Properties

Label 3150.2.d
Level 3150
Weight 2
Character orbit d
Rep. character \(\chi_{3150}(3149,\cdot)\)
Character field \(\Q\)
Dimension 48
Newform subspaces 6
Sturm bound 1440
Trace bound 13

Related objects

Downloads

Learn more about

Defining parameters

Level: \( N \) = \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 3150.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 105 \)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(1440\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(11\), \(13\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(3150, [\chi])\).

Total New Old
Modular forms 768 48 720
Cusp forms 672 48 624
Eisenstein series 96 0 96

Trace form

\( 48q + 48q^{4} + O(q^{10}) \) \( 48q + 48q^{4} + 48q^{16} - 16q^{46} + 32q^{49} + 48q^{64} + 64q^{79} + 64q^{91} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(3150, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
3150.2.d.a \(8\) \(25.153\) 8.0.7442857984.4 None \(-8\) \(0\) \(0\) \(0\) \(q-q^{2}+q^{4}-\beta _{1}q^{7}-q^{8}-\beta _{3}q^{11}+\cdots\)
3150.2.d.b \(8\) \(25.153\) 8.0.40960000.1 None \(-8\) \(0\) \(0\) \(0\) \(q-q^{2}+q^{4}+(\beta _{1}+\beta _{2})q^{7}-q^{8}+\beta _{2}q^{11}+\cdots\)
3150.2.d.c \(8\) \(25.153\) 8.0.7442857984.4 None \(-8\) \(0\) \(0\) \(0\) \(q-q^{2}+q^{4}+\beta _{2}q^{7}-q^{8}-\beta _{3}q^{11}+\cdots\)
3150.2.d.d \(8\) \(25.153\) 8.0.7442857984.4 None \(8\) \(0\) \(0\) \(0\) \(q+q^{2}+q^{4}-\beta _{2}q^{7}+q^{8}-\beta _{3}q^{11}+\cdots\)
3150.2.d.e \(8\) \(25.153\) 8.0.40960000.1 None \(8\) \(0\) \(0\) \(0\) \(q+q^{2}+q^{4}+(\beta _{1}-\beta _{2})q^{7}+q^{8}+\beta _{2}q^{11}+\cdots\)
3150.2.d.f \(8\) \(25.153\) 8.0.7442857984.4 None \(8\) \(0\) \(0\) \(0\) \(q+q^{2}+q^{4}+\beta _{1}q^{7}+q^{8}-\beta _{3}q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(3150, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(3150, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(315, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(525, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(630, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1050, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1575, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + T )^{8} \))(\( ( 1 + T )^{8} \))(\( ( 1 + T )^{8} \))(\( ( 1 - T )^{8} \))(\( ( 1 - T )^{8} \))(\( ( 1 - T )^{8} \))
$3$ (\( \))(\( \))(\( \))(\( \))(\( \))(\( \))
$5$ (\( \))(\( \))(\( \))(\( \))(\( \))(\( \))
$7$ (\( 1 + 2 T^{2} + 24 T^{3} + 2 T^{4} + 168 T^{5} + 98 T^{6} + 2401 T^{8} \))(\( ( 1 - 6 T^{2} + 49 T^{4} )^{2} \))(\( 1 + 2 T^{2} - 24 T^{3} + 2 T^{4} - 168 T^{5} + 98 T^{6} + 2401 T^{8} \))(\( 1 + 2 T^{2} + 24 T^{3} + 2 T^{4} + 168 T^{5} + 98 T^{6} + 2401 T^{8} \))(\( ( 1 - 6 T^{2} + 49 T^{4} )^{2} \))(\( 1 + 2 T^{2} - 24 T^{3} + 2 T^{4} - 168 T^{5} + 98 T^{6} + 2401 T^{8} \))
$11$ (\( 1 - 36 T^{2} + 776 T^{4} - 11916 T^{6} + 146094 T^{8} - 1441836 T^{10} + 11361416 T^{12} - 63776196 T^{14} + 214358881 T^{16} \))(\( ( 1 - 20 T^{2} + 121 T^{4} )^{4} \))(\( 1 - 36 T^{2} + 776 T^{4} - 11916 T^{6} + 146094 T^{8} - 1441836 T^{10} + 11361416 T^{12} - 63776196 T^{14} + 214358881 T^{16} \))(\( 1 - 36 T^{2} + 776 T^{4} - 11916 T^{6} + 146094 T^{8} - 1441836 T^{10} + 11361416 T^{12} - 63776196 T^{14} + 214358881 T^{16} \))(\( ( 1 - 20 T^{2} + 121 T^{4} )^{4} \))(\( 1 - 36 T^{2} + 776 T^{4} - 11916 T^{6} + 146094 T^{8} - 1441836 T^{10} + 11361416 T^{12} - 63776196 T^{14} + 214358881 T^{16} \))
$13$ (\( ( 1 + 4 T + 30 T^{2} + 132 T^{3} + 514 T^{4} + 1716 T^{5} + 5070 T^{6} + 8788 T^{7} + 28561 T^{8} )^{2} \))(\( ( 1 + 22 T^{2} + 259 T^{4} + 3718 T^{6} + 28561 T^{8} )^{2} \))(\( ( 1 - 4 T + 30 T^{2} - 132 T^{3} + 514 T^{4} - 1716 T^{5} + 5070 T^{6} - 8788 T^{7} + 28561 T^{8} )^{2} \))(\( ( 1 + 4 T + 30 T^{2} + 132 T^{3} + 514 T^{4} + 1716 T^{5} + 5070 T^{6} + 8788 T^{7} + 28561 T^{8} )^{2} \))(\( ( 1 + 22 T^{2} + 259 T^{4} + 3718 T^{6} + 28561 T^{8} )^{2} \))(\( ( 1 - 4 T + 30 T^{2} - 132 T^{3} + 514 T^{4} - 1716 T^{5} + 5070 T^{6} - 8788 T^{7} + 28561 T^{8} )^{2} \))
$17$ (\( 1 - 36 T^{2} + 248 T^{4} + 5364 T^{6} - 133842 T^{8} + 1550196 T^{10} + 20713208 T^{12} - 868952484 T^{14} + 6975757441 T^{16} \))(\( ( 1 - 29 T^{2} + 289 T^{4} )^{4} \))(\( 1 - 36 T^{2} + 248 T^{4} + 5364 T^{6} - 133842 T^{8} + 1550196 T^{10} + 20713208 T^{12} - 868952484 T^{14} + 6975757441 T^{16} \))(\( 1 - 36 T^{2} + 248 T^{4} + 5364 T^{6} - 133842 T^{8} + 1550196 T^{10} + 20713208 T^{12} - 868952484 T^{14} + 6975757441 T^{16} \))(\( ( 1 - 29 T^{2} + 289 T^{4} )^{4} \))(\( 1 - 36 T^{2} + 248 T^{4} + 5364 T^{6} - 133842 T^{8} + 1550196 T^{10} + 20713208 T^{12} - 868952484 T^{14} + 6975757441 T^{16} \))
$19$ (\( 1 - 44 T^{2} + 1256 T^{4} - 31140 T^{6} + 702382 T^{8} - 11241540 T^{10} + 163683176 T^{12} - 2070018764 T^{14} + 16983563041 T^{16} \))(\( ( 1 - 16 T^{2} - 14 T^{4} - 5776 T^{6} + 130321 T^{8} )^{2} \))(\( 1 - 44 T^{2} + 1256 T^{4} - 31140 T^{6} + 702382 T^{8} - 11241540 T^{10} + 163683176 T^{12} - 2070018764 T^{14} + 16983563041 T^{16} \))(\( 1 - 44 T^{2} + 1256 T^{4} - 31140 T^{6} + 702382 T^{8} - 11241540 T^{10} + 163683176 T^{12} - 2070018764 T^{14} + 16983563041 T^{16} \))(\( ( 1 - 16 T^{2} - 14 T^{4} - 5776 T^{6} + 130321 T^{8} )^{2} \))(\( 1 - 44 T^{2} + 1256 T^{4} - 31140 T^{6} + 702382 T^{8} - 11241540 T^{10} + 163683176 T^{12} - 2070018764 T^{14} + 16983563041 T^{16} \))
$23$ (\( ( 1 - 4 T + 48 T^{2} - 180 T^{3} + 1438 T^{4} - 4140 T^{5} + 25392 T^{6} - 48668 T^{7} + 279841 T^{8} )^{2} \))(\( ( 1 + T + 23 T^{2} )^{8} \))(\( ( 1 - 4 T + 48 T^{2} - 180 T^{3} + 1438 T^{4} - 4140 T^{5} + 25392 T^{6} - 48668 T^{7} + 279841 T^{8} )^{2} \))(\( ( 1 + 4 T + 48 T^{2} + 180 T^{3} + 1438 T^{4} + 4140 T^{5} + 25392 T^{6} + 48668 T^{7} + 279841 T^{8} )^{2} \))(\( ( 1 - T + 23 T^{2} )^{8} \))(\( ( 1 + 4 T + 48 T^{2} + 180 T^{3} + 1438 T^{4} + 4140 T^{5} + 25392 T^{6} + 48668 T^{7} + 279841 T^{8} )^{2} \))
$29$ (\( 1 - 128 T^{2} + 8732 T^{4} - 399744 T^{6} + 13409894 T^{8} - 336184704 T^{10} + 6175977692 T^{12} - 76137385088 T^{14} + 500246412961 T^{16} \))(\( ( 1 - 30 T^{2} + 107 T^{4} - 25230 T^{6} + 707281 T^{8} )^{2} \))(\( 1 - 128 T^{2} + 8732 T^{4} - 399744 T^{6} + 13409894 T^{8} - 336184704 T^{10} + 6175977692 T^{12} - 76137385088 T^{14} + 500246412961 T^{16} \))(\( 1 - 128 T^{2} + 8732 T^{4} - 399744 T^{6} + 13409894 T^{8} - 336184704 T^{10} + 6175977692 T^{12} - 76137385088 T^{14} + 500246412961 T^{16} \))(\( ( 1 - 30 T^{2} + 107 T^{4} - 25230 T^{6} + 707281 T^{8} )^{2} \))(\( 1 - 128 T^{2} + 8732 T^{4} - 399744 T^{6} + 13409894 T^{8} - 336184704 T^{10} + 6175977692 T^{12} - 76137385088 T^{14} + 500246412961 T^{16} \))
$31$ (\( 1 - 32 T^{2} + 572 T^{4} - 21984 T^{6} + 1650310 T^{8} - 21126624 T^{10} + 528254012 T^{12} - 28400117792 T^{14} + 852891037441 T^{16} \))(\( ( 1 - 34 T^{2} + 1411 T^{4} - 32674 T^{6} + 923521 T^{8} )^{2} \))(\( 1 - 32 T^{2} + 572 T^{4} - 21984 T^{6} + 1650310 T^{8} - 21126624 T^{10} + 528254012 T^{12} - 28400117792 T^{14} + 852891037441 T^{16} \))(\( 1 - 32 T^{2} + 572 T^{4} - 21984 T^{6} + 1650310 T^{8} - 21126624 T^{10} + 528254012 T^{12} - 28400117792 T^{14} + 852891037441 T^{16} \))(\( ( 1 - 34 T^{2} + 1411 T^{4} - 32674 T^{6} + 923521 T^{8} )^{2} \))(\( 1 - 32 T^{2} + 572 T^{4} - 21984 T^{6} + 1650310 T^{8} - 21126624 T^{10} + 528254012 T^{12} - 28400117792 T^{14} + 852891037441 T^{16} \))
$37$ (\( 1 - 152 T^{2} + 10004 T^{4} - 394632 T^{6} + 13477382 T^{8} - 540251208 T^{10} + 18749106644 T^{12} - 389990414168 T^{14} + 3512479453921 T^{16} \))(\( ( 1 - 66 T^{2} + 1369 T^{4} )^{4} \))(\( 1 - 152 T^{2} + 10004 T^{4} - 394632 T^{6} + 13477382 T^{8} - 540251208 T^{10} + 18749106644 T^{12} - 389990414168 T^{14} + 3512479453921 T^{16} \))(\( 1 - 152 T^{2} + 10004 T^{4} - 394632 T^{6} + 13477382 T^{8} - 540251208 T^{10} + 18749106644 T^{12} - 389990414168 T^{14} + 3512479453921 T^{16} \))(\( ( 1 - 66 T^{2} + 1369 T^{4} )^{4} \))(\( 1 - 152 T^{2} + 10004 T^{4} - 394632 T^{6} + 13477382 T^{8} - 540251208 T^{10} + 18749106644 T^{12} - 389990414168 T^{14} + 3512479453921 T^{16} \))
$41$ (\( ( 1 - 4 T + 142 T^{2} - 468 T^{3} + 8354 T^{4} - 19188 T^{5} + 238702 T^{6} - 275684 T^{7} + 2825761 T^{8} )^{2} \))(\( ( 1 + 74 T^{2} + 3931 T^{4} + 124394 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 + 4 T + 142 T^{2} + 468 T^{3} + 8354 T^{4} + 19188 T^{5} + 238702 T^{6} + 275684 T^{7} + 2825761 T^{8} )^{2} \))(\( ( 1 + 4 T + 142 T^{2} + 468 T^{3} + 8354 T^{4} + 19188 T^{5} + 238702 T^{6} + 275684 T^{7} + 2825761 T^{8} )^{2} \))(\( ( 1 + 74 T^{2} + 3931 T^{4} + 124394 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 - 4 T + 142 T^{2} - 468 T^{3} + 8354 T^{4} - 19188 T^{5} + 238702 T^{6} - 275684 T^{7} + 2825761 T^{8} )^{2} \))
$43$ (\( 1 - 228 T^{2} + 23256 T^{4} - 1476524 T^{6} + 70398222 T^{8} - 2730092876 T^{10} + 79507636056 T^{12} - 1441270775172 T^{14} + 11688200277601 T^{16} \))(\( ( 1 - 118 T^{2} + 6979 T^{4} - 218182 T^{6} + 3418801 T^{8} )^{2} \))(\( 1 - 228 T^{2} + 23256 T^{4} - 1476524 T^{6} + 70398222 T^{8} - 2730092876 T^{10} + 79507636056 T^{12} - 1441270775172 T^{14} + 11688200277601 T^{16} \))(\( 1 - 228 T^{2} + 23256 T^{4} - 1476524 T^{6} + 70398222 T^{8} - 2730092876 T^{10} + 79507636056 T^{12} - 1441270775172 T^{14} + 11688200277601 T^{16} \))(\( ( 1 - 118 T^{2} + 6979 T^{4} - 218182 T^{6} + 3418801 T^{8} )^{2} \))(\( 1 - 228 T^{2} + 23256 T^{4} - 1476524 T^{6} + 70398222 T^{8} - 2730092876 T^{10} + 79507636056 T^{12} - 1441270775172 T^{14} + 11688200277601 T^{16} \))
$47$ (\( 1 + 68 T^{2} + 10040 T^{4} + 446124 T^{6} + 34404910 T^{8} + 985487916 T^{10} + 48991997240 T^{12} + 732986642372 T^{14} + 23811286661761 T^{16} \))(\( ( 1 - 128 T^{2} + 7714 T^{4} - 282752 T^{6} + 4879681 T^{8} )^{2} \))(\( 1 + 68 T^{2} + 10040 T^{4} + 446124 T^{6} + 34404910 T^{8} + 985487916 T^{10} + 48991997240 T^{12} + 732986642372 T^{14} + 23811286661761 T^{16} \))(\( 1 + 68 T^{2} + 10040 T^{4} + 446124 T^{6} + 34404910 T^{8} + 985487916 T^{10} + 48991997240 T^{12} + 732986642372 T^{14} + 23811286661761 T^{16} \))(\( ( 1 - 128 T^{2} + 7714 T^{4} - 282752 T^{6} + 4879681 T^{8} )^{2} \))(\( 1 + 68 T^{2} + 10040 T^{4} + 446124 T^{6} + 34404910 T^{8} + 985487916 T^{10} + 48991997240 T^{12} + 732986642372 T^{14} + 23811286661761 T^{16} \))
$53$ (\( ( 1 - 4 T + 160 T^{2} - 636 T^{3} + 11630 T^{4} - 33708 T^{5} + 449440 T^{6} - 595508 T^{7} + 7890481 T^{8} )^{2} \))(\( ( 1 + 2 T + 57 T^{2} + 106 T^{3} + 2809 T^{4} )^{4} \))(\( ( 1 - 4 T + 160 T^{2} - 636 T^{3} + 11630 T^{4} - 33708 T^{5} + 449440 T^{6} - 595508 T^{7} + 7890481 T^{8} )^{2} \))(\( ( 1 + 4 T + 160 T^{2} + 636 T^{3} + 11630 T^{4} + 33708 T^{5} + 449440 T^{6} + 595508 T^{7} + 7890481 T^{8} )^{2} \))(\( ( 1 - 2 T + 57 T^{2} - 106 T^{3} + 2809 T^{4} )^{4} \))(\( ( 1 + 4 T + 160 T^{2} + 636 T^{3} + 11630 T^{4} + 33708 T^{5} + 449440 T^{6} + 595508 T^{7} + 7890481 T^{8} )^{2} \))
$59$ (\( ( 1 + 42 T^{2} + 312 T^{3} + 3106 T^{4} + 18408 T^{5} + 146202 T^{6} + 12117361 T^{8} )^{2} \))(\( ( 1 + 46 T^{2} + 5691 T^{4} + 160126 T^{6} + 12117361 T^{8} )^{2} \))(\( ( 1 + 42 T^{2} - 312 T^{3} + 3106 T^{4} - 18408 T^{5} + 146202 T^{6} + 12117361 T^{8} )^{2} \))(\( ( 1 + 42 T^{2} - 312 T^{3} + 3106 T^{4} - 18408 T^{5} + 146202 T^{6} + 12117361 T^{8} )^{2} \))(\( ( 1 + 46 T^{2} + 5691 T^{4} + 160126 T^{6} + 12117361 T^{8} )^{2} \))(\( ( 1 + 42 T^{2} + 312 T^{3} + 3106 T^{4} + 18408 T^{5} + 146202 T^{6} + 12117361 T^{8} )^{2} \))
$61$ (\( 1 - 236 T^{2} + 31208 T^{4} - 2804772 T^{6} + 194357422 T^{8} - 10436556612 T^{10} + 432101005928 T^{12} - 12158808349196 T^{14} + 191707312997281 T^{16} \))(\( ( 1 - 214 T^{2} + 18691 T^{4} - 796294 T^{6} + 13845841 T^{8} )^{2} \))(\( 1 - 236 T^{2} + 31208 T^{4} - 2804772 T^{6} + 194357422 T^{8} - 10436556612 T^{10} + 432101005928 T^{12} - 12158808349196 T^{14} + 191707312997281 T^{16} \))(\( 1 - 236 T^{2} + 31208 T^{4} - 2804772 T^{6} + 194357422 T^{8} - 10436556612 T^{10} + 432101005928 T^{12} - 12158808349196 T^{14} + 191707312997281 T^{16} \))(\( ( 1 - 214 T^{2} + 18691 T^{4} - 796294 T^{6} + 13845841 T^{8} )^{2} \))(\( 1 - 236 T^{2} + 31208 T^{4} - 2804772 T^{6} + 194357422 T^{8} - 10436556612 T^{10} + 432101005928 T^{12} - 12158808349196 T^{14} + 191707312997281 T^{16} \))
$67$ (\( 1 - 180 T^{2} + 22904 T^{4} - 1953756 T^{6} + 148114446 T^{8} - 8770410684 T^{10} + 461541275384 T^{12} - 16282508790420 T^{14} + 406067677556641 T^{16} \))(\( ( 1 - 67 T^{2} )^{8} \))(\( 1 - 180 T^{2} + 22904 T^{4} - 1953756 T^{6} + 148114446 T^{8} - 8770410684 T^{10} + 461541275384 T^{12} - 16282508790420 T^{14} + 406067677556641 T^{16} \))(\( 1 - 180 T^{2} + 22904 T^{4} - 1953756 T^{6} + 148114446 T^{8} - 8770410684 T^{10} + 461541275384 T^{12} - 16282508790420 T^{14} + 406067677556641 T^{16} \))(\( ( 1 - 67 T^{2} )^{8} \))(\( 1 - 180 T^{2} + 22904 T^{4} - 1953756 T^{6} + 148114446 T^{8} - 8770410684 T^{10} + 461541275384 T^{12} - 16282508790420 T^{14} + 406067677556641 T^{16} \))
$71$ (\( 1 - 460 T^{2} + 98600 T^{4} - 12828132 T^{6} + 1106047246 T^{8} - 64666613412 T^{10} + 2505591746600 T^{12} - 58926130603660 T^{14} + 645753531245761 T^{16} \))(\( ( 1 - 20 T^{2} - 2618 T^{4} - 100820 T^{6} + 25411681 T^{8} )^{2} \))(\( 1 - 460 T^{2} + 98600 T^{4} - 12828132 T^{6} + 1106047246 T^{8} - 64666613412 T^{10} + 2505591746600 T^{12} - 58926130603660 T^{14} + 645753531245761 T^{16} \))(\( 1 - 460 T^{2} + 98600 T^{4} - 12828132 T^{6} + 1106047246 T^{8} - 64666613412 T^{10} + 2505591746600 T^{12} - 58926130603660 T^{14} + 645753531245761 T^{16} \))(\( ( 1 - 20 T^{2} - 2618 T^{4} - 100820 T^{6} + 25411681 T^{8} )^{2} \))(\( 1 - 460 T^{2} + 98600 T^{4} - 12828132 T^{6} + 1106047246 T^{8} - 64666613412 T^{10} + 2505591746600 T^{12} - 58926130603660 T^{14} + 645753531245761 T^{16} \))
$73$ (\( ( 1 - 24 T + 404 T^{2} - 4680 T^{3} + 45878 T^{4} - 341640 T^{5} + 2152916 T^{6} - 9336408 T^{7} + 28398241 T^{8} )^{2} \))(\( ( 1 + 72 T^{2} + 4754 T^{4} + 383688 T^{6} + 28398241 T^{8} )^{2} \))(\( ( 1 + 24 T + 404 T^{2} + 4680 T^{3} + 45878 T^{4} + 341640 T^{5} + 2152916 T^{6} + 9336408 T^{7} + 28398241 T^{8} )^{2} \))(\( ( 1 - 24 T + 404 T^{2} - 4680 T^{3} + 45878 T^{4} - 341640 T^{5} + 2152916 T^{6} - 9336408 T^{7} + 28398241 T^{8} )^{2} \))(\( ( 1 + 72 T^{2} + 4754 T^{4} + 383688 T^{6} + 28398241 T^{8} )^{2} \))(\( ( 1 + 24 T + 404 T^{2} + 4680 T^{3} + 45878 T^{4} + 341640 T^{5} + 2152916 T^{6} + 9336408 T^{7} + 28398241 T^{8} )^{2} \))
$79$ (\( ( 1 + 4 T + 48 T^{2} + 532 T^{3} + 9470 T^{4} + 42028 T^{5} + 299568 T^{6} + 1972156 T^{7} + 38950081 T^{8} )^{2} \))(\( ( 1 - 12 T + 144 T^{2} - 948 T^{3} + 6241 T^{4} )^{4} \))(\( ( 1 + 4 T + 48 T^{2} + 532 T^{3} + 9470 T^{4} + 42028 T^{5} + 299568 T^{6} + 1972156 T^{7} + 38950081 T^{8} )^{2} \))(\( ( 1 + 4 T + 48 T^{2} + 532 T^{3} + 9470 T^{4} + 42028 T^{5} + 299568 T^{6} + 1972156 T^{7} + 38950081 T^{8} )^{2} \))(\( ( 1 - 12 T + 144 T^{2} - 948 T^{3} + 6241 T^{4} )^{4} \))(\( ( 1 + 4 T + 48 T^{2} + 532 T^{3} + 9470 T^{4} + 42028 T^{5} + 299568 T^{6} + 1972156 T^{7} + 38950081 T^{8} )^{2} \))
$83$ (\( 1 - 424 T^{2} + 86492 T^{4} - 11335896 T^{6} + 1081357798 T^{8} - 78092987544 T^{10} + 4104765099932 T^{12} - 138622718308456 T^{14} + 2252292232139041 T^{16} \))(\( ( 1 - 62 T^{2} + 9739 T^{4} - 427118 T^{6} + 47458321 T^{8} )^{2} \))(\( 1 - 424 T^{2} + 86492 T^{4} - 11335896 T^{6} + 1081357798 T^{8} - 78092987544 T^{10} + 4104765099932 T^{12} - 138622718308456 T^{14} + 2252292232139041 T^{16} \))(\( 1 - 424 T^{2} + 86492 T^{4} - 11335896 T^{6} + 1081357798 T^{8} - 78092987544 T^{10} + 4104765099932 T^{12} - 138622718308456 T^{14} + 2252292232139041 T^{16} \))(\( ( 1 - 62 T^{2} + 9739 T^{4} - 427118 T^{6} + 47458321 T^{8} )^{2} \))(\( 1 - 424 T^{2} + 86492 T^{4} - 11335896 T^{6} + 1081357798 T^{8} - 78092987544 T^{10} + 4104765099932 T^{12} - 138622718308456 T^{14} + 2252292232139041 T^{16} \))
$89$ (\( ( 1 + 4 T + 118 T^{2} - 1308 T^{3} - 670 T^{4} - 116412 T^{5} + 934678 T^{6} + 2819876 T^{7} + 62742241 T^{8} )^{2} \))(\( ( 1 + 116 T^{2} + 6406 T^{4} + 918836 T^{6} + 62742241 T^{8} )^{2} \))(\( ( 1 - 4 T + 118 T^{2} + 1308 T^{3} - 670 T^{4} + 116412 T^{5} + 934678 T^{6} - 2819876 T^{7} + 62742241 T^{8} )^{2} \))(\( ( 1 - 4 T + 118 T^{2} + 1308 T^{3} - 670 T^{4} + 116412 T^{5} + 934678 T^{6} - 2819876 T^{7} + 62742241 T^{8} )^{2} \))(\( ( 1 + 116 T^{2} + 6406 T^{4} + 918836 T^{6} + 62742241 T^{8} )^{2} \))(\( ( 1 + 4 T + 118 T^{2} - 1308 T^{3} - 670 T^{4} - 116412 T^{5} + 934678 T^{6} + 2819876 T^{7} + 62742241 T^{8} )^{2} \))
$97$ (\( ( 1 + 12 T + 176 T^{2} + 2148 T^{3} + 29438 T^{4} + 208356 T^{5} + 1655984 T^{6} + 10952076 T^{7} + 88529281 T^{8} )^{2} \))(\( ( 1 + 48 T^{2} - 9406 T^{4} + 451632 T^{6} + 88529281 T^{8} )^{2} \))(\( ( 1 - 12 T + 176 T^{2} - 2148 T^{3} + 29438 T^{4} - 208356 T^{5} + 1655984 T^{6} - 10952076 T^{7} + 88529281 T^{8} )^{2} \))(\( ( 1 + 12 T + 176 T^{2} + 2148 T^{3} + 29438 T^{4} + 208356 T^{5} + 1655984 T^{6} + 10952076 T^{7} + 88529281 T^{8} )^{2} \))(\( ( 1 + 48 T^{2} - 9406 T^{4} + 451632 T^{6} + 88529281 T^{8} )^{2} \))(\( ( 1 - 12 T + 176 T^{2} - 2148 T^{3} + 29438 T^{4} - 208356 T^{5} + 1655984 T^{6} - 10952076 T^{7} + 88529281 T^{8} )^{2} \))
show more
show less