Properties

Label 756.2.b
Level 756
Weight 2
Character orbit b
Rep. character \(\chi_{756}(55,\cdot)\)
Character field \(\Q\)
Dimension 64
Newform subspaces 6
Sturm bound 288
Trace bound 14

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Defining parameters

Level: \( N \) = \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 756.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 28 \)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(288\)
Trace bound: \(14\)
Distinguishing \(T_p\): \(5\), \(19\), \(47\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(756, [\chi])\).

Total New Old
Modular forms 156 64 92
Cusp forms 132 64 68
Eisenstein series 24 0 24

Trace form

\( 64q + O(q^{10}) \) \( 64q + 16q^{16} + 4q^{22} - 56q^{25} - 2q^{28} + 8q^{37} + 36q^{46} - 8q^{49} + 44q^{58} + 24q^{64} - 78q^{70} + 8q^{85} - 28q^{88} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(756, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
756.2.b.a \(4\) \(6.037\) \(\Q(\sqrt{-2}, \sqrt{7})\) \(\Q(\sqrt{-21}) \) \(0\) \(0\) \(0\) \(0\) \(q-\beta _{2}q^{2}-2q^{4}+(\beta _{1}-\beta _{2})q^{5}-\beta _{3}q^{7}+\cdots\)
756.2.b.b \(4\) \(6.037\) \(\Q(\sqrt{-2}, \sqrt{7})\) \(\Q(\sqrt{-21}) \) \(0\) \(0\) \(0\) \(0\) \(q+\beta _{2}q^{2}-2q^{4}+(\beta _{1}-\beta _{2})q^{5}+\beta _{3}q^{7}+\cdots\)
756.2.b.c \(12\) \(6.037\) 12.0.\(\cdots\).1 None \(0\) \(0\) \(0\) \(-2\) \(q+\beta _{7}q^{2}+(1+\beta _{11})q^{4}-\beta _{8}q^{5}+(-\beta _{6}+\cdots)q^{7}+\cdots\)
756.2.b.d \(12\) \(6.037\) 12.0.\(\cdots\).1 None \(0\) \(0\) \(0\) \(2\) \(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}-\beta _{8}q^{5}+(\beta _{6}+\cdots)q^{7}+\cdots\)
756.2.b.e \(16\) \(6.037\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{2}+\beta _{2}q^{4}-\beta _{11}q^{5}+\beta _{6}q^{7}+\cdots\)
756.2.b.f \(16\) \(6.037\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{14}q^{2}+\beta _{8}q^{4}+\beta _{10}q^{5}+\beta _{6}q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(756, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(756, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + 2 T^{2} )^{2} \))(\( ( 1 + 2 T^{2} )^{2} \))(\( 1 - 4 T^{2} + 11 T^{4} - 26 T^{6} + 44 T^{8} - 64 T^{10} + 64 T^{12} \))(\( 1 - 4 T^{2} + 11 T^{4} - 26 T^{6} + 44 T^{8} - 64 T^{10} + 64 T^{12} \))(\( 1 - T^{4} - 4 T^{6} - 4 T^{8} - 16 T^{10} - 16 T^{12} + 256 T^{16} \))(\( 1 - T^{4} - 4 T^{6} - 4 T^{8} - 16 T^{10} - 16 T^{12} + 256 T^{16} \))
$3$ 1
$5$ (\( 1 - 4 T^{2} - 9 T^{4} - 100 T^{6} + 625 T^{8} \))(\( 1 - 4 T^{2} - 9 T^{4} - 100 T^{6} + 625 T^{8} \))(\( ( 1 - 16 T^{2} + 154 T^{4} - 918 T^{6} + 3850 T^{8} - 10000 T^{10} + 15625 T^{12} )^{2} \))(\( ( 1 - 16 T^{2} + 154 T^{4} - 918 T^{6} + 3850 T^{8} - 10000 T^{10} + 15625 T^{12} )^{2} \))(\( ( 1 - 15 T^{2} + 109 T^{4} - 642 T^{6} + 3506 T^{8} - 16050 T^{10} + 68125 T^{12} - 234375 T^{14} + 390625 T^{16} )^{2} \))(\( ( 1 - 15 T^{2} + 109 T^{4} - 642 T^{6} + 3506 T^{8} - 16050 T^{10} + 68125 T^{12} - 234375 T^{14} + 390625 T^{16} )^{2} \))
$7$ (\( ( 1 - 7 T^{2} )^{2} \))(\( ( 1 - 7 T^{2} )^{2} \))(\( ( 1 + T + 5 T^{2} + 2 T^{3} + 35 T^{4} + 49 T^{5} + 343 T^{6} )^{2} \))(\( ( 1 - T + 5 T^{2} - 2 T^{3} + 35 T^{4} - 49 T^{5} + 343 T^{6} )^{2} \))(\( 1 + 7 T^{2} + 62 T^{4} - 343 T^{6} - 1662 T^{8} - 16807 T^{10} + 148862 T^{12} + 823543 T^{14} + 5764801 T^{16} \))(\( 1 + 7 T^{2} + 62 T^{4} - 343 T^{6} - 1662 T^{8} - 16807 T^{10} + 148862 T^{12} + 823543 T^{14} + 5764801 T^{16} \))
$11$ (\( 1 + 20 T^{2} + 279 T^{4} + 2420 T^{6} + 14641 T^{8} \))(\( 1 + 20 T^{2} + 279 T^{4} + 2420 T^{6} + 14641 T^{8} \))(\( ( 1 - 40 T^{2} + 722 T^{4} - 8858 T^{6} + 87362 T^{8} - 585640 T^{10} + 1771561 T^{12} )^{2} \))(\( ( 1 - 40 T^{2} + 722 T^{4} - 8858 T^{6} + 87362 T^{8} - 585640 T^{10} + 1771561 T^{12} )^{2} \))(\( ( 1 - 45 T^{2} + 1070 T^{4} - 17563 T^{6} + 218210 T^{8} - 2125123 T^{10} + 15665870 T^{12} - 79720245 T^{14} + 214358881 T^{16} )^{2} \))(\( ( 1 - 45 T^{2} + 1070 T^{4} - 17563 T^{6} + 218210 T^{8} - 2125123 T^{10} + 15665870 T^{12} - 79720245 T^{14} + 214358881 T^{16} )^{2} \))
$13$ (\( ( 1 - 13 T^{2} )^{4} \))(\( ( 1 - 13 T^{2} )^{4} \))(\( ( 1 - 23 T^{2} + 331 T^{4} - 3102 T^{6} + 55939 T^{8} - 656903 T^{10} + 4826809 T^{12} )^{2} \))(\( ( 1 - 23 T^{2} + 331 T^{4} - 3102 T^{6} + 55939 T^{8} - 656903 T^{10} + 4826809 T^{12} )^{2} \))(\( ( 1 - 43 T^{2} + 1050 T^{4} - 19597 T^{6} + 291626 T^{8} - 3311893 T^{10} + 29989050 T^{12} - 207552787 T^{14} + 815730721 T^{16} )^{2} \))(\( ( 1 - 43 T^{2} + 1050 T^{4} - 19597 T^{6} + 291626 T^{8} - 3311893 T^{10} + 29989050 T^{12} - 207552787 T^{14} + 815730721 T^{16} )^{2} \))
$17$ (\( ( 1 - 20 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 20 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 50 T^{2} + 1419 T^{4} - 28884 T^{6} + 410091 T^{8} - 4176050 T^{10} + 24137569 T^{12} )^{2} \))(\( ( 1 - 50 T^{2} + 1419 T^{4} - 28884 T^{6} + 410091 T^{8} - 4176050 T^{10} + 24137569 T^{12} )^{2} \))(\( ( 1 - 55 T^{2} + 2117 T^{4} - 53882 T^{6} + 1068186 T^{8} - 15571898 T^{10} + 176813957 T^{12} - 1327566295 T^{14} + 6975757441 T^{16} )^{2} \))(\( ( 1 - 55 T^{2} + 2117 T^{4} - 53882 T^{6} + 1068186 T^{8} - 15571898 T^{10} + 176813957 T^{12} - 1327566295 T^{14} + 6975757441 T^{16} )^{2} \))
$19$ (\( ( 1 - 12 T + 67 T^{2} - 228 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 + 12 T + 67 T^{2} + 228 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 - 3 T + 46 T^{2} - 117 T^{3} + 874 T^{4} - 1083 T^{5} + 6859 T^{6} )^{4} \))(\( ( 1 + 3 T + 46 T^{2} + 117 T^{3} + 874 T^{4} + 1083 T^{5} + 6859 T^{6} )^{4} \))(\( ( 1 + 84 T^{2} + 3780 T^{4} + 114828 T^{6} + 2535190 T^{8} + 41452908 T^{10} + 492613380 T^{12} + 3951854004 T^{14} + 16983563041 T^{16} )^{2} \))(\( ( 1 + 84 T^{2} + 3780 T^{4} + 114828 T^{6} + 2535190 T^{8} + 41452908 T^{10} + 492613380 T^{12} + 3951854004 T^{14} + 16983563041 T^{16} )^{2} \))
$23$ (\( 1 - 4 T^{2} - 513 T^{4} - 2116 T^{6} + 279841 T^{8} \))(\( 1 - 4 T^{2} - 513 T^{4} - 2116 T^{6} + 279841 T^{8} \))(\( ( 1 - 112 T^{2} + 5594 T^{4} - 163154 T^{6} + 2959226 T^{8} - 31342192 T^{10} + 148035889 T^{12} )^{2} \))(\( ( 1 - 112 T^{2} + 5594 T^{4} - 163154 T^{6} + 2959226 T^{8} - 31342192 T^{10} + 148035889 T^{12} )^{2} \))(\( ( 1 - 87 T^{2} + 3842 T^{4} - 111721 T^{6} + 2697242 T^{8} - 59100409 T^{10} + 1075149122 T^{12} - 12879122343 T^{14} + 78310985281 T^{16} )^{2} \))(\( ( 1 - 87 T^{2} + 3842 T^{4} - 111721 T^{6} + 2697242 T^{8} - 59100409 T^{10} + 1075149122 T^{12} - 12879122343 T^{14} + 78310985281 T^{16} )^{2} \))
$29$ (\( ( 1 + 29 T^{2} )^{4} \))(\( ( 1 + 29 T^{2} )^{4} \))(\( ( 1 + 42 T^{2} + 1155 T^{4} + 14916 T^{6} + 971355 T^{8} + 29705802 T^{10} + 594823321 T^{12} )^{2} \))(\( ( 1 + 42 T^{2} + 1155 T^{4} + 14916 T^{6} + 971355 T^{8} + 29705802 T^{10} + 594823321 T^{12} )^{2} \))(\( ( 1 + 133 T^{2} + 9622 T^{4} + 456771 T^{6} + 15547890 T^{8} + 384144411 T^{10} + 6805457782 T^{12} + 79111501693 T^{14} + 500246412961 T^{16} )^{2} \))(\( ( 1 + 133 T^{2} + 9622 T^{4} + 456771 T^{6} + 15547890 T^{8} + 384144411 T^{10} + 6805457782 T^{12} + 79111501693 T^{14} + 500246412961 T^{16} )^{2} \))
$31$ (\( ( 1 + 6 T + 43 T^{2} + 186 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 - 6 T + 43 T^{2} - 186 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 + 6 T + 48 T^{2} + 264 T^{3} + 1488 T^{4} + 5766 T^{5} + 29791 T^{6} )^{4} \))(\( ( 1 - 6 T + 48 T^{2} - 264 T^{3} + 1488 T^{4} - 5766 T^{5} + 29791 T^{6} )^{4} \))(\( ( 1 + 124 T^{2} + 8479 T^{4} + 404384 T^{6} + 14327960 T^{8} + 388613024 T^{10} + 7830534559 T^{12} + 110050456444 T^{14} + 852891037441 T^{16} )^{2} \))(\( ( 1 + 124 T^{2} + 8479 T^{4} + 404384 T^{6} + 14327960 T^{8} + 388613024 T^{10} + 7830534559 T^{12} + 110050456444 T^{14} + 852891037441 T^{16} )^{2} \))
$37$ (\( ( 1 + 8 T + 27 T^{2} + 296 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 + 8 T + 27 T^{2} + 296 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 - 3 T + 58 T^{2} - 39 T^{3} + 2146 T^{4} - 4107 T^{5} + 50653 T^{6} )^{4} \))(\( ( 1 - 3 T + 58 T^{2} - 39 T^{3} + 2146 T^{4} - 4107 T^{5} + 50653 T^{6} )^{4} \))(\( ( 1 - 2 T + 77 T^{2} - 134 T^{3} + 4112 T^{4} - 4958 T^{5} + 105413 T^{6} - 101306 T^{7} + 1874161 T^{8} )^{4} \))(\( ( 1 - 2 T + 77 T^{2} - 134 T^{3} + 4112 T^{4} - 4958 T^{5} + 105413 T^{6} - 101306 T^{7} + 1874161 T^{8} )^{4} \))
$41$ (\( 1 + 68 T^{2} + 2943 T^{4} + 114308 T^{6} + 2825761 T^{8} \))(\( 1 + 68 T^{2} + 2943 T^{4} + 114308 T^{6} + 2825761 T^{8} \))(\( ( 1 - 112 T^{2} + 5698 T^{4} - 227382 T^{6} + 9578338 T^{8} - 316485232 T^{10} + 4750104241 T^{12} )^{2} \))(\( ( 1 - 112 T^{2} + 5698 T^{4} - 227382 T^{6} + 9578338 T^{8} - 316485232 T^{10} + 4750104241 T^{12} )^{2} \))(\( ( 1 - 210 T^{2} + 21937 T^{4} - 1498866 T^{6} + 72486980 T^{8} - 2519593746 T^{10} + 61988719057 T^{12} - 997521890610 T^{14} + 7984925229121 T^{16} )^{2} \))(\( ( 1 - 210 T^{2} + 21937 T^{4} - 1498866 T^{6} + 72486980 T^{8} - 2519593746 T^{10} + 61988719057 T^{12} - 997521890610 T^{14} + 7984925229121 T^{16} )^{2} \))
$43$ (\( ( 1 - 43 T^{2} )^{4} \))(\( ( 1 - 43 T^{2} )^{4} \))(\( ( 1 - 154 T^{2} + 11947 T^{4} - 616020 T^{6} + 22090003 T^{8} - 526495354 T^{10} + 6321363049 T^{12} )^{2} \))(\( ( 1 - 154 T^{2} + 11947 T^{4} - 616020 T^{6} + 22090003 T^{8} - 526495354 T^{10} + 6321363049 T^{12} )^{2} \))(\( ( 1 - 159 T^{2} + 14377 T^{4} - 922458 T^{6} + 45070550 T^{8} - 1705624842 T^{10} + 49152101977 T^{12} - 1005096724791 T^{14} + 11688200277601 T^{16} )^{2} \))(\( ( 1 - 159 T^{2} + 14377 T^{4} - 922458 T^{6} + 45070550 T^{8} - 1705624842 T^{10} + 49152101977 T^{12} - 1005096724791 T^{14} + 11688200277601 T^{16} )^{2} \))
$47$ (\( ( 1 + 47 T^{2} )^{4} \))(\( ( 1 + 47 T^{2} )^{4} \))(\( ( 1 + 102 T^{2} + 9163 T^{4} + 465404 T^{6} + 20241067 T^{8} + 497727462 T^{10} + 10779215329 T^{12} )^{2} \))(\( ( 1 + 102 T^{2} + 9163 T^{4} + 465404 T^{6} + 20241067 T^{8} + 497727462 T^{10} + 10779215329 T^{12} )^{2} \))(\( ( 1 + 2 T + 73 T^{2} - 126 T^{3} + 2120 T^{4} - 5922 T^{5} + 161257 T^{6} + 207646 T^{7} + 4879681 T^{8} )^{4} \))(\( ( 1 - 2 T + 73 T^{2} + 126 T^{3} + 2120 T^{4} + 5922 T^{5} + 161257 T^{6} - 207646 T^{7} + 4879681 T^{8} )^{4} \))
$53$ (\( ( 1 + 53 T^{2} )^{4} \))(\( ( 1 + 53 T^{2} )^{4} \))(\( ( 1 + 86 T^{2} + 7463 T^{4} + 484852 T^{6} + 20963567 T^{8} + 678581366 T^{10} + 22164361129 T^{12} )^{2} \))(\( ( 1 + 86 T^{2} + 7463 T^{4} + 484852 T^{6} + 20963567 T^{8} + 678581366 T^{10} + 22164361129 T^{12} )^{2} \))(\( ( 1 + 348 T^{2} + 56327 T^{4} + 5511008 T^{6} + 355579928 T^{8} + 15480421472 T^{10} + 444447123287 T^{12} + 7713197672892 T^{14} + 62259690411361 T^{16} )^{2} \))(\( ( 1 + 348 T^{2} + 56327 T^{4} + 5511008 T^{6} + 355579928 T^{8} + 15480421472 T^{10} + 444447123287 T^{12} + 7713197672892 T^{14} + 62259690411361 T^{16} )^{2} \))
$59$ (\( ( 1 + 59 T^{2} )^{4} \))(\( ( 1 + 59 T^{2} )^{4} \))(\( ( 1 - 74 T^{2} + 12051 T^{4} - 523684 T^{6} + 41949531 T^{8} - 896684714 T^{10} + 42180533641 T^{12} )^{2} \))(\( ( 1 - 74 T^{2} + 12051 T^{4} - 523684 T^{6} + 41949531 T^{8} - 896684714 T^{10} + 42180533641 T^{12} )^{2} \))(\( ( 1 + 11 T + 159 T^{2} + 1104 T^{3} + 10756 T^{4} + 65136 T^{5} + 553479 T^{6} + 2259169 T^{7} + 12117361 T^{8} )^{4} \))(\( ( 1 - 11 T + 159 T^{2} - 1104 T^{3} + 10756 T^{4} - 65136 T^{5} + 553479 T^{6} - 2259169 T^{7} + 12117361 T^{8} )^{4} \))
$61$ (\( ( 1 - 61 T^{2} )^{4} \))(\( ( 1 - 61 T^{2} )^{4} \))(\( ( 1 - 155 T^{2} + 12259 T^{4} - 719586 T^{6} + 45615739 T^{8} - 2146105355 T^{10} + 51520374361 T^{12} )^{2} \))(\( ( 1 - 155 T^{2} + 12259 T^{4} - 719586 T^{6} + 45615739 T^{8} - 2146105355 T^{10} + 51520374361 T^{12} )^{2} \))(\( ( 1 - 268 T^{2} + 40068 T^{4} - 3979924 T^{6} + 284320790 T^{8} - 14809297204 T^{10} + 554775157188 T^{12} - 13807460328748 T^{14} + 191707312997281 T^{16} )^{2} \))(\( ( 1 - 268 T^{2} + 40068 T^{4} - 3979924 T^{6} + 284320790 T^{8} - 14809297204 T^{10} + 554775157188 T^{12} - 13807460328748 T^{14} + 191707312997281 T^{16} )^{2} \))
$67$ (\( ( 1 - 67 T^{2} )^{4} \))(\( ( 1 - 67 T^{2} )^{4} \))(\( ( 1 - 347 T^{2} + 53251 T^{4} - 4619670 T^{6} + 239043739 T^{8} - 6992438987 T^{10} + 90458382169 T^{12} )^{2} \))(\( ( 1 - 347 T^{2} + 53251 T^{4} - 4619670 T^{6} + 239043739 T^{8} - 6992438987 T^{10} + 90458382169 T^{12} )^{2} \))(\( ( 1 - 325 T^{2} + 52002 T^{4} - 5466211 T^{6} + 421211930 T^{8} - 24537821179 T^{10} + 1047898594242 T^{12} - 29398974204925 T^{14} + 406067677556641 T^{16} )^{2} \))(\( ( 1 - 325 T^{2} + 52002 T^{4} - 5466211 T^{6} + 421211930 T^{8} - 24537821179 T^{10} + 1047898594242 T^{12} - 29398974204925 T^{14} + 406067677556641 T^{16} )^{2} \))
$71$ (\( 1 - 100 T^{2} + 4959 T^{4} - 504100 T^{6} + 25411681 T^{8} \))(\( 1 - 100 T^{2} + 4959 T^{4} - 504100 T^{6} + 25411681 T^{8} \))(\( ( 1 - 276 T^{2} + 38818 T^{4} - 3386178 T^{6} + 195681538 T^{8} - 7013623956 T^{10} + 128100283921 T^{12} )^{2} \))(\( ( 1 - 276 T^{2} + 38818 T^{4} - 3386178 T^{6} + 195681538 T^{8} - 7013623956 T^{10} + 128100283921 T^{12} )^{2} \))(\( ( 1 - 419 T^{2} + 83894 T^{4} - 10494221 T^{6} + 893482514 T^{8} - 52901368061 T^{10} + 2131887565814 T^{12} - 53674018962899 T^{14} + 645753531245761 T^{16} )^{2} \))(\( ( 1 - 419 T^{2} + 83894 T^{4} - 10494221 T^{6} + 893482514 T^{8} - 52901368061 T^{10} + 2131887565814 T^{12} - 53674018962899 T^{14} + 645753531245761 T^{16} )^{2} \))
$73$ (\( ( 1 - 73 T^{2} )^{4} \))(\( ( 1 - 73 T^{2} )^{4} \))(\( ( 1 - 247 T^{2} + 30171 T^{4} - 2531358 T^{6} + 160781259 T^{8} - 7014365527 T^{10} + 151334226289 T^{12} )^{2} \))(\( ( 1 - 247 T^{2} + 30171 T^{4} - 2531358 T^{6} + 160781259 T^{8} - 7014365527 T^{10} + 151334226289 T^{12} )^{2} \))(\( ( 1 - 105 T^{2} + 4878 T^{4} - 673927 T^{6} + 86417250 T^{8} - 3591356983 T^{10} + 138526619598 T^{12} - 15890093760345 T^{14} + 806460091894081 T^{16} )^{2} \))(\( ( 1 - 105 T^{2} + 4878 T^{4} - 673927 T^{6} + 86417250 T^{8} - 3591356983 T^{10} + 138526619598 T^{12} - 15890093760345 T^{14} + 806460091894081 T^{16} )^{2} \))
$79$ (\( ( 1 - 79 T^{2} )^{4} \))(\( ( 1 - 79 T^{2} )^{4} \))(\( ( 1 - 59 T^{2} + 13987 T^{4} - 673302 T^{6} + 87292867 T^{8} - 2298054779 T^{10} + 243087455521 T^{12} )^{2} \))(\( ( 1 - 59 T^{2} + 13987 T^{4} - 673302 T^{6} + 87292867 T^{8} - 2298054779 T^{10} + 243087455521 T^{12} )^{2} \))(\( ( 1 - 406 T^{2} + 84777 T^{4} - 11389882 T^{6} + 1067641484 T^{8} - 71084253562 T^{10} + 3302071016937 T^{12} - 98693506941526 T^{14} + 1517108809906561 T^{16} )^{2} \))(\( ( 1 - 406 T^{2} + 84777 T^{4} - 11389882 T^{6} + 1067641484 T^{8} - 71084253562 T^{10} + 3302071016937 T^{12} - 98693506941526 T^{14} + 1517108809906561 T^{16} )^{2} \))
$83$ (\( ( 1 + 83 T^{2} )^{4} \))(\( ( 1 + 83 T^{2} )^{4} \))(\( ( 1 + 102 T^{2} + 16579 T^{4} + 1402844 T^{6} + 114212731 T^{8} + 4840748742 T^{10} + 326940373369 T^{12} )^{2} \))(\( ( 1 + 102 T^{2} + 16579 T^{4} + 1402844 T^{6} + 114212731 T^{8} + 4840748742 T^{10} + 326940373369 T^{12} )^{2} \))(\( ( 1 - T + 175 T^{2} + 192 T^{3} + 17036 T^{4} + 15936 T^{5} + 1205575 T^{6} - 571787 T^{7} + 47458321 T^{8} )^{4} \))(\( ( 1 + T + 175 T^{2} - 192 T^{3} + 17036 T^{4} - 15936 T^{5} + 1205575 T^{6} + 571787 T^{7} + 47458321 T^{8} )^{4} \))
$89$ (\( 1 - 172 T^{2} + 21663 T^{4} - 1362412 T^{6} + 62742241 T^{8} \))(\( 1 - 172 T^{2} + 21663 T^{4} - 1362412 T^{6} + 62742241 T^{8} \))(\( ( 1 - 88 T^{2} + 17938 T^{4} - 1434078 T^{6} + 142086898 T^{8} - 5521317208 T^{10} + 496981290961 T^{12} )^{2} \))(\( ( 1 - 88 T^{2} + 17938 T^{4} - 1434078 T^{6} + 142086898 T^{8} - 5521317208 T^{10} + 496981290961 T^{12} )^{2} \))(\( ( 1 - 93 T^{2} + 15598 T^{4} - 511251 T^{6} + 94398818 T^{8} - 4049619171 T^{10} + 978653475118 T^{12} - 46219260059373 T^{14} + 3936588805702081 T^{16} )^{2} \))(\( ( 1 - 93 T^{2} + 15598 T^{4} - 511251 T^{6} + 94398818 T^{8} - 4049619171 T^{10} + 978653475118 T^{12} - 46219260059373 T^{14} + 3936588805702081 T^{16} )^{2} \))
$97$ (\( ( 1 - 97 T^{2} )^{4} \))(\( ( 1 - 97 T^{2} )^{4} \))(\( ( 1 - 371 T^{2} + 67195 T^{4} - 7803234 T^{6} + 632237755 T^{8} - 32844363251 T^{10} + 832972004929 T^{12} )^{2} \))(\( ( 1 - 371 T^{2} + 67195 T^{4} - 7803234 T^{6} + 632237755 T^{8} - 32844363251 T^{10} + 832972004929 T^{12} )^{2} \))(\( ( 1 + 47 T^{2} + 17382 T^{4} + 1290401 T^{6} + 195527762 T^{8} + 12141383009 T^{10} + 1538815962342 T^{12} + 39149684231663 T^{14} + 7837433594376961 T^{16} )^{2} \))(\( ( 1 + 47 T^{2} + 17382 T^{4} + 1290401 T^{6} + 195527762 T^{8} + 12141383009 T^{10} + 1538815962342 T^{12} + 39149684231663 T^{14} + 7837433594376961 T^{16} )^{2} \))
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