Properties

Label 6048.2.h
Level 6048
Weight 2
Character orbit h
Rep. character \(\chi_{6048}(2591,\cdot)\)
Character field \(\Q\)
Dimension 96
Newform subspaces 9
Sturm bound 2304
Trace bound 13

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Defining parameters

Level: \( N \) = \( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6048.h (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 12 \)
Character field: \(\Q\)
Newform subspaces: \( 9 \)
Sturm bound: \(2304\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(6048, [\chi])\).

Total New Old
Modular forms 1200 96 1104
Cusp forms 1104 96 1008
Eisenstein series 96 0 96

Trace form

\( 96q + O(q^{10}) \) \( 96q - 96q^{25} + 64q^{37} - 96q^{49} - 64q^{61} - 80q^{85} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(6048, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
6048.2.h.a \(8\) \(48.294\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q+(-\zeta_{24}+\zeta_{24}^{2}+\zeta_{24}^{7})q^{5}+\zeta_{24}q^{7}+\cdots\)
6048.2.h.b \(8\) \(48.294\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q+(2\zeta_{24}^{3}+3\zeta_{24}^{5})q^{5}-\zeta_{24}q^{7}+(3\zeta_{24}^{6}+\cdots)q^{11}+\cdots\)
6048.2.h.c \(8\) \(48.294\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q+(\zeta_{24}^{2}+\zeta_{24}^{3})q^{5}-\zeta_{24}q^{7}+\zeta_{24}^{6}q^{11}+\cdots\)
6048.2.h.d \(8\) \(48.294\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q+(-\zeta_{24}^{2}-\zeta_{24}^{3})q^{5}-\zeta_{24}q^{7}+(2\zeta_{24}^{4}+\cdots)q^{11}+\cdots\)
6048.2.h.e \(8\) \(48.294\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q+2\zeta_{24}^{3}q^{5}+\zeta_{24}q^{7}+(2\zeta_{24}^{4}+2\zeta_{24}^{6}+\cdots)q^{11}+\cdots\)
6048.2.h.f \(8\) \(48.294\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q+(-2\zeta_{24}^{4}-\zeta_{24}^{7})q^{5}+\zeta_{24}q^{7}+\cdots\)
6048.2.h.g \(8\) \(48.294\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q+(-\zeta_{24}+\zeta_{24}^{2}+\zeta_{24}^{3}+\zeta_{24}^{5}+\cdots)q^{5}+\cdots\)
6048.2.h.h \(16\) \(48.294\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{12}q^{5}+\beta _{3}q^{7}+(-\beta _{11}+\beta _{13}+\cdots)q^{11}+\cdots\)
6048.2.h.i \(24\) \(48.294\) None \(0\) \(0\) \(0\) \(0\)

Decomposition of \(S_{2}^{\mathrm{old}}(6048, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(6048, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(144, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(288, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(336, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(432, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(672, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(756, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(864, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1008, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(2016, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(3024, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))
$3$ (\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))
$5$ (\( 1 - 16 T^{2} + 174 T^{4} - 1280 T^{6} + 7379 T^{8} - 32000 T^{10} + 108750 T^{12} - 250000 T^{14} + 390625 T^{16} \))(\( ( 1 + 8 T^{2} + 39 T^{4} + 200 T^{6} + 625 T^{8} )^{2} \))(\( ( 1 - 10 T^{2} + 51 T^{4} - 250 T^{6} + 625 T^{8} )^{2} \))(\( ( 1 - 10 T^{2} + 51 T^{4} - 250 T^{6} + 625 T^{8} )^{2} \))(\( ( 1 - 2 T^{2} + 25 T^{4} )^{4} \))(\( ( 1 - 8 T^{2} + 39 T^{4} - 200 T^{6} + 625 T^{8} )^{2} \))(\( 1 - 16 T^{2} + 174 T^{4} - 1280 T^{6} + 7379 T^{8} - 32000 T^{10} + 108750 T^{12} - 250000 T^{14} + 390625 T^{16} \))(\( ( 1 - 16 T^{2} + 111 T^{4} - 400 T^{6} + 625 T^{8} )^{4} \))
$7$ (\( ( 1 + T^{2} )^{4} \))(\( ( 1 + T^{2} )^{4} \))(\( ( 1 + T^{2} )^{4} \))(\( ( 1 + T^{2} )^{4} \))(\( ( 1 + T^{2} )^{4} \))(\( ( 1 + T^{2} )^{4} \))(\( ( 1 + T^{2} )^{4} \))(\( ( 1 + T^{2} )^{8} \))
$11$ (\( ( 1 + 4 T + 32 T^{2} + 136 T^{3} + 463 T^{4} + 1496 T^{5} + 3872 T^{6} + 5324 T^{7} + 14641 T^{8} )^{2} \))(\( ( 1 + 16 T^{2} + 231 T^{4} + 1936 T^{6} + 14641 T^{8} )^{2} \))(\( ( 1 + 20 T^{2} + 121 T^{4} )^{4} \))(\( ( 1 + 16 T^{2} + 210 T^{4} + 1936 T^{6} + 14641 T^{8} )^{2} \))(\( ( 1 + 4 T^{2} - 138 T^{4} + 484 T^{6} + 14641 T^{8} )^{2} \))(\( ( 1 + 32 T^{2} + 471 T^{4} + 3872 T^{6} + 14641 T^{8} )^{2} \))(\( ( 1 - 4 T + 32 T^{2} - 136 T^{3} + 463 T^{4} - 1496 T^{5} + 3872 T^{6} - 5324 T^{7} + 14641 T^{8} )^{2} \))(\( ( 1 + 32 T^{2} + 622 T^{4} + 8576 T^{6} + 99379 T^{8} + 1037696 T^{10} + 9106702 T^{12} + 56689952 T^{14} + 214358881 T^{16} )^{2} \))
$13$ (\( ( 1 + 4 T + 36 T^{2} + 116 T^{3} + 602 T^{4} + 1508 T^{5} + 6084 T^{6} + 8788 T^{7} + 28561 T^{8} )^{2} \))(\( ( 1 + 6 T + 32 T^{2} + 78 T^{3} + 169 T^{4} )^{4} \))(\( ( 1 + 4 T + 24 T^{2} + 52 T^{3} + 169 T^{4} )^{4} \))(\( ( 1 + 20 T^{2} + 169 T^{4} )^{4} \))(\( ( 1 - 6 T + 29 T^{2} - 78 T^{3} + 169 T^{4} )^{4} \))(\( ( 1 - 4 T + 13 T^{2} )^{8} \))(\( ( 1 + 4 T + 36 T^{2} + 116 T^{3} + 602 T^{4} + 1508 T^{5} + 6084 T^{6} + 8788 T^{7} + 28561 T^{8} )^{2} \))(\( ( 1 - 2 T + 26 T^{2} - 78 T^{3} + 434 T^{4} - 1014 T^{5} + 4394 T^{6} - 4394 T^{7} + 28561 T^{8} )^{4} \))
$17$ (\( ( 1 - 56 T^{2} + 1330 T^{4} - 16184 T^{6} + 83521 T^{8} )^{2} \))(\( ( 1 - 40 T^{2} + 786 T^{4} - 11560 T^{6} + 83521 T^{8} )^{2} \))(\( ( 1 - 26 T^{2} + 531 T^{4} - 7514 T^{6} + 83521 T^{8} )^{2} \))(\( ( 1 - 58 T^{2} + 1395 T^{4} - 16762 T^{6} + 83521 T^{8} )^{2} \))(\( ( 1 - 58 T^{2} + 1395 T^{4} - 16762 T^{6} + 83521 T^{8} )^{2} \))(\( ( 1 - 16 T^{2} + 450 T^{4} - 4624 T^{6} + 83521 T^{8} )^{2} \))(\( ( 1 - 56 T^{2} + 1330 T^{4} - 16184 T^{6} + 83521 T^{8} )^{2} \))(\( ( 1 - 24 T^{2} + 788 T^{4} - 17736 T^{6} + 292518 T^{8} - 5125704 T^{10} + 65814548 T^{12} - 579301656 T^{14} + 6975757441 T^{16} )^{2} \))
$19$ (\( ( 1 - 26 T^{2} + 795 T^{4} - 9386 T^{6} + 130321 T^{8} )^{2} \))(\( ( 1 - 13 T^{2} + 361 T^{4} )^{4} \))(\( ( 1 - 32 T^{2} + 361 T^{4} )^{4} \))(\( ( 1 - 56 T^{2} + 1410 T^{4} - 20216 T^{6} + 130321 T^{8} )^{2} \))(\( ( 1 - 22 T^{2} + 361 T^{4} )^{4} \))(\( ( 1 - 37 T^{2} + 361 T^{4} )^{4} \))(\( ( 1 - 26 T^{2} + 795 T^{4} - 9386 T^{6} + 130321 T^{8} )^{2} \))(\( ( 1 - 36 T^{2} + 746 T^{4} - 8304 T^{6} + 72147 T^{8} - 2997744 T^{10} + 97219466 T^{12} - 1693651716 T^{14} + 16983563041 T^{16} )^{2} \))
$23$ (\( ( 1 + 4 T + 32 T^{2} + 40 T^{3} + 655 T^{4} + 920 T^{5} + 16928 T^{6} + 48668 T^{7} + 279841 T^{8} )^{2} \))(\( ( 1 + 88 T^{2} + 2991 T^{4} + 46552 T^{6} + 279841 T^{8} )^{2} \))(\( ( 1 + 14 T^{2} + 529 T^{4} )^{4} \))(\( ( 1 + 38 T^{2} + 529 T^{4} )^{4} \))(\( ( 1 + 82 T^{2} + 2715 T^{4} + 43378 T^{6} + 279841 T^{8} )^{2} \))(\( ( 1 + 40 T^{2} + 783 T^{4} + 21160 T^{6} + 279841 T^{8} )^{2} \))(\( ( 1 - 4 T + 32 T^{2} - 40 T^{3} + 655 T^{4} - 920 T^{5} + 16928 T^{6} - 48668 T^{7} + 279841 T^{8} )^{2} \))(\( ( 1 + 64 T^{2} + 2055 T^{4} + 33856 T^{6} + 279841 T^{8} )^{4} \))
$29$ (\( 1 - 64 T^{2} + 4452 T^{4} - 163520 T^{6} + 6050150 T^{8} - 137520320 T^{10} + 3148815012 T^{12} - 38068692544 T^{14} + 500246412961 T^{16} \))(\( ( 1 - 40 T^{2} + 841 T^{4} )^{4} \))(\( ( 1 - 16 T^{2} + 1362 T^{4} - 13456 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 - 56 T^{2} + 1602 T^{4} - 47096 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 - 31 T^{2} + 841 T^{4} )^{4} \))(\( ( 1 - 52 T^{2} + 1590 T^{4} - 43732 T^{6} + 707281 T^{8} )^{2} \))(\( 1 - 64 T^{2} + 4452 T^{4} - 163520 T^{6} + 6050150 T^{8} - 137520320 T^{10} + 3148815012 T^{12} - 38068692544 T^{14} + 500246412961 T^{16} \))(\( ( 1 - 180 T^{2} + 15368 T^{4} - 805116 T^{6} + 28222446 T^{8} - 677102556 T^{10} + 10869494408 T^{12} - 107068197780 T^{14} + 500246412961 T^{16} )^{2} \))
$31$ (\( 1 - 116 T^{2} + 7098 T^{4} - 315760 T^{6} + 11036483 T^{8} - 303445360 T^{10} + 6555152058 T^{12} - 102950426996 T^{14} + 852891037441 T^{16} \))(\( ( 1 - 2 T^{2} - 429 T^{4} - 1922 T^{6} + 923521 T^{8} )^{2} \))(\( ( 1 - 80 T^{2} + 3138 T^{4} - 76880 T^{6} + 923521 T^{8} )^{2} \))(\( ( 1 - 104 T^{2} + 4530 T^{4} - 99944 T^{6} + 923521 T^{8} )^{2} \))(\( ( 1 + 70 T^{2} + 2763 T^{4} + 67270 T^{6} + 923521 T^{8} )^{2} \))(\( ( 1 - 110 T^{2} + 4899 T^{4} - 105710 T^{6} + 923521 T^{8} )^{2} \))(\( 1 - 116 T^{2} + 7098 T^{4} - 315760 T^{6} + 11036483 T^{8} - 303445360 T^{10} + 6555152058 T^{12} - 102950426996 T^{14} + 852891037441 T^{16} \))(\( ( 1 - 112 T^{2} + 4690 T^{4} - 84064 T^{6} + 1025299 T^{8} - 80785504 T^{10} + 4331313490 T^{12} - 99400412272 T^{14} + 852891037441 T^{16} )^{2} \))
$37$ (\( ( 1 - 10 T + 91 T^{2} - 370 T^{3} + 1369 T^{4} )^{4} \))(\( ( 1 + 2 T + 63 T^{2} + 74 T^{3} + 1369 T^{4} )^{4} \))(\( ( 1 - 9 T + 37 T^{2} )^{8} \))(\( ( 1 + 7 T + 37 T^{2} )^{8} \))(\( ( 1 + 8 T + 66 T^{2} + 296 T^{3} + 1369 T^{4} )^{4} \))(\( ( 1 + 20 T + 171 T^{2} + 740 T^{3} + 1369 T^{4} )^{4} \))(\( ( 1 - 10 T + 91 T^{2} - 370 T^{3} + 1369 T^{4} )^{4} \))(\( ( 1 - 10 T + 146 T^{2} - 1056 T^{3} + 8039 T^{4} - 39072 T^{5} + 199874 T^{6} - 506530 T^{7} + 1874161 T^{8} )^{4} \))
$41$ (\( 1 - 288 T^{2} + 37342 T^{4} - 2869632 T^{6} + 143812995 T^{8} - 4823851392 T^{10} + 105519567262 T^{12} - 1368030021408 T^{14} + 7984925229121 T^{16} \))(\( ( 1 - 8 T^{2} + 2055 T^{4} - 13448 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 - 10 T^{2} + 2787 T^{4} - 16810 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 - 122 T^{2} + 6867 T^{4} - 205082 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 - 70 T^{2} + 1681 T^{4} )^{4} \))(\( ( 1 - 16 T^{2} + 3279 T^{4} - 26896 T^{6} + 2825761 T^{8} )^{2} \))(\( 1 - 288 T^{2} + 37342 T^{4} - 2869632 T^{6} + 143812995 T^{8} - 4823851392 T^{10} + 105519567262 T^{12} - 1368030021408 T^{14} + 7984925229121 T^{16} \))(\( ( 1 - 168 T^{2} + 14246 T^{4} - 874752 T^{6} + 41345307 T^{8} - 1470458112 T^{10} + 40255791206 T^{12} - 798017512488 T^{14} + 7984925229121 T^{16} )^{2} \))
$43$ (\( 1 - 40 T^{2} + 5668 T^{4} - 165496 T^{6} + 14065894 T^{8} - 306002104 T^{10} + 19377764068 T^{12} - 252854521960 T^{14} + 11688200277601 T^{16} \))(\( ( 1 - 148 T^{2} + 9066 T^{4} - 273652 T^{6} + 3418801 T^{8} )^{2} \))(\( ( 1 - 62 T^{2} + 4443 T^{4} - 114638 T^{6} + 3418801 T^{8} )^{2} \))(\( ( 1 - 46 T^{2} + 2283 T^{4} - 85054 T^{6} + 3418801 T^{8} )^{2} \))(\( ( 1 - 142 T^{2} + 8523 T^{4} - 262558 T^{6} + 3418801 T^{8} )^{2} \))(\( ( 1 - 116 T^{2} + 6294 T^{4} - 214484 T^{6} + 3418801 T^{8} )^{2} \))(\( 1 - 40 T^{2} + 5668 T^{4} - 165496 T^{6} + 14065894 T^{8} - 306002104 T^{10} + 19377764068 T^{12} - 252854521960 T^{14} + 11688200277601 T^{16} \))(\( ( 1 - 264 T^{2} + 32912 T^{4} - 2530824 T^{6} + 131089566 T^{8} - 4679493576 T^{10} + 112519578512 T^{12} - 1668839844936 T^{14} + 11688200277601 T^{16} )^{2} \))
$47$ (\( ( 1 - 4 T + 96 T^{2} - 188 T^{3} + 2209 T^{4} )^{4} \))(\( ( 1 + 112 T^{2} + 6786 T^{4} + 247408 T^{6} + 4879681 T^{8} )^{2} \))(\( ( 1 + 70 T^{2} + 2187 T^{4} + 154630 T^{6} + 4879681 T^{8} )^{2} \))(\( ( 1 + 118 T^{2} + 7515 T^{4} + 260662 T^{6} + 4879681 T^{8} )^{2} \))(\( ( 1 + 28 T^{2} - 1530 T^{4} + 61852 T^{6} + 4879681 T^{8} )^{2} \))(\( ( 1 + 76 T^{2} + 5430 T^{4} + 167884 T^{6} + 4879681 T^{8} )^{2} \))(\( ( 1 + 4 T + 96 T^{2} + 188 T^{3} + 2209 T^{4} )^{4} \))(\( ( 1 + 140 T^{2} + 12088 T^{4} + 786212 T^{6} + 39897838 T^{8} + 1736742308 T^{10} + 58985583928 T^{12} + 1509090146060 T^{14} + 23811286661761 T^{16} )^{2} \))
$53$ (\( 1 - 136 T^{2} + 6300 T^{4} + 118408 T^{6} - 20822362 T^{8} + 332608072 T^{10} + 49710030300 T^{12} - 3014353113544 T^{14} + 62259690411361 T^{16} \))(\( ( 1 - 100 T^{2} + 7686 T^{4} - 280900 T^{6} + 7890481 T^{8} )^{2} \))(\( ( 1 - 44 T^{2} + 2646 T^{4} - 123596 T^{6} + 7890481 T^{8} )^{2} \))(\( ( 1 - 172 T^{2} + 12630 T^{4} - 483148 T^{6} + 7890481 T^{8} )^{2} \))(\( ( 1 - 142 T^{2} + 9795 T^{4} - 398878 T^{6} + 7890481 T^{8} )^{2} \))(\( ( 1 - 98 T^{2} + 2809 T^{4} )^{4} \))(\( 1 - 136 T^{2} + 6300 T^{4} + 118408 T^{6} - 20822362 T^{8} + 332608072 T^{10} + 49710030300 T^{12} - 3014353113544 T^{14} + 62259690411361 T^{16} \))(\( ( 1 - 264 T^{2} + 32204 T^{4} - 2520504 T^{6} + 149086662 T^{8} - 7080095736 T^{10} + 254105050124 T^{12} - 5851391338056 T^{14} + 62259690411361 T^{16} )^{2} \))
$59$ (\( ( 1 + 32 T + 584 T^{2} + 7136 T^{3} + 63778 T^{4} + 421024 T^{5} + 2032904 T^{6} + 6572128 T^{7} + 12117361 T^{8} )^{2} \))(\( ( 1 + 184 T^{2} + 15234 T^{4} + 640504 T^{6} + 12117361 T^{8} )^{2} \))(\( ( 1 + 70 T^{2} + 5787 T^{4} + 243670 T^{6} + 12117361 T^{8} )^{2} \))(\( ( 1 + 70 T^{2} + 5787 T^{4} + 243670 T^{6} + 12117361 T^{8} )^{2} \))(\( ( 1 + 70 T^{2} + 5787 T^{4} + 243670 T^{6} + 12117361 T^{8} )^{2} \))(\( ( 1 + 220 T^{2} + 19014 T^{4} + 765820 T^{6} + 12117361 T^{8} )^{2} \))(\( ( 1 - 32 T + 584 T^{2} - 7136 T^{3} + 63778 T^{4} - 421024 T^{5} + 2032904 T^{6} - 6572128 T^{7} + 12117361 T^{8} )^{2} \))(\( ( 1 + 260 T^{2} + 30952 T^{4} + 2376716 T^{6} + 147668398 T^{8} + 8273348396 T^{10} + 375056557672 T^{12} + 10966938746660 T^{14} + 146830437604321 T^{16} )^{2} \))
$61$ (\( ( 1 + 4 T + 78 T^{2} + 244 T^{3} + 3721 T^{4} )^{4} \))(\( ( 1 - 4 T + 78 T^{2} - 244 T^{3} + 3721 T^{4} )^{4} \))(\( ( 1 + 16 T + 180 T^{2} + 976 T^{3} + 3721 T^{4} )^{4} \))(\( ( 1 - 4 T + 72 T^{2} - 244 T^{3} + 3721 T^{4} )^{4} \))(\( ( 1 - 16 T + 162 T^{2} - 976 T^{3} + 3721 T^{4} )^{4} \))(\( ( 1 + 4 T + 18 T^{2} + 244 T^{3} + 3721 T^{4} )^{4} \))(\( ( 1 + 4 T + 78 T^{2} + 244 T^{3} + 3721 T^{4} )^{4} \))(\( ( 1 + 80 T^{2} - 528 T^{3} + 2286 T^{4} - 32208 T^{5} + 297680 T^{6} + 13845841 T^{8} )^{4} \))
$67$ (\( 1 - 200 T^{2} + 27492 T^{4} - 2608600 T^{6} + 201429158 T^{8} - 11710005400 T^{10} + 553994618532 T^{12} - 18091676433800 T^{14} + 406067677556641 T^{16} \))(\( ( 1 - 116 T^{2} + 12042 T^{4} - 520724 T^{6} + 20151121 T^{8} )^{2} \))(\( ( 1 - 44 T^{2} + 3318 T^{4} - 197516 T^{6} + 20151121 T^{8} )^{2} \))(\( ( 1 - 70 T^{2} + 4489 T^{4} )^{4} \))(\( ( 1 - 254 T^{2} + 25083 T^{4} - 1140206 T^{6} + 20151121 T^{8} )^{2} \))(\( ( 1 - 140 T^{2} + 10806 T^{4} - 628460 T^{6} + 20151121 T^{8} )^{2} \))(\( 1 - 200 T^{2} + 27492 T^{4} - 2608600 T^{6} + 201429158 T^{8} - 11710005400 T^{10} + 553994618532 T^{12} - 18091676433800 T^{14} + 406067677556641 T^{16} \))(\( ( 1 - 352 T^{2} + 62320 T^{4} - 7098256 T^{6} + 563157790 T^{8} - 31864071184 T^{10} + 1255817860720 T^{12} - 31841350523488 T^{14} + 406067677556641 T^{16} )^{2} \))
$71$ (\( ( 1 - 36 T + 760 T^{2} - 10392 T^{3} + 103479 T^{4} - 737832 T^{5} + 3831160 T^{6} - 12884796 T^{7} + 25411681 T^{8} )^{2} \))(\( ( 1 + 128 T^{2} + 8103 T^{4} + 645248 T^{6} + 25411681 T^{8} )^{2} \))(\( ( 1 + 110 T^{2} + 5041 T^{4} )^{4} \))(\( ( 1 + 71 T^{2} )^{8} \))(\( ( 1 + 242 T^{2} + 24507 T^{4} + 1219922 T^{6} + 25411681 T^{8} )^{2} \))(\( ( 1 + 248 T^{2} + 25215 T^{4} + 1250168 T^{6} + 25411681 T^{8} )^{2} \))(\( ( 1 + 36 T + 760 T^{2} + 10392 T^{3} + 103479 T^{4} + 737832 T^{5} + 3831160 T^{6} + 12884796 T^{7} + 25411681 T^{8} )^{2} \))(\( ( 1 + 216 T^{2} + 28598 T^{4} + 2870208 T^{6} + 226923531 T^{8} + 14468718528 T^{10} + 726723253238 T^{12} + 27669661326936 T^{14} + 645753531245761 T^{16} )^{2} \))
$73$ (\( ( 1 - 12 T + 228 T^{2} - 1644 T^{3} + 20234 T^{4} - 120012 T^{5} + 1215012 T^{6} - 4668204 T^{7} + 28398241 T^{8} )^{2} \))(\( ( 1 - 10 T + 144 T^{2} - 730 T^{3} + 5329 T^{4} )^{4} \))(\( ( 1 + 50 T^{2} + 5329 T^{4} )^{4} \))(\( ( 1 - 16 T + 186 T^{2} - 1168 T^{3} + 5329 T^{4} )^{4} \))(\( ( 1 + 4 T + 73 T^{2} )^{8} \))(\( ( 1 + 8 T + 54 T^{2} + 584 T^{3} + 5329 T^{4} )^{4} \))(\( ( 1 - 12 T + 228 T^{2} - 1644 T^{3} + 20234 T^{4} - 120012 T^{5} + 1215012 T^{6} - 4668204 T^{7} + 28398241 T^{8} )^{2} \))(\( ( 1 + 14 T + 278 T^{2} + 2994 T^{3} + 29954 T^{4} + 218562 T^{5} + 1481462 T^{6} + 5446238 T^{7} + 28398241 T^{8} )^{4} \))
$79$ (\( 1 - 424 T^{2} + 86788 T^{4} - 11475640 T^{6} + 1070068678 T^{8} - 71619469240 T^{10} + 3380399629828 T^{12} - 103069081140904 T^{14} + 1517108809906561 T^{16} \))(\( ( 1 - 164 T^{2} + 13914 T^{4} - 1023524 T^{6} + 38950081 T^{8} )^{2} \))(\( ( 1 - 110 T^{2} + 4923 T^{4} - 686510 T^{6} + 38950081 T^{8} )^{2} \))(\( ( 1 - 62 T^{2} + 10539 T^{4} - 386942 T^{6} + 38950081 T^{8} )^{2} \))(\( ( 1 - 58 T^{2} + 6241 T^{4} )^{4} \))(\( ( 1 - 260 T^{2} + 28614 T^{4} - 1622660 T^{6} + 38950081 T^{8} )^{2} \))(\( 1 - 424 T^{2} + 86788 T^{4} - 11475640 T^{6} + 1070068678 T^{8} - 71619469240 T^{10} + 3380399629828 T^{12} - 103069081140904 T^{14} + 1517108809906561 T^{16} \))(\( ( 1 + 56 T^{2} + 14896 T^{4} + 945080 T^{6} + 108009022 T^{8} + 5898244280 T^{10} + 580200406576 T^{12} + 13612897509176 T^{14} + 1517108809906561 T^{16} )^{2} \))
$83$ (\( ( 1 - 16 T + 88 T^{2} - 560 T^{3} + 8322 T^{4} - 46480 T^{5} + 606232 T^{6} - 9148592 T^{7} + 47458321 T^{8} )^{2} \))(\( ( 1 - 40 T^{2} - 1374 T^{4} - 275560 T^{6} + 47458321 T^{8} )^{2} \))(\( ( 1 + 262 T^{2} + 30075 T^{4} + 1804918 T^{6} + 47458321 T^{8} )^{2} \))(\( ( 1 + 182 T^{2} + 21195 T^{4} + 1253798 T^{6} + 47458321 T^{8} )^{2} \))(\( ( 1 + 154 T^{2} + 6889 T^{4} )^{4} \))(\( ( 1 + 124 T^{2} + 14550 T^{4} + 854236 T^{6} + 47458321 T^{8} )^{2} \))(\( ( 1 + 16 T + 88 T^{2} + 560 T^{3} + 8322 T^{4} + 46480 T^{5} + 606232 T^{6} + 9148592 T^{7} + 47458321 T^{8} )^{2} \))(\( ( 1 + 516 T^{2} + 125336 T^{4} + 18717900 T^{6} + 1875461838 T^{8} + 128947613100 T^{10} + 5948236120856 T^{12} + 168701232658404 T^{14} + 2252292232139041 T^{16} )^{2} \))
$89$ (\( 1 - 496 T^{2} + 119790 T^{4} - 18269696 T^{6} + 1929877235 T^{8} - 144714262016 T^{10} + 7515893049390 T^{12} - 246502720316656 T^{14} + 3936588805702081 T^{16} \))(\( ( 1 - 352 T^{2} + 46815 T^{4} - 2788192 T^{6} + 62742241 T^{8} )^{2} \))(\( ( 1 - 116 T^{2} + 5382 T^{4} - 918836 T^{6} + 62742241 T^{8} )^{2} \))(\( ( 1 + 22 T^{2} + 7921 T^{4} )^{4} \))(\( ( 1 - 58 T^{2} + 15507 T^{4} - 459418 T^{6} + 62742241 T^{8} )^{2} \))(\( ( 1 - 64 T^{2} + 15999 T^{4} - 506944 T^{6} + 62742241 T^{8} )^{2} \))(\( 1 - 496 T^{2} + 119790 T^{4} - 18269696 T^{6} + 1929877235 T^{8} - 144714262016 T^{10} + 7515893049390 T^{12} - 246502720316656 T^{14} + 3936588805702081 T^{16} \))(\( ( 1 - 336 T^{2} + 65102 T^{4} - 8829312 T^{6} + 891379059 T^{8} - 69936980352 T^{10} + 4084645373582 T^{12} - 166985713762896 T^{14} + 3936588805702081 T^{16} )^{2} \))
$97$ (\( ( 1 - 24 T + 444 T^{2} - 5928 T^{3} + 63110 T^{4} - 575016 T^{5} + 4177596 T^{6} - 21904152 T^{7} + 88529281 T^{8} )^{2} \))(\( ( 1 + 12 T + 182 T^{2} + 1164 T^{3} + 9409 T^{4} )^{4} \))(\( ( 1 + 4 T + 48 T^{2} + 388 T^{3} + 9409 T^{4} )^{4} \))(\( ( 1 + 44 T^{2} + 9409 T^{4} )^{4} \))(\( ( 1 + 12 T + 134 T^{2} + 1164 T^{3} + 9409 T^{4} )^{4} \))(\( ( 1 + 8 T + 162 T^{2} + 776 T^{3} + 9409 T^{4} )^{4} \))(\( ( 1 - 24 T + 444 T^{2} - 5928 T^{3} + 63110 T^{4} - 575016 T^{5} + 4177596 T^{6} - 21904152 T^{7} + 88529281 T^{8} )^{2} \))(\( ( 1 + 4 T + 344 T^{2} + 1068 T^{3} + 48302 T^{4} + 103596 T^{5} + 3236696 T^{6} + 3650692 T^{7} + 88529281 T^{8} )^{4} \))
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