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

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