Properties

Label 1680.2.cz
Level 1680
Weight 2
Character orbit cz
Rep. character \(\chi_{1680}(97,\cdot)\)
Character field \(\Q(\zeta_{4})\)
Dimension 96
Newform subspaces 6
Sturm bound 768
Trace bound 7

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Defining parameters

Level: \( N \) \(=\) \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1680.cz (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 35 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 6 \)
Sturm bound: \(768\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1680, [\chi])\).

Total New Old
Modular forms 816 96 720
Cusp forms 720 96 624
Eisenstein series 96 0 96

Trace form

\( 96q + O(q^{10}) \) \( 96q - 48q^{23} + 24q^{35} + 48q^{67} - 64q^{71} - 96q^{81} - 16q^{91} - 16q^{93} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1680, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1680.2.cz.a \(8\) \(13.415\) 8.0.1698758656.6 None \(0\) \(0\) \(0\) \(-8\) \(q-\beta _{1}q^{3}-\beta _{2}q^{5}+(-1-\beta _{3}+\beta _{4}+\cdots)q^{7}+\cdots\)
1680.2.cz.b \(8\) \(13.415\) 8.0.1698758656.6 None \(0\) \(0\) \(0\) \(-4\) \(q+\beta _{1}q^{3}+\beta _{2}q^{5}+(-\beta _{1}+\beta _{3}-\beta _{5}+\cdots)q^{7}+\cdots\)
1680.2.cz.c \(16\) \(13.415\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{7}q^{3}+(1+\beta _{1}+\beta _{2}-\beta _{3}+\beta _{5}+\cdots)q^{5}+\cdots\)
1680.2.cz.d \(16\) \(13.415\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(8\) \(q+\beta _{10}q^{3}+(1+\beta _{1}-\beta _{2}+\beta _{3}+\beta _{4}+\cdots)q^{5}+\cdots\)
1680.2.cz.e \(24\) \(13.415\) None \(0\) \(0\) \(0\) \(0\)
1680.2.cz.f \(24\) \(13.415\) None \(0\) \(0\) \(0\) \(4\)

Decomposition of \(S_{2}^{\mathrm{old}}(1680, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1680, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(140, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(280, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(420, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(560, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(840, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{4} )^{4} \))(\( ( 1 + T^{4} )^{4} \))
$5$ (\( 1 + 2 T^{2} + 16 T^{3} + 2 T^{4} + 80 T^{5} + 50 T^{6} + 625 T^{8} \))(\( 1 + 2 T^{2} - 16 T^{3} + 2 T^{4} - 80 T^{5} + 50 T^{6} + 625 T^{8} \))(\( 1 - 8 T^{2} - 4 T^{4} + 200 T^{6} - 986 T^{8} + 5000 T^{10} - 2500 T^{12} - 125000 T^{14} + 390625 T^{16} \))(\( 1 + 28 T^{4} - 256 T^{6} - 26 T^{8} - 6400 T^{10} + 17500 T^{12} + 390625 T^{16} \))
$7$ (\( 1 + 8 T + 34 T^{2} + 112 T^{3} + 322 T^{4} + 784 T^{5} + 1666 T^{6} + 2744 T^{7} + 2401 T^{8} \))(\( 1 + 4 T + 6 T^{2} - 28 T^{3} - 126 T^{4} - 196 T^{5} + 294 T^{6} + 1372 T^{7} + 2401 T^{8} \))(\( 1 - 32 T^{3} + 36 T^{4} - 96 T^{5} + 512 T^{6} - 1216 T^{7} + 3334 T^{8} - 8512 T^{9} + 25088 T^{10} - 32928 T^{11} + 86436 T^{12} - 537824 T^{13} + 5764801 T^{16} \))(\( 1 - 8 T + 32 T^{2} - 88 T^{3} + 196 T^{4} - 248 T^{5} - 416 T^{6} + 2840 T^{7} - 8634 T^{8} + 19880 T^{9} - 20384 T^{10} - 85064 T^{11} + 470596 T^{12} - 1479016 T^{13} + 3764768 T^{14} - 6588344 T^{15} + 5764801 T^{16} \))
$11$ (\( ( 1 + 4 T + 12 T^{2} - 4 T^{3} + 18 T^{4} - 44 T^{5} + 1452 T^{6} + 5324 T^{7} + 14641 T^{8} )^{2} \))(\( ( 1 + 4 T + 12 T^{2} - 4 T^{3} + 18 T^{4} - 44 T^{5} + 1452 T^{6} + 5324 T^{7} + 14641 T^{8} )^{2} \))(\( ( 1 + 4 T + 36 T^{2} + 116 T^{3} + 546 T^{4} + 1276 T^{5} + 4356 T^{6} + 5324 T^{7} + 14641 T^{8} )^{4} \))(\( ( 1 - 4 T + 32 T^{2} - 68 T^{3} + 402 T^{4} - 748 T^{5} + 3872 T^{6} - 5324 T^{7} + 14641 T^{8} )^{4} \))
$13$ (\( ( 1 - 8 T + 32 T^{2} - 136 T^{3} + 562 T^{4} - 1768 T^{5} + 5408 T^{6} - 17576 T^{7} + 28561 T^{8} )^{2} \))(\( ( 1 + 8 T + 32 T^{2} + 136 T^{3} + 562 T^{4} + 1768 T^{5} + 5408 T^{6} + 17576 T^{7} + 28561 T^{8} )^{2} \))(\( 1 - 536 T^{4} + 97308 T^{8} - 1310248 T^{12} - 1567451002 T^{16} - 37421993128 T^{20} + 79377124999068 T^{24} - 12487773625649816 T^{28} + 665416609183179841 T^{32} \))(\( 1 + 424 T^{4} + 47004 T^{8} - 12160488 T^{12} - 4129271418 T^{16} - 347315697768 T^{20} + 38342606809884 T^{24} + 9878388091931944 T^{28} + 665416609183179841 T^{32} \))
$17$ (\( 1 + 12 T + 72 T^{2} + 388 T^{3} + 1696 T^{4} + 4732 T^{5} + 9944 T^{6} + 2420 T^{7} - 98754 T^{8} + 41140 T^{9} + 2873816 T^{10} + 23248316 T^{11} + 141651616 T^{12} + 550904516 T^{13} + 1737904968 T^{14} + 4924064076 T^{15} + 6975757441 T^{16} \))(\( 1 - 12 T + 72 T^{2} - 388 T^{3} + 1696 T^{4} - 4732 T^{5} + 9944 T^{6} - 2420 T^{7} - 98754 T^{8} - 41140 T^{9} + 2873816 T^{10} - 23248316 T^{11} + 141651616 T^{12} - 550904516 T^{13} + 1737904968 T^{14} - 4924064076 T^{15} + 6975757441 T^{16} \))(\( 1 - 904 T^{4} + 210332 T^{8} + 67692360 T^{12} - 44568692154 T^{16} + 5653733599560 T^{20} + 1467225014080412 T^{24} - 526690502455703944 T^{28} + 48661191875666868481 T^{32} \))(\( 1 + 120 T^{4} + 166556 T^{8} - 9625784 T^{12} + 12237871174 T^{16} - 803955105464 T^{20} + 1161854256343196 T^{24} + 69914668467571320 T^{28} + 48661191875666868481 T^{32} \))
$19$ (\( ( 1 + 4 T + 62 T^{2} + 212 T^{3} + 1666 T^{4} + 4028 T^{5} + 22382 T^{6} + 27436 T^{7} + 130321 T^{8} )^{2} \))(\( ( 1 - 4 T + 62 T^{2} - 212 T^{3} + 1666 T^{4} - 4028 T^{5} + 22382 T^{6} - 27436 T^{7} + 130321 T^{8} )^{2} \))(\( ( 1 + 120 T^{2} + 6692 T^{4} + 228264 T^{6} + 5227942 T^{8} + 82403304 T^{10} + 872108132 T^{12} + 5645505720 T^{14} + 16983563041 T^{16} )^{2} \))(\( ( 1 + 48 T^{2} + 1524 T^{4} + 40016 T^{6} + 818246 T^{8} + 14445776 T^{10} + 198609204 T^{12} + 2258202288 T^{14} + 16983563041 T^{16} )^{2} \))
$23$ (\( 1 + 16 T + 128 T^{2} + 656 T^{3} + 3300 T^{4} + 21520 T^{5} + 137088 T^{6} + 691344 T^{7} + 3238918 T^{8} + 15900912 T^{9} + 72519552 T^{10} + 261833840 T^{11} + 923475300 T^{12} + 4222241008 T^{13} + 18948593792 T^{14} + 54477207152 T^{15} + 78310985281 T^{16} \))(\( 1 + 16 T + 128 T^{2} + 656 T^{3} + 3300 T^{4} + 21520 T^{5} + 137088 T^{6} + 691344 T^{7} + 3238918 T^{8} + 15900912 T^{9} + 72519552 T^{10} + 261833840 T^{11} + 923475300 T^{12} + 4222241008 T^{13} + 18948593792 T^{14} + 54477207152 T^{15} + 78310985281 T^{16} \))(\( ( 1 + 4 T + 8 T^{2} - 4 T^{3} - 1548 T^{4} - 3956 T^{5} - 3432 T^{6} + 53556 T^{7} + 1130086 T^{8} + 1231788 T^{9} - 1815528 T^{10} - 48132652 T^{11} - 433193868 T^{12} - 25745372 T^{13} + 1184287112 T^{14} + 13619301788 T^{15} + 78310985281 T^{16} )^{2} \))(\( ( 1 - 20 T + 200 T^{2} - 1516 T^{3} + 10388 T^{4} - 65340 T^{5} + 378328 T^{6} - 2076676 T^{7} + 10539814 T^{8} - 47763548 T^{9} + 200135512 T^{10} - 794991780 T^{11} + 2906988308 T^{12} - 9757495988 T^{13} + 29607177800 T^{14} - 68096508940 T^{15} + 78310985281 T^{16} )^{2} \))
$29$ (\( 1 - 4 T^{2} + 200 T^{4} - 5548 T^{6} + 1325934 T^{8} - 4665868 T^{10} + 141456200 T^{12} - 2379293284 T^{14} + 500246412961 T^{16} \))(\( 1 - 4 T^{2} + 200 T^{4} - 5548 T^{6} + 1325934 T^{8} - 4665868 T^{10} + 141456200 T^{12} - 2379293284 T^{14} + 500246412961 T^{16} \))(\( ( 1 - 136 T^{2} + 8924 T^{4} - 381688 T^{6} + 12377958 T^{8} - 320999608 T^{10} + 6311775644 T^{12} - 80895971656 T^{14} + 500246412961 T^{16} )^{2} \))(\( ( 1 - 184 T^{2} + 15868 T^{4} - 835400 T^{6} + 29324070 T^{8} - 702571400 T^{10} + 11223134908 T^{12} - 109447491064 T^{14} + 500246412961 T^{16} )^{2} \))
$31$ (\( 1 - 148 T^{2} + 11192 T^{4} - 565212 T^{6} + 20507310 T^{8} - 543168732 T^{10} + 10336047032 T^{12} - 131350544788 T^{14} + 852891037441 T^{16} \))(\( 1 - 148 T^{2} + 11192 T^{4} - 565212 T^{6} + 20507310 T^{8} - 543168732 T^{10} + 10336047032 T^{12} - 131350544788 T^{14} + 852891037441 T^{16} \))(\( ( 1 - 136 T^{2} + 8612 T^{4} - 354840 T^{6} + 11712390 T^{8} - 341001240 T^{10} + 7953362852 T^{12} - 120700500616 T^{14} + 852891037441 T^{16} )^{2} \))(\( ( 1 - 128 T^{2} + 9396 T^{4} - 463040 T^{6} + 16703398 T^{8} - 444981440 T^{10} + 8677403316 T^{12} - 113600471168 T^{14} + 852891037441 T^{16} )^{2} \))
$37$ (\( 1 + 28 T + 392 T^{2} + 4180 T^{3} + 38912 T^{4} + 311708 T^{5} + 2210520 T^{6} + 14742164 T^{7} + 92944158 T^{8} + 545460068 T^{9} + 3026201880 T^{10} + 15788945324 T^{11} + 72927352832 T^{12} + 289857740260 T^{13} + 1005764752328 T^{14} + 2658092559724 T^{15} + 3512479453921 T^{16} \))(\( 1 + 28 T + 392 T^{2} + 4180 T^{3} + 38912 T^{4} + 311708 T^{5} + 2210520 T^{6} + 14742164 T^{7} + 92944158 T^{8} + 545460068 T^{9} + 3026201880 T^{10} + 15788945324 T^{11} + 72927352832 T^{12} + 289857740260 T^{13} + 1005764752328 T^{14} + 2658092559724 T^{15} + 3512479453921 T^{16} \))(\( ( 1 - 8 T + 32 T^{2} - 56 T^{3} - 2980 T^{4} + 15992 T^{5} - 31008 T^{6} - 263416 T^{7} + 5332518 T^{8} - 9746392 T^{9} - 42449952 T^{10} + 810042776 T^{11} - 5584999780 T^{12} - 3883261592 T^{13} + 82103245088 T^{14} - 759455017064 T^{15} + 3512479453921 T^{16} )^{2} \))(\( ( 1 - 16 T + 128 T^{2} - 944 T^{3} + 8860 T^{4} - 73552 T^{5} + 488320 T^{6} - 3175280 T^{7} + 20212134 T^{8} - 117485360 T^{9} + 668510080 T^{10} - 3725629456 T^{11} + 16605066460 T^{12} - 65460695408 T^{13} + 328412980352 T^{14} - 1518910034128 T^{15} + 3512479453921 T^{16} )^{2} \))
$41$ (\( 1 - 176 T^{2} + 13468 T^{4} - 643920 T^{6} + 26131718 T^{8} - 1082429520 T^{10} + 38057349148 T^{12} - 836018346416 T^{14} + 7984925229121 T^{16} \))(\( 1 - 176 T^{2} + 13468 T^{4} - 643920 T^{6} + 26131718 T^{8} - 1082429520 T^{10} + 38057349148 T^{12} - 836018346416 T^{14} + 7984925229121 T^{16} \))(\( ( 1 - 200 T^{2} + 20668 T^{4} - 1400376 T^{6} + 67532294 T^{8} - 2354032056 T^{10} + 58402828348 T^{12} - 950020848200 T^{14} + 7984925229121 T^{16} )^{2} \))(\( ( 1 - 144 T^{2} + 12956 T^{4} - 823792 T^{6} + 38320198 T^{8} - 1384794352 T^{10} + 36610559516 T^{12} - 684015010704 T^{14} + 7984925229121 T^{16} )^{2} \))
$43$ (\( 1 + 32 T^{3} - 6588 T^{4} + 1632 T^{5} + 512 T^{6} - 117440 T^{7} + 17663206 T^{8} - 5049920 T^{9} + 946688 T^{10} + 129755424 T^{11} - 22523060988 T^{12} + 4704270176 T^{13} + 11688200277601 T^{16} \))(\( 1 + 32 T^{3} - 6588 T^{4} + 1632 T^{5} + 512 T^{6} - 117440 T^{7} + 17663206 T^{8} - 5049920 T^{9} + 946688 T^{10} + 129755424 T^{11} - 22523060988 T^{12} + 4704270176 T^{13} + 11688200277601 T^{16} \))(\( ( 1 + 24 T + 288 T^{2} + 2504 T^{3} + 19620 T^{4} + 145992 T^{5} + 988256 T^{6} + 5831128 T^{7} + 34926502 T^{8} + 250738504 T^{9} + 1827285344 T^{10} + 11607385944 T^{11} + 67076875620 T^{12} + 368109141272 T^{13} + 1820552558112 T^{14} + 6523646666568 T^{15} + 11688200277601 T^{16} )^{2} \))(\( ( 1 - 8 T + 32 T^{2} - 280 T^{3} + 2788 T^{4} - 14232 T^{5} + 63840 T^{6} - 569416 T^{7} + 5017638 T^{8} - 24484888 T^{9} + 118040160 T^{10} - 1131543624 T^{11} + 9531617188 T^{12} - 41162364040 T^{13} + 202283617568 T^{14} - 2174548888856 T^{15} + 11688200277601 T^{16} )^{2} \))
$47$ (\( 1 - 24 T + 288 T^{2} - 2728 T^{3} + 26308 T^{4} - 243208 T^{5} + 1981280 T^{6} - 14819768 T^{7} + 104930310 T^{8} - 696529096 T^{9} + 4376647520 T^{10} - 25250584184 T^{11} + 128374647748 T^{12} - 625653179096 T^{13} + 3104414014752 T^{14} - 12158954891112 T^{15} + 23811286661761 T^{16} \))(\( 1 + 24 T + 288 T^{2} + 2728 T^{3} + 26308 T^{4} + 243208 T^{5} + 1981280 T^{6} + 14819768 T^{7} + 104930310 T^{8} + 696529096 T^{9} + 4376647520 T^{10} + 25250584184 T^{11} + 128374647748 T^{12} + 625653179096 T^{13} + 3104414014752 T^{14} + 12158954891112 T^{15} + 23811286661761 T^{16} \))(\( 1 + 2312 T^{4} - 7152868 T^{8} - 8327985864 T^{12} + 35781048692550 T^{16} - 40637914388829384 T^{20} - \)\(17\!\cdots\!48\)\( T^{24} + \)\(26\!\cdots\!92\)\( T^{28} + \)\(56\!\cdots\!21\)\( T^{32} \))(\( 1 + 3784 T^{4} + 1124764 T^{8} + 9138019192 T^{12} + 63538455194182 T^{16} + 44590618628837752 T^{20} + 26782078030828949404 T^{24} + \)\(43\!\cdots\!44\)\( T^{28} + \)\(56\!\cdots\!21\)\( T^{32} \))
$53$ (\( 1 + 8 T + 32 T^{2} + 264 T^{3} + 5828 T^{4} + 46136 T^{5} + 217440 T^{6} + 2204664 T^{7} + 21970150 T^{8} + 116847192 T^{9} + 610788960 T^{10} + 6868589272 T^{11} + 45985723268 T^{12} + 110403610152 T^{13} + 709259556128 T^{14} + 9397689118696 T^{15} + 62259690411361 T^{16} \))(\( 1 + 8 T + 32 T^{2} + 264 T^{3} + 5828 T^{4} + 46136 T^{5} + 217440 T^{6} + 2204664 T^{7} + 21970150 T^{8} + 116847192 T^{9} + 610788960 T^{10} + 6868589272 T^{11} + 45985723268 T^{12} + 110403610152 T^{13} + 709259556128 T^{14} + 9397689118696 T^{15} + 62259690411361 T^{16} \))(\( ( 1 + 20 T + 200 T^{2} + 1284 T^{3} + 2996 T^{4} - 21028 T^{5} - 195432 T^{6} - 604308 T^{7} - 304634 T^{8} - 32028324 T^{9} - 548968488 T^{10} - 3130585556 T^{11} + 23639881076 T^{12} + 536963013012 T^{13} + 4432872225800 T^{14} + 23494222796740 T^{15} + 62259690411361 T^{16} )^{2} \))(\( ( 1 - 12 T + 72 T^{2} - 572 T^{3} + 1780 T^{4} + 1436 T^{5} + 18200 T^{6} - 441076 T^{7} + 6254598 T^{8} - 23377028 T^{9} + 51123800 T^{10} + 213787372 T^{11} + 14045056180 T^{12} - 239207821996 T^{13} + 1595834001288 T^{14} - 14096533678044 T^{15} + 62259690411361 T^{16} )^{2} \))
$59$ (\( ( 1 - 4 T + 70 T^{2} - 116 T^{3} + 802 T^{4} - 6844 T^{5} + 243670 T^{6} - 821516 T^{7} + 12117361 T^{8} )^{2} \))(\( ( 1 + 4 T + 70 T^{2} + 116 T^{3} + 802 T^{4} + 6844 T^{5} + 243670 T^{6} + 821516 T^{7} + 12117361 T^{8} )^{2} \))(\( ( 1 + 280 T^{2} + 33212 T^{4} + 2343784 T^{6} + 136126182 T^{8} + 8158712104 T^{10} + 402441793532 T^{12} + 11810549419480 T^{14} + 146830437604321 T^{16} )^{2} \))(\( ( 1 + 312 T^{2} + 49660 T^{4} + 5040712 T^{6} + 354176614 T^{8} + 17546718472 T^{10} + 601748147260 T^{12} + 13160326495992 T^{14} + 146830437604321 T^{16} )^{2} \))
$61$ (\( 1 - 264 T^{2} + 37532 T^{4} - 3615096 T^{6} + 255247078 T^{8} - 13451772216 T^{10} + 519662104412 T^{12} - 13601378831304 T^{14} + 191707312997281 T^{16} \))(\( 1 - 264 T^{2} + 37532 T^{4} - 3615096 T^{6} + 255247078 T^{8} - 13451772216 T^{10} + 519662104412 T^{12} - 13601378831304 T^{14} + 191707312997281 T^{16} \))(\( ( 1 - 312 T^{2} + 47324 T^{4} - 4644168 T^{6} + 328520806 T^{8} - 17280949128 T^{10} + 655240579484 T^{12} - 16074356800632 T^{14} + 191707312997281 T^{16} )^{2} \))(\( ( 1 - 200 T^{2} + 17532 T^{4} - 857912 T^{6} + 37932838 T^{8} - 3192290552 T^{10} + 242745284412 T^{12} - 10304074872200 T^{14} + 191707312997281 T^{16} )^{2} \))
$67$ (\( 1 + 64 T^{3} + 9348 T^{4} - 10560 T^{5} + 2048 T^{6} - 428416 T^{7} + 51599014 T^{8} - 28703872 T^{9} + 9193472 T^{10} - 3176057280 T^{11} + 188372679108 T^{12} + 86408006848 T^{13} + 406067677556641 T^{16} \))(\( 1 + 64 T^{3} + 9348 T^{4} - 10560 T^{5} + 2048 T^{6} - 428416 T^{7} + 51599014 T^{8} - 28703872 T^{9} + 9193472 T^{10} - 3176057280 T^{11} + 188372679108 T^{12} + 86408006848 T^{13} + 406067677556641 T^{16} \))(\( ( 1 - 24 T + 288 T^{2} - 2984 T^{3} + 22788 T^{4} - 56616 T^{5} - 752032 T^{6} + 14609960 T^{7} - 156156506 T^{8} + 978867320 T^{9} - 3375871648 T^{10} - 17027998008 T^{11} + 459203745348 T^{12} - 4028773319288 T^{13} + 26052014064672 T^{14} - 145457078527752 T^{15} + 406067677556641 T^{16} )^{2} \))(\( ( 1 - 16 T + 128 T^{2} - 1424 T^{3} + 22436 T^{4} - 211280 T^{5} + 1522560 T^{6} - 15870032 T^{7} + 163564774 T^{8} - 1063292144 T^{9} + 6834771840 T^{10} - 63545206640 T^{11} + 452110550756 T^{12} - 1922578152368 T^{13} + 11578672917632 T^{14} - 96971385685168 T^{15} + 406067677556641 T^{16} )^{2} \))
$71$ (\( ( 1 + 4 T + 158 T^{2} - 220 T^{3} + 10018 T^{4} - 15620 T^{5} + 796478 T^{6} + 1431644 T^{7} + 25411681 T^{8} )^{2} \))(\( ( 1 + 4 T + 158 T^{2} - 220 T^{3} + 10018 T^{4} - 15620 T^{5} + 796478 T^{6} + 1431644 T^{7} + 25411681 T^{8} )^{2} \))(\( ( 1 - 8 T + 140 T^{2} - 1024 T^{3} + 14962 T^{4} - 72704 T^{5} + 705740 T^{6} - 2863288 T^{7} + 25411681 T^{8} )^{4} \))(\( ( 1 + 16 T + 232 T^{2} + 2376 T^{3} + 21730 T^{4} + 168696 T^{5} + 1169512 T^{6} + 5726576 T^{7} + 25411681 T^{8} )^{4} \))
$73$ (\( 1 - 28 T + 392 T^{2} - 4756 T^{3} + 46736 T^{4} - 302508 T^{5} + 1459480 T^{6} - 3207460 T^{7} - 21483554 T^{8} - 234144580 T^{9} + 7777568920 T^{10} - 117680754636 T^{11} + 1327220191376 T^{12} - 9859528496308 T^{13} + 59323016705288 T^{14} - 309327158534716 T^{15} + 806460091894081 T^{16} \))(\( 1 + 28 T + 392 T^{2} + 4756 T^{3} + 46736 T^{4} + 302508 T^{5} + 1459480 T^{6} + 3207460 T^{7} - 21483554 T^{8} + 234144580 T^{9} + 7777568920 T^{10} + 117680754636 T^{11} + 1327220191376 T^{12} + 9859528496308 T^{13} + 59323016705288 T^{14} + 309327158534716 T^{15} + 806460091894081 T^{16} \))(\( 1 - 9816 T^{4} + 48230108 T^{8} - 385344995944 T^{12} + 2907384584774982 T^{16} - 10943120062961734504 T^{20} + \)\(38\!\cdots\!48\)\( T^{24} - \)\(22\!\cdots\!36\)\( T^{28} + \)\(65\!\cdots\!61\)\( T^{32} \))(\( 1 - 15256 T^{4} + 80862300 T^{8} + 94053698264 T^{12} - 2362018367550906 T^{16} + 2670959590242353624 T^{20} + \)\(65\!\cdots\!00\)\( T^{24} - \)\(34\!\cdots\!76\)\( T^{28} + \)\(65\!\cdots\!61\)\( T^{32} \))
$79$ (\( 1 - 272 T^{2} + 39644 T^{4} - 4069360 T^{6} + 348070342 T^{8} - 25396875760 T^{10} + 1544137011164 T^{12} - 66119787901712 T^{14} + 1517108809906561 T^{16} \))(\( 1 - 272 T^{2} + 39644 T^{4} - 4069360 T^{6} + 348070342 T^{8} - 25396875760 T^{10} + 1544137011164 T^{12} - 66119787901712 T^{14} + 1517108809906561 T^{16} \))(\( ( 1 - 248 T^{2} + 39836 T^{4} - 4744648 T^{6} + 419330758 T^{8} - 29611348168 T^{10} + 1551615426716 T^{12} - 60285688969208 T^{14} + 1517108809906561 T^{16} )^{2} \))(\( ( 1 - 312 T^{2} + 58396 T^{4} - 7293320 T^{6} + 672141766 T^{8} - 45517610120 T^{10} + 2274528930076 T^{12} - 75843286122552 T^{14} + 1517108809906561 T^{16} )^{2} \))
$83$ (\( 1 + 16 T + 128 T^{2} + 976 T^{3} + 16964 T^{4} + 235664 T^{5} + 2075520 T^{6} + 16972368 T^{7} + 135669670 T^{8} + 1408706544 T^{9} + 14298257280 T^{10} + 134749611568 T^{11} + 805082957444 T^{12} + 3844503667568 T^{13} + 41848367791232 T^{14} + 434176815834032 T^{15} + 2252292232139041 T^{16} \))(\( 1 - 16 T + 128 T^{2} - 976 T^{3} + 16964 T^{4} - 235664 T^{5} + 2075520 T^{6} - 16972368 T^{7} + 135669670 T^{8} - 1408706544 T^{9} + 14298257280 T^{10} - 134749611568 T^{11} + 805082957444 T^{12} - 3844503667568 T^{13} + 41848367791232 T^{14} - 434176815834032 T^{15} + 2252292232139041 T^{16} \))(\( 1 + 11592 T^{4} + 91463132 T^{8} + 885997919352 T^{12} + 6666213559678726 T^{16} + 42047973661939327992 T^{20} + \)\(20\!\cdots\!12\)\( T^{24} + \)\(12\!\cdots\!12\)\( T^{28} + \)\(50\!\cdots\!81\)\( T^{32} \))(\( 1 + 5000 T^{4} + 95818588 T^{8} + 596752860728 T^{12} + 4571727903671302 T^{16} + 28320888822097717688 T^{20} + \)\(21\!\cdots\!08\)\( T^{24} + \)\(53\!\cdots\!00\)\( T^{28} + \)\(50\!\cdots\!81\)\( T^{32} \))
$89$ (\( ( 1 - 32 T + 628 T^{2} - 8320 T^{3} + 88630 T^{4} - 740480 T^{5} + 4974388 T^{6} - 22559008 T^{7} + 62742241 T^{8} )^{2} \))(\( ( 1 + 32 T + 628 T^{2} + 8320 T^{3} + 88630 T^{4} + 740480 T^{5} + 4974388 T^{6} + 22559008 T^{7} + 62742241 T^{8} )^{2} \))(\( ( 1 + 328 T^{2} + 42716 T^{4} + 2964280 T^{6} + 188103750 T^{8} + 23480061880 T^{10} + 2680097566556 T^{12} + 163009863435208 T^{14} + 3936588805702081 T^{16} )^{2} \))(\( ( 1 + 576 T^{2} + 155068 T^{4} + 25338304 T^{6} + 2740378246 T^{8} + 200704705984 T^{10} + 9729313827388 T^{12} + 286261223593536 T^{14} + 3936588805702081 T^{16} )^{2} \))
$97$ (\( 1 - 28 T + 392 T^{2} - 4900 T^{3} + 61488 T^{4} - 704956 T^{5} + 7640472 T^{6} - 89011524 T^{7} + 981394654 T^{8} - 8634117828 T^{9} + 71889201048 T^{10} - 643394307388 T^{11} + 5443488430128 T^{12} - 42077967259300 T^{13} + 326525025932168 T^{14} - 2262351965387164 T^{15} + 7837433594376961 T^{16} \))(\( 1 + 28 T + 392 T^{2} + 4900 T^{3} + 61488 T^{4} + 704956 T^{5} + 7640472 T^{6} + 89011524 T^{7} + 981394654 T^{8} + 8634117828 T^{9} + 71889201048 T^{10} + 643394307388 T^{11} + 5443488430128 T^{12} + 42077967259300 T^{13} + 326525025932168 T^{14} + 2262351965387164 T^{15} + 7837433594376961 T^{16} \))(\( 1 - 1624 T^{4} + 150045276 T^{8} - 609733450088 T^{12} + 17981575892543942 T^{16} - 53979263937940026728 T^{20} + \)\(11\!\cdots\!36\)\( T^{24} - \)\(11\!\cdots\!84\)\( T^{28} + \)\(61\!\cdots\!21\)\( T^{32} \))(\( 1 - 55064 T^{4} + 1465436892 T^{8} - 24336256217256 T^{12} + 274732504520067270 T^{16} - \)\(21\!\cdots\!36\)\( T^{20} + \)\(11\!\cdots\!12\)\( T^{24} - \)\(38\!\cdots\!24\)\( T^{28} + \)\(61\!\cdots\!21\)\( T^{32} \))
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