Properties

Label 525.2.q
Level 525
Weight 2
Character orbit q
Rep. character \(\chi_{525}(299,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 88
Newform subspaces 7
Sturm bound 160
Trace bound 19

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Defining parameters

Level: \( N \) = \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 525.q (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 105 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 7 \)
Sturm bound: \(160\)
Trace bound: \(19\)
Distinguishing \(T_p\): \(2\), \(11\), \(13\)

Dimensions

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

Total New Old
Modular forms 184 104 80
Cusp forms 136 88 48
Eisenstein series 48 16 32

Trace form

\( 88q - 36q^{4} + 10q^{9} + O(q^{10}) \) \( 88q - 36q^{4} + 10q^{9} - 28q^{16} + 18q^{19} - 72q^{24} - 24q^{31} - 96q^{36} + 6q^{39} - 56q^{46} + 2q^{49} + 18q^{51} + 102q^{54} - 78q^{61} + 32q^{64} - 12q^{66} - 32q^{79} + 14q^{81} - 30q^{84} + 18q^{91} + 84q^{94} + 234q^{96} + 20q^{99} + O(q^{100}) \)

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.q.a \(4\) \(4.192\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q+(-\zeta_{12}-\zeta_{12}^{3})q^{2}+(2\zeta_{12}-\zeta_{12}^{3})q^{3}+\cdots\)
525.2.q.b \(4\) \(4.192\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q+(-\zeta_{12}-\zeta_{12}^{3})q^{2}+(\zeta_{12}+\zeta_{12}^{3})q^{3}+\cdots\)
525.2.q.c \(4\) \(4.192\) \(\Q(\zeta_{12})\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(q+(-\zeta_{12}-\zeta_{12}^{3})q^{3}+2\zeta_{12}^{2}q^{4}+\cdots\)
525.2.q.d \(4\) \(4.192\) \(\Q(\zeta_{12})\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(q+(-\zeta_{12}-\zeta_{12}^{3})q^{3}+2\zeta_{12}^{2}q^{4}+\cdots\)
525.2.q.e \(16\) \(4.192\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{1}q^{2}+(-\beta _{1}+\beta _{8}+\beta _{14})q^{3}+(\beta _{5}+\cdots)q^{4}+\cdots\)
525.2.q.f \(16\) \(4.192\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{1}q^{2}+(-\beta _{1}+\beta _{7}-\beta _{8}+\beta _{12}+\cdots)q^{3}+\cdots\)
525.2.q.g \(40\) \(4.192\) None \(0\) \(0\) \(0\) \(0\)

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

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - T^{2} - 3 T^{4} - 4 T^{6} + 16 T^{8} \))(\( 1 - T^{2} - 3 T^{4} - 4 T^{6} + 16 T^{8} \))(\( ( 1 - 2 T^{2} + 4 T^{4} )^{2} \))(\( ( 1 - 2 T^{2} + 4 T^{4} )^{2} \))(\( 1 - 5 T^{2} + 9 T^{4} + 8 T^{6} - 64 T^{8} + 120 T^{10} - 32 T^{12} - 368 T^{14} + 1104 T^{16} - 1472 T^{18} - 512 T^{20} + 7680 T^{22} - 16384 T^{24} + 8192 T^{26} + 36864 T^{28} - 81920 T^{30} + 65536 T^{32} \))(\( 1 - 5 T^{2} + 9 T^{4} + 8 T^{6} - 64 T^{8} + 120 T^{10} - 32 T^{12} - 368 T^{14} + 1104 T^{16} - 1472 T^{18} - 512 T^{20} + 7680 T^{22} - 16384 T^{24} + 8192 T^{26} + 36864 T^{28} - 81920 T^{30} + 65536 T^{32} \))
$3$ (\( ( 1 - 3 T^{2} )^{2} \))(\( 1 + 3 T^{2} + 9 T^{4} \))(\( 1 + 3 T^{2} + 9 T^{4} \))(\( 1 + 3 T^{2} + 9 T^{4} \))(\( 1 + 4 T^{2} - 6 T^{4} - 32 T^{6} - 29 T^{8} - 288 T^{10} - 486 T^{12} + 2916 T^{14} + 6561 T^{16} \))(\( 1 - 5 T^{2} + 33 T^{4} - 110 T^{6} + 430 T^{8} - 990 T^{10} + 2673 T^{12} - 3645 T^{14} + 6561 T^{16} \))
$5$ 1
$7$ (\( 1 + 11 T^{2} + 49 T^{4} \))(\( 1 + 11 T^{2} + 49 T^{4} \))(\( 1 + 2 T^{2} + 49 T^{4} \))(\( 1 - 13 T^{2} + 49 T^{4} \))(\( 1 - 4 T^{2} - 26 T^{4} - 244 T^{6} + 3907 T^{8} - 11956 T^{10} - 62426 T^{12} - 470596 T^{14} + 5764801 T^{16} \))(\( 1 - 4 T^{2} - 26 T^{4} - 244 T^{6} + 3907 T^{8} - 11956 T^{10} - 62426 T^{12} - 470596 T^{14} + 5764801 T^{16} \))
$11$ (\( ( 1 - 6 T + 23 T^{2} - 66 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 + 6 T + 23 T^{2} + 66 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 + 11 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 + 11 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 + 16 T^{2} - 2 T^{4} - 30 T^{5} + 268 T^{6} - 1548 T^{7} + 21079 T^{8} - 17028 T^{9} + 32428 T^{10} - 39930 T^{11} - 29282 T^{12} + 28344976 T^{14} + 214358881 T^{16} )^{2} \))(\( ( 1 + 16 T^{2} - 2 T^{4} + 30 T^{5} + 268 T^{6} + 1548 T^{7} + 21079 T^{8} + 17028 T^{9} + 32428 T^{10} + 39930 T^{11} - 29282 T^{12} + 28344976 T^{14} + 214358881 T^{16} )^{2} \))
$13$ (\( ( 1 + 14 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 + 14 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 22 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 + 23 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 + 83 T^{2} + 3217 T^{4} + 76058 T^{6} + 1197778 T^{8} + 12853802 T^{10} + 91880737 T^{12} + 400625147 T^{14} + 815730721 T^{16} )^{2} \))(\( ( 1 + 83 T^{2} + 3217 T^{4} + 76058 T^{6} + 1197778 T^{8} + 12853802 T^{10} + 91880737 T^{12} + 400625147 T^{14} + 815730721 T^{16} )^{2} \))
$17$ (\( 1 - 2 T^{2} - 285 T^{4} - 578 T^{6} + 83521 T^{8} \))(\( 1 - 2 T^{2} - 285 T^{4} - 578 T^{6} + 83521 T^{8} \))(\( ( 1 + 17 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 + 17 T^{2} + 289 T^{4} )^{2} \))(\( 1 + 76 T^{2} + 3024 T^{4} + 74528 T^{6} + 1179962 T^{8} + 10465632 T^{10} - 4424000 T^{12} - 1753904276 T^{14} - 37267323453 T^{16} - 506878335764 T^{18} - 369496904000 T^{20} + 252614914528608 T^{22} + 8231128701597242 T^{24} + 150247993412663072 T^{26} + 1761849645382797264 T^{28} + 12796714818514470604 T^{30} + 48661191875666868481 T^{32} \))(\( 1 + 76 T^{2} + 3024 T^{4} + 74528 T^{6} + 1179962 T^{8} + 10465632 T^{10} - 4424000 T^{12} - 1753904276 T^{14} - 37267323453 T^{16} - 506878335764 T^{18} - 369496904000 T^{20} + 252614914528608 T^{22} + 8231128701597242 T^{24} + 150247993412663072 T^{26} + 1761849645382797264 T^{28} + 12796714818514470604 T^{30} + 48661191875666868481 T^{32} \))
$19$ (\( ( 1 - 12 T + 67 T^{2} - 228 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 - 12 T + 67 T^{2} - 228 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 - T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2} \))(\( ( 1 - 8 T + 19 T^{2} )^{2}( 1 - T + 19 T^{2} )^{2} \))(\( ( 1 + 9 T + 100 T^{2} + 657 T^{3} + 4723 T^{4} + 26244 T^{5} + 148996 T^{6} + 704196 T^{7} + 3331528 T^{8} + 13379724 T^{9} + 53787556 T^{10} + 180007596 T^{11} + 615506083 T^{12} + 1626797043 T^{13} + 4704588100 T^{14} + 8044845651 T^{15} + 16983563041 T^{16} )^{2} \))(\( ( 1 + 9 T + 100 T^{2} + 657 T^{3} + 4723 T^{4} + 26244 T^{5} + 148996 T^{6} + 704196 T^{7} + 3331528 T^{8} + 13379724 T^{9} + 53787556 T^{10} + 180007596 T^{11} + 615506083 T^{12} + 1626797043 T^{13} + 4704588100 T^{14} + 8044845651 T^{15} + 16983563041 T^{16} )^{2} \))
$23$ (\( 1 - 43 T^{2} + 1320 T^{4} - 22747 T^{6} + 279841 T^{8} \))(\( 1 - 43 T^{2} + 1320 T^{4} - 22747 T^{6} + 279841 T^{8} \))(\( ( 1 - 23 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 - 23 T^{2} + 529 T^{4} )^{2} \))(\( 1 - 53 T^{2} + 399 T^{4} + 18308 T^{6} - 55297 T^{8} - 3303501 T^{10} - 170695622 T^{12} - 2962951637 T^{14} + 281105182152 T^{16} - 1567401415973 T^{18} - 47767633556102 T^{20} - 489036707347389 T^{22} - 4330362553083457 T^{24} + 758436567299485892 T^{26} + 8743935148376108079 T^{28} - \)\(61\!\cdots\!77\)\( T^{30} + \)\(61\!\cdots\!61\)\( T^{32} \))(\( 1 - 53 T^{2} + 399 T^{4} + 18308 T^{6} - 55297 T^{8} - 3303501 T^{10} - 170695622 T^{12} - 2962951637 T^{14} + 281105182152 T^{16} - 1567401415973 T^{18} - 47767633556102 T^{20} - 489036707347389 T^{22} - 4330362553083457 T^{24} + 758436567299485892 T^{26} + 8743935148376108079 T^{28} - \)\(61\!\cdots\!77\)\( T^{30} + \)\(61\!\cdots\!61\)\( T^{32} \))
$29$ (\( ( 1 - 55 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 55 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 29 T^{2} )^{4} \))(\( ( 1 - 29 T^{2} )^{4} \))(\( ( 1 - 53 T^{2} + 3250 T^{4} - 128951 T^{6} + 4063174 T^{8} - 108447791 T^{10} + 2298663250 T^{12} - 31525636013 T^{14} + 500246412961 T^{16} )^{2} \))(\( ( 1 - 53 T^{2} + 3250 T^{4} - 128951 T^{6} + 4063174 T^{8} - 108447791 T^{10} + 2298663250 T^{12} - 31525636013 T^{14} + 500246412961 T^{16} )^{2} \))
$31$ (\( ( 1 + 6 T + 43 T^{2} + 186 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 + 6 T + 43 T^{2} + 186 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 + 4 T + 31 T^{2} )^{2}( 1 + 11 T + 31 T^{2} )^{2} \))(\( ( 1 - 11 T + 31 T^{2} )^{2}( 1 - 4 T + 31 T^{2} )^{2} \))(\( ( 1 + 21 T + 262 T^{2} + 2415 T^{3} + 17293 T^{4} + 101304 T^{5} + 505090 T^{6} + 2328618 T^{7} + 11769748 T^{8} + 72187158 T^{9} + 485391490 T^{10} + 3017947464 T^{11} + 15970448653 T^{12} + 69139399665 T^{13} + 232525964422 T^{14} + 577764896331 T^{15} + 852891037441 T^{16} )^{2} \))(\( ( 1 + 21 T + 262 T^{2} + 2415 T^{3} + 17293 T^{4} + 101304 T^{5} + 505090 T^{6} + 2328618 T^{7} + 11769748 T^{8} + 72187158 T^{9} + 485391490 T^{10} + 3017947464 T^{11} + 15970448653 T^{12} + 69139399665 T^{13} + 232525964422 T^{14} + 577764896331 T^{15} + 852891037441 T^{16} )^{2} \))
$37$ (\( 1 + 58 T^{2} + 1995 T^{4} + 79402 T^{6} + 1874161 T^{8} \))(\( 1 + 58 T^{2} + 1995 T^{4} + 79402 T^{6} + 1874161 T^{8} \))(\( ( 1 - 73 T^{2} + 1369 T^{4} )( 1 + 26 T^{2} + 1369 T^{4} ) \))(\( ( 1 + 26 T^{2} + 1369 T^{4} )( 1 + 47 T^{2} + 1369 T^{4} ) \))(\( 1 + 97 T^{2} + 5532 T^{4} + 186305 T^{6} + 3327797 T^{8} + 12794040 T^{10} + 1243533586 T^{12} + 183679377646 T^{14} + 10522943320200 T^{16} + 251457067997374 T^{18} + 2330582149071346 T^{20} + 32826006305802360 T^{22} + 11688818589319942037 T^{24} + \)\(89\!\cdots\!45\)\( T^{26} + \)\(36\!\cdots\!92\)\( T^{28} + \)\(87\!\cdots\!33\)\( T^{30} + \)\(12\!\cdots\!41\)\( T^{32} \))(\( 1 + 97 T^{2} + 5532 T^{4} + 186305 T^{6} + 3327797 T^{8} + 12794040 T^{10} + 1243533586 T^{12} + 183679377646 T^{14} + 10522943320200 T^{16} + 251457067997374 T^{18} + 2330582149071346 T^{20} + 32826006305802360 T^{22} + 11688818589319942037 T^{24} + \)\(89\!\cdots\!45\)\( T^{26} + \)\(36\!\cdots\!92\)\( T^{28} + \)\(87\!\cdots\!33\)\( T^{30} + \)\(12\!\cdots\!41\)\( T^{32} \))
$41$ (\( ( 1 - 3 T + 41 T^{2} )^{4} \))(\( ( 1 + 3 T + 41 T^{2} )^{4} \))(\( ( 1 + 41 T^{2} )^{4} \))(\( ( 1 + 41 T^{2} )^{4} \))(\( ( 1 - 15 T + 218 T^{2} - 1791 T^{3} + 14136 T^{4} - 73431 T^{5} + 366458 T^{6} - 1033815 T^{7} + 2825761 T^{8} )^{4} \))(\( ( 1 + 15 T + 218 T^{2} + 1791 T^{3} + 14136 T^{4} + 73431 T^{5} + 366458 T^{6} + 1033815 T^{7} + 2825761 T^{8} )^{4} \))
$43$ (\( ( 1 - 85 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 85 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 61 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 61 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 304 T^{2} + 41758 T^{4} - 3395548 T^{6} + 179098699 T^{8} - 6278368252 T^{10} + 142762292158 T^{12} - 1921694366896 T^{14} + 11688200277601 T^{16} )^{2} \))(\( ( 1 - 304 T^{2} + 41758 T^{4} - 3395548 T^{6} + 179098699 T^{8} - 6278368252 T^{10} + 142762292158 T^{12} - 1921694366896 T^{14} + 11688200277601 T^{16} )^{2} \))
$47$ (\( ( 1 + 47 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 47 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 47 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 47 T^{2} + 2209 T^{4} )^{2} \))(\( 1 + 268 T^{2} + 36732 T^{4} + 3628040 T^{6} + 294880202 T^{8} + 20635855860 T^{10} + 1272375350416 T^{12} + 70342526859604 T^{14} + 3494142383383395 T^{16} + 155386641832865236 T^{18} + 6208785822293297296 T^{20} + \)\(22\!\cdots\!40\)\( T^{22} + \)\(70\!\cdots\!22\)\( T^{24} + \)\(19\!\cdots\!60\)\( T^{26} + \)\(42\!\cdots\!12\)\( T^{28} + \)\(68\!\cdots\!92\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \))(\( 1 + 268 T^{2} + 36732 T^{4} + 3628040 T^{6} + 294880202 T^{8} + 20635855860 T^{10} + 1272375350416 T^{12} + 70342526859604 T^{14} + 3494142383383395 T^{16} + 155386641832865236 T^{18} + 6208785822293297296 T^{20} + \)\(22\!\cdots\!40\)\( T^{22} + \)\(70\!\cdots\!22\)\( T^{24} + \)\(19\!\cdots\!60\)\( T^{26} + \)\(42\!\cdots\!12\)\( T^{28} + \)\(68\!\cdots\!92\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \))
$53$ (\( ( 1 - 53 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 53 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 53 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 53 T^{2} + 2809 T^{4} )^{2} \))(\( 1 - 104 T^{2} + 3204 T^{4} + 8144 T^{6} - 2651158 T^{8} - 214376472 T^{10} + 18555084688 T^{12} + 949831131304 T^{14} - 128029390162317 T^{16} + 2668075647832936 T^{18} + 146408543184054928 T^{20} - 4751517542968956888 T^{22} - \)\(16\!\cdots\!38\)\( T^{24} + \)\(14\!\cdots\!56\)\( T^{26} + \)\(15\!\cdots\!64\)\( T^{28} - \)\(14\!\cdots\!76\)\( T^{30} + \)\(38\!\cdots\!21\)\( T^{32} \))(\( 1 - 104 T^{2} + 3204 T^{4} + 8144 T^{6} - 2651158 T^{8} - 214376472 T^{10} + 18555084688 T^{12} + 949831131304 T^{14} - 128029390162317 T^{16} + 2668075647832936 T^{18} + 146408543184054928 T^{20} - 4751517542968956888 T^{22} - \)\(16\!\cdots\!38\)\( T^{24} + \)\(14\!\cdots\!56\)\( T^{26} + \)\(15\!\cdots\!64\)\( T^{28} - \)\(14\!\cdots\!76\)\( T^{30} + \)\(38\!\cdots\!21\)\( T^{32} \))
$59$ (\( ( 1 - 59 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 - 59 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 - 59 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 - 59 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 12 T - 80 T^{2} - 1164 T^{3} + 7690 T^{4} + 80082 T^{5} - 434420 T^{6} - 1772232 T^{7} + 28861927 T^{8} - 104561688 T^{9} - 1512216020 T^{10} + 16447161078 T^{11} + 93182506090 T^{12} - 832171884036 T^{13} - 3374442691280 T^{14} + 29863817817828 T^{15} + 146830437604321 T^{16} )^{2} \))(\( ( 1 - 12 T - 80 T^{2} + 1164 T^{3} + 7690 T^{4} - 80082 T^{5} - 434420 T^{6} + 1772232 T^{7} + 28861927 T^{8} + 104561688 T^{9} - 1512216020 T^{10} - 16447161078 T^{11} + 93182506090 T^{12} + 832171884036 T^{13} - 3374442691280 T^{14} - 29863817817828 T^{15} + 146830437604321 T^{16} )^{2} \))
$61$ (\( ( 1 + 9 T + 88 T^{2} + 549 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 + 9 T + 88 T^{2} + 549 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 - 14 T + 61 T^{2} )^{2}( 1 - 13 T + 61 T^{2} )^{2} \))(\( ( 1 - 13 T + 61 T^{2} )^{2}( 1 + T + 61 T^{2} )^{2} \))(\( ( 1 - 15 T + 223 T^{2} - 2220 T^{3} + 19711 T^{4} - 141723 T^{5} + 816310 T^{6} - 5175267 T^{7} + 31433836 T^{8} - 315691287 T^{9} + 3037489510 T^{10} - 32168428263 T^{11} + 272915371951 T^{12} - 1875003788220 T^{13} + 11489043482503 T^{14} - 47141142540315 T^{15} + 191707312997281 T^{16} )^{2} \))(\( ( 1 - 15 T + 223 T^{2} - 2220 T^{3} + 19711 T^{4} - 141723 T^{5} + 816310 T^{6} - 5175267 T^{7} + 31433836 T^{8} - 315691287 T^{9} + 3037489510 T^{10} - 32168428263 T^{11} + 272915371951 T^{12} - 1875003788220 T^{13} + 11489043482503 T^{14} - 47141142540315 T^{15} + 191707312997281 T^{16} )^{2} \))
$67$ (\( 1 - 35 T^{2} - 3264 T^{4} - 157115 T^{6} + 20151121 T^{8} \))(\( 1 - 35 T^{2} - 3264 T^{4} - 157115 T^{6} + 20151121 T^{8} \))(\( ( 1 - 109 T^{2} + 4489 T^{4} )( 1 - 13 T^{2} + 4489 T^{4} ) \))(\( ( 1 - 109 T^{2} + 4489 T^{4} )( 1 + 122 T^{2} + 4489 T^{4} ) \))(\( 1 + 484 T^{2} + 128562 T^{4} + 23999600 T^{6} + 3478000361 T^{8} + 410568162660 T^{10} + 40599889512562 T^{12} + 3416547531270040 T^{14} + 246775367829948324 T^{16} + 15336881867871209560 T^{18} + \)\(81\!\cdots\!02\)\( T^{20} + \)\(37\!\cdots\!40\)\( T^{22} + \)\(14\!\cdots\!01\)\( T^{24} + \)\(43\!\cdots\!00\)\( T^{26} + \)\(10\!\cdots\!82\)\( T^{28} + \)\(17\!\cdots\!36\)\( T^{30} + \)\(16\!\cdots\!81\)\( T^{32} \))(\( 1 + 484 T^{2} + 128562 T^{4} + 23999600 T^{6} + 3478000361 T^{8} + 410568162660 T^{10} + 40599889512562 T^{12} + 3416547531270040 T^{14} + 246775367829948324 T^{16} + 15336881867871209560 T^{18} + \)\(81\!\cdots\!02\)\( T^{20} + \)\(37\!\cdots\!40\)\( T^{22} + \)\(14\!\cdots\!01\)\( T^{24} + \)\(43\!\cdots\!00\)\( T^{26} + \)\(10\!\cdots\!82\)\( T^{28} + \)\(17\!\cdots\!36\)\( T^{30} + \)\(16\!\cdots\!81\)\( T^{32} \))
$71$ (\( ( 1 - 94 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 94 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 71 T^{2} )^{4} \))(\( ( 1 - 71 T^{2} )^{4} \))(\( ( 1 - 464 T^{2} + 99532 T^{4} - 12936548 T^{6} + 1114829374 T^{8} - 65213138468 T^{10} + 2529275433292 T^{12} - 59438531739344 T^{14} + 645753531245761 T^{16} )^{2} \))(\( ( 1 - 464 T^{2} + 99532 T^{4} - 12936548 T^{6} + 1114829374 T^{8} - 65213138468 T^{10} + 2529275433292 T^{12} - 59438531739344 T^{14} + 645753531245761 T^{16} )^{2} \))
$73$ (\( 1 - 134 T^{2} + 12627 T^{4} - 714086 T^{6} + 28398241 T^{8} \))(\( 1 - 134 T^{2} + 12627 T^{4} - 714086 T^{6} + 28398241 T^{8} \))(\( ( 1 - 97 T^{2} + 5329 T^{4} )( 1 - 46 T^{2} + 5329 T^{4} ) \))(\( ( 1 - 46 T^{2} + 5329 T^{4} )( 1 + 143 T^{2} + 5329 T^{4} ) \))(\( 1 - 335 T^{2} + 63708 T^{4} - 7606879 T^{6} + 587709533 T^{8} - 19770442248 T^{10} - 1697574900854 T^{12} + 348751235496070 T^{14} - 32202831466789560 T^{16} + 1858495333958557030 T^{18} - 48208141150002997814 T^{20} - \)\(29\!\cdots\!72\)\( T^{22} + \)\(47\!\cdots\!73\)\( T^{24} - \)\(32\!\cdots\!71\)\( T^{26} + \)\(14\!\cdots\!68\)\( T^{28} - \)\(40\!\cdots\!15\)\( T^{30} + \)\(65\!\cdots\!61\)\( T^{32} \))(\( 1 - 335 T^{2} + 63708 T^{4} - 7606879 T^{6} + 587709533 T^{8} - 19770442248 T^{10} - 1697574900854 T^{12} + 348751235496070 T^{14} - 32202831466789560 T^{16} + 1858495333958557030 T^{18} - 48208141150002997814 T^{20} - \)\(29\!\cdots\!72\)\( T^{22} + \)\(47\!\cdots\!73\)\( T^{24} - \)\(32\!\cdots\!71\)\( T^{26} + \)\(14\!\cdots\!68\)\( T^{28} - \)\(40\!\cdots\!15\)\( T^{30} + \)\(65\!\cdots\!61\)\( T^{32} \))
$79$ (\( ( 1 + 16 T + 177 T^{2} + 1264 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 + 16 T + 177 T^{2} + 1264 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 - 13 T + 79 T^{2} )^{2}( 1 - 4 T + 79 T^{2} )^{2} \))(\( ( 1 - 4 T + 79 T^{2} )^{2}( 1 + 17 T + 79 T^{2} )^{2} \))(\( ( 1 - 29 T + 294 T^{2} - 1975 T^{3} + 27377 T^{4} - 260496 T^{5} + 598654 T^{6} - 2403434 T^{7} + 77714340 T^{8} - 189871286 T^{9} + 3736199614 T^{10} - 128434687344 T^{11} + 1066336367537 T^{12} - 6077186388025 T^{13} + 71467711923174 T^{14} - 556913360598611 T^{15} + 1517108809906561 T^{16} )^{2} \))(\( ( 1 - 29 T + 294 T^{2} - 1975 T^{3} + 27377 T^{4} - 260496 T^{5} + 598654 T^{6} - 2403434 T^{7} + 77714340 T^{8} - 189871286 T^{9} + 3736199614 T^{10} - 128434687344 T^{11} + 1066336367537 T^{12} - 6077186388025 T^{13} + 71467711923174 T^{14} - 556913360598611 T^{15} + 1517108809906561 T^{16} )^{2} \))
$83$ (\( ( 1 - 85 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 85 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 83 T^{2} )^{4} \))(\( ( 1 - 83 T^{2} )^{4} \))(\( ( 1 - 535 T^{2} + 130150 T^{4} - 19183249 T^{6} + 1908109846 T^{8} - 132153402361 T^{10} + 6176700478150 T^{12} - 174913099752415 T^{14} + 2252292232139041 T^{16} )^{2} \))(\( ( 1 - 535 T^{2} + 130150 T^{4} - 19183249 T^{6} + 1908109846 T^{8} - 132153402361 T^{10} + 6176700478150 T^{12} - 174913099752415 T^{14} + 2252292232139041 T^{16} )^{2} \))
$89$ (\( ( 1 + 3 T - 80 T^{2} + 267 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 3 T - 80 T^{2} - 267 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 89 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 89 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 + 3 T - 53 T^{2} + 2820 T^{3} + 14227 T^{4} - 160275 T^{5} + 3467116 T^{6} + 26593569 T^{7} - 193500020 T^{8} + 2366827641 T^{9} + 27463025836 T^{10} - 112988906475 T^{11} + 892633862707 T^{12} + 15747047646180 T^{13} - 26340008420933 T^{14} + 132694004686587 T^{15} + 3936588805702081 T^{16} )^{2} \))(\( ( 1 - 3 T - 53 T^{2} - 2820 T^{3} + 14227 T^{4} + 160275 T^{5} + 3467116 T^{6} - 26593569 T^{7} - 193500020 T^{8} - 2366827641 T^{9} + 27463025836 T^{10} + 112988906475 T^{11} + 892633862707 T^{12} - 15747047646180 T^{13} - 26340008420933 T^{14} - 132694004686587 T^{15} + 3936588805702081 T^{16} )^{2} \))
$97$ (\( ( 1 + 86 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 + 86 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 + 167 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 + 2 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 + 368 T^{2} + 81676 T^{4} + 12257504 T^{6} + 1385094598 T^{8} + 115330855136 T^{10} + 7230717554956 T^{12} + 306533697813872 T^{14} + 7837433594376961 T^{16} )^{2} \))(\( ( 1 + 368 T^{2} + 81676 T^{4} + 12257504 T^{6} + 1385094598 T^{8} + 115330855136 T^{10} + 7230717554956 T^{12} + 306533697813872 T^{14} + 7837433594376961 T^{16} )^{2} \))
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