Properties

Label 1350.2.c
Level 1350
Weight 2
Character orbit c
Rep. character \(\chi_{1350}(649,\cdot)\)
Character field \(\Q\)
Dimension 24
Newform subspaces 12
Sturm bound 540
Trace bound 29

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Defining parameters

Level: \( N \) = \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1350.c (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 12 \)
Sturm bound: \(540\)
Trace bound: \(29\)
Distinguishing \(T_p\): \(7\), \(11\), \(13\), \(17\), \(29\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1350, [\chi])\).

Total New Old
Modular forms 306 24 282
Cusp forms 234 24 210
Eisenstein series 72 0 72

Trace form

\( 24q - 24q^{4} + O(q^{10}) \) \( 24q - 24q^{4} + 24q^{16} - 12q^{19} + 24q^{34} + 48q^{49} - 12q^{61} - 24q^{64} + 12q^{76} + 36q^{79} + 96q^{91} - 72q^{94} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1350, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1350.2.c.a \(2\) \(10.780\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+iq^{2}-q^{4}+2iq^{7}-iq^{8}-3q^{11}+\cdots\)
1350.2.c.b \(2\) \(10.780\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+iq^{2}-q^{4}+iq^{7}-iq^{8}-3q^{11}+\cdots\)
1350.2.c.c \(2\) \(10.780\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q-iq^{2}-q^{4}+2iq^{7}+iq^{8}-3q^{11}+\cdots\)
1350.2.c.d \(2\) \(10.780\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q-iq^{2}-q^{4}+2iq^{7}+iq^{8}-3q^{11}+\cdots\)
1350.2.c.e \(2\) \(10.780\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q-iq^{2}-q^{4}+4iq^{7}+iq^{8}-3q^{11}+\cdots\)
1350.2.c.f \(2\) \(10.780\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+iq^{2}-q^{4}+iq^{7}-iq^{8}+2iq^{13}+\cdots\)
1350.2.c.g \(2\) \(10.780\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q-iq^{2}-q^{4}+iq^{7}+iq^{8}+2iq^{13}+\cdots\)
1350.2.c.h \(2\) \(10.780\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+iq^{2}-q^{4}+4iq^{7}-iq^{8}+3q^{11}+\cdots\)
1350.2.c.i \(2\) \(10.780\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+iq^{2}-q^{4}+2iq^{7}-iq^{8}+3q^{11}+\cdots\)
1350.2.c.j \(2\) \(10.780\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+iq^{2}-q^{4}+2iq^{7}-iq^{8}+3q^{11}+\cdots\)
1350.2.c.k \(2\) \(10.780\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q-iq^{2}-q^{4}+iq^{7}+iq^{8}+3q^{11}+\cdots\)
1350.2.c.l \(2\) \(10.780\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q-iq^{2}-q^{4}+2iq^{7}+iq^{8}+3q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1350, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1350, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(135, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(270, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(450, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(675, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))
$3$ (\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))
$5$ (\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))
$7$ (\( 1 - 10 T^{2} + 49 T^{4} \))(\( 1 - 13 T^{2} + 49 T^{4} \))(\( 1 - 10 T^{2} + 49 T^{4} \))(\( 1 - 10 T^{2} + 49 T^{4} \))(\( 1 + 2 T^{2} + 49 T^{4} \))(\( 1 - 13 T^{2} + 49 T^{4} \))(\( 1 - 13 T^{2} + 49 T^{4} \))(\( 1 + 2 T^{2} + 49 T^{4} \))(\( 1 - 10 T^{2} + 49 T^{4} \))(\( 1 - 10 T^{2} + 49 T^{4} \))(\( 1 - 13 T^{2} + 49 T^{4} \))(\( 1 - 10 T^{2} + 49 T^{4} \))
$11$ (\( ( 1 + 3 T + 11 T^{2} )^{2} \))(\( ( 1 + 3 T + 11 T^{2} )^{2} \))(\( ( 1 + 3 T + 11 T^{2} )^{2} \))(\( ( 1 + 3 T + 11 T^{2} )^{2} \))(\( ( 1 + 3 T + 11 T^{2} )^{2} \))(\( ( 1 + 11 T^{2} )^{2} \))(\( ( 1 + 11 T^{2} )^{2} \))(\( ( 1 - 3 T + 11 T^{2} )^{2} \))(\( ( 1 - 3 T + 11 T^{2} )^{2} \))(\( ( 1 - 3 T + 11 T^{2} )^{2} \))(\( ( 1 - 3 T + 11 T^{2} )^{2} \))(\( ( 1 - 3 T + 11 T^{2} )^{2} \))
$13$ (\( 1 - 25 T^{2} + 169 T^{4} \))(\( ( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} ) \))(\( 1 - T^{2} + 169 T^{4} \))(\( 1 - T^{2} + 169 T^{4} \))(\( 1 - 25 T^{2} + 169 T^{4} \))(\( 1 - 22 T^{2} + 169 T^{4} \))(\( 1 - 22 T^{2} + 169 T^{4} \))(\( 1 - 25 T^{2} + 169 T^{4} \))(\( 1 - T^{2} + 169 T^{4} \))(\( 1 - T^{2} + 169 T^{4} \))(\( ( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} ) \))(\( 1 - 25 T^{2} + 169 T^{4} \))
$17$ (\( 1 - 25 T^{2} + 289 T^{4} \))(\( ( 1 - 17 T^{2} )^{2} \))(\( 1 - 25 T^{2} + 289 T^{4} \))(\( 1 + 2 T^{2} + 289 T^{4} \))(\( ( 1 - 17 T^{2} )^{2} \))(\( 1 + 2 T^{2} + 289 T^{4} \))(\( 1 + 2 T^{2} + 289 T^{4} \))(\( ( 1 - 17 T^{2} )^{2} \))(\( 1 + 2 T^{2} + 289 T^{4} \))(\( 1 - 25 T^{2} + 289 T^{4} \))(\( ( 1 - 17 T^{2} )^{2} \))(\( 1 - 25 T^{2} + 289 T^{4} \))
$19$ (\( ( 1 + 8 T + 19 T^{2} )^{2} \))(\( ( 1 + 2 T + 19 T^{2} )^{2} \))(\( ( 1 - 4 T + 19 T^{2} )^{2} \))(\( ( 1 - 4 T + 19 T^{2} )^{2} \))(\( ( 1 + 2 T + 19 T^{2} )^{2} \))(\( ( 1 - T + 19 T^{2} )^{2} \))(\( ( 1 - T + 19 T^{2} )^{2} \))(\( ( 1 + 2 T + 19 T^{2} )^{2} \))(\( ( 1 - 4 T + 19 T^{2} )^{2} \))(\( ( 1 - 4 T + 19 T^{2} )^{2} \))(\( ( 1 + 2 T + 19 T^{2} )^{2} \))(\( ( 1 + 8 T + 19 T^{2} )^{2} \))
$23$ (\( 1 - 37 T^{2} + 529 T^{4} \))(\( 1 - 10 T^{2} + 529 T^{4} \))(\( 1 + 35 T^{2} + 529 T^{4} \))(\( 1 - 37 T^{2} + 529 T^{4} \))(\( 1 + 35 T^{2} + 529 T^{4} \))(\( 1 - 10 T^{2} + 529 T^{4} \))(\( 1 - 10 T^{2} + 529 T^{4} \))(\( 1 + 35 T^{2} + 529 T^{4} \))(\( 1 - 37 T^{2} + 529 T^{4} \))(\( 1 + 35 T^{2} + 529 T^{4} \))(\( 1 - 10 T^{2} + 529 T^{4} \))(\( 1 - 37 T^{2} + 529 T^{4} \))
$29$ (\( ( 1 - 9 T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 3 T + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 - 3 T + 29 T^{2} )^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 9 T + 29 T^{2} )^{2} \))
$31$ (\( ( 1 + 7 T + 31 T^{2} )^{2} \))(\( ( 1 - 5 T + 31 T^{2} )^{2} \))(\( ( 1 - 5 T + 31 T^{2} )^{2} \))(\( ( 1 - 2 T + 31 T^{2} )^{2} \))(\( ( 1 + 10 T + 31 T^{2} )^{2} \))(\( ( 1 - 5 T + 31 T^{2} )^{2} \))(\( ( 1 - 5 T + 31 T^{2} )^{2} \))(\( ( 1 + 10 T + 31 T^{2} )^{2} \))(\( ( 1 - 2 T + 31 T^{2} )^{2} \))(\( ( 1 - 5 T + 31 T^{2} )^{2} \))(\( ( 1 - 5 T + 31 T^{2} )^{2} \))(\( ( 1 + 7 T + 31 T^{2} )^{2} \))
$37$ (\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( 1 + 26 T^{2} + 1369 T^{4} \))(\( 1 + 47 T^{2} + 1369 T^{4} \))(\( 1 - 25 T^{2} + 1369 T^{4} \))(\( 1 - 25 T^{2} + 1369 T^{4} \))(\( 1 - 25 T^{2} + 1369 T^{4} \))(\( 1 - 25 T^{2} + 1369 T^{4} \))(\( 1 + 47 T^{2} + 1369 T^{4} \))(\( 1 + 26 T^{2} + 1369 T^{4} \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))
$41$ (\( ( 1 + 12 T + 41 T^{2} )^{2} \))(\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 12 T + 41 T^{2} )^{2} \))(\( ( 1 - 12 T + 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 12 T + 41 T^{2} )^{2} \))
$43$ (\( 1 - 37 T^{2} + 1849 T^{4} \))(\( 1 + 14 T^{2} + 1849 T^{4} \))(\( 1 - 85 T^{2} + 1849 T^{4} \))(\( 1 - 70 T^{2} + 1849 T^{4} \))(\( 1 + 14 T^{2} + 1849 T^{4} \))(\( 1 + 35 T^{2} + 1849 T^{4} \))(\( 1 + 35 T^{2} + 1849 T^{4} \))(\( 1 + 14 T^{2} + 1849 T^{4} \))(\( 1 - 70 T^{2} + 1849 T^{4} \))(\( 1 - 85 T^{2} + 1849 T^{4} \))(\( 1 + 14 T^{2} + 1849 T^{4} \))(\( 1 - 37 T^{2} + 1849 T^{4} \))
$47$ (\( 1 - 85 T^{2} + 2209 T^{4} \))(\( 1 - 58 T^{2} + 2209 T^{4} \))(\( 1 - 13 T^{2} + 2209 T^{4} \))(\( 1 - 85 T^{2} + 2209 T^{4} \))(\( 1 - 13 T^{2} + 2209 T^{4} \))(\( 1 + 50 T^{2} + 2209 T^{4} \))(\( 1 + 50 T^{2} + 2209 T^{4} \))(\( 1 - 13 T^{2} + 2209 T^{4} \))(\( 1 - 85 T^{2} + 2209 T^{4} \))(\( 1 - 13 T^{2} + 2209 T^{4} \))(\( 1 - 58 T^{2} + 2209 T^{4} \))(\( 1 - 85 T^{2} + 2209 T^{4} \))
$53$ (\( 1 + 38 T^{2} + 2809 T^{4} \))(\( 1 - 25 T^{2} + 2809 T^{4} \))(\( 1 + 38 T^{2} + 2809 T^{4} \))(\( 1 + 38 T^{2} + 2809 T^{4} \))(\( 1 - 70 T^{2} + 2809 T^{4} \))(\( ( 1 - 53 T^{2} )^{2} \))(\( ( 1 - 53 T^{2} )^{2} \))(\( 1 - 70 T^{2} + 2809 T^{4} \))(\( 1 + 38 T^{2} + 2809 T^{4} \))(\( 1 + 38 T^{2} + 2809 T^{4} \))(\( 1 - 25 T^{2} + 2809 T^{4} \))(\( 1 + 38 T^{2} + 2809 T^{4} \))
$59$ (\( ( 1 + 12 T + 59 T^{2} )^{2} \))(\( ( 1 + 12 T + 59 T^{2} )^{2} \))(\( ( 1 - 12 T + 59 T^{2} )^{2} \))(\( ( 1 + 9 T + 59 T^{2} )^{2} \))(\( ( 1 - 3 T + 59 T^{2} )^{2} \))(\( ( 1 + 6 T + 59 T^{2} )^{2} \))(\( ( 1 - 6 T + 59 T^{2} )^{2} \))(\( ( 1 + 3 T + 59 T^{2} )^{2} \))(\( ( 1 - 9 T + 59 T^{2} )^{2} \))(\( ( 1 + 12 T + 59 T^{2} )^{2} \))(\( ( 1 - 12 T + 59 T^{2} )^{2} \))(\( ( 1 - 12 T + 59 T^{2} )^{2} \))
$61$ (\( ( 1 + 10 T + 61 T^{2} )^{2} \))(\( ( 1 - 8 T + 61 T^{2} )^{2} \))(\( ( 1 - 2 T + 61 T^{2} )^{2} \))(\( ( 1 - 11 T + 61 T^{2} )^{2} \))(\( ( 1 + 7 T + 61 T^{2} )^{2} \))(\( ( 1 + 7 T + 61 T^{2} )^{2} \))(\( ( 1 + 7 T + 61 T^{2} )^{2} \))(\( ( 1 + 7 T + 61 T^{2} )^{2} \))(\( ( 1 - 11 T + 61 T^{2} )^{2} \))(\( ( 1 - 2 T + 61 T^{2} )^{2} \))(\( ( 1 - 8 T + 61 T^{2} )^{2} \))(\( ( 1 + 10 T + 61 T^{2} )^{2} \))
$67$ (\( 1 - 118 T^{2} + 4489 T^{4} \))(\( 1 + 62 T^{2} + 4489 T^{4} \))(\( 1 - 118 T^{2} + 4489 T^{4} \))(\( 1 + 62 T^{2} + 4489 T^{4} \))(\( 1 - 118 T^{2} + 4489 T^{4} \))(\( 1 - 118 T^{2} + 4489 T^{4} \))(\( 1 - 118 T^{2} + 4489 T^{4} \))(\( 1 - 118 T^{2} + 4489 T^{4} \))(\( 1 + 62 T^{2} + 4489 T^{4} \))(\( 1 - 118 T^{2} + 4489 T^{4} \))(\( 1 + 62 T^{2} + 4489 T^{4} \))(\( 1 - 118 T^{2} + 4489 T^{4} \))
$71$ (\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 15 T + 71 T^{2} )^{2} \))(\( ( 1 - 15 T + 71 T^{2} )^{2} \))(\( ( 1 + 6 T + 71 T^{2} )^{2} \))(\( ( 1 - 6 T + 71 T^{2} )^{2} \))(\( ( 1 + 15 T + 71 T^{2} )^{2} \))(\( ( 1 - 15 T + 71 T^{2} )^{2} \))(\( ( 1 - 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))
$73$ (\( 1 - 142 T^{2} + 5329 T^{4} \))(\( 1 - 97 T^{2} + 5329 T^{4} \))(\( 1 - 46 T^{2} + 5329 T^{4} \))(\( 1 - 142 T^{2} + 5329 T^{4} \))(\( 1 - 142 T^{2} + 5329 T^{4} \))(\( 1 - 97 T^{2} + 5329 T^{4} \))(\( 1 - 97 T^{2} + 5329 T^{4} \))(\( 1 - 142 T^{2} + 5329 T^{4} \))(\( 1 - 142 T^{2} + 5329 T^{4} \))(\( 1 - 46 T^{2} + 5329 T^{4} \))(\( 1 - 97 T^{2} + 5329 T^{4} \))(\( 1 - 142 T^{2} + 5329 T^{4} \))
$79$ (\( ( 1 - T + 79 T^{2} )^{2} \))(\( ( 1 + 8 T + 79 T^{2} )^{2} \))(\( ( 1 - 13 T + 79 T^{2} )^{2} \))(\( ( 1 - 10 T + 79 T^{2} )^{2} \))(\( ( 1 + 8 T + 79 T^{2} )^{2} \))(\( ( 1 - T + 79 T^{2} )^{2} \))(\( ( 1 - T + 79 T^{2} )^{2} \))(\( ( 1 + 8 T + 79 T^{2} )^{2} \))(\( ( 1 - 10 T + 79 T^{2} )^{2} \))(\( ( 1 - 13 T + 79 T^{2} )^{2} \))(\( ( 1 + 8 T + 79 T^{2} )^{2} \))(\( ( 1 - T + 79 T^{2} )^{2} \))
$83$ (\( 1 + 158 T^{2} + 6889 T^{4} \))(\( 1 - 157 T^{2} + 6889 T^{4} \))(\( 1 - 130 T^{2} + 6889 T^{4} \))(\( ( 1 - 83 T^{2} )^{2} \))(\( 1 - 22 T^{2} + 6889 T^{4} \))(\( 1 - 130 T^{2} + 6889 T^{4} \))(\( 1 - 130 T^{2} + 6889 T^{4} \))(\( 1 - 22 T^{2} + 6889 T^{4} \))(\( ( 1 - 83 T^{2} )^{2} \))(\( 1 - 130 T^{2} + 6889 T^{4} \))(\( 1 - 157 T^{2} + 6889 T^{4} \))(\( 1 + 158 T^{2} + 6889 T^{4} \))
$89$ (\( ( 1 + 89 T^{2} )^{2} \))(\( ( 1 - 18 T + 89 T^{2} )^{2} \))(\( ( 1 + 12 T + 89 T^{2} )^{2} \))(\( ( 1 + 6 T + 89 T^{2} )^{2} \))(\( ( 1 + 12 T + 89 T^{2} )^{2} \))(\( ( 1 + 6 T + 89 T^{2} )^{2} \))(\( ( 1 - 6 T + 89 T^{2} )^{2} \))(\( ( 1 - 12 T + 89 T^{2} )^{2} \))(\( ( 1 - 6 T + 89 T^{2} )^{2} \))(\( ( 1 - 12 T + 89 T^{2} )^{2} \))(\( ( 1 + 18 T + 89 T^{2} )^{2} \))(\( ( 1 + 89 T^{2} )^{2} \))
$97$ (\( 1 + 2 T^{2} + 9409 T^{4} \))(\( 1 - 193 T^{2} + 9409 T^{4} \))(\( 1 - 190 T^{2} + 9409 T^{4} \))(\( 1 - 145 T^{2} + 9409 T^{4} \))(\( 1 + 167 T^{2} + 9409 T^{4} \))(\( 1 - 169 T^{2} + 9409 T^{4} \))(\( 1 - 169 T^{2} + 9409 T^{4} \))(\( 1 + 167 T^{2} + 9409 T^{4} \))(\( 1 - 145 T^{2} + 9409 T^{4} \))(\( 1 - 190 T^{2} + 9409 T^{4} \))(\( 1 - 193 T^{2} + 9409 T^{4} \))(\( 1 + 2 T^{2} + 9409 T^{4} \))
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