Properties

Label 2100.2.f
Level 2100
Weight 2
Character orbit f
Rep. character \(\chi_{2100}(1049,\cdot)\)
Character field \(\Q\)
Dimension 48
Newform subspaces 9
Sturm bound 960
Trace bound 21

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Defining parameters

Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2100.f (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 105 \)
Character field: \(\Q\)
Newform subspaces: \( 9 \)
Sturm bound: \(960\)
Trace bound: \(21\)
Distinguishing \(T_p\): \(11\), \(13\), \(23\), \(41\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(2100, [\chi])\).

Total New Old
Modular forms 516 48 468
Cusp forms 444 48 396
Eisenstein series 72 0 72

Trace form

\( 48q - 12q^{9} + O(q^{10}) \) \( 48q - 12q^{9} + 2q^{21} - 34q^{49} + 28q^{51} + 88q^{79} + 20q^{81} + 10q^{91} + 60q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(2100, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
2100.2.f.a \(4\) \(16.769\) \(\Q(i, \sqrt{5})\) None \(0\) \(-2\) \(0\) \(-6\) \(q+\beta _{1}q^{3}+(-1+\beta _{2}-\beta _{3})q^{7}+(-\beta _{1}+\cdots)q^{9}+\cdots\)
2100.2.f.b \(4\) \(16.769\) \(\Q(i, \sqrt{5})\) None \(0\) \(-2\) \(0\) \(-6\) \(q+\beta _{1}q^{3}+(-2-\beta _{1}-\beta _{2})q^{7}+(-\beta _{1}+\cdots)q^{9}+\cdots\)
2100.2.f.c \(4\) \(16.769\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q+(-\zeta_{12}+2\zeta_{12}^{3})q^{3}+(-2\zeta_{12}+3\zeta_{12}^{3})q^{7}+\cdots\)
2100.2.f.d \(4\) \(16.769\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q+(-\zeta_{12}+2\zeta_{12}^{3})q^{3}+(-2\zeta_{12}-\zeta_{12}^{3})q^{7}+\cdots\)
2100.2.f.e \(4\) \(16.769\) \(\Q(\zeta_{12})\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{12}q^{3}+(-\zeta_{12}-\zeta_{12}^{3})q^{7}+3q^{9}+\cdots\)
2100.2.f.f \(4\) \(16.769\) \(\Q(\zeta_{12})\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(q+(2\zeta_{12}-\zeta_{12}^{3})q^{3}+(3\zeta_{12}-\zeta_{12}^{3})q^{7}+\cdots\)
2100.2.f.g \(4\) \(16.769\) \(\Q(i, \sqrt{5})\) None \(0\) \(2\) \(0\) \(6\) \(q-\beta _{1}q^{3}+(1-\beta _{2}+\beta _{3})q^{7}+(-\beta _{1}+\cdots)q^{9}+\cdots\)
2100.2.f.h \(4\) \(16.769\) \(\Q(i, \sqrt{5})\) None \(0\) \(2\) \(0\) \(6\) \(q-\beta _{1}q^{3}+(2+\beta _{1}+\beta _{2})q^{7}+(-\beta _{1}+\cdots)q^{9}+\cdots\)
2100.2.f.i \(16\) \(16.769\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{3}q^{3}+(\beta _{1}-\beta _{8})q^{7}+(-1-\beta _{7}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(2100, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(2100, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(420, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(525, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1050, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4} \))(\( 1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4} \))(\( 1 + 3 T^{2} + 9 T^{4} \))(\( 1 + 3 T^{2} + 9 T^{4} \))(\( ( 1 - 3 T^{2} )^{2} \))(\( ( 1 - 3 T^{2} )^{2} \))(\( 1 - 2 T + 2 T^{2} - 6 T^{3} + 9 T^{4} \))(\( 1 - 2 T + 2 T^{2} - 6 T^{3} + 9 T^{4} \))(\( ( 1 + 6 T^{2} + 22 T^{4} + 54 T^{6} + 81 T^{8} )^{2} \))
$5$ 1
$7$ (\( 1 + 6 T + 18 T^{2} + 42 T^{3} + 49 T^{4} \))(\( 1 + 6 T + 18 T^{2} + 42 T^{3} + 49 T^{4} \))(\( 1 + 2 T^{2} + 49 T^{4} \))(\( 1 + 2 T^{2} + 49 T^{4} \))(\( 1 + 2 T^{2} + 49 T^{4} \))(\( 1 - 13 T^{2} + 49 T^{4} \))(\( 1 - 6 T + 18 T^{2} - 42 T^{3} + 49 T^{4} \))(\( 1 - 6 T + 18 T^{2} - 42 T^{3} + 49 T^{4} \))(\( ( 1 + 6 T^{2} + 49 T^{4} )^{4} \))
$11$ (\( 1 - 16 T^{2} + 126 T^{4} - 1936 T^{6} + 14641 T^{8} \))(\( 1 - 16 T^{2} + 126 T^{4} - 1936 T^{6} + 14641 T^{8} \))(\( ( 1 + 5 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 + 5 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 11 T^{2} )^{4} \))(\( ( 1 - 11 T^{2} )^{4} \))(\( 1 - 16 T^{2} + 126 T^{4} - 1936 T^{6} + 14641 T^{8} \))(\( 1 - 16 T^{2} + 126 T^{4} - 1936 T^{6} + 14641 T^{8} \))(\( ( 1 - 28 T^{2} + 393 T^{4} - 3388 T^{6} + 14641 T^{8} )^{4} \))
$13$ (\( ( 1 + 2 T + 22 T^{2} + 26 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 + 2 T + 22 T^{2} + 26 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 + 23 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 + 23 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 22 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 + 23 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 2 T + 22 T^{2} - 26 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 - 2 T + 22 T^{2} - 26 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 + 16 T^{2} + 82 T^{4} + 2704 T^{6} + 28561 T^{8} )^{4} \))
$17$ (\( ( 1 - 14 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 14 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 25 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 25 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 17 T^{2} )^{4} \))(\( ( 1 - 17 T^{2} )^{4} \))(\( ( 1 - 14 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 14 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 46 T^{2} + 1062 T^{4} - 13294 T^{6} + 83521 T^{8} )^{4} \))
$19$ (\( 1 - 16 T^{2} + 286 T^{4} - 5776 T^{6} + 130321 T^{8} \))(\( 1 - 16 T^{2} + 286 T^{4} - 5776 T^{6} + 130321 T^{8} \))(\( ( 1 - 8 T + 19 T^{2} )^{2}( 1 + 8 T + 19 T^{2} )^{2} \))(\( ( 1 - 8 T + 19 T^{2} )^{2}( 1 + 8 T + 19 T^{2} )^{2} \))(\( ( 1 - 8 T + 19 T^{2} )^{2}( 1 + 8 T + 19 T^{2} )^{2} \))(\( ( 1 - 7 T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2} \))(\( 1 - 16 T^{2} + 286 T^{4} - 5776 T^{6} + 130321 T^{8} \))(\( 1 - 16 T^{2} + 286 T^{4} - 5776 T^{6} + 130321 T^{8} \))(\( ( 1 - 46 T^{2} + 1126 T^{4} - 16606 T^{6} + 130321 T^{8} )^{4} \))
$23$ (\( ( 1 - 2 T + 2 T^{2} - 46 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 + 2 T + 2 T^{2} + 46 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 + 23 T^{2} )^{4} \))(\( ( 1 + 23 T^{2} )^{4} \))(\( ( 1 + 23 T^{2} )^{4} \))(\( ( 1 + 23 T^{2} )^{4} \))(\( ( 1 + 2 T + 2 T^{2} + 46 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 - 2 T + 2 T^{2} - 46 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 + 12 T^{2} - 31 T^{4} + 6348 T^{6} + 279841 T^{8} )^{4} \))
$29$ (\( ( 1 - 10 T + 29 T^{2} )^{2}( 1 + 10 T + 29 T^{2} )^{2} \))(\( ( 1 - 10 T + 29 T^{2} )^{2}( 1 + 10 T + 29 T^{2} )^{2} \))(\( ( 1 - 31 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 31 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 29 T^{2} )^{4} \))(\( ( 1 - 29 T^{2} )^{4} \))(\( ( 1 - 10 T + 29 T^{2} )^{2}( 1 + 10 T + 29 T^{2} )^{2} \))(\( ( 1 - 10 T + 29 T^{2} )^{2}( 1 + 10 T + 29 T^{2} )^{2} \))(\( ( 1 - 12 T^{2} + 1313 T^{4} - 10092 T^{6} + 707281 T^{8} )^{4} \))
$31$ (\( 1 - 112 T^{2} + 5038 T^{4} - 107632 T^{6} + 923521 T^{8} \))(\( 1 - 112 T^{2} + 5038 T^{4} - 107632 T^{6} + 923521 T^{8} \))(\( ( 1 - 4 T + 31 T^{2} )^{2}( 1 + 4 T + 31 T^{2} )^{2} \))(\( ( 1 - 4 T + 31 T^{2} )^{2}( 1 + 4 T + 31 T^{2} )^{2} \))(\( ( 1 - 4 T + 31 T^{2} )^{2}( 1 + 4 T + 31 T^{2} )^{2} \))(\( ( 1 - 11 T + 31 T^{2} )^{2}( 1 + 11 T + 31 T^{2} )^{2} \))(\( 1 - 112 T^{2} + 5038 T^{4} - 107632 T^{6} + 923521 T^{8} \))(\( 1 - 112 T^{2} + 5038 T^{4} - 107632 T^{6} + 923521 T^{8} \))(\( ( 1 - 94 T^{2} + 4006 T^{4} - 90334 T^{6} + 923521 T^{8} )^{4} \))
$37$ (\( ( 1 - 54 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 - 54 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 - 10 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 - 10 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 + 26 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 + 26 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 - 54 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 - 54 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 - 69 T^{2} + 1369 T^{4} )^{8} \))
$41$ (\( ( 1 - 16 T + 126 T^{2} - 656 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 16 T + 126 T^{2} + 656 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{4} \))(\( ( 1 + 6 T + 41 T^{2} )^{4} \))(\( ( 1 + 41 T^{2} )^{4} \))(\( ( 1 + 41 T^{2} )^{4} \))(\( ( 1 - 16 T + 126 T^{2} - 656 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 16 T + 126 T^{2} + 656 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 74 T^{2} + 4606 T^{4} + 124394 T^{6} + 2825761 T^{8} )^{4} \))
$43$ (\( 1 - 112 T^{2} + 6334 T^{4} - 207088 T^{6} + 3418801 T^{8} \))(\( 1 - 112 T^{2} + 6334 T^{4} - 207088 T^{6} + 3418801 T^{8} \))(\( ( 1 + 14 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 + 14 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 22 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 + 83 T^{2} + 1849 T^{4} )^{2} \))(\( 1 - 112 T^{2} + 6334 T^{4} - 207088 T^{6} + 3418801 T^{8} \))(\( 1 - 112 T^{2} + 6334 T^{4} - 207088 T^{6} + 3418801 T^{8} \))(\( ( 1 - 82 T^{2} + 3379 T^{4} - 151618 T^{6} + 3418801 T^{8} )^{4} \))
$47$ (\( 1 - 176 T^{2} + 12142 T^{4} - 388784 T^{6} + 4879681 T^{8} \))(\( 1 - 176 T^{2} + 12142 T^{4} - 388784 T^{6} + 4879681 T^{8} \))(\( ( 1 - 85 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 - 85 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 - 47 T^{2} )^{4} \))(\( ( 1 - 47 T^{2} )^{4} \))(\( 1 - 176 T^{2} + 12142 T^{4} - 388784 T^{6} + 4879681 T^{8} \))(\( 1 - 176 T^{2} + 12142 T^{4} - 388784 T^{6} + 4879681 T^{8} \))(\( ( 1 + 34 T^{2} + 1582 T^{4} + 75106 T^{6} + 4879681 T^{8} )^{4} \))
$53$ (\( ( 1 - 14 T + 150 T^{2} - 742 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 + 14 T + 150 T^{2} + 742 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 - 2 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 2 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 + 53 T^{2} )^{4} \))(\( ( 1 + 53 T^{2} )^{4} \))(\( ( 1 + 14 T + 150 T^{2} + 742 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 - 14 T + 150 T^{2} - 742 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 + 32 T^{2} + 5374 T^{4} + 89888 T^{6} + 7890481 T^{8} )^{4} \))
$59$ (\( ( 1 + 38 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 38 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 6 T + 59 T^{2} )^{4} \))(\( ( 1 - 6 T + 59 T^{2} )^{4} \))(\( ( 1 + 59 T^{2} )^{4} \))(\( ( 1 + 59 T^{2} )^{4} \))(\( ( 1 + 38 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 38 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 76 T^{2} + 3906 T^{4} + 264556 T^{6} + 12117361 T^{8} )^{4} \))
$61$ (\( 1 - 52 T^{2} + 2998 T^{4} - 193492 T^{6} + 13845841 T^{8} \))(\( 1 - 52 T^{2} + 2998 T^{4} - 193492 T^{6} + 13845841 T^{8} \))(\( ( 1 - 14 T + 61 T^{2} )^{2}( 1 + 14 T + 61 T^{2} )^{2} \))(\( ( 1 - 14 T + 61 T^{2} )^{2}( 1 + 14 T + 61 T^{2} )^{2} \))(\( ( 1 - 14 T + 61 T^{2} )^{2}( 1 + 14 T + 61 T^{2} )^{2} \))(\( ( 1 - T + 61 T^{2} )^{2}( 1 + T + 61 T^{2} )^{2} \))(\( 1 - 52 T^{2} + 2998 T^{4} - 193492 T^{6} + 13845841 T^{8} \))(\( 1 - 52 T^{2} + 2998 T^{4} - 193492 T^{6} + 13845841 T^{8} \))(\( ( 1 - 94 T^{2} + 6526 T^{4} - 349774 T^{6} + 13845841 T^{8} )^{4} \))
$67$ (\( 1 - 160 T^{2} + 13758 T^{4} - 718240 T^{6} + 20151121 T^{8} \))(\( 1 - 160 T^{2} + 13758 T^{4} - 718240 T^{6} + 20151121 T^{8} \))(\( ( 1 - 130 T^{2} + 4489 T^{4} )^{2} \))(\( ( 1 - 130 T^{2} + 4489 T^{4} )^{2} \))(\( ( 1 + 122 T^{2} + 4489 T^{4} )^{2} \))(\( ( 1 - 13 T^{2} + 4489 T^{4} )^{2} \))(\( 1 - 160 T^{2} + 13758 T^{4} - 718240 T^{6} + 20151121 T^{8} \))(\( 1 - 160 T^{2} + 13758 T^{4} - 718240 T^{6} + 20151121 T^{8} \))(\( ( 1 - 89 T^{2} + 4489 T^{4} )^{8} \))
$71$ (\( 1 - 32 T^{2} + 9838 T^{4} - 161312 T^{6} + 25411681 T^{8} \))(\( 1 - 32 T^{2} + 9838 T^{4} - 161312 T^{6} + 25411681 T^{8} \))(\( ( 1 - 34 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 34 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 71 T^{2} )^{4} \))(\( ( 1 - 71 T^{2} )^{4} \))(\( 1 - 32 T^{2} + 9838 T^{4} - 161312 T^{6} + 25411681 T^{8} \))(\( 1 - 32 T^{2} + 9838 T^{4} - 161312 T^{6} + 25411681 T^{8} \))(\( ( 1 - 164 T^{2} + 13681 T^{4} - 826724 T^{6} + 25411681 T^{8} )^{4} \))
$73$ (\( ( 1 + 18 T + 222 T^{2} + 1314 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 + 18 T + 222 T^{2} + 1314 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 + 98 T^{2} + 5329 T^{4} )^{2} \))(\( ( 1 + 98 T^{2} + 5329 T^{4} )^{2} \))(\( ( 1 - 46 T^{2} + 5329 T^{4} )^{2} \))(\( ( 1 - 46 T^{2} + 5329 T^{4} )^{2} \))(\( ( 1 - 18 T + 222 T^{2} - 1314 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 - 18 T + 222 T^{2} - 1314 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 + 144 T^{2} + 5329 T^{4} )^{8} \))
$79$ (\( ( 1 + 12 T + 174 T^{2} + 948 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 + 12 T + 174 T^{2} + 948 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 - 13 T + 79 T^{2} )^{4} \))(\( ( 1 - 13 T + 79 T^{2} )^{4} \))(\( ( 1 - 4 T + 79 T^{2} )^{4} \))(\( ( 1 - 4 T + 79 T^{2} )^{4} \))(\( ( 1 + 12 T + 174 T^{2} + 948 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 + 12 T + 174 T^{2} + 948 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 - 6 T + 147 T^{2} - 474 T^{3} + 6241 T^{4} )^{8} \))
$83$ (\( 1 - 272 T^{2} + 31774 T^{4} - 1873808 T^{6} + 47458321 T^{8} \))(\( 1 - 272 T^{2} + 31774 T^{4} - 1873808 T^{6} + 47458321 T^{8} \))(\( ( 1 - 22 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 22 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 83 T^{2} )^{4} \))(\( ( 1 - 83 T^{2} )^{4} \))(\( 1 - 272 T^{2} + 31774 T^{4} - 1873808 T^{6} + 47458321 T^{8} \))(\( 1 - 272 T^{2} + 31774 T^{4} - 1873808 T^{6} + 47458321 T^{8} \))(\( ( 1 - 242 T^{2} + 28294 T^{4} - 1667138 T^{6} + 47458321 T^{8} )^{4} \))
$89$ (\( ( 1 + 16 T + 222 T^{2} + 1424 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 16 T + 222 T^{2} - 1424 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 89 T^{2} )^{4} \))(\( ( 1 + 89 T^{2} )^{4} \))(\( ( 1 + 89 T^{2} )^{4} \))(\( ( 1 + 89 T^{2} )^{4} \))(\( ( 1 + 16 T + 222 T^{2} + 1424 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 16 T + 222 T^{2} - 1424 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 116 T^{2} + 6706 T^{4} + 918836 T^{6} + 62742241 T^{8} )^{4} \))
$97$ (\( ( 1 - 6 T + 78 T^{2} - 582 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 6 T + 78 T^{2} - 582 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 + 191 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 + 191 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 + 2 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 + 167 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 + 6 T + 78 T^{2} + 582 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 + 6 T + 78 T^{2} + 582 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 + 294 T^{2} + 38222 T^{4} + 2766246 T^{6} + 88529281 T^{8} )^{4} \))
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