238 T^{3} + 12569 T^{4} - 15946 T^{5} + 713751 T^{6} + 300763 T^{7} + 20151121 T^{8} )^{4}$$)
$71$ ($$( 1 + 4 T + 118 T^{2} + 284 T^{3} + 5041 T^{4} )^{4}$$)($$( 1 - 16 T + 185 T^{2} - 1824 T^{3} + 16049 T^{4} - 129504 T^{5} + 932585 T^{6} - 5726576 T^{7} + 25411681 T^{8} )^{2}$$)($$( 1 - T + 71 T^{2} )^{8}$$)($$( 1 + 8 T + 146 T^{2} + 568 T^{3} + 5041 T^{4} )^{4}$$)($$( 1 + 14 T + 215 T^{2} + 2156 T^{3} + 23821 T^{4} + 153076 T^{5} + 1083815 T^{6} + 5010754 T^{7} + 25411681 T^{8} )^{4}$$)($$( 1 - 11 T + 215 T^{2} - 1274 T^{3} + 17569 T^{4} - 90454 T^{5} + 1083815 T^{6} - 3937021 T^{7} + 25411681 T^{8} )^{4}$$)
$73$ ($$( 1 + 154 T^{2} + 14564 T^{4} + 820666 T^{6} + 28398241 T^{8} )^{2}$$)($$( 1 + 73 T^{2} )^{8}$$)($$( 1 + 222 T^{2} + 22854 T^{4} + 1183038 T^{6} + 28398241 T^{8} )^{2}$$)($$( 1 + 124 T^{2} + 5329 T^{4} )^{4}$$)($$( 1 + 270 T^{2} + 41394 T^{4} + 4349916 T^{6} + 358079783 T^{8} + 23180702364 T^{10} + 1175516787954 T^{12} + 40860241098030 T^{14} + 806460091894081 T^{16} )^{2}$$)($$( 1 + 349 T^{2} + 60102 T^{4} + 6782387 T^{6} + 565496465 T^{8} + 36143340323 T^{10} + 1706791080582 T^{12} + 52815644974861 T^{14} + 806460091894081 T^{16} )^{2}$$)
$79$ ($$( 1 - 32 T^{2} + 5570 T^{4} - 199712 T^{6} + 38950081 T^{8} )^{2}$$)($$( 1 - 8 T - 15 T^{2} + 752 T^{3} - 4831 T^{4} + 59408 T^{5} - 93615 T^{6} - 3944312 T^{7} + 38950081 T^{8} )( 1 + 8 T - 15 T^{2} - 752 T^{3} - 4831 T^{4} - 59408 T^{5} - 93615 T^{6} + 3944312 T^{7} + 38950081 T^{8} )$$)($$( 1 - 140 T^{2} + 6241 T^{4} )^{4}$$)($$( 1 - 232 T^{2} + 24210 T^{4} - 1447912 T^{6} + 38950081 T^{8} )^{2}$$)($$( 1 - 272 T^{2} + 42763 T^{4} - 4853036 T^{6} + 423367825 T^{8} - 30287797676 T^{10} + 1665622313803 T^{12} - 66119787901712 T^{14} + 1517108809906561 T^{16} )^{2}$$)($$( 1 - 444 T^{2} + 91050 T^{4} - 11682681 T^{6} + 1069340489 T^{8} - 72911612121 T^{10} + 3546404875050 T^{12} - 107930830251324 T^{14} + 1517108809906561 T^{16} )^{2}$$)
$83$ ($$( 1 + 292 T^{2} + 34982 T^{4} + 2011588 T^{6} + 47458321 T^{8} )^{2}$$)($$( 1 + 83 T^{2} )^{8}$$)($$( 1 + 186 T^{2} + 22422 T^{4} + 1281354 T^{6} + 47458321 T^{8} )^{2}$$)($$( 1 + 78 T^{2} + 6889 T^{4} )^{4}$$)($$( 1 + 462 T^{2} + 104778 T^{4} + 15055836 T^{6} + 1489084799 T^{8} + 103719654204 T^{10} + 4972587957738 T^{12} + 151046452496478 T^{14} + 2252292232139041 T^{16} )^{2}$$)($$( 1 + 429 T^{2} + 93792 T^{4} + 13224747 T^{6} + 1299816125 T^{8} + 91105282083 T^{10} + 4451210843232 T^{12} + 140257420175301 T^{14} + 2252292232139041 T^{16} )^{2}$$)
$89$ ($$( 1 - 140 T^{2} + 11670 T^{4} - 1108940 T^{6} + 62742241 T^{8} )^{2}$$)($$( 1 - 89 T^{2} )^{8}$$)($$( 1 + 84 T^{2} + 2481 T^{4} + 665364 T^{6} + 62742241 T^{8} )^{2}$$)($$( 1 - 145 T^{2} + 7921 T^{4} )^{4}$$)($$( 1 - 186 T^{2} + 27441 T^{4} - 3244218 T^{6} + 324982517 T^{8} - 25697450778 T^{10} + 1721709835281 T^{12} - 92438520118746 T^{14} + 3936588805702081 T^{16} )^{2}$$)($$( 1 - 547 T^{2} + 136828 T^{4} - 20978449 T^{6} + 2212310245 T^{8} - 166170294529 T^{10} + 8584895351548 T^{12} - 271848766155667 T^{14} + 3936588805702081 T^{16} )^{2}$$)
$97$ ($$( 1 - 152 T^{2} + 16502 T^{4} - 1430168 T^{6} + 88529281 T^{8} )^{2}$$)($$( 1 - 97 T^{2} )^{8}$$)($$( 1 - 356 T^{2} + 50257 T^{4} - 3349604 T^{6} + 88529281 T^{8} )^{2}$$)($$( 1 - 102 T^{2} + 15611 T^{4} - 959718 T^{6} + 88529281 T^{8} )^{2}$$)($$( 1 - 502 T^{2} + 117402 T^{4} - 17438396 T^{6} + 1914009407 T^{8} - 164077867964 T^{10} + 10393514647962 T^{12} - 418151946474358 T^{14} + 7837433594376961 T^{16} )^{2}$$)($$( 1 - 291 T^{2} + 62572 T^{4} - 8875143 T^{6} + 1004261265 T^{8} - 83506220487 T^{10} + 5539454170732 T^{12} - 242394853434339 T^{14} + 7837433594376961 T^{16} )^{2}$$)