Properties

Label 4410.2.b
Level $4410$
Weight $2$
Character orbit 4410.b
Rep. character $\chi_{4410}(881,\cdot)$
Character field $\Q$
Dimension $48$
Newform subspaces $6$
Sturm bound $2016$
Trace bound $22$

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Defining parameters

Level: \( N \) \(=\) \( 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4410.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 21 \)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(2016\)
Trace bound: \(22\)
Distinguishing \(T_p\): \(11\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(4410, [\chi])\).

Total New Old
Modular forms 1072 48 1024
Cusp forms 944 48 896
Eisenstein series 128 0 128

Trace form

\( 48q - 48q^{4} + O(q^{10}) \) \( 48q - 48q^{4} + 48q^{16} + 48q^{25} - 96q^{37} - 96q^{43} + 32q^{46} - 48q^{64} + 32q^{67} + 96q^{79} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(4410, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
4410.2.b.a \(8\) \(35.214\) \(\Q(\zeta_{16})\) None \(0\) \(0\) \(-8\) \(0\) \(q-\zeta_{16}q^{2}-q^{4}-q^{5}+\zeta_{16}q^{8}+\zeta_{16}q^{10}+\cdots\)
4410.2.b.b \(8\) \(35.214\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(-8\) \(0\) \(q-\zeta_{24}q^{2}-q^{4}-q^{5}+\zeta_{24}q^{8}+\zeta_{24}q^{10}+\cdots\)
4410.2.b.c \(8\) \(35.214\) \(\Q(\zeta_{16})\) None \(0\) \(0\) \(-8\) \(0\) \(q+\zeta_{16}q^{2}-q^{4}-q^{5}-\zeta_{16}q^{8}-\zeta_{16}q^{10}+\cdots\)
4410.2.b.d \(8\) \(35.214\) \(\Q(\zeta_{16})\) None \(0\) \(0\) \(8\) \(0\) \(q-\zeta_{16}q^{2}-q^{4}+q^{5}+\zeta_{16}q^{8}-\zeta_{16}q^{10}+\cdots\)
4410.2.b.e \(8\) \(35.214\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(8\) \(0\) \(q-\zeta_{24}q^{2}-q^{4}+q^{5}+\zeta_{24}q^{8}-\zeta_{24}q^{10}+\cdots\)
4410.2.b.f \(8\) \(35.214\) \(\Q(\zeta_{16})\) None \(0\) \(0\) \(8\) \(0\) \(q+\zeta_{16}q^{2}-q^{4}+q^{5}-\zeta_{16}q^{8}+\zeta_{16}q^{10}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(4410, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(4410, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(147, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(294, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(315, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(441, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(630, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(735, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(882, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1470, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(2205, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + T^{2} )^{4} \))(\( ( 1 + T^{2} )^{4} \))(\( ( 1 + T^{2} )^{4} \))(\( ( 1 + T^{2} )^{4} \))(\( ( 1 + T^{2} )^{4} \))(\( ( 1 + T^{2} )^{4} \))
$3$ 1
$5$ (\( ( 1 + T )^{8} \))(\( ( 1 + T )^{8} \))(\( ( 1 + T )^{8} \))(\( ( 1 - T )^{8} \))(\( ( 1 - T )^{8} \))(\( ( 1 - T )^{8} \))
$7$ 1
$11$ (\( 1 - 64 T^{2} + 1952 T^{4} - 37312 T^{6} + 489570 T^{8} - 4514752 T^{10} + 28579232 T^{12} - 113379904 T^{14} + 214358881 T^{16} \))(\( 1 - 32 T^{2} + 670 T^{4} - 9920 T^{6} + 121411 T^{8} - 1200320 T^{10} + 9809470 T^{12} - 56689952 T^{14} + 214358881 T^{16} \))(\( 1 - 32 T^{2} + 544 T^{4} - 8352 T^{6} + 109154 T^{8} - 1010592 T^{10} + 7964704 T^{12} - 56689952 T^{14} + 214358881 T^{16} \))(\( 1 - 64 T^{2} + 1952 T^{4} - 37312 T^{6} + 489570 T^{8} - 4514752 T^{10} + 28579232 T^{12} - 113379904 T^{14} + 214358881 T^{16} \))(\( 1 - 32 T^{2} + 670 T^{4} - 9920 T^{6} + 121411 T^{8} - 1200320 T^{10} + 9809470 T^{12} - 56689952 T^{14} + 214358881 T^{16} \))(\( 1 - 32 T^{2} + 544 T^{4} - 8352 T^{6} + 109154 T^{8} - 1010592 T^{10} + 7964704 T^{12} - 56689952 T^{14} + 214358881 T^{16} \))
$13$ (\( 1 - 16 T^{2} + 308 T^{4} - 5424 T^{6} + 50118 T^{8} - 916656 T^{10} + 8796788 T^{12} - 77228944 T^{14} + 815730721 T^{16} \))(\( 1 - 80 T^{2} + 3006 T^{4} - 69568 T^{6} + 1087139 T^{8} - 11756992 T^{10} + 85854366 T^{12} - 386144720 T^{14} + 815730721 T^{16} \))(\( 1 - 80 T^{2} + 2996 T^{4} - 69104 T^{6} + 1078470 T^{8} - 11678576 T^{10} + 85568756 T^{12} - 386144720 T^{14} + 815730721 T^{16} \))(\( 1 - 16 T^{2} + 308 T^{4} - 5424 T^{6} + 50118 T^{8} - 916656 T^{10} + 8796788 T^{12} - 77228944 T^{14} + 815730721 T^{16} \))(\( 1 - 80 T^{2} + 3006 T^{4} - 69568 T^{6} + 1087139 T^{8} - 11756992 T^{10} + 85854366 T^{12} - 386144720 T^{14} + 815730721 T^{16} \))(\( 1 - 80 T^{2} + 2996 T^{4} - 69104 T^{6} + 1078470 T^{8} - 11678576 T^{10} + 85568756 T^{12} - 386144720 T^{14} + 815730721 T^{16} \))
$17$ (\( ( 1 + 28 T^{2} - 72 T^{3} + 468 T^{4} - 1224 T^{5} + 8092 T^{6} + 83521 T^{8} )^{2} \))(\( ( 1 + 28 T^{2} + 96 T^{3} + 342 T^{4} + 1632 T^{5} + 8092 T^{6} + 83521 T^{8} )^{2} \))(\( ( 1 + 60 T^{2} + 8 T^{3} + 1460 T^{4} + 136 T^{5} + 17340 T^{6} + 83521 T^{8} )^{2} \))(\( ( 1 + 28 T^{2} + 72 T^{3} + 468 T^{4} + 1224 T^{5} + 8092 T^{6} + 83521 T^{8} )^{2} \))(\( ( 1 + 28 T^{2} - 96 T^{3} + 342 T^{4} - 1632 T^{5} + 8092 T^{6} + 83521 T^{8} )^{2} \))(\( ( 1 + 60 T^{2} - 8 T^{3} + 1460 T^{4} - 136 T^{5} + 17340 T^{6} + 83521 T^{8} )^{2} \))
$19$ (\( 1 - 112 T^{2} + 5956 T^{4} - 197328 T^{6} + 4474022 T^{8} - 71235408 T^{10} + 776191876 T^{12} - 5269138672 T^{14} + 16983563041 T^{16} \))(\( 1 - 116 T^{2} + 6330 T^{4} - 213136 T^{6} + 4859555 T^{8} - 76942096 T^{10} + 824931930 T^{12} - 5457322196 T^{14} + 16983563041 T^{16} \))(\( 1 - 48 T^{2} + 1092 T^{4} - 20496 T^{6} + 392870 T^{8} - 7399056 T^{10} + 142310532 T^{12} - 2258202288 T^{14} + 16983563041 T^{16} \))(\( 1 - 112 T^{2} + 5956 T^{4} - 197328 T^{6} + 4474022 T^{8} - 71235408 T^{10} + 776191876 T^{12} - 5269138672 T^{14} + 16983563041 T^{16} \))(\( 1 - 116 T^{2} + 6330 T^{4} - 213136 T^{6} + 4859555 T^{8} - 76942096 T^{10} + 824931930 T^{12} - 5457322196 T^{14} + 16983563041 T^{16} \))(\( 1 - 48 T^{2} + 1092 T^{4} - 20496 T^{6} + 392870 T^{8} - 7399056 T^{10} + 142310532 T^{12} - 2258202288 T^{14} + 16983563041 T^{16} \))
$23$ (\( 1 + 48 T^{2} + 980 T^{4} - 4592 T^{6} - 398298 T^{8} - 2429168 T^{10} + 274244180 T^{12} + 7105722672 T^{14} + 78310985281 T^{16} \))(\( ( 1 - 78 T^{2} + 2531 T^{4} - 41262 T^{6} + 279841 T^{8} )^{2} \))(\( 1 - 144 T^{2} + 9428 T^{4} - 376368 T^{6} + 10282534 T^{8} - 199098672 T^{10} + 2638340948 T^{12} - 21317168016 T^{14} + 78310985281 T^{16} \))(\( 1 + 48 T^{2} + 980 T^{4} - 4592 T^{6} - 398298 T^{8} - 2429168 T^{10} + 274244180 T^{12} + 7105722672 T^{14} + 78310985281 T^{16} \))(\( ( 1 - 78 T^{2} + 2531 T^{4} - 41262 T^{6} + 279841 T^{8} )^{2} \))(\( 1 - 144 T^{2} + 9428 T^{4} - 376368 T^{6} + 10282534 T^{8} - 199098672 T^{10} + 2638340948 T^{12} - 21317168016 T^{14} + 78310985281 T^{16} \))
$29$ (\( 1 - 144 T^{2} + 8384 T^{4} - 272784 T^{6} + 7329058 T^{8} - 229411344 T^{10} + 5929843904 T^{12} - 85654558224 T^{14} + 500246412961 T^{16} \))(\( ( 1 - 36 T^{2} + 470 T^{4} - 30276 T^{6} + 707281 T^{8} )^{2} \))(\( 1 - 112 T^{2} + 7488 T^{4} - 339056 T^{6} + 11412770 T^{8} - 285146096 T^{10} + 5296120128 T^{12} - 66620211952 T^{14} + 500246412961 T^{16} \))(\( 1 - 144 T^{2} + 8384 T^{4} - 272784 T^{6} + 7329058 T^{8} - 229411344 T^{10} + 5929843904 T^{12} - 85654558224 T^{14} + 500246412961 T^{16} \))(\( ( 1 - 36 T^{2} + 470 T^{4} - 30276 T^{6} + 707281 T^{8} )^{2} \))(\( 1 - 112 T^{2} + 7488 T^{4} - 339056 T^{6} + 11412770 T^{8} - 285146096 T^{10} + 5296120128 T^{12} - 66620211952 T^{14} + 500246412961 T^{16} \))
$31$ (\( 1 - 112 T^{2} + 5984 T^{4} - 201840 T^{6} + 5984898 T^{8} - 193968240 T^{10} + 5526349664 T^{12} - 99400412272 T^{14} + 852891037441 T^{16} \))(\( ( 1 - 100 T^{2} + 4294 T^{4} - 96100 T^{6} + 923521 T^{8} )^{2} \))(\( 1 - 176 T^{2} + 14816 T^{4} - 791216 T^{6} + 29218434 T^{8} - 760358576 T^{10} + 13682887136 T^{12} - 156200647856 T^{14} + 852891037441 T^{16} \))(\( 1 - 112 T^{2} + 5984 T^{4} - 201840 T^{6} + 5984898 T^{8} - 193968240 T^{10} + 5526349664 T^{12} - 99400412272 T^{14} + 852891037441 T^{16} \))(\( ( 1 - 100 T^{2} + 4294 T^{4} - 96100 T^{6} + 923521 T^{8} )^{2} \))(\( 1 - 176 T^{2} + 14816 T^{4} - 791216 T^{6} + 29218434 T^{8} - 760358576 T^{10} + 13682887136 T^{12} - 156200647856 T^{14} + 852891037441 T^{16} \))
$37$ (\( ( 1 + 8 T + 132 T^{2} + 784 T^{3} + 7188 T^{4} + 29008 T^{5} + 180708 T^{6} + 405224 T^{7} + 1874161 T^{8} )^{2} \))(\( ( 1 + 24 T + 360 T^{2} + 3480 T^{3} + 25055 T^{4} + 128760 T^{5} + 492840 T^{6} + 1215672 T^{7} + 1874161 T^{8} )^{2} \))(\( ( 1 - 8 T + 52 T^{2} - 432 T^{3} + 3764 T^{4} - 15984 T^{5} + 71188 T^{6} - 405224 T^{7} + 1874161 T^{8} )^{2} \))(\( ( 1 + 8 T + 132 T^{2} + 784 T^{3} + 7188 T^{4} + 29008 T^{5} + 180708 T^{6} + 405224 T^{7} + 1874161 T^{8} )^{2} \))(\( ( 1 + 24 T + 360 T^{2} + 3480 T^{3} + 25055 T^{4} + 128760 T^{5} + 492840 T^{6} + 1215672 T^{7} + 1874161 T^{8} )^{2} \))(\( ( 1 - 8 T + 52 T^{2} - 432 T^{3} + 3764 T^{4} - 15984 T^{5} + 71188 T^{6} - 405224 T^{7} + 1874161 T^{8} )^{2} \))
$41$ (\( ( 1 + 24 T + 372 T^{2} + 3720 T^{3} + 28158 T^{4} + 152520 T^{5} + 625332 T^{6} + 1654104 T^{7} + 2825761 T^{8} )^{2} \))(\( ( 1 - 16 T + 232 T^{2} - 2000 T^{3} + 15591 T^{4} - 82000 T^{5} + 389992 T^{6} - 1102736 T^{7} + 2825761 T^{8} )^{2} \))(\( ( 1 - 8 T + 116 T^{2} - 664 T^{3} + 6526 T^{4} - 27224 T^{5} + 194996 T^{6} - 551368 T^{7} + 2825761 T^{8} )^{2} \))(\( ( 1 - 24 T + 372 T^{2} - 3720 T^{3} + 28158 T^{4} - 152520 T^{5} + 625332 T^{6} - 1654104 T^{7} + 2825761 T^{8} )^{2} \))(\( ( 1 + 16 T + 232 T^{2} + 2000 T^{3} + 15591 T^{4} + 82000 T^{5} + 389992 T^{6} + 1102736 T^{7} + 2825761 T^{8} )^{2} \))(\( ( 1 + 8 T + 116 T^{2} + 664 T^{3} + 6526 T^{4} + 27224 T^{5} + 194996 T^{6} + 551368 T^{7} + 2825761 T^{8} )^{2} \))
$43$ (\( ( 1 + 8 T + 76 T^{2} + 256 T^{3} + 1492 T^{4} + 11008 T^{5} + 140524 T^{6} + 636056 T^{7} + 3418801 T^{8} )^{2} \))(\( ( 1 + 8 T + 180 T^{2} + 1000 T^{3} + 11750 T^{4} + 43000 T^{5} + 332820 T^{6} + 636056 T^{7} + 3418801 T^{8} )^{2} \))(\( ( 1 + 8 T + 124 T^{2} + 720 T^{3} + 7508 T^{4} + 30960 T^{5} + 229276 T^{6} + 636056 T^{7} + 3418801 T^{8} )^{2} \))(\( ( 1 + 8 T + 76 T^{2} + 256 T^{3} + 1492 T^{4} + 11008 T^{5} + 140524 T^{6} + 636056 T^{7} + 3418801 T^{8} )^{2} \))(\( ( 1 + 8 T + 180 T^{2} + 1000 T^{3} + 11750 T^{4} + 43000 T^{5} + 332820 T^{6} + 636056 T^{7} + 3418801 T^{8} )^{2} \))(\( ( 1 + 8 T + 124 T^{2} + 720 T^{3} + 7508 T^{4} + 30960 T^{5} + 229276 T^{6} + 636056 T^{7} + 3418801 T^{8} )^{2} \))
$47$ (\( ( 1 - 24 T + 380 T^{2} - 3968 T^{3} + 31844 T^{4} - 186496 T^{5} + 839420 T^{6} - 2491752 T^{7} + 4879681 T^{8} )^{2} \))(\( ( 1 - 8 T + 142 T^{2} - 688 T^{3} + 8355 T^{4} - 32336 T^{5} + 313678 T^{6} - 830584 T^{7} + 4879681 T^{8} )^{2} \))(\( ( 1 + 8 T + 156 T^{2} + 880 T^{3} + 10404 T^{4} + 41360 T^{5} + 344604 T^{6} + 830584 T^{7} + 4879681 T^{8} )^{2} \))(\( ( 1 + 24 T + 380 T^{2} + 3968 T^{3} + 31844 T^{4} + 186496 T^{5} + 839420 T^{6} + 2491752 T^{7} + 4879681 T^{8} )^{2} \))(\( ( 1 + 8 T + 142 T^{2} + 688 T^{3} + 8355 T^{4} + 32336 T^{5} + 313678 T^{6} + 830584 T^{7} + 4879681 T^{8} )^{2} \))(\( ( 1 - 8 T + 156 T^{2} - 880 T^{3} + 10404 T^{4} - 41360 T^{5} + 344604 T^{6} - 830584 T^{7} + 4879681 T^{8} )^{2} \))
$53$ (\( 1 - 120 T^{2} + 14060 T^{4} - 973896 T^{6} + 61968070 T^{8} - 2735673864 T^{10} + 110940162860 T^{12} - 2659723335480 T^{14} + 62259690411361 T^{16} \))(\( 1 - 148 T^{2} + 10938 T^{4} - 528080 T^{6} + 25559075 T^{8} - 1483376720 T^{10} + 86306081178 T^{12} - 3280325447092 T^{14} + 62259690411361 T^{16} \))(\( 1 - 248 T^{2} + 34028 T^{4} - 3010504 T^{6} + 189188806 T^{8} - 8456505736 T^{10} + 268497287468 T^{12} - 5496761559992 T^{14} + 62259690411361 T^{16} \))(\( 1 - 120 T^{2} + 14060 T^{4} - 973896 T^{6} + 61968070 T^{8} - 2735673864 T^{10} + 110940162860 T^{12} - 2659723335480 T^{14} + 62259690411361 T^{16} \))(\( 1 - 148 T^{2} + 10938 T^{4} - 528080 T^{6} + 25559075 T^{8} - 1483376720 T^{10} + 86306081178 T^{12} - 3280325447092 T^{14} + 62259690411361 T^{16} \))(\( 1 - 248 T^{2} + 34028 T^{4} - 3010504 T^{6} + 189188806 T^{8} - 8456505736 T^{10} + 268497287468 T^{12} - 5496761559992 T^{14} + 62259690411361 T^{16} \))
$59$ (\( ( 1 + 8 T + 176 T^{2} + 888 T^{3} + 12914 T^{4} + 52392 T^{5} + 612656 T^{6} + 1643032 T^{7} + 12117361 T^{8} )^{2} \))(\( ( 1 - 24 T + 352 T^{2} - 3528 T^{3} + 29874 T^{4} - 208152 T^{5} + 1225312 T^{6} - 4929096 T^{7} + 12117361 T^{8} )^{2} \))(\( ( 1 + 24 T + 432 T^{2} + 4904 T^{3} + 44786 T^{4} + 289336 T^{5} + 1503792 T^{6} + 4929096 T^{7} + 12117361 T^{8} )^{2} \))(\( ( 1 - 8 T + 176 T^{2} - 888 T^{3} + 12914 T^{4} - 52392 T^{5} + 612656 T^{6} - 1643032 T^{7} + 12117361 T^{8} )^{2} \))(\( ( 1 + 24 T + 352 T^{2} + 3528 T^{3} + 29874 T^{4} + 208152 T^{5} + 1225312 T^{6} + 4929096 T^{7} + 12117361 T^{8} )^{2} \))(\( ( 1 - 24 T + 432 T^{2} - 4904 T^{3} + 44786 T^{4} - 289336 T^{5} + 1503792 T^{6} - 4929096 T^{7} + 12117361 T^{8} )^{2} \))
$61$ (\( 1 - 304 T^{2} + 47620 T^{4} - 4871952 T^{6} + 350453414 T^{8} - 18128533392 T^{10} + 659338948420 T^{12} - 15662193805744 T^{14} + 191707312997281 T^{16} \))(\( 1 - 248 T^{2} + 33756 T^{4} - 3125128 T^{6} + 217206374 T^{8} - 11628601288 T^{10} + 467380208796 T^{12} - 12777052841528 T^{14} + 191707312997281 T^{16} \))(\( 1 - 432 T^{2} + 84740 T^{4} - 9833616 T^{6} + 737501350 T^{8} - 36590885136 T^{10} + 1173296566340 T^{12} - 22256801723952 T^{14} + 191707312997281 T^{16} \))(\( 1 - 304 T^{2} + 47620 T^{4} - 4871952 T^{6} + 350453414 T^{8} - 18128533392 T^{10} + 659338948420 T^{12} - 15662193805744 T^{14} + 191707312997281 T^{16} \))(\( 1 - 248 T^{2} + 33756 T^{4} - 3125128 T^{6} + 217206374 T^{8} - 11628601288 T^{10} + 467380208796 T^{12} - 12777052841528 T^{14} + 191707312997281 T^{16} \))(\( 1 - 432 T^{2} + 84740 T^{4} - 9833616 T^{6} + 737501350 T^{8} - 36590885136 T^{10} + 1173296566340 T^{12} - 22256801723952 T^{14} + 191707312997281 T^{16} \))
$67$ (\( ( 1 + 164 T^{2} - 200 T^{3} + 14052 T^{4} - 13400 T^{5} + 736196 T^{6} + 20151121 T^{8} )^{2} \))(\( ( 1 - 24 T + 244 T^{2} - 1080 T^{3} + 2694 T^{4} - 72360 T^{5} + 1095316 T^{6} - 7218312 T^{7} + 20151121 T^{8} )^{2} \))(\( ( 1 + 16 T + 276 T^{2} + 2936 T^{3} + 27492 T^{4} + 196712 T^{5} + 1238964 T^{6} + 4812208 T^{7} + 20151121 T^{8} )^{2} \))(\( ( 1 + 164 T^{2} - 200 T^{3} + 14052 T^{4} - 13400 T^{5} + 736196 T^{6} + 20151121 T^{8} )^{2} \))(\( ( 1 - 24 T + 244 T^{2} - 1080 T^{3} + 2694 T^{4} - 72360 T^{5} + 1095316 T^{6} - 7218312 T^{7} + 20151121 T^{8} )^{2} \))(\( ( 1 + 16 T + 276 T^{2} + 2936 T^{3} + 27492 T^{4} + 196712 T^{5} + 1238964 T^{6} + 4812208 T^{7} + 20151121 T^{8} )^{2} \))
$71$ (\( 1 - 352 T^{2} + 54788 T^{4} - 5305888 T^{6} + 402214086 T^{8} - 26746981408 T^{10} + 1392255178628 T^{12} - 45091299940192 T^{14} + 645753531245761 T^{16} \))(\( 1 - 280 T^{2} + 41532 T^{4} - 4371368 T^{6} + 352979654 T^{8} - 22036066088 T^{10} + 1055397935292 T^{12} - 35868079497880 T^{14} + 645753531245761 T^{16} \))(\( 1 - 416 T^{2} + 80900 T^{4} - 9781984 T^{6} + 820517062 T^{8} - 49310981344 T^{10} + 2055804992900 T^{12} - 53289718111136 T^{14} + 645753531245761 T^{16} \))(\( 1 - 352 T^{2} + 54788 T^{4} - 5305888 T^{6} + 402214086 T^{8} - 26746981408 T^{10} + 1392255178628 T^{12} - 45091299940192 T^{14} + 645753531245761 T^{16} \))(\( 1 - 280 T^{2} + 41532 T^{4} - 4371368 T^{6} + 352979654 T^{8} - 22036066088 T^{10} + 1055397935292 T^{12} - 35868079497880 T^{14} + 645753531245761 T^{16} \))(\( 1 - 416 T^{2} + 80900 T^{4} - 9781984 T^{6} + 820517062 T^{8} - 49310981344 T^{10} + 2055804992900 T^{12} - 53289718111136 T^{14} + 645753531245761 T^{16} \))
$73$ (\( 1 - 184 T^{2} + 31436 T^{4} - 3096008 T^{6} + 278954470 T^{8} - 16498626632 T^{10} + 892727104076 T^{12} - 27845497637176 T^{14} + 806460091894081 T^{16} \))(\( ( 1 - 156 T^{2} + 12134 T^{4} - 831324 T^{6} + 28398241 T^{8} )^{2} \))(\( 1 - 248 T^{2} + 27212 T^{4} - 1705864 T^{6} + 97521382 T^{8} - 9090549256 T^{10} + 772772934092 T^{12} - 37530888119672 T^{14} + 806460091894081 T^{16} \))(\( 1 - 184 T^{2} + 31436 T^{4} - 3096008 T^{6} + 278954470 T^{8} - 16498626632 T^{10} + 892727104076 T^{12} - 27845497637176 T^{14} + 806460091894081 T^{16} \))(\( ( 1 - 156 T^{2} + 12134 T^{4} - 831324 T^{6} + 28398241 T^{8} )^{2} \))(\( 1 - 248 T^{2} + 27212 T^{4} - 1705864 T^{6} + 97521382 T^{8} - 9090549256 T^{10} + 772772934092 T^{12} - 37530888119672 T^{14} + 806460091894081 T^{16} \))
$79$ (\( ( 1 + 126 T^{2} + 6241 T^{4} )^{4} \))(\( ( 1 - 24 T + 324 T^{2} - 3096 T^{3} + 25622 T^{4} - 244584 T^{5} + 2022084 T^{6} - 11832936 T^{7} + 38950081 T^{8} )^{2} \))(\( ( 1 + 188 T^{2} + 19270 T^{4} + 1173308 T^{6} + 38950081 T^{8} )^{2} \))(\( ( 1 + 126 T^{2} + 6241 T^{4} )^{4} \))(\( ( 1 - 24 T + 324 T^{2} - 3096 T^{3} + 25622 T^{4} - 244584 T^{5} + 2022084 T^{6} - 11832936 T^{7} + 38950081 T^{8} )^{2} \))(\( ( 1 + 188 T^{2} + 19270 T^{4} + 1173308 T^{6} + 38950081 T^{8} )^{2} \))
$83$ (\( ( 1 + 8 T + 324 T^{2} + 1896 T^{3} + 40022 T^{4} + 157368 T^{5} + 2232036 T^{6} + 4574296 T^{7} + 47458321 T^{8} )^{2} \))(\( ( 1 + 8 T + 148 T^{2} + 1192 T^{3} + 18438 T^{4} + 98936 T^{5} + 1019572 T^{6} + 4574296 T^{7} + 47458321 T^{8} )^{2} \))(\( ( 1 - 8 T + 68 T^{2} - 360 T^{3} + 7766 T^{4} - 29880 T^{5} + 468452 T^{6} - 4574296 T^{7} + 47458321 T^{8} )^{2} \))(\( ( 1 - 8 T + 324 T^{2} - 1896 T^{3} + 40022 T^{4} - 157368 T^{5} + 2232036 T^{6} - 4574296 T^{7} + 47458321 T^{8} )^{2} \))(\( ( 1 - 8 T + 148 T^{2} - 1192 T^{3} + 18438 T^{4} - 98936 T^{5} + 1019572 T^{6} - 4574296 T^{7} + 47458321 T^{8} )^{2} \))(\( ( 1 + 8 T + 68 T^{2} + 360 T^{3} + 7766 T^{4} + 29880 T^{5} + 468452 T^{6} + 4574296 T^{7} + 47458321 T^{8} )^{2} \))
$89$ (\( ( 1 - 24 T + 428 T^{2} - 5032 T^{3} + 54470 T^{4} - 447848 T^{5} + 3390188 T^{6} - 16919256 T^{7} + 62742241 T^{8} )^{2} \))(\( ( 1 - 16 T + 304 T^{2} - 3344 T^{3} + 36834 T^{4} - 297616 T^{5} + 2407984 T^{6} - 11279504 T^{7} + 62742241 T^{8} )^{2} \))(\( ( 1 + 40 T + 876 T^{2} + 12824 T^{3} + 139590 T^{4} + 1141336 T^{5} + 6938796 T^{6} + 28198760 T^{7} + 62742241 T^{8} )^{2} \))(\( ( 1 + 24 T + 428 T^{2} + 5032 T^{3} + 54470 T^{4} + 447848 T^{5} + 3390188 T^{6} + 16919256 T^{7} + 62742241 T^{8} )^{2} \))(\( ( 1 + 16 T + 304 T^{2} + 3344 T^{3} + 36834 T^{4} + 297616 T^{5} + 2407984 T^{6} + 11279504 T^{7} + 62742241 T^{8} )^{2} \))(\( ( 1 - 40 T + 876 T^{2} - 12824 T^{3} + 139590 T^{4} - 1141336 T^{5} + 6938796 T^{6} - 28198760 T^{7} + 62742241 T^{8} )^{2} \))
$97$ (\( 1 - 504 T^{2} + 122652 T^{4} - 19352520 T^{6} + 2189485382 T^{8} - 182087860680 T^{10} + 10858293373212 T^{12} - 419817890484216 T^{14} + 7837433594376961 T^{16} \))(\( 1 + 24 T^{2} + 19036 T^{4} - 653400 T^{6} + 157960902 T^{8} - 6147840600 T^{10} + 1685243393116 T^{12} + 19991328118296 T^{14} + 7837433594376961 T^{16} \))(\( ( 1 - 316 T^{2} + 43270 T^{4} - 2973244 T^{6} + 88529281 T^{8} )^{2} \))(\( 1 - 504 T^{2} + 122652 T^{4} - 19352520 T^{6} + 2189485382 T^{8} - 182087860680 T^{10} + 10858293373212 T^{12} - 419817890484216 T^{14} + 7837433594376961 T^{16} \))(\( 1 + 24 T^{2} + 19036 T^{4} - 653400 T^{6} + 157960902 T^{8} - 6147840600 T^{10} + 1685243393116 T^{12} + 19991328118296 T^{14} + 7837433594376961 T^{16} \))(\( ( 1 - 316 T^{2} + 43270 T^{4} - 2973244 T^{6} + 88529281 T^{8} )^{2} \))
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