Properties

Label 525.2.r
Level 525
Weight 2
Character orbit r
Rep. character \(\chi_{525}(424,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 48
Newform subspaces 8
Sturm bound 160
Trace bound 6

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Defining parameters

Level: \( N \) = \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 525.r (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 35 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 8 \)
Sturm bound: \(160\)
Trace bound: \(6\)
Distinguishing \(T_p\): \(2\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(525, [\chi])\).

Total New Old
Modular forms 184 48 136
Cusp forms 136 48 88
Eisenstein series 48 0 48

Trace form

\( 48q + 24q^{4} + 8q^{6} + 24q^{9} + O(q^{10}) \) \( 48q + 24q^{4} + 8q^{6} + 24q^{9} - 4q^{11} + 36q^{14} - 40q^{16} + 22q^{19} - 2q^{21} + 12q^{24} + 24q^{26} - 32q^{29} - 32q^{31} - 32q^{34} + 48q^{36} - 2q^{39} - 8q^{41} + 36q^{44} - 8q^{46} + 74q^{49} - 8q^{51} + 4q^{54} - 108q^{56} + 8q^{59} - 2q^{61} - 208q^{64} - 40q^{66} - 48q^{69} + 80q^{71} - 8q^{74} - 8q^{76} - 12q^{79} - 24q^{81} - 72q^{84} + 12q^{86} - 32q^{89} + 58q^{91} + 48q^{94} - 8q^{96} - 8q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(525, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
525.2.r.a \(4\) \(4.192\) \(\Q(\zeta_{12})\) None \(-6\) \(0\) \(0\) \(10\) \(q+(-1+\zeta_{12}-\zeta_{12}^{2})q^{2}+(\zeta_{12}-\zeta_{12}^{3})q^{3}+\cdots\)
525.2.r.b \(4\) \(4.192\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{12}q^{3}+(-2+2\zeta_{12}^{2})q^{4}+(-\zeta_{12}+\cdots)q^{7}+\cdots\)
525.2.r.c \(4\) \(4.192\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{12}q^{2}+(\zeta_{12}-\zeta_{12}^{3})q^{3}-\zeta_{12}^{2}q^{4}+\cdots\)
525.2.r.d \(4\) \(4.192\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q+2\zeta_{12}q^{2}+(-\zeta_{12}+\zeta_{12}^{3})q^{3}+2\zeta_{12}^{2}q^{4}+\cdots\)
525.2.r.e \(4\) \(4.192\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q+2\zeta_{12}q^{2}+(\zeta_{12}-\zeta_{12}^{3})q^{3}+2\zeta_{12}^{2}q^{4}+\cdots\)
525.2.r.f \(4\) \(4.192\) \(\Q(\zeta_{12})\) None \(6\) \(0\) \(0\) \(-10\) \(q+(2-\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}-\zeta_{12}q^{3}+\cdots\)
525.2.r.g \(8\) \(4.192\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{24}^{5}q^{2}-\zeta_{24}q^{3}+\zeta_{24}^{7}q^{6}+(\zeta_{24}+\cdots)q^{7}+\cdots\)
525.2.r.h \(16\) \(4.192\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{2}+\beta _{12}q^{3}+(1+\beta _{3}+2\beta _{5}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(525, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(525, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(175, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + 2 T + 2 T^{2} )^{2}( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} ) \))(\( ( 1 + 2 T^{2} + 4 T^{4} )^{2} \))(\( 1 + 3 T^{2} + 5 T^{4} + 12 T^{6} + 16 T^{8} \))(\( ( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} )( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} ) \))(\( ( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} )( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} ) \))(\( ( 1 - 2 T + 2 T^{2} )^{2}( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} ) \))(\( ( 1 + 2 T^{2} )^{4}( 1 - 2 T^{2} + 4 T^{4} )^{2} \))(\( 1 + T^{2} + 2 T^{4} - 7 T^{6} - 22 T^{8} - 31 T^{10} - 15 T^{12} + 132 T^{14} + 220 T^{16} + 528 T^{18} - 240 T^{20} - 1984 T^{22} - 5632 T^{24} - 7168 T^{26} + 8192 T^{28} + 16384 T^{30} + 65536 T^{32} \))
$3$ (\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( ( 1 - T^{2} + T^{4} )^{2} \))(\( ( 1 - T^{2} + T^{4} )^{4} \))
$5$ 1
$7$ (\( ( 1 - 5 T + 7 T^{2} )^{2} \))(\( 1 + 11 T^{2} + 49 T^{4} \))(\( 1 - 13 T^{2} + 49 T^{4} \))(\( 1 - 13 T^{2} + 49 T^{4} \))(\( 1 + 11 T^{2} + 49 T^{4} \))(\( ( 1 + 5 T + 7 T^{2} )^{2} \))(\( 1 + 10 T^{2} + 51 T^{4} + 490 T^{6} + 2401 T^{8} \))(\( 1 - 21 T^{2} + 329 T^{4} - 3318 T^{6} + 27414 T^{8} - 162582 T^{10} + 789929 T^{12} - 2470629 T^{14} + 5764801 T^{16} \))
$11$ (\( 1 - 2 T - 16 T^{2} + 4 T^{3} + 235 T^{4} + 44 T^{5} - 1936 T^{6} - 2662 T^{7} + 14641 T^{8} \))(\( ( 1 - 11 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 11 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} )^{2} \))(\( 1 - 2 T - 16 T^{2} + 4 T^{3} + 235 T^{4} + 44 T^{5} - 1936 T^{6} - 2662 T^{7} + 14641 T^{8} \))(\( ( 1 + 4 T - 8 T^{2} + 8 T^{3} + 279 T^{4} + 88 T^{5} - 968 T^{6} + 5324 T^{7} + 14641 T^{8} )^{2} \))(\( ( 1 + 8 T + 28 T^{2} + 120 T^{3} + 402 T^{4} + 396 T^{5} + 384 T^{6} - 2384 T^{7} - 29421 T^{8} - 26224 T^{9} + 46464 T^{10} + 527076 T^{11} + 5885682 T^{12} + 19326120 T^{13} + 49603708 T^{14} + 155897368 T^{15} + 214358881 T^{16} )^{2} \))
$13$ (\( 1 - 14 T^{2} + 195 T^{4} - 2366 T^{6} + 28561 T^{8} \))(\( ( 1 - 25 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 17 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 17 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 25 T^{2} + 169 T^{4} )^{2} \))(\( 1 - 14 T^{2} + 195 T^{4} - 2366 T^{6} + 28561 T^{8} \))(\( ( 1 - 30 T^{2} + 491 T^{4} - 5070 T^{6} + 28561 T^{8} )^{2} \))(\( ( 1 - 17 T^{2} + 425 T^{4} - 6726 T^{6} + 86438 T^{8} - 1136694 T^{10} + 12138425 T^{12} - 82055753 T^{14} + 815730721 T^{16} )^{2} \))
$17$ (\( 1 - 6 T + 24 T^{2} - 72 T^{3} + 59 T^{4} - 1224 T^{5} + 6936 T^{6} - 29478 T^{7} + 83521 T^{8} \))(\( 1 - 2 T^{2} - 285 T^{4} - 578 T^{6} + 83521 T^{8} \))(\( ( 1 - 8 T + 47 T^{2} - 136 T^{3} + 289 T^{4} )( 1 + 8 T + 47 T^{2} + 136 T^{3} + 289 T^{4} ) \))(\( 1 + 18 T^{2} + 35 T^{4} + 5202 T^{6} + 83521 T^{8} \))(\( ( 1 + 17 T^{2} + 289 T^{4} )^{2} \))(\( 1 + 6 T + 24 T^{2} + 72 T^{3} + 59 T^{4} + 1224 T^{5} + 6936 T^{6} + 29478 T^{7} + 83521 T^{8} \))(\( 1 + 24 T^{2} + 142 T^{4} - 3456 T^{6} - 61629 T^{8} - 998784 T^{10} + 11859982 T^{12} + 579301656 T^{14} + 6975757441 T^{16} \))(\( 1 + 60 T^{2} + 1484 T^{4} + 20616 T^{6} + 278218 T^{8} + 6808500 T^{10} + 123849488 T^{12} + 666359940 T^{14} - 6309930269 T^{16} + 192578022660 T^{18} + 10344033087248 T^{20} + 164340638536500 T^{22} + 1940781283720138 T^{24} + 41561730251656584 T^{26} + 864611400048965324 T^{28} + 10102669593564055740 T^{30} + 48661191875666868481 T^{32} \))
$19$ (\( 1 - 2 T - 23 T^{2} + 22 T^{3} + 292 T^{4} + 418 T^{5} - 8303 T^{6} - 13718 T^{7} + 130321 T^{8} \))(\( ( 1 - 5 T + 6 T^{2} - 95 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 - 8 T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2} \))(\( ( 1 - 8 T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2} \))(\( ( 1 - 8 T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2} \))(\( 1 - 2 T - 23 T^{2} + 22 T^{3} + 292 T^{4} + 418 T^{5} - 8303 T^{6} - 13718 T^{7} + 130321 T^{8} \))(\( ( 1 + 2 T - 3 T^{2} - 62 T^{3} - 388 T^{4} - 1178 T^{5} - 1083 T^{6} + 13718 T^{7} + 130321 T^{8} )^{2} \))(\( ( 1 - 3 T - 30 T^{2} - 125 T^{3} + 969 T^{4} + 3616 T^{5} + 6406 T^{6} - 66486 T^{7} - 238148 T^{8} - 1263234 T^{9} + 2312566 T^{10} + 24802144 T^{11} + 126281049 T^{12} - 309512375 T^{13} - 1411376430 T^{14} - 2681615217 T^{15} + 16983563041 T^{16} )^{2} \))
$23$ (\( 1 - 6 T + 52 T^{2} - 240 T^{3} + 1347 T^{4} - 5520 T^{5} + 27508 T^{6} - 73002 T^{7} + 279841 T^{8} \))(\( 1 + 10 T^{2} - 429 T^{4} + 5290 T^{6} + 279841 T^{8} \))(\( 1 + 42 T^{2} + 1235 T^{4} + 22218 T^{6} + 279841 T^{8} \))(\( 1 + 30 T^{2} + 371 T^{4} + 15870 T^{6} + 279841 T^{8} \))(\( ( 1 + 23 T^{2} + 529 T^{4} )^{2} \))(\( 1 + 6 T + 52 T^{2} + 240 T^{3} + 1347 T^{4} + 5520 T^{5} + 27508 T^{6} + 73002 T^{7} + 279841 T^{8} \))(\( 1 + 48 T^{2} + 958 T^{4} + 13824 T^{6} + 313059 T^{8} + 7312896 T^{10} + 268087678 T^{12} + 7105722672 T^{14} + 78310985281 T^{16} \))(\( 1 + 44 T^{2} - 388 T^{4} - 24248 T^{6} + 734474 T^{8} + 14717524 T^{10} - 547791216 T^{12} - 1264808844 T^{14} + 414286345891 T^{16} - 669083878476 T^{18} - 153294441676656 T^{20} + 2178721749218836 T^{22} + 57517382603277194 T^{24} - 1004510043908560952 T^{26} - 8502874279623884548 T^{28} + \)\(51\!\cdots\!96\)\( T^{30} + \)\(61\!\cdots\!61\)\( T^{32} \))
$29$ (\( ( 1 + 2 T + 32 T^{2} + 58 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{4} \))(\( ( 1 - 8 T + 29 T^{2} )^{4} \))(\( ( 1 - 8 T + 29 T^{2} )^{4} \))(\( ( 1 + 4 T + 29 T^{2} )^{4} \))(\( ( 1 + 2 T + 32 T^{2} + 58 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 + 8 T + 56 T^{2} + 232 T^{3} + 841 T^{4} )^{4} \))(\( ( 1 + 4 T + 29 T^{2} )^{16} \))
$31$ (\( 1 + 6 T - 23 T^{2} - 18 T^{3} + 1404 T^{4} - 558 T^{5} - 22103 T^{6} + 178746 T^{7} + 923521 T^{8} \))(\( ( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 - 8 T + 33 T^{2} - 248 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 + T - 30 T^{2} + 31 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 + 9 T + 50 T^{2} + 279 T^{3} + 961 T^{4} )^{2} \))(\( 1 + 6 T - 23 T^{2} - 18 T^{3} + 1404 T^{4} - 558 T^{5} - 22103 T^{6} + 178746 T^{7} + 923521 T^{8} \))(\( ( 1 - 6 T - 27 T^{2} - 6 T^{3} + 1892 T^{4} - 186 T^{5} - 25947 T^{6} - 178746 T^{7} + 923521 T^{8} )^{2} \))(\( ( 1 + 9 T + 13 T^{2} + 90 T^{3} + 749 T^{4} - 1629 T^{5} + 22230 T^{6} + 154161 T^{7} - 324118 T^{8} + 4778991 T^{9} + 21363030 T^{10} - 48529539 T^{11} + 691717229 T^{12} + 2576623590 T^{13} + 11537547853 T^{14} + 247613526999 T^{15} + 852891037441 T^{16} )^{2} \))
$37$ (\( 1 + 18 T + 205 T^{2} + 1746 T^{3} + 12036 T^{4} + 64602 T^{5} + 280645 T^{6} + 911754 T^{7} + 1874161 T^{8} \))(\( 1 + 25 T^{2} - 744 T^{4} + 34225 T^{6} + 1874161 T^{8} \))(\( 1 + 25 T^{2} - 744 T^{4} + 34225 T^{6} + 1874161 T^{8} \))(\( 1 + 25 T^{2} - 744 T^{4} + 34225 T^{6} + 1874161 T^{8} \))(\( 1 + 65 T^{2} + 2856 T^{4} + 88985 T^{6} + 1874161 T^{8} \))(\( 1 - 18 T + 205 T^{2} - 1746 T^{3} + 12036 T^{4} - 64602 T^{5} + 280645 T^{6} - 911754 T^{7} + 1874161 T^{8} \))(\( 1 + 46 T^{2} - 759 T^{4} + 6302 T^{6} + 3494660 T^{8} + 8627438 T^{10} - 1422488199 T^{12} + 118023414814 T^{14} + 3512479453921 T^{16} \))(\( 1 + 108 T^{2} + 7374 T^{4} + 467096 T^{6} + 22703481 T^{8} + 873938840 T^{10} + 33932606830 T^{12} + 1175343473028 T^{14} + 38616591751924 T^{16} + 1609045214575332 T^{18} + 63595168349119630 T^{20} + 2242287961638825560 T^{22} + 79745510544985799001 T^{24} + \)\(22\!\cdots\!04\)\( T^{26} + \)\(48\!\cdots\!94\)\( T^{28} + \)\(97\!\cdots\!12\)\( T^{30} + \)\(12\!\cdots\!41\)\( T^{32} \))
$41$ (\( ( 1 - 2 T + 80 T^{2} - 82 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 - 12 T + 41 T^{2} )^{4} \))(\( ( 1 + 41 T^{2} )^{4} \))(\( ( 1 + 6 T + 41 T^{2} )^{4} \))(\( ( 1 + 10 T + 41 T^{2} )^{4} \))(\( ( 1 - 2 T + 80 T^{2} - 82 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 4 T + 68 T^{2} + 164 T^{3} + 1681 T^{4} )^{4} \))(\( ( 1 - 4 T + 104 T^{2} - 256 T^{3} + 4966 T^{4} - 10496 T^{5} + 174824 T^{6} - 275684 T^{7} + 2825761 T^{8} )^{4} \))
$43$ (\( 1 - 110 T^{2} + 6291 T^{4} - 203390 T^{6} + 3418801 T^{8} \))(\( ( 1 - 85 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 22 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 85 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 61 T^{2} + 1849 T^{4} )^{2} \))(\( 1 - 110 T^{2} + 6291 T^{4} - 203390 T^{6} + 3418801 T^{8} \))(\( ( 1 - 6 T^{2} + 3059 T^{4} - 11094 T^{6} + 3418801 T^{8} )^{2} \))(\( ( 1 - 215 T^{2} + 21194 T^{4} - 1322001 T^{6} + 62692682 T^{8} - 2444379849 T^{10} + 72458068394 T^{12} - 1359093055535 T^{14} + 11688200277601 T^{16} )^{2} \))
$47$ (\( 1 + 90 T^{2} + 5891 T^{4} + 198810 T^{6} + 4879681 T^{8} \))(\( 1 + 58 T^{2} + 1155 T^{4} + 128122 T^{6} + 4879681 T^{8} \))(\( 1 - 6 T^{2} - 2173 T^{4} - 13254 T^{6} + 4879681 T^{8} \))(\( 1 + 90 T^{2} + 5891 T^{4} + 198810 T^{6} + 4879681 T^{8} \))(\( 1 + 58 T^{2} + 1155 T^{4} + 128122 T^{6} + 4879681 T^{8} \))(\( 1 + 90 T^{2} + 5891 T^{4} + 198810 T^{6} + 4879681 T^{8} \))(\( 1 - 76 T^{2} + 1962 T^{4} + 45904 T^{6} - 3092269 T^{8} + 101401936 T^{10} + 9573934122 T^{12} - 819220365004 T^{14} + 23811286661761 T^{16} \))(\( 1 + 212 T^{2} + 20956 T^{4} + 1454040 T^{6} + 90802154 T^{8} + 5296739420 T^{10} + 289183381584 T^{12} + 15685077059564 T^{14} + 793076937068387 T^{16} + 34648335224576876 T^{18} + 1411122652631194704 T^{20} + 57094694749782569180 T^{22} + \)\(21\!\cdots\!94\)\( T^{24} + \)\(76\!\cdots\!60\)\( T^{26} + \)\(24\!\cdots\!96\)\( T^{28} + \)\(54\!\cdots\!28\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \))
$53$ (\( 1 - 36 T + 642 T^{2} - 7560 T^{3} + 64187 T^{4} - 400680 T^{5} + 1803378 T^{6} - 5359572 T^{7} + 7890481 T^{8} \))(\( ( 1 + 53 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 4 T - 37 T^{2} - 212 T^{3} + 2809 T^{4} )( 1 + 4 T - 37 T^{2} + 212 T^{3} + 2809 T^{4} ) \))(\( ( 1 - 14 T + 143 T^{2} - 742 T^{3} + 2809 T^{4} )( 1 + 14 T + 143 T^{2} + 742 T^{3} + 2809 T^{4} ) \))(\( 1 - 38 T^{2} - 1365 T^{4} - 106742 T^{6} + 7890481 T^{8} \))(\( 1 + 36 T + 642 T^{2} + 7560 T^{3} + 64187 T^{4} + 400680 T^{5} + 1803378 T^{6} + 5359572 T^{7} + 7890481 T^{8} \))(\( 1 + 164 T^{2} + 15066 T^{4} + 1018768 T^{6} + 56507555 T^{8} + 2861719312 T^{10} + 118877986746 T^{12} + 3634955225156 T^{14} + 62259690411361 T^{16} \))(\( 1 + 236 T^{2} + 25324 T^{4} + 1967208 T^{6} + 148062890 T^{8} + 10417342148 T^{10} + 635434140816 T^{12} + 36036607646708 T^{14} + 1967961208523651 T^{16} + 101226830879602772 T^{18} + 5013881014859972496 T^{20} + \)\(23\!\cdots\!92\)\( T^{22} + \)\(92\!\cdots\!90\)\( T^{24} + \)\(34\!\cdots\!92\)\( T^{26} + \)\(12\!\cdots\!84\)\( T^{28} + \)\(32\!\cdots\!84\)\( T^{30} + \)\(38\!\cdots\!21\)\( T^{32} \))
$59$ (\( 1 - 10 T - 16 T^{2} + 20 T^{3} + 4075 T^{4} + 1180 T^{5} - 55696 T^{6} - 2053790 T^{7} + 12117361 T^{8} \))(\( ( 1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 - 10 T + 41 T^{2} - 590 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 + 8 T + 5 T^{2} + 472 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4} )^{2} \))(\( 1 - 10 T - 16 T^{2} + 20 T^{3} + 4075 T^{4} + 1180 T^{5} - 55696 T^{6} - 2053790 T^{7} + 12117361 T^{8} \))(\( ( 1 - 116 T^{2} + 9975 T^{4} - 403796 T^{6} + 12117361 T^{8} )^{2} \))(\( ( 1 - 10 T - 96 T^{2} + 1492 T^{3} + 4998 T^{4} - 116970 T^{5} + 158320 T^{6} + 3153670 T^{7} - 19452177 T^{8} + 186066530 T^{9} + 551111920 T^{10} - 24023181630 T^{11} + 60562570278 T^{12} + 1066667054108 T^{13} - 4049331229536 T^{14} - 24886514848190 T^{15} + 146830437604321 T^{16} )^{2} \))
$61$ (\( ( 1 + 4 T - 45 T^{2} + 244 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 + 7 T - 12 T^{2} + 427 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 - 13 T + 61 T^{2} )^{2}( 1 - T + 61 T^{2} )^{2} \))(\( ( 1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 + 4 T - 45 T^{2} + 244 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 + 8 T - 2 T^{2} - 448 T^{3} - 3269 T^{4} - 27328 T^{5} - 7442 T^{6} + 1815848 T^{7} + 13845841 T^{8} )^{2} \))(\( ( 1 - 5 T - 36 T^{2} - 305 T^{3} + 3721 T^{4} )^{8} \))
$67$ (\( 1 - 30 T + 473 T^{2} - 5190 T^{3} + 45540 T^{4} - 347730 T^{5} + 2123297 T^{6} - 9022890 T^{7} + 20151121 T^{8} \))(\( 1 + 85 T^{2} + 2736 T^{4} + 381565 T^{6} + 20151121 T^{8} \))(\( ( 1 - 13 T^{2} + 4489 T^{4} )( 1 + 122 T^{2} + 4489 T^{4} ) \))(\( 1 + 85 T^{2} + 2736 T^{4} + 381565 T^{6} + 20151121 T^{8} \))(\( ( 1 - 13 T^{2} + 4489 T^{4} )( 1 + 122 T^{2} + 4489 T^{4} ) \))(\( 1 + 30 T + 473 T^{2} + 5190 T^{3} + 45540 T^{4} + 347730 T^{5} + 2123297 T^{6} + 9022890 T^{7} + 20151121 T^{8} \))(\( 1 - 58 T^{2} - 5807 T^{4} - 11194 T^{6} + 48855124 T^{8} - 50249866 T^{10} - 117017559647 T^{12} - 5246586165802 T^{14} + 406067677556641 T^{16} \))(\( 1 + 501 T^{2} + 138952 T^{4} + 26939913 T^{6} + 4029956509 T^{8} + 487571679936 T^{10} + 49074273490354 T^{12} + 4173403064575434 T^{14} + 302514422246172016 T^{16} + 18734406356879123226 T^{18} + \)\(98\!\cdots\!34\)\( T^{20} + \)\(44\!\cdots\!84\)\( T^{22} + \)\(16\!\cdots\!69\)\( T^{24} + \)\(49\!\cdots\!37\)\( T^{26} + \)\(11\!\cdots\!72\)\( T^{28} + \)\(18\!\cdots\!29\)\( T^{30} + \)\(16\!\cdots\!81\)\( T^{32} \))
$71$ (\( ( 1 - 2 T + 116 T^{2} - 142 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 - 12 T + 71 T^{2} )^{4} \))(\( ( 1 + 12 T + 71 T^{2} )^{4} \))(\( ( 1 - 6 T + 71 T^{2} )^{4} \))(\( ( 1 + 6 T + 71 T^{2} )^{4} \))(\( ( 1 - 2 T + 116 T^{2} - 142 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 + 4 T + 144 T^{2} + 284 T^{3} + 5041 T^{4} )^{4} \))(\( ( 1 - 22 T + 252 T^{2} - 1806 T^{3} + 12966 T^{4} - 128226 T^{5} + 1270332 T^{6} - 7874042 T^{7} + 25411681 T^{8} )^{4} \))
$73$ (\( 1 - 30 T + 505 T^{2} - 6150 T^{3} + 58596 T^{4} - 448950 T^{5} + 2691145 T^{6} - 11670510 T^{7} + 28398241 T^{8} \))(\( 1 + 25 T^{2} - 4704 T^{4} + 133225 T^{6} + 28398241 T^{8} \))(\( 1 + 25 T^{2} - 4704 T^{4} + 133225 T^{6} + 28398241 T^{8} \))(\( 1 + 145 T^{2} + 15696 T^{4} + 772705 T^{6} + 28398241 T^{8} \))(\( 1 + 137 T^{2} + 13440 T^{4} + 730073 T^{6} + 28398241 T^{8} \))(\( 1 + 30 T + 505 T^{2} + 6150 T^{3} + 58596 T^{4} + 448950 T^{5} + 2691145 T^{6} + 11670510 T^{7} + 28398241 T^{8} \))(\( 1 + 142 T^{2} + 9465 T^{4} + 5822 T^{6} - 21383596 T^{8} + 31025438 T^{10} + 268789351065 T^{12} + 21489460133038 T^{14} + 806460091894081 T^{16} \))(\( 1 + 324 T^{2} + 50502 T^{4} + 5409800 T^{6} + 473911185 T^{8} + 35078829272 T^{10} + 2162596802518 T^{12} + 122333701830108 T^{14} + 7885309757584996 T^{16} + 651916297052645532 T^{18} + 61413945183735570838 T^{20} + \)\(53\!\cdots\!08\)\( T^{22} + \)\(38\!\cdots\!85\)\( T^{24} + \)\(23\!\cdots\!00\)\( T^{26} + \)\(11\!\cdots\!42\)\( T^{28} + \)\(39\!\cdots\!16\)\( T^{30} + \)\(65\!\cdots\!61\)\( T^{32} \))
$79$ (\( 1 - 6 T - 23 T^{2} + 594 T^{3} - 5604 T^{4} + 46926 T^{5} - 143543 T^{6} - 2958234 T^{7} + 38950081 T^{8} \))(\( ( 1 - 4 T + 79 T^{2} )^{2}( 1 + 17 T + 79 T^{2} )^{2} \))(\( ( 1 + 7 T - 30 T^{2} + 553 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} )^{2} \))(\( 1 - 6 T - 23 T^{2} + 594 T^{3} - 5604 T^{4} + 46926 T^{5} - 143543 T^{6} - 2958234 T^{7} + 38950081 T^{8} \))(\( ( 1 - 2 T - 123 T^{2} + 62 T^{3} + 9572 T^{4} + 4898 T^{5} - 767643 T^{6} - 986078 T^{7} + 38950081 T^{8} )^{2} \))(\( ( 1 - 8 T - 182 T^{2} + 1560 T^{3} + 19865 T^{4} - 148676 T^{5} - 1461174 T^{6} + 5521756 T^{7} + 104973092 T^{8} + 436218724 T^{9} - 9119186934 T^{10} - 73303066364 T^{11} + 773743359065 T^{12} + 4800207982440 T^{13} - 44241916904822 T^{14} - 153631271889272 T^{15} + 1517108809906561 T^{16} )^{2} \))
$83$ (\( 1 - 20 T^{2} + 8586 T^{4} - 137780 T^{6} + 47458321 T^{8} \))(\( ( 1 - 22 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 30 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 162 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 130 T^{2} + 6889 T^{4} )^{2} \))(\( 1 - 20 T^{2} + 8586 T^{4} - 137780 T^{6} + 47458321 T^{8} \))(\( ( 1 - 296 T^{2} + 35554 T^{4} - 2039144 T^{6} + 47458321 T^{8} )^{2} \))(\( ( 1 - 436 T^{2} + 89684 T^{4} - 11804316 T^{6} + 1127629270 T^{8} - 81319932924 T^{10} + 4256252060564 T^{12} - 142546002788884 T^{14} + 2252292232139041 T^{16} )^{2} \))
$89$ (\( 1 - 6 T - 4 T^{2} + 828 T^{3} - 9525 T^{4} + 73692 T^{5} - 31684 T^{6} - 4229814 T^{7} + 62742241 T^{8} \))(\( ( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 12 T + 55 T^{2} + 1068 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 16 T + 167 T^{2} - 1424 T^{3} + 7921 T^{4} )^{2} \))(\( 1 - 6 T - 4 T^{2} + 828 T^{3} - 9525 T^{4} + 73692 T^{5} - 31684 T^{6} - 4229814 T^{7} + 62742241 T^{8} \))(\( ( 1 + 16 T + 32 T^{2} + 736 T^{3} + 19471 T^{4} + 65504 T^{5} + 253472 T^{6} + 11279504 T^{7} + 62742241 T^{8} )^{2} \))(\( ( 1 + 10 T - 260 T^{2} - 1572 T^{3} + 55478 T^{4} + 195874 T^{5} - 7285344 T^{6} - 5340058 T^{7} + 785732359 T^{8} - 475265162 T^{9} - 57707209824 T^{10} + 138085097906 T^{11} + 3480814046198 T^{12} - 8778141453828 T^{13} - 129215135649860 T^{14} + 442313348955290 T^{15} + 3936588805702081 T^{16} )^{2} \))
$97$ (\( 1 - 164 T^{2} + 13254 T^{4} - 1543076 T^{6} + 88529281 T^{8} \))(\( ( 1 - 94 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 113 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 158 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 158 T^{2} + 9409 T^{4} )^{2} \))(\( 1 - 164 T^{2} + 13254 T^{4} - 1543076 T^{6} + 88529281 T^{8} \))(\( ( 1 - 244 T^{2} + 31654 T^{4} - 2295796 T^{6} + 88529281 T^{8} )^{2} \))(\( ( 1 - 516 T^{2} + 120042 T^{4} - 17454416 T^{6} + 1892032563 T^{8} - 164228600144 T^{10} + 10627231949802 T^{12} - 429813554543364 T^{14} + 7837433594376961 T^{16} )^{2} \))
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