Properties

Label 2400.2.d
Level 2400
Weight 2
Character orbit d
Rep. character \(\chi_{2400}(49,\cdot)\)
Character field \(\Q\)
Dimension 36
Newform subspaces 8
Sturm bound 960
Trace bound 13

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Defining parameters

Level: \( N \) \(=\) \( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2400.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 40 \)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(960\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(7\), \(11\), \(13\), \(37\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(2400, [\chi])\).

Total New Old
Modular forms 528 36 492
Cusp forms 432 36 396
Eisenstein series 96 0 96

Trace form

\( 36q + 36q^{9} + O(q^{10}) \) \( 36q + 36q^{9} - 40q^{31} - 16q^{39} + 8q^{41} - 36q^{49} + 64q^{71} + 24q^{79} + 36q^{81} + 40q^{89} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(2400, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
2400.2.d.a \(2\) \(19.164\) \(\Q(\sqrt{-1}) \) None \(0\) \(-2\) \(0\) \(0\) \(q-q^{3}+iq^{7}+q^{9}+2iq^{11}+3iq^{17}+\cdots\)
2400.2.d.b \(2\) \(19.164\) \(\Q(\sqrt{-1}) \) None \(0\) \(-2\) \(0\) \(0\) \(q-q^{3}+iq^{7}+q^{9}+4q^{13}-iq^{17}+\cdots\)
2400.2.d.c \(2\) \(19.164\) \(\Q(\sqrt{-1}) \) None \(0\) \(2\) \(0\) \(0\) \(q+q^{3}+iq^{7}+q^{9}-4q^{13}-iq^{17}+\cdots\)
2400.2.d.d \(2\) \(19.164\) \(\Q(\sqrt{-1}) \) None \(0\) \(2\) \(0\) \(0\) \(q+q^{3}+iq^{7}+q^{9}-2iq^{11}+3iq^{17}+\cdots\)
2400.2.d.e \(6\) \(19.164\) 6.0.399424.1 None \(0\) \(-6\) \(0\) \(0\) \(q-q^{3}-\beta _{5}q^{7}+q^{9}+(\beta _{2}-\beta _{3}-\beta _{5})q^{11}+\cdots\)
2400.2.d.f \(6\) \(19.164\) 6.0.399424.1 None \(0\) \(6\) \(0\) \(0\) \(q+q^{3}-\beta _{5}q^{7}+q^{9}+(-\beta _{2}+\beta _{3}+\beta _{5})q^{11}+\cdots\)
2400.2.d.g \(8\) \(19.164\) 8.0.214798336.3 None \(0\) \(-8\) \(0\) \(0\) \(q-q^{3}+(\beta _{2}+\beta _{7})q^{7}+q^{9}+\beta _{5}q^{11}+\cdots\)
2400.2.d.h \(8\) \(19.164\) 8.0.214798336.3 None \(0\) \(8\) \(0\) \(0\) \(q+q^{3}+(\beta _{2}+\beta _{7})q^{7}+q^{9}-\beta _{5}q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(2400, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(2400, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(480, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(600, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(800, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{6} \))(\( ( 1 - T )^{6} \))(\( ( 1 + T )^{8} \))(\( ( 1 - T )^{8} \))
$5$ 1
$7$ (\( 1 - 10 T^{2} + 49 T^{4} \))(\( 1 - 10 T^{2} + 49 T^{4} \))(\( 1 - 10 T^{2} + 49 T^{4} \))(\( 1 - 10 T^{2} + 49 T^{4} \))(\( 1 - 6 T^{2} + 47 T^{4} - 500 T^{6} + 2303 T^{8} - 14406 T^{10} + 117649 T^{12} \))(\( 1 - 6 T^{2} + 47 T^{4} - 500 T^{6} + 2303 T^{8} - 14406 T^{10} + 117649 T^{12} \))(\( 1 - 28 T^{2} + 330 T^{4} - 2224 T^{6} + 13203 T^{8} - 108976 T^{10} + 792330 T^{12} - 3294172 T^{14} + 5764801 T^{16} \))(\( 1 - 28 T^{2} + 330 T^{4} - 2224 T^{6} + 13203 T^{8} - 108976 T^{10} + 792330 T^{12} - 3294172 T^{14} + 5764801 T^{16} \))
$11$ (\( 1 - 6 T^{2} + 121 T^{4} \))(\( ( 1 - 11 T^{2} )^{2} \))(\( ( 1 - 11 T^{2} )^{2} \))(\( 1 - 6 T^{2} + 121 T^{4} \))(\( 1 - 2 T^{2} + 87 T^{4} + 4 T^{6} + 10527 T^{8} - 29282 T^{10} + 1771561 T^{12} \))(\( 1 - 2 T^{2} + 87 T^{4} + 4 T^{6} + 10527 T^{8} - 29282 T^{10} + 1771561 T^{12} \))(\( 1 - 56 T^{2} + 1612 T^{4} - 29896 T^{6} + 388998 T^{8} - 3617416 T^{10} + 23601292 T^{12} - 99207416 T^{14} + 214358881 T^{16} \))(\( 1 - 56 T^{2} + 1612 T^{4} - 29896 T^{6} + 388998 T^{8} - 3617416 T^{10} + 23601292 T^{12} - 99207416 T^{14} + 214358881 T^{16} \))
$13$ (\( ( 1 + 13 T^{2} )^{2} \))(\( ( 1 - 4 T + 13 T^{2} )^{2} \))(\( ( 1 + 4 T + 13 T^{2} )^{2} \))(\( ( 1 + 13 T^{2} )^{2} \))(\( ( 1 + 11 T^{2} + 16 T^{3} + 143 T^{4} + 2197 T^{6} )^{2} \))(\( ( 1 + 11 T^{2} - 16 T^{3} + 143 T^{4} + 2197 T^{6} )^{2} \))(\( ( 1 + 30 T^{2} - 32 T^{3} + 451 T^{4} - 416 T^{5} + 5070 T^{6} + 28561 T^{8} )^{2} \))(\( ( 1 + 30 T^{2} + 32 T^{3} + 451 T^{4} + 416 T^{5} + 5070 T^{6} + 28561 T^{8} )^{2} \))
$17$ (\( 1 + 2 T^{2} + 289 T^{4} \))(\( ( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} ) \))(\( ( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} ) \))(\( 1 + 2 T^{2} + 289 T^{4} \))(\( 1 - 34 T^{2} + 351 T^{4} - 1084 T^{6} + 101439 T^{8} - 2839714 T^{10} + 24137569 T^{12} \))(\( 1 - 34 T^{2} + 351 T^{4} - 1084 T^{6} + 101439 T^{8} - 2839714 T^{10} + 24137569 T^{12} \))(\( 1 - 56 T^{2} + 1484 T^{4} - 24968 T^{6} + 374950 T^{8} - 7215752 T^{10} + 123945164 T^{12} - 1351703864 T^{14} + 6975757441 T^{16} \))(\( 1 - 56 T^{2} + 1484 T^{4} - 24968 T^{6} + 374950 T^{8} - 7215752 T^{10} + 123945164 T^{12} - 1351703864 T^{14} + 6975757441 T^{16} \))
$19$ (\( 1 - 22 T^{2} + 361 T^{4} \))(\( 1 - 22 T^{2} + 361 T^{4} \))(\( 1 - 22 T^{2} + 361 T^{4} \))(\( 1 - 22 T^{2} + 361 T^{4} \))(\( 1 - 74 T^{2} + 2647 T^{4} - 60620 T^{6} + 955567 T^{8} - 9643754 T^{10} + 47045881 T^{12} \))(\( 1 - 74 T^{2} + 2647 T^{4} - 60620 T^{6} + 955567 T^{8} - 9643754 T^{10} + 47045881 T^{12} \))(\( 1 - 36 T^{2} + 1546 T^{4} - 35120 T^{6} + 832243 T^{8} - 12678320 T^{10} + 201476266 T^{12} - 1693651716 T^{14} + 16983563041 T^{16} \))(\( 1 - 36 T^{2} + 1546 T^{4} - 35120 T^{6} + 832243 T^{8} - 12678320 T^{10} + 201476266 T^{12} - 1693651716 T^{14} + 16983563041 T^{16} \))
$23$ (\( 1 - 30 T^{2} + 529 T^{4} \))(\( 1 - 30 T^{2} + 529 T^{4} \))(\( 1 - 30 T^{2} + 529 T^{4} \))(\( 1 - 30 T^{2} + 529 T^{4} \))(\( 1 - 98 T^{2} + 4527 T^{4} - 128636 T^{6} + 2394783 T^{8} - 27424418 T^{10} + 148035889 T^{12} \))(\( 1 - 98 T^{2} + 4527 T^{4} - 128636 T^{6} + 2394783 T^{8} - 27424418 T^{10} + 148035889 T^{12} \))(\( 1 - 56 T^{2} + 1324 T^{4} - 36616 T^{6} + 1094310 T^{8} - 19369864 T^{10} + 370509484 T^{12} - 8290009784 T^{14} + 78310985281 T^{16} \))(\( 1 - 56 T^{2} + 1324 T^{4} - 36616 T^{6} + 1094310 T^{8} - 19369864 T^{10} + 370509484 T^{12} - 8290009784 T^{14} + 78310985281 T^{16} \))
$29$ (\( 1 - 22 T^{2} + 841 T^{4} \))(\( 1 - 22 T^{2} + 841 T^{4} \))(\( 1 - 22 T^{2} + 841 T^{4} \))(\( 1 - 22 T^{2} + 841 T^{4} \))(\( ( 1 - 54 T^{2} + 841 T^{4} )^{3} \))(\( ( 1 - 54 T^{2} + 841 T^{4} )^{3} \))(\( 1 - 88 T^{2} + 4780 T^{4} - 171048 T^{6} + 5385990 T^{8} - 143851368 T^{10} + 3380803180 T^{12} - 52344452248 T^{14} + 500246412961 T^{16} \))(\( 1 - 88 T^{2} + 4780 T^{4} - 171048 T^{6} + 5385990 T^{8} - 143851368 T^{10} + 3380803180 T^{12} - 52344452248 T^{14} + 500246412961 T^{16} \))
$31$ (\( ( 1 + 10 T + 31 T^{2} )^{2} \))(\( ( 1 + 2 T + 31 T^{2} )^{2} \))(\( ( 1 + 2 T + 31 T^{2} )^{2} \))(\( ( 1 + 10 T + 31 T^{2} )^{2} \))(\( ( 1 - 6 T + 77 T^{2} - 308 T^{3} + 2387 T^{4} - 5766 T^{5} + 29791 T^{6} )^{2} \))(\( ( 1 - 6 T + 77 T^{2} - 308 T^{3} + 2387 T^{4} - 5766 T^{5} + 29791 T^{6} )^{2} \))(\( ( 1 + 4 T + 54 T^{2} + 168 T^{3} + 2099 T^{4} + 5208 T^{5} + 51894 T^{6} + 119164 T^{7} + 923521 T^{8} )^{2} \))(\( ( 1 + 4 T + 54 T^{2} + 168 T^{3} + 2099 T^{4} + 5208 T^{5} + 51894 T^{6} + 119164 T^{7} + 923521 T^{8} )^{2} \))
$37$ (\( ( 1 + 4 T + 37 T^{2} )^{2} \))(\( ( 1 - 8 T + 37 T^{2} )^{2} \))(\( ( 1 + 8 T + 37 T^{2} )^{2} \))(\( ( 1 - 4 T + 37 T^{2} )^{2} \))(\( ( 1 - 4 T + 51 T^{2} - 40 T^{3} + 1887 T^{4} - 5476 T^{5} + 50653 T^{6} )^{2} \))(\( ( 1 + 4 T + 51 T^{2} + 40 T^{3} + 1887 T^{4} + 5476 T^{5} + 50653 T^{6} )^{2} \))(\( ( 1 + 84 T^{2} - 128 T^{3} + 3542 T^{4} - 4736 T^{5} + 114996 T^{6} + 1874161 T^{8} )^{2} \))(\( ( 1 + 84 T^{2} + 128 T^{3} + 3542 T^{4} + 4736 T^{5} + 114996 T^{6} + 1874161 T^{8} )^{2} \))
$41$ (\( ( 1 - 10 T + 41 T^{2} )^{2} \))(\( ( 1 - 2 T + 41 T^{2} )^{2} \))(\( ( 1 - 2 T + 41 T^{2} )^{2} \))(\( ( 1 - 10 T + 41 T^{2} )^{2} \))(\( ( 1 + 10 T + 87 T^{2} + 588 T^{3} + 3567 T^{4} + 16810 T^{5} + 68921 T^{6} )^{2} \))(\( ( 1 + 10 T + 87 T^{2} + 588 T^{3} + 3567 T^{4} + 16810 T^{5} + 68921 T^{6} )^{2} \))(\( ( 1 + 100 T^{2} + 56 T^{3} + 5166 T^{4} + 2296 T^{5} + 168100 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 + 100 T^{2} + 56 T^{3} + 5166 T^{4} + 2296 T^{5} + 168100 T^{6} + 2825761 T^{8} )^{2} \))
$43$ (\( ( 1 - 4 T + 43 T^{2} )^{2} \))(\( ( 1 + 4 T + 43 T^{2} )^{2} \))(\( ( 1 - 4 T + 43 T^{2} )^{2} \))(\( ( 1 + 4 T + 43 T^{2} )^{2} \))(\( ( 1 - 4 T + 65 T^{2} - 216 T^{3} + 2795 T^{4} - 7396 T^{5} + 79507 T^{6} )^{2} \))(\( ( 1 + 4 T + 65 T^{2} + 216 T^{3} + 2795 T^{4} + 7396 T^{5} + 79507 T^{6} )^{2} \))(\( ( 1 + 4 T + 58 T^{2} + 336 T^{3} + 3379 T^{4} + 14448 T^{5} + 107242 T^{6} + 318028 T^{7} + 3418801 T^{8} )^{2} \))(\( ( 1 - 4 T + 58 T^{2} - 336 T^{3} + 3379 T^{4} - 14448 T^{5} + 107242 T^{6} - 318028 T^{7} + 3418801 T^{8} )^{2} \))
$47$ (\( 1 - 78 T^{2} + 2209 T^{4} \))(\( 1 + 50 T^{2} + 2209 T^{4} \))(\( 1 + 50 T^{2} + 2209 T^{4} \))(\( 1 - 78 T^{2} + 2209 T^{4} \))(\( 1 - 82 T^{2} + 7967 T^{4} - 348252 T^{6} + 17599103 T^{8} - 400133842 T^{10} + 10779215329 T^{12} \))(\( 1 - 82 T^{2} + 7967 T^{4} - 348252 T^{6} + 17599103 T^{8} - 400133842 T^{10} + 10779215329 T^{12} \))(\( 1 - 232 T^{2} + 26076 T^{4} - 1910872 T^{6} + 102863686 T^{8} - 4221116248 T^{10} + 127242561756 T^{12} - 2500777956328 T^{14} + 23811286661761 T^{16} \))(\( 1 - 232 T^{2} + 26076 T^{4} - 1910872 T^{6} + 102863686 T^{8} - 4221116248 T^{10} + 127242561756 T^{12} - 2500777956328 T^{14} + 23811286661761 T^{16} \))
$53$ (\( ( 1 + 10 T + 53 T^{2} )^{2} \))(\( ( 1 + 6 T + 53 T^{2} )^{2} \))(\( ( 1 - 6 T + 53 T^{2} )^{2} \))(\( ( 1 - 10 T + 53 T^{2} )^{2} \))(\( ( 1 - 2 T + 53 T^{2} )^{6} \))(\( ( 1 + 2 T + 53 T^{2} )^{6} \))(\( ( 1 + 4 T + 92 T^{2} - 44 T^{3} + 3982 T^{4} - 2332 T^{5} + 258428 T^{6} + 595508 T^{7} + 7890481 T^{8} )^{2} \))(\( ( 1 - 4 T + 92 T^{2} + 44 T^{3} + 3982 T^{4} + 2332 T^{5} + 258428 T^{6} - 595508 T^{7} + 7890481 T^{8} )^{2} \))
$59$ (\( 1 - 54 T^{2} + 3481 T^{4} \))(\( 1 - 102 T^{2} + 3481 T^{4} \))(\( 1 - 102 T^{2} + 3481 T^{4} \))(\( 1 - 54 T^{2} + 3481 T^{4} \))(\( 1 - 274 T^{2} + 33911 T^{4} - 2503644 T^{6} + 118044191 T^{8} - 3320156914 T^{10} + 42180533641 T^{12} \))(\( 1 - 274 T^{2} + 33911 T^{4} - 2503644 T^{6} + 118044191 T^{8} - 3320156914 T^{10} + 42180533641 T^{12} \))(\( 1 - 40 T^{2} + 2364 T^{4} - 112984 T^{6} + 20250598 T^{8} - 393297304 T^{10} + 28645441404 T^{12} - 1687221345640 T^{14} + 146830437604321 T^{16} \))(\( 1 - 40 T^{2} + 2364 T^{4} - 112984 T^{6} + 20250598 T^{8} - 393297304 T^{10} + 28645441404 T^{12} - 1687221345640 T^{14} + 146830437604321 T^{16} \))
$61$ (\( 1 - 58 T^{2} + 3721 T^{4} \))(\( ( 1 - 61 T^{2} )^{2} \))(\( ( 1 - 61 T^{2} )^{2} \))(\( 1 - 58 T^{2} + 3721 T^{4} \))(\( 1 - 110 T^{2} + 10759 T^{4} - 685796 T^{6} + 40034239 T^{8} - 1523042510 T^{10} + 51520374361 T^{12} \))(\( 1 - 110 T^{2} + 10759 T^{4} - 685796 T^{6} + 40034239 T^{8} - 1523042510 T^{10} + 51520374361 T^{12} \))(\( 1 - 252 T^{2} + 32650 T^{4} - 2942672 T^{6} + 202734451 T^{8} - 10949682512 T^{10} + 452066708650 T^{12} - 12983134338972 T^{14} + 191707312997281 T^{16} \))(\( 1 - 252 T^{2} + 32650 T^{4} - 2942672 T^{6} + 202734451 T^{8} - 10949682512 T^{10} + 452066708650 T^{12} - 12983134338972 T^{14} + 191707312997281 T^{16} \))
$67$ (\( ( 1 + 12 T + 67 T^{2} )^{2} \))(\( ( 1 - 12 T + 67 T^{2} )^{2} \))(\( ( 1 + 12 T + 67 T^{2} )^{2} \))(\( ( 1 - 12 T + 67 T^{2} )^{2} \))(\( ( 1 + 4 T + 67 T^{2} )^{6} \))(\( ( 1 - 4 T + 67 T^{2} )^{6} \))(\( ( 1 - 12 T + 154 T^{2} - 1520 T^{3} + 16755 T^{4} - 101840 T^{5} + 691306 T^{6} - 3609156 T^{7} + 20151121 T^{8} )^{2} \))(\( ( 1 + 12 T + 154 T^{2} + 1520 T^{3} + 16755 T^{4} + 101840 T^{5} + 691306 T^{6} + 3609156 T^{7} + 20151121 T^{8} )^{2} \))
$71$ (\( ( 1 - 4 T + 71 T^{2} )^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( ( 1 - 4 T + 71 T^{2} )^{2} \))(\( ( 1 - 4 T + 101 T^{2} - 632 T^{3} + 7171 T^{4} - 20164 T^{5} + 357911 T^{6} )^{2} \))(\( ( 1 - 4 T + 101 T^{2} - 632 T^{3} + 7171 T^{4} - 20164 T^{5} + 357911 T^{6} )^{2} \))(\( ( 1 - 20 T + 380 T^{2} - 4188 T^{3} + 43342 T^{4} - 297348 T^{5} + 1915580 T^{6} - 7158220 T^{7} + 25411681 T^{8} )^{2} \))(\( ( 1 - 20 T + 380 T^{2} - 4188 T^{3} + 43342 T^{4} - 297348 T^{5} + 1915580 T^{6} - 7158220 T^{7} + 25411681 T^{8} )^{2} \))
$73$ (\( 1 - 46 T^{2} + 5329 T^{4} \))(\( ( 1 - 16 T + 73 T^{2} )( 1 + 16 T + 73 T^{2} ) \))(\( ( 1 - 16 T + 73 T^{2} )( 1 + 16 T + 73 T^{2} ) \))(\( 1 - 46 T^{2} + 5329 T^{4} \))(\( ( 1 - 16 T + 73 T^{2} )^{3}( 1 + 16 T + 73 T^{2} )^{3} \))(\( ( 1 - 16 T + 73 T^{2} )^{3}( 1 + 16 T + 73 T^{2} )^{3} \))(\( 1 - 184 T^{2} + 15196 T^{4} - 1235336 T^{6} + 104948486 T^{8} - 6583105544 T^{10} + 431539670236 T^{12} - 27845497637176 T^{14} + 806460091894081 T^{16} \))(\( 1 - 184 T^{2} + 15196 T^{4} - 1235336 T^{6} + 104948486 T^{8} - 6583105544 T^{10} + 431539670236 T^{12} - 27845497637176 T^{14} + 806460091894081 T^{16} \))
$79$ (\( ( 1 + 14 T + 79 T^{2} )^{2} \))(\( ( 1 - 10 T + 79 T^{2} )^{2} \))(\( ( 1 - 10 T + 79 T^{2} )^{2} \))(\( ( 1 + 14 T + 79 T^{2} )^{2} \))(\( ( 1 - 18 T + 317 T^{2} - 2908 T^{3} + 25043 T^{4} - 112338 T^{5} + 493039 T^{6} )^{2} \))(\( ( 1 - 18 T + 317 T^{2} - 2908 T^{3} + 25043 T^{4} - 112338 T^{5} + 493039 T^{6} )^{2} \))(\( ( 1 + 8 T + 132 T^{2} + 1032 T^{3} + 16454 T^{4} + 81528 T^{5} + 823812 T^{6} + 3944312 T^{7} + 38950081 T^{8} )^{2} \))(\( ( 1 + 8 T + 132 T^{2} + 1032 T^{3} + 16454 T^{4} + 81528 T^{5} + 823812 T^{6} + 3944312 T^{7} + 38950081 T^{8} )^{2} \))
$83$ (\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 - 16 T + 83 T^{2} )^{2} \))(\( ( 1 + 16 T + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 16 T + 265 T^{2} + 2400 T^{3} + 21995 T^{4} + 110224 T^{5} + 571787 T^{6} )^{2} \))(\( ( 1 - 16 T + 265 T^{2} - 2400 T^{3} + 21995 T^{4} - 110224 T^{5} + 571787 T^{6} )^{2} \))(\( ( 1 + 16 T + 276 T^{2} + 2576 T^{3} + 30070 T^{4} + 213808 T^{5} + 1901364 T^{6} + 9148592 T^{7} + 47458321 T^{8} )^{2} \))(\( ( 1 - 16 T + 276 T^{2} - 2576 T^{3} + 30070 T^{4} - 213808 T^{5} + 1901364 T^{6} - 9148592 T^{7} + 47458321 T^{8} )^{2} \))
$89$ (\( ( 1 + 14 T + 89 T^{2} )^{2} \))(\( ( 1 - 10 T + 89 T^{2} )^{2} \))(\( ( 1 - 10 T + 89 T^{2} )^{2} \))(\( ( 1 + 14 T + 89 T^{2} )^{2} \))(\( ( 1 - 14 T + 263 T^{2} - 2308 T^{3} + 23407 T^{4} - 110894 T^{5} + 704969 T^{6} )^{2} \))(\( ( 1 - 14 T + 263 T^{2} - 2308 T^{3} + 23407 T^{4} - 110894 T^{5} + 704969 T^{6} )^{2} \))(\( ( 1 + 132 T^{2} - 64 T^{3} + 18534 T^{4} - 5696 T^{5} + 1045572 T^{6} + 62742241 T^{8} )^{2} \))(\( ( 1 + 132 T^{2} - 64 T^{3} + 18534 T^{4} - 5696 T^{5} + 1045572 T^{6} + 62742241 T^{8} )^{2} \))
$97$ (\( 1 - 94 T^{2} + 9409 T^{4} \))(\( 1 - 190 T^{2} + 9409 T^{4} \))(\( 1 - 190 T^{2} + 9409 T^{4} \))(\( 1 - 94 T^{2} + 9409 T^{4} \))(\( 1 - 250 T^{2} + 24143 T^{4} - 1697004 T^{6} + 227161487 T^{8} - 22132320250 T^{10} + 832972004929 T^{12} \))(\( 1 - 250 T^{2} + 24143 T^{4} - 1697004 T^{6} + 227161487 T^{8} - 22132320250 T^{10} + 832972004929 T^{12} \))(\( 1 - 356 T^{2} + 80362 T^{4} - 11927760 T^{6} + 1353370099 T^{8} - 112228293840 T^{10} + 7114390079722 T^{12} - 296538033754724 T^{14} + 7837433594376961 T^{16} \))(\( 1 - 356 T^{2} + 80362 T^{4} - 11927760 T^{6} + 1353370099 T^{8} - 112228293840 T^{10} + 7114390079722 T^{12} - 296538033754724 T^{14} + 7837433594376961 T^{16} \))
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