Properties

 Label 1050.2.m Level 1050 Weight 2 Character orbit m Rep. character $$\chi_{1050}(307,\cdot)$$ Character field $$\Q(\zeta_{4})$$ Dimension 48 Newform subspaces 6 Sturm bound 480 Trace bound 19

Related objects

Defining parameters

 Level: $$N$$ = $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 1050.m (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$35$$ Character field: $$\Q(i)$$ Newform subspaces: $$6$$ Sturm bound: $$480$$ Trace bound: $$19$$ Distinguishing $$T_p$$: $$11$$, $$13$$, $$19$$

Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(1050, [\chi])$$.

Total New Old
Modular forms 528 48 480
Cusp forms 432 48 384
Eisenstein series 96 0 96

Trace form

 $$48q - 12q^{7} + O(q^{10})$$ $$48q - 12q^{7} - 32q^{11} - 48q^{16} - 8q^{21} - 16q^{22} - 32q^{23} + 12q^{28} - 48q^{36} + 56q^{37} + 4q^{42} + 32q^{46} - 64q^{51} + 16q^{53} - 16q^{56} + 8q^{57} + 24q^{58} + 12q^{63} - 32q^{71} - 32q^{78} - 48q^{81} - 32q^{86} + 16q^{88} + 112q^{91} - 32q^{92} + 8q^{93} + 16q^{98} + O(q^{100})$$

Decomposition of $$S_{2}^{\mathrm{new}}(1050, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
1050.2.m.a $$8$$ $$8.384$$ 8.0.1698758656.6 None $$0$$ $$0$$ $$0$$ $$-8$$ $$q-\beta _{3}q^{2}+\beta _{3}q^{3}-\beta _{5}q^{4}+\beta _{5}q^{6}+\cdots$$
1050.2.m.b $$8$$ $$8.384$$ 8.0.1698758656.6 None $$0$$ $$0$$ $$0$$ $$-4$$ $$q-\beta _{3}q^{2}-\beta _{3}q^{3}-\beta _{5}q^{4}-\beta _{5}q^{6}+\cdots$$
1050.2.m.c $$8$$ $$8.384$$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{24}q^{2}-\zeta_{24}q^{3}+\zeta_{24}^{3}q^{4}-\zeta_{24}^{3}q^{6}+\cdots$$
1050.2.m.d $$8$$ $$8.384$$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{24}q^{2}+\zeta_{24}q^{3}+\zeta_{24}^{3}q^{4}+\zeta_{24}^{3}q^{6}+\cdots$$
1050.2.m.e $$8$$ $$8.384$$ 8.0.1871773696.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{1}q^{2}+\beta _{1}q^{3}+\beta _{2}q^{4}-\beta _{2}q^{6}+\cdots$$
1050.2.m.f $$8$$ $$8.384$$ 8.0.1871773696.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+\beta _{1}q^{3}+\beta _{2}q^{4}+\beta _{2}q^{6}+\cdots$$

Decomposition of $$S_{2}^{\mathrm{old}}(1050, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(1050, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(35, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(70, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(105, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(175, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(210, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(350, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(525, [\chi])$$$$^{\oplus 2}$$

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)
$3$ ($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)
$5$ 1
$7$ ($$1 + 8 T + 34 T^{2} + 112 T^{3} + 322 T^{4} + 784 T^{5} + 1666 T^{6} + 2744 T^{7} + 2401 T^{8}$$)($$1 + 4 T + 6 T^{2} - 28 T^{3} - 126 T^{4} - 196 T^{5} + 294 T^{6} + 1372 T^{7} + 2401 T^{8}$$)($$1 - 94 T^{4} + 2401 T^{8}$$)($$1 - 94 T^{4} + 2401 T^{8}$$)($$( 1 + 12 T^{2} + 49 T^{4} )^{2}$$)($$( 1 - 12 T^{2} + 49 T^{4} )^{2}$$)
$11$ ($$( 1 - 4 T + 12 T^{2} + 4 T^{3} + 18 T^{4} + 44 T^{5} + 1452 T^{6} - 5324 T^{7} + 14641 T^{8} )^{2}$$)($$( 1 - 4 T + 12 T^{2} + 4 T^{3} + 18 T^{4} + 44 T^{5} + 1452 T^{6} - 5324 T^{7} + 14641 T^{8} )^{2}$$)($$( 1 + 4 T + 14 T^{2} + 44 T^{3} + 121 T^{4} )^{4}$$)($$( 1 + 4 T + 14 T^{2} + 44 T^{3} + 121 T^{4} )^{4}$$)($$( 1 + 2 T + 10 T^{2} + 22 T^{3} + 121 T^{4} )^{4}$$)($$( 1 + 2 T + 10 T^{2} + 22 T^{3} + 121 T^{4} )^{4}$$)
$13$ ($$( 1 + 8 T + 32 T^{2} + 136 T^{3} + 562 T^{4} + 1768 T^{5} + 5408 T^{6} + 17576 T^{7} + 28561 T^{8} )^{2}$$)($$( 1 - 8 T + 32 T^{2} - 136 T^{3} + 562 T^{4} - 1768 T^{5} + 5408 T^{6} - 17576 T^{7} + 28561 T^{8} )^{2}$$)($$1 + 142 T^{4} - 7149 T^{8} + 4055662 T^{12} + 815730721 T^{16}$$)($$1 + 142 T^{4} - 7149 T^{8} + 4055662 T^{12} + 815730721 T^{16}$$)($$1 + 292 T^{4} + 48486 T^{8} + 8339812 T^{12} + 815730721 T^{16}$$)($$1 + 292 T^{4} + 48486 T^{8} + 8339812 T^{12} + 815730721 T^{16}$$)
$17$ ($$1 - 12 T + 72 T^{2} - 388 T^{3} + 1696 T^{4} - 4732 T^{5} + 9944 T^{6} - 2420 T^{7} - 98754 T^{8} - 41140 T^{9} + 2873816 T^{10} - 23248316 T^{11} + 141651616 T^{12} - 550904516 T^{13} + 1737904968 T^{14} - 4924064076 T^{15} + 6975757441 T^{16}$$)($$1 + 12 T + 72 T^{2} + 388 T^{3} + 1696 T^{4} + 4732 T^{5} + 9944 T^{6} + 2420 T^{7} - 98754 T^{8} + 41140 T^{9} + 2873816 T^{10} + 23248316 T^{11} + 141651616 T^{12} + 550904516 T^{13} + 1737904968 T^{14} + 4924064076 T^{15} + 6975757441 T^{16}$$)($$( 1 - 497 T^{4} + 83521 T^{8} )^{2}$$)($$( 1 - 497 T^{4} + 83521 T^{8} )^{2}$$)($$1 - 252 T^{4} + 99718 T^{8} - 21047292 T^{12} + 6975757441 T^{16}$$)($$1 - 252 T^{4} + 99718 T^{8} - 21047292 T^{12} + 6975757441 T^{16}$$)
$19$ ($$( 1 - 4 T + 62 T^{2} - 212 T^{3} + 1666 T^{4} - 4028 T^{5} + 22382 T^{6} - 27436 T^{7} + 130321 T^{8} )^{2}$$)($$( 1 + 4 T + 62 T^{2} + 212 T^{3} + 1666 T^{4} + 4028 T^{5} + 22382 T^{6} + 27436 T^{7} + 130321 T^{8} )^{2}$$)($$( 1 + 8 T + 42 T^{2} + 152 T^{3} + 361 T^{4} )^{4}$$)($$( 1 - 8 T + 42 T^{2} - 152 T^{3} + 361 T^{4} )^{4}$$)($$( 1 - 2 T + 26 T^{2} - 38 T^{3} + 361 T^{4} )^{4}$$)($$( 1 + 2 T + 26 T^{2} + 38 T^{3} + 361 T^{4} )^{4}$$)
$23$ ($$1 + 16 T + 128 T^{2} + 656 T^{3} + 3300 T^{4} + 21520 T^{5} + 137088 T^{6} + 691344 T^{7} + 3238918 T^{8} + 15900912 T^{9} + 72519552 T^{10} + 261833840 T^{11} + 923475300 T^{12} + 4222241008 T^{13} + 18948593792 T^{14} + 54477207152 T^{15} + 78310985281 T^{16}$$)($$1 + 16 T + 128 T^{2} + 656 T^{3} + 3300 T^{4} + 21520 T^{5} + 137088 T^{6} + 691344 T^{7} + 3238918 T^{8} + 15900912 T^{9} + 72519552 T^{10} + 261833840 T^{11} + 923475300 T^{12} + 4222241008 T^{13} + 18948593792 T^{14} + 54477207152 T^{15} + 78310985281 T^{16}$$)($$( 1 + 967 T^{4} + 279841 T^{8} )^{2}$$)($$( 1 + 967 T^{4} + 279841 T^{8} )^{2}$$)($$1 - 252 T^{4} + 242758 T^{8} - 70519932 T^{12} + 78310985281 T^{16}$$)($$1 - 252 T^{4} + 242758 T^{8} - 70519932 T^{12} + 78310985281 T^{16}$$)
$29$ ($$1 - 4 T^{2} + 200 T^{4} - 5548 T^{6} + 1325934 T^{8} - 4665868 T^{10} + 141456200 T^{12} - 2379293284 T^{14} + 500246412961 T^{16}$$)($$1 - 4 T^{2} + 200 T^{4} - 5548 T^{6} + 1325934 T^{8} - 4665868 T^{10} + 141456200 T^{12} - 2379293284 T^{14} + 500246412961 T^{16}$$)($$( 1 - 38 T^{2} + 1611 T^{4} - 31958 T^{6} + 707281 T^{8} )^{2}$$)($$( 1 - 38 T^{2} + 1611 T^{4} - 31958 T^{6} + 707281 T^{8} )^{2}$$)($$( 1 - 29 T^{2} )^{8}$$)($$( 1 - 29 T^{2} )^{8}$$)
$31$ ($$1 - 148 T^{2} + 11192 T^{4} - 565212 T^{6} + 20507310 T^{8} - 543168732 T^{10} + 10336047032 T^{12} - 131350544788 T^{14} + 852891037441 T^{16}$$)($$1 - 148 T^{2} + 11192 T^{4} - 565212 T^{6} + 20507310 T^{8} - 543168732 T^{10} + 10336047032 T^{12} - 131350544788 T^{14} + 852891037441 T^{16}$$)($$( 1 - 26 T^{2} + 1899 T^{4} - 24986 T^{6} + 923521 T^{8} )^{2}$$)($$( 1 - 26 T^{2} + 1899 T^{4} - 24986 T^{6} + 923521 T^{8} )^{2}$$)($$( 1 - 12 T^{2} + 1126 T^{4} - 11532 T^{6} + 923521 T^{8} )^{2}$$)($$( 1 - 12 T^{2} + 1126 T^{4} - 11532 T^{6} + 923521 T^{8} )^{2}$$)
$37$ ($$1 - 28 T + 392 T^{2} - 4180 T^{3} + 38912 T^{4} - 311708 T^{5} + 2210520 T^{6} - 14742164 T^{7} + 92944158 T^{8} - 545460068 T^{9} + 3026201880 T^{10} - 15788945324 T^{11} + 72927352832 T^{12} - 289857740260 T^{13} + 1005764752328 T^{14} - 2658092559724 T^{15} + 3512479453921 T^{16}$$)($$1 - 28 T + 392 T^{2} - 4180 T^{3} + 38912 T^{4} - 311708 T^{5} + 2210520 T^{6} - 14742164 T^{7} + 92944158 T^{8} - 545460068 T^{9} + 3026201880 T^{10} - 15788945324 T^{11} + 72927352832 T^{12} - 289857740260 T^{13} + 1005764752328 T^{14} - 2658092559724 T^{15} + 3512479453921 T^{16}$$)($$( 1 - 2062 T^{4} + 1874161 T^{8} )^{2}$$)($$( 1 - 2062 T^{4} + 1874161 T^{8} )^{2}$$)($$( 1 - 1294 T^{4} + 1874161 T^{8} )^{2}$$)($$( 1 - 1294 T^{4} + 1874161 T^{8} )^{2}$$)
$41$ ($$1 - 176 T^{2} + 13468 T^{4} - 643920 T^{6} + 26131718 T^{8} - 1082429520 T^{10} + 38057349148 T^{12} - 836018346416 T^{14} + 7984925229121 T^{16}$$)($$1 - 176 T^{2} + 13468 T^{4} - 643920 T^{6} + 26131718 T^{8} - 1082429520 T^{10} + 38057349148 T^{12} - 836018346416 T^{14} + 7984925229121 T^{16}$$)($$( 1 - 126 T^{2} + 7139 T^{4} - 211806 T^{6} + 2825761 T^{8} )^{2}$$)($$( 1 - 126 T^{2} + 7139 T^{4} - 211806 T^{6} + 2825761 T^{8} )^{2}$$)($$( 1 - 40 T^{2} + 1214 T^{4} - 67240 T^{6} + 2825761 T^{8} )^{2}$$)($$( 1 - 40 T^{2} + 1214 T^{4} - 67240 T^{6} + 2825761 T^{8} )^{2}$$)
$43$ ($$1 + 32 T^{3} - 6588 T^{4} + 1632 T^{5} + 512 T^{6} - 117440 T^{7} + 17663206 T^{8} - 5049920 T^{9} + 946688 T^{10} + 129755424 T^{11} - 22523060988 T^{12} + 4704270176 T^{13} + 11688200277601 T^{16}$$)($$1 + 32 T^{3} - 6588 T^{4} + 1632 T^{5} + 512 T^{6} - 117440 T^{7} + 17663206 T^{8} - 5049920 T^{9} + 946688 T^{10} + 129755424 T^{11} - 22523060988 T^{12} + 4704270176 T^{13} + 11688200277601 T^{16}$$)($$1 + 5182 T^{4} + 12352611 T^{8} + 17716226782 T^{12} + 11688200277601 T^{16}$$)($$1 + 5182 T^{4} + 12352611 T^{8} + 17716226782 T^{12} + 11688200277601 T^{16}$$)($$1 - 188 T^{4} - 5134362 T^{8} - 642734588 T^{12} + 11688200277601 T^{16}$$)($$1 - 188 T^{4} - 5134362 T^{8} - 642734588 T^{12} + 11688200277601 T^{16}$$)
$47$ ($$1 - 24 T + 288 T^{2} - 2728 T^{3} + 26308 T^{4} - 243208 T^{5} + 1981280 T^{6} - 14819768 T^{7} + 104930310 T^{8} - 696529096 T^{9} + 4376647520 T^{10} - 25250584184 T^{11} + 128374647748 T^{12} - 625653179096 T^{13} + 3104414014752 T^{14} - 12158954891112 T^{15} + 23811286661761 T^{16}$$)($$1 + 24 T + 288 T^{2} + 2728 T^{3} + 26308 T^{4} + 243208 T^{5} + 1981280 T^{6} + 14819768 T^{7} + 104930310 T^{8} + 696529096 T^{9} + 4376647520 T^{10} + 25250584184 T^{11} + 128374647748 T^{12} + 625653179096 T^{13} + 3104414014752 T^{14} + 12158954891112 T^{15} + 23811286661761 T^{16}$$)($$1 - 4732 T^{4} + 13117830 T^{8} - 23090650492 T^{12} + 23811286661761 T^{16}$$)($$1 - 4732 T^{4} + 13117830 T^{8} - 23090650492 T^{12} + 23811286661761 T^{16}$$)($$1 + 4068 T^{4} + 12565318 T^{8} + 19850542308 T^{12} + 23811286661761 T^{16}$$)($$1 + 4068 T^{4} + 12565318 T^{8} + 19850542308 T^{12} + 23811286661761 T^{16}$$)
$53$ ($$1 - 8 T + 32 T^{2} - 264 T^{3} + 5828 T^{4} - 46136 T^{5} + 217440 T^{6} - 2204664 T^{7} + 21970150 T^{8} - 116847192 T^{9} + 610788960 T^{10} - 6868589272 T^{11} + 45985723268 T^{12} - 110403610152 T^{13} + 709259556128 T^{14} - 9397689118696 T^{15} + 62259690411361 T^{16}$$)($$1 - 8 T + 32 T^{2} - 264 T^{3} + 5828 T^{4} - 46136 T^{5} + 217440 T^{6} - 2204664 T^{7} + 21970150 T^{8} - 116847192 T^{9} + 610788960 T^{10} - 6868589272 T^{11} + 45985723268 T^{12} - 110403610152 T^{13} + 709259556128 T^{14} - 9397689118696 T^{15} + 62259690411361 T^{16}$$)($$( 1 - 56 T^{2} + 327 T^{4} - 157304 T^{6} + 7890481 T^{8} )( 1 + 56 T^{2} + 327 T^{4} + 157304 T^{6} + 7890481 T^{8} )$$)($$( 1 - 56 T^{2} + 327 T^{4} - 157304 T^{6} + 7890481 T^{8} )( 1 + 56 T^{2} + 327 T^{4} + 157304 T^{6} + 7890481 T^{8} )$$)($$1 - 1692 T^{4} - 2725850 T^{8} - 13350693852 T^{12} + 62259690411361 T^{16}$$)($$1 - 1692 T^{4} - 2725850 T^{8} - 13350693852 T^{12} + 62259690411361 T^{16}$$)
$59$ ($$( 1 + 4 T + 70 T^{2} + 116 T^{3} + 802 T^{4} + 6844 T^{5} + 243670 T^{6} + 821516 T^{7} + 12117361 T^{8} )^{2}$$)($$( 1 - 4 T + 70 T^{2} - 116 T^{3} + 802 T^{4} - 6844 T^{5} + 243670 T^{6} - 821516 T^{7} + 12117361 T^{8} )^{2}$$)($$( 1 + 4 T + 95 T^{2} + 236 T^{3} + 3481 T^{4} )^{4}$$)($$( 1 - 4 T + 95 T^{2} - 236 T^{3} + 3481 T^{4} )^{4}$$)($$( 1 - 10 T + 130 T^{2} - 590 T^{3} + 3481 T^{4} )^{4}$$)($$( 1 + 10 T + 130 T^{2} + 590 T^{3} + 3481 T^{4} )^{4}$$)
$61$ ($$1 - 264 T^{2} + 37532 T^{4} - 3615096 T^{6} + 255247078 T^{8} - 13451772216 T^{10} + 519662104412 T^{12} - 13601378831304 T^{14} + 191707312997281 T^{16}$$)($$1 - 264 T^{2} + 37532 T^{4} - 3615096 T^{6} + 255247078 T^{8} - 13451772216 T^{10} + 519662104412 T^{12} - 13601378831304 T^{14} + 191707312997281 T^{16}$$)($$( 1 - 218 T^{2} + 19275 T^{4} - 811178 T^{6} + 13845841 T^{8} )^{2}$$)($$( 1 - 218 T^{2} + 19275 T^{4} - 811178 T^{6} + 13845841 T^{8} )^{2}$$)($$( 1 + 24 T^{2} + 1294 T^{4} + 89304 T^{6} + 13845841 T^{8} )^{2}$$)($$( 1 + 24 T^{2} + 1294 T^{4} + 89304 T^{6} + 13845841 T^{8} )^{2}$$)
$67$ ($$1 + 64 T^{3} + 9348 T^{4} - 10560 T^{5} + 2048 T^{6} - 428416 T^{7} + 51599014 T^{8} - 28703872 T^{9} + 9193472 T^{10} - 3176057280 T^{11} + 188372679108 T^{12} + 86408006848 T^{13} + 406067677556641 T^{16}$$)($$1 + 64 T^{3} + 9348 T^{4} - 10560 T^{5} + 2048 T^{6} - 428416 T^{7} + 51599014 T^{8} - 28703872 T^{9} + 9193472 T^{10} - 3176057280 T^{11} + 188372679108 T^{12} + 86408006848 T^{13} + 406067677556641 T^{16}$$)($$1 - 3932 T^{4} + 41402598 T^{8} - 79234207772 T^{12} + 406067677556641 T^{16}$$)($$1 - 3932 T^{4} + 41402598 T^{8} - 79234207772 T^{12} + 406067677556641 T^{16}$$)($$1 - 2492 T^{4} - 10980570 T^{8} - 50216593532 T^{12} + 406067677556641 T^{16}$$)($$1 - 2492 T^{4} - 10980570 T^{8} - 50216593532 T^{12} + 406067677556641 T^{16}$$)
$71$ ($$( 1 - 4 T + 158 T^{2} + 220 T^{3} + 10018 T^{4} + 15620 T^{5} + 796478 T^{6} - 1431644 T^{7} + 25411681 T^{8} )^{2}$$)($$( 1 - 4 T + 158 T^{2} + 220 T^{3} + 10018 T^{4} + 15620 T^{5} + 796478 T^{6} - 1431644 T^{7} + 25411681 T^{8} )^{2}$$)($$( 1 + 16 T + 194 T^{2} + 1136 T^{3} + 5041 T^{4} )^{4}$$)($$( 1 + 16 T + 194 T^{2} + 1136 T^{3} + 5041 T^{4} )^{4}$$)($$( 1 - 10 T + 154 T^{2} - 710 T^{3} + 5041 T^{4} )^{4}$$)($$( 1 - 10 T + 154 T^{2} - 710 T^{3} + 5041 T^{4} )^{4}$$)
$73$ ($$1 + 28 T + 392 T^{2} + 4756 T^{3} + 46736 T^{4} + 302508 T^{5} + 1459480 T^{6} + 3207460 T^{7} - 21483554 T^{8} + 234144580 T^{9} + 7777568920 T^{10} + 117680754636 T^{11} + 1327220191376 T^{12} + 9859528496308 T^{13} + 59323016705288 T^{14} + 309327158534716 T^{15} + 806460091894081 T^{16}$$)($$1 - 28 T + 392 T^{2} - 4756 T^{3} + 46736 T^{4} - 302508 T^{5} + 1459480 T^{6} - 3207460 T^{7} - 21483554 T^{8} - 234144580 T^{9} + 7777568920 T^{10} - 117680754636 T^{11} + 1327220191376 T^{12} - 9859528496308 T^{13} + 59323016705288 T^{14} - 309327158534716 T^{15} + 806460091894081 T^{16}$$)($$1 - 15548 T^{4} + 109241286 T^{8} - 441535851068 T^{12} + 806460091894081 T^{16}$$)($$1 - 15548 T^{4} + 109241286 T^{8} - 441535851068 T^{12} + 806460091894081 T^{16}$$)($$1 + 388 T^{4} - 43924410 T^{8} + 11018517508 T^{12} + 806460091894081 T^{16}$$)($$1 + 388 T^{4} - 43924410 T^{8} + 11018517508 T^{12} + 806460091894081 T^{16}$$)
$79$ ($$1 - 272 T^{2} + 39644 T^{4} - 4069360 T^{6} + 348070342 T^{8} - 25396875760 T^{10} + 1544137011164 T^{12} - 66119787901712 T^{14} + 1517108809906561 T^{16}$$)($$1 - 272 T^{2} + 39644 T^{4} - 4069360 T^{6} + 348070342 T^{8} - 25396875760 T^{10} + 1544137011164 T^{12} - 66119787901712 T^{14} + 1517108809906561 T^{16}$$)($$( 1 - 92 T^{2} + 12870 T^{4} - 574172 T^{6} + 38950081 T^{8} )^{2}$$)($$( 1 - 92 T^{2} + 12870 T^{4} - 574172 T^{6} + 38950081 T^{8} )^{2}$$)($$( 1 - 140 T^{2} + 9894 T^{4} - 873740 T^{6} + 38950081 T^{8} )^{2}$$)($$( 1 - 140 T^{2} + 9894 T^{4} - 873740 T^{6} + 38950081 T^{8} )^{2}$$)
$83$ ($$1 + 16 T + 128 T^{2} + 976 T^{3} + 16964 T^{4} + 235664 T^{5} + 2075520 T^{6} + 16972368 T^{7} + 135669670 T^{8} + 1408706544 T^{9} + 14298257280 T^{10} + 134749611568 T^{11} + 805082957444 T^{12} + 3844503667568 T^{13} + 41848367791232 T^{14} + 434176815834032 T^{15} + 2252292232139041 T^{16}$$)($$1 - 16 T + 128 T^{2} - 976 T^{3} + 16964 T^{4} - 235664 T^{5} + 2075520 T^{6} - 16972368 T^{7} + 135669670 T^{8} - 1408706544 T^{9} + 14298257280 T^{10} - 134749611568 T^{11} + 805082957444 T^{12} - 3844503667568 T^{13} + 41848367791232 T^{14} - 434176815834032 T^{15} + 2252292232139041 T^{16}$$)($$1 + 15358 T^{4} + 117552483 T^{8} + 728864893918 T^{12} + 2252292232139041 T^{16}$$)($$1 + 15358 T^{4} + 117552483 T^{8} + 728864893918 T^{12} + 2252292232139041 T^{16}$$)($$( 1 - 13294 T^{4} + 47458321 T^{8} )^{2}$$)($$( 1 - 13294 T^{4} + 47458321 T^{8} )^{2}$$)
$89$ ($$( 1 - 32 T + 628 T^{2} - 8320 T^{3} + 88630 T^{4} - 740480 T^{5} + 4974388 T^{6} - 22559008 T^{7} + 62742241 T^{8} )^{2}$$)($$( 1 + 32 T + 628 T^{2} + 8320 T^{3} + 88630 T^{4} + 740480 T^{5} + 4974388 T^{6} + 22559008 T^{7} + 62742241 T^{8} )^{2}$$)($$( 1 + 12 T + 89 T^{2} )^{8}$$)($$( 1 - 12 T + 89 T^{2} )^{8}$$)($$( 1 - 6 T + 70 T^{2} - 534 T^{3} + 7921 T^{4} )^{4}$$)($$( 1 + 6 T + 70 T^{2} + 534 T^{3} + 7921 T^{4} )^{4}$$)
$97$ ($$1 + 28 T + 392 T^{2} + 4900 T^{3} + 61488 T^{4} + 704956 T^{5} + 7640472 T^{6} + 89011524 T^{7} + 981394654 T^{8} + 8634117828 T^{9} + 71889201048 T^{10} + 643394307388 T^{11} + 5443488430128 T^{12} + 42077967259300 T^{13} + 326525025932168 T^{14} + 2262351965387164 T^{15} + 7837433594376961 T^{16}$$)($$1 - 28 T + 392 T^{2} - 4900 T^{3} + 61488 T^{4} - 704956 T^{5} + 7640472 T^{6} - 89011524 T^{7} + 981394654 T^{8} - 8634117828 T^{9} + 71889201048 T^{10} - 643394307388 T^{11} + 5443488430128 T^{12} - 42077967259300 T^{13} + 326525025932168 T^{14} - 2262351965387164 T^{15} + 7837433594376961 T^{16}$$)($$1 + 4996 T^{4} + 20789766 T^{8} + 442292287876 T^{12} + 7837433594376961 T^{16}$$)($$1 + 4996 T^{4} + 20789766 T^{8} + 442292287876 T^{12} + 7837433594376961 T^{16}$$)($$( 1 - 144 T^{2} + 9409 T^{4} )^{2}( 1 + 144 T^{2} + 9409 T^{4} )^{2}$$)($$( 1 - 144 T^{2} + 9409 T^{4} )^{2}( 1 + 144 T^{2} + 9409 T^{4} )^{2}$$)