Properties

Label 1350.2.a
Level 1350
Weight 2
Character orbit a
Rep. character \(\chi_{1350}(1,\cdot)\)
Character field \(\Q\)
Dimension 26
Newform subspaces 24
Sturm bound 540
Trace bound 17

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Defining parameters

Level: \( N \) = \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1350.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 24 \)
Sturm bound: \(540\)
Trace bound: \(17\)
Distinguishing \(T_p\): \(7\), \(11\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1350))\).

Total New Old
Modular forms 306 26 280
Cusp forms 235 26 209
Eisenstein series 71 0 71

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(3\)\(5\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(3\)
\(+\)\(+\)\(-\)\(-\)\(4\)
\(+\)\(-\)\(+\)\(-\)\(3\)
\(+\)\(-\)\(-\)\(+\)\(3\)
\(-\)\(+\)\(+\)\(-\)\(4\)
\(-\)\(+\)\(-\)\(+\)\(2\)
\(-\)\(-\)\(+\)\(+\)\(2\)
\(-\)\(-\)\(-\)\(-\)\(5\)
Plus space\(+\)\(10\)
Minus space\(-\)\(16\)

Trace form

\( 26q + 26q^{4} - 6q^{7} + O(q^{10}) \) \( 26q + 26q^{4} - 6q^{7} + 26q^{16} - 6q^{22} - 6q^{28} + 2q^{31} + 8q^{34} + 12q^{37} + 36q^{43} - 4q^{46} + 60q^{49} + 36q^{58} + 4q^{61} + 26q^{64} - 12q^{67} + 30q^{73} + 12q^{79} + 12q^{82} - 6q^{88} + 24q^{91} + 28q^{94} - 30q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1350))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 5
1350.2.a.a \(1\) \(10.780\) \(\Q\) None \(-1\) \(0\) \(0\) \(-4\) \(+\) \(-\) \(-\) \(q-q^{2}+q^{4}-4q^{7}-q^{8}+3q^{11}-q^{13}+\cdots\)
1350.2.a.b \(1\) \(10.780\) \(\Q\) None \(-1\) \(0\) \(0\) \(-4\) \(+\) \(+\) \(-\) \(q-q^{2}+q^{4}-4q^{7}-q^{8}+5q^{11}+3q^{13}+\cdots\)
1350.2.a.c \(1\) \(10.780\) \(\Q\) None \(-1\) \(0\) \(0\) \(-2\) \(+\) \(+\) \(+\) \(q-q^{2}+q^{4}-2q^{7}-q^{8}-3q^{11}+q^{13}+\cdots\)
1350.2.a.d \(1\) \(10.780\) \(\Q\) None \(-1\) \(0\) \(0\) \(-2\) \(+\) \(-\) \(+\) \(q-q^{2}+q^{4}-2q^{7}-q^{8}+3q^{11}-5q^{13}+\cdots\)
1350.2.a.e \(1\) \(10.780\) \(\Q\) None \(-1\) \(0\) \(0\) \(-2\) \(+\) \(+\) \(+\) \(q-q^{2}+q^{4}-2q^{7}-q^{8}+3q^{11}-5q^{13}+\cdots\)
1350.2.a.f \(1\) \(10.780\) \(\Q\) None \(-1\) \(0\) \(0\) \(-1\) \(+\) \(-\) \(-\) \(q-q^{2}+q^{4}-q^{7}-q^{8}+2q^{13}+q^{14}+\cdots\)
1350.2.a.g \(1\) \(10.780\) \(\Q\) None \(-1\) \(0\) \(0\) \(1\) \(+\) \(+\) \(+\) \(q-q^{2}+q^{4}+q^{7}-q^{8}-2q^{13}-q^{14}+\cdots\)
1350.2.a.h \(1\) \(10.780\) \(\Q\) None \(-1\) \(0\) \(0\) \(1\) \(+\) \(-\) \(+\) \(q-q^{2}+q^{4}+q^{7}-q^{8}+3q^{11}+4q^{13}+\cdots\)
1350.2.a.i \(1\) \(10.780\) \(\Q\) None \(-1\) \(0\) \(0\) \(2\) \(+\) \(+\) \(-\) \(q-q^{2}+q^{4}+2q^{7}-q^{8}-3q^{11}+5q^{13}+\cdots\)
1350.2.a.j \(1\) \(10.780\) \(\Q\) None \(-1\) \(0\) \(0\) \(4\) \(+\) \(-\) \(-\) \(q-q^{2}+q^{4}+4q^{7}-q^{8}-5q^{11}-3q^{13}+\cdots\)
1350.2.a.k \(1\) \(10.780\) \(\Q\) None \(-1\) \(0\) \(0\) \(4\) \(+\) \(-\) \(+\) \(q-q^{2}+q^{4}+4q^{7}-q^{8}-3q^{11}+q^{13}+\cdots\)
1350.2.a.l \(1\) \(10.780\) \(\Q\) None \(1\) \(0\) \(0\) \(-4\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{4}-4q^{7}+q^{8}-5q^{11}+3q^{13}+\cdots\)
1350.2.a.m \(1\) \(10.780\) \(\Q\) None \(1\) \(0\) \(0\) \(-4\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{4}-4q^{7}+q^{8}-3q^{11}-q^{13}+\cdots\)
1350.2.a.n \(1\) \(10.780\) \(\Q\) None \(1\) \(0\) \(0\) \(-2\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{4}-2q^{7}+q^{8}-3q^{11}-5q^{13}+\cdots\)
1350.2.a.o \(1\) \(10.780\) \(\Q\) None \(1\) \(0\) \(0\) \(-2\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{4}-2q^{7}+q^{8}-3q^{11}-5q^{13}+\cdots\)
1350.2.a.p \(1\) \(10.780\) \(\Q\) None \(1\) \(0\) \(0\) \(-2\) \(-\) \(+\) \(+\) \(q+q^{2}+q^{4}-2q^{7}+q^{8}+3q^{11}+q^{13}+\cdots\)
1350.2.a.q \(1\) \(10.780\) \(\Q\) None \(1\) \(0\) \(0\) \(-1\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{4}-q^{7}+q^{8}+2q^{13}-q^{14}+\cdots\)
1350.2.a.r \(1\) \(10.780\) \(\Q\) None \(1\) \(0\) \(0\) \(1\) \(-\) \(+\) \(+\) \(q+q^{2}+q^{4}+q^{7}+q^{8}-3q^{11}+4q^{13}+\cdots\)
1350.2.a.s \(1\) \(10.780\) \(\Q\) None \(1\) \(0\) \(0\) \(1\) \(-\) \(+\) \(+\) \(q+q^{2}+q^{4}+q^{7}+q^{8}-2q^{13}+q^{14}+\cdots\)
1350.2.a.t \(1\) \(10.780\) \(\Q\) None \(1\) \(0\) \(0\) \(2\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{4}+2q^{7}+q^{8}+3q^{11}+5q^{13}+\cdots\)
1350.2.a.u \(1\) \(10.780\) \(\Q\) None \(1\) \(0\) \(0\) \(4\) \(-\) \(+\) \(+\) \(q+q^{2}+q^{4}+4q^{7}+q^{8}+3q^{11}+q^{13}+\cdots\)
1350.2.a.v \(1\) \(10.780\) \(\Q\) None \(1\) \(0\) \(0\) \(4\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{4}+4q^{7}+q^{8}+5q^{11}-3q^{13}+\cdots\)
1350.2.a.w \(2\) \(10.780\) \(\Q(\sqrt{19}) \) None \(-2\) \(0\) \(0\) \(0\) \(+\) \(+\) \(-\) \(q-q^{2}+q^{4}+\beta q^{7}-q^{8}+\beta q^{11}-\beta q^{14}+\cdots\)
1350.2.a.x \(2\) \(10.780\) \(\Q(\sqrt{19}) \) None \(2\) \(0\) \(0\) \(0\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{4}+\beta q^{7}+q^{8}-\beta q^{11}+\beta q^{14}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1350))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1350)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(135))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(270))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(450))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(675))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))
$3$ (\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))
$5$ (\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))
$7$ (\( 1 + 4 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 - 2 T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 - 2 T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 - 5 T^{2} + 49 T^{4} \))(\( 1 - 5 T^{2} + 49 T^{4} \))
$11$ (\( 1 - 3 T + 11 T^{2} \))(\( 1 - 5 T + 11 T^{2} \))(\( 1 + 3 T + 11 T^{2} \))(\( 1 - 3 T + 11 T^{2} \))(\( 1 - 3 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - 3 T + 11 T^{2} \))(\( 1 + 3 T + 11 T^{2} \))(\( 1 + 5 T + 11 T^{2} \))(\( 1 + 3 T + 11 T^{2} \))(\( 1 + 5 T + 11 T^{2} \))(\( 1 + 3 T + 11 T^{2} \))(\( 1 + 3 T + 11 T^{2} \))(\( 1 + 3 T + 11 T^{2} \))(\( 1 - 3 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 3 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - 3 T + 11 T^{2} \))(\( 1 - 3 T + 11 T^{2} \))(\( 1 - 5 T + 11 T^{2} \))(\( 1 + 3 T^{2} + 121 T^{4} \))(\( 1 + 3 T^{2} + 121 T^{4} \))
$13$ (\( 1 + T + 13 T^{2} \))(\( 1 - 3 T + 13 T^{2} \))(\( 1 - T + 13 T^{2} \))(\( 1 + 5 T + 13 T^{2} \))(\( 1 + 5 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 - 5 T + 13 T^{2} \))(\( 1 + 3 T + 13 T^{2} \))(\( 1 - T + 13 T^{2} \))(\( 1 - 3 T + 13 T^{2} \))(\( 1 + T + 13 T^{2} \))(\( 1 + 5 T + 13 T^{2} \))(\( 1 + 5 T + 13 T^{2} \))(\( 1 - T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 - 5 T + 13 T^{2} \))(\( 1 - T + 13 T^{2} \))(\( 1 + 3 T + 13 T^{2} \))(\( ( 1 + 13 T^{2} )^{2} \))(\( ( 1 + 13 T^{2} )^{2} \))
$17$ (\( 1 + 17 T^{2} \))(\( 1 + T + 17 T^{2} \))(\( 1 - 3 T + 17 T^{2} \))(\( 1 - 3 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 + T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 - T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 + 3 T + 17 T^{2} \))(\( 1 + 3 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 - T + 17 T^{2} \))(\( ( 1 + 4 T + 17 T^{2} )^{2} \))(\( ( 1 - 4 T + 17 T^{2} )^{2} \))
$19$ (\( 1 - 2 T + 19 T^{2} \))(\( 1 + 6 T + 19 T^{2} \))(\( 1 - 8 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 + T + 19 T^{2} \))(\( 1 + T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 + 6 T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 + 6 T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 - 8 T + 19 T^{2} \))(\( 1 + T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 + T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 + 6 T + 19 T^{2} \))(\( ( 1 - 6 T + 19 T^{2} )^{2} \))(\( ( 1 - 6 T + 19 T^{2} )^{2} \))
$23$ (\( 1 - 9 T + 23 T^{2} \))(\( 1 + T + 23 T^{2} \))(\( 1 + 3 T + 23 T^{2} \))(\( 1 - 9 T + 23 T^{2} \))(\( 1 + 3 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 + 3 T + 23 T^{2} \))(\( 1 + T + 23 T^{2} \))(\( 1 - 9 T + 23 T^{2} \))(\( 1 - T + 23 T^{2} \))(\( 1 + 9 T + 23 T^{2} \))(\( 1 - 3 T + 23 T^{2} \))(\( 1 + 9 T + 23 T^{2} \))(\( 1 - 3 T + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 - 3 T + 23 T^{2} \))(\( 1 + 9 T + 23 T^{2} \))(\( 1 - T + 23 T^{2} \))(\( ( 1 - 2 T + 23 T^{2} )^{2} \))(\( ( 1 + 2 T + 23 T^{2} )^{2} \))
$29$ (\( 1 + 6 T + 29 T^{2} \))(\( 1 - 9 T + 29 T^{2} \))(\( 1 + 9 T + 29 T^{2} \))(\( 1 + 3 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 + 9 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 + 9 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 - 3 T + 29 T^{2} \))(\( 1 - 9 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 - 9 T + 29 T^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))
$31$ (\( 1 + 10 T + 31 T^{2} \))(\( 1 + 5 T + 31 T^{2} \))(\( 1 + 7 T + 31 T^{2} \))(\( 1 - 5 T + 31 T^{2} \))(\( 1 - 2 T + 31 T^{2} \))(\( 1 - 5 T + 31 T^{2} \))(\( 1 - 5 T + 31 T^{2} \))(\( 1 - 5 T + 31 T^{2} \))(\( 1 - 2 T + 31 T^{2} \))(\( 1 + 5 T + 31 T^{2} \))(\( 1 + 10 T + 31 T^{2} \))(\( 1 + 5 T + 31 T^{2} \))(\( 1 + 10 T + 31 T^{2} \))(\( 1 - 2 T + 31 T^{2} \))(\( 1 - 5 T + 31 T^{2} \))(\( 1 + 7 T + 31 T^{2} \))(\( 1 - 5 T + 31 T^{2} \))(\( 1 - 5 T + 31 T^{2} \))(\( 1 - 5 T + 31 T^{2} \))(\( 1 - 2 T + 31 T^{2} \))(\( 1 + 10 T + 31 T^{2} \))(\( 1 + 5 T + 31 T^{2} \))(\( ( 1 - 7 T + 31 T^{2} )^{2} \))(\( ( 1 - 7 T + 31 T^{2} )^{2} \))
$37$ (\( 1 + 7 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 - 10 T + 37 T^{2} \))(\( 1 + 11 T + 37 T^{2} \))(\( 1 + 7 T + 37 T^{2} \))(\( 1 - 7 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 - 11 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 - 7 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 + 7 T + 37 T^{2} \))(\( 1 + 11 T + 37 T^{2} \))(\( 1 - 10 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 + 7 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 - 7 T + 37 T^{2} \))(\( 1 - 11 T + 37 T^{2} \))(\( 1 - 7 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 - 2 T^{2} + 1369 T^{4} \))(\( 1 - 2 T^{2} + 1369 T^{4} \))
$41$ (\( 1 - 6 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 + 12 T + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 12 T + 41 T^{2} \))(\( 1 - 12 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 - 12 T + 41 T^{2} \))(\( 1 - 12 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 12 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 + 6 T^{2} + 1681 T^{4} \))(\( 1 + 6 T^{2} + 1681 T^{4} \))
$43$ (\( 1 + 10 T + 43 T^{2} \))(\( 1 + T + 43 T^{2} \))(\( 1 - 7 T + 43 T^{2} \))(\( 1 - T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 11 T + 43 T^{2} \))(\( 1 + 11 T + 43 T^{2} \))(\( 1 - 10 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 - T + 43 T^{2} \))(\( 1 - 10 T + 43 T^{2} \))(\( 1 + T + 43 T^{2} \))(\( 1 + 10 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - T + 43 T^{2} \))(\( 1 - 7 T + 43 T^{2} \))(\( 1 - 11 T + 43 T^{2} \))(\( 1 - 10 T + 43 T^{2} \))(\( 1 + 11 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 - 10 T + 43 T^{2} \))(\( 1 - T + 43 T^{2} \))(\( 1 + 10 T^{2} + 1849 T^{4} \))(\( 1 + 10 T^{2} + 1849 T^{4} \))
$47$ (\( 1 + 9 T + 47 T^{2} \))(\( 1 - 13 T + 47 T^{2} \))(\( 1 - 3 T + 47 T^{2} \))(\( 1 + 9 T + 47 T^{2} \))(\( 1 - 3 T + 47 T^{2} \))(\( 1 + 12 T + 47 T^{2} \))(\( 1 + 12 T + 47 T^{2} \))(\( 1 - 6 T + 47 T^{2} \))(\( 1 - 3 T + 47 T^{2} \))(\( 1 - 13 T + 47 T^{2} \))(\( 1 + 9 T + 47 T^{2} \))(\( 1 + 13 T + 47 T^{2} \))(\( 1 - 9 T + 47 T^{2} \))(\( 1 + 3 T + 47 T^{2} \))(\( 1 - 9 T + 47 T^{2} \))(\( 1 + 3 T + 47 T^{2} \))(\( 1 - 12 T + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 - 12 T + 47 T^{2} \))(\( 1 + 3 T + 47 T^{2} \))(\( 1 - 9 T + 47 T^{2} \))(\( 1 + 13 T + 47 T^{2} \))(\( ( 1 + 2 T + 47 T^{2} )^{2} \))(\( ( 1 - 2 T + 47 T^{2} )^{2} \))
$53$ (\( 1 + 6 T + 53 T^{2} \))(\( 1 + 53 T^{2} \))(\( 1 + 12 T + 53 T^{2} \))(\( 1 - 12 T + 53 T^{2} \))(\( 1 + 12 T + 53 T^{2} \))(\( 1 + 53 T^{2} \))(\( 1 + 53 T^{2} \))(\( 1 - 9 T + 53 T^{2} \))(\( 1 + 12 T + 53 T^{2} \))(\( 1 + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 - 12 T + 53 T^{2} \))(\( 1 + 12 T + 53 T^{2} \))(\( 1 - 12 T + 53 T^{2} \))(\( 1 + 53 T^{2} \))(\( 1 + 9 T + 53 T^{2} \))(\( 1 + 53 T^{2} \))(\( 1 - 12 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 + 53 T^{2} \))(\( ( 1 - 3 T + 53 T^{2} )^{2} \))(\( ( 1 + 3 T + 53 T^{2} )^{2} \))
$59$ (\( 1 - 3 T + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 + 9 T + 59 T^{2} \))(\( 1 - 6 T + 59 T^{2} \))(\( 1 + 6 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 - 9 T + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 + 3 T + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 + 3 T + 59 T^{2} \))(\( 1 - 9 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 + 6 T + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 - 6 T + 59 T^{2} \))(\( 1 + 9 T + 59 T^{2} \))(\( 1 - 3 T + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( 1 + 42 T^{2} + 3481 T^{4} \))(\( 1 + 42 T^{2} + 3481 T^{4} \))
$61$ (\( 1 + 7 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 - 2 T + 61 T^{2} \))(\( 1 - 11 T + 61 T^{2} \))(\( 1 + 7 T + 61 T^{2} \))(\( 1 + 7 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 - 11 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 + 7 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 + 7 T + 61 T^{2} \))(\( 1 - 11 T + 61 T^{2} \))(\( 1 - 2 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 + 7 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 + 7 T + 61 T^{2} \))(\( 1 - 11 T + 61 T^{2} \))(\( 1 + 7 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( ( 1 + 4 T + 61 T^{2} )^{2} \))(\( ( 1 + 4 T + 61 T^{2} )^{2} \))
$67$ (\( 1 + 4 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 + 14 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 + 14 T + 67 T^{2} \))(\( 1 - 14 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 + 14 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 + 14 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 - 14 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 + 58 T^{2} + 4489 T^{4} \))(\( 1 + 58 T^{2} + 4489 T^{4} \))
$71$ (\( 1 + 15 T + 71 T^{2} \))(\( 1 - 6 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 - 12 T + 71 T^{2} \))(\( 1 - 15 T + 71 T^{2} \))(\( 1 + 6 T + 71 T^{2} \))(\( 1 - 6 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 15 T + 71 T^{2} \))(\( 1 + 6 T + 71 T^{2} \))(\( 1 - 15 T + 71 T^{2} \))(\( 1 + 6 T + 71 T^{2} \))(\( 1 - 15 T + 71 T^{2} \))(\( 1 + 15 T + 71 T^{2} \))(\( 1 + 12 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 - 6 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 6 T + 71 T^{2} \))(\( 1 - 15 T + 71 T^{2} \))(\( 1 + 15 T + 71 T^{2} \))(\( 1 - 6 T + 71 T^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))
$73$ (\( 1 - 2 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 + 7 T + 73 T^{2} \))(\( 1 - 7 T + 73 T^{2} \))(\( 1 - 7 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 + 7 T + 73 T^{2} \))(\( 1 - 7 T + 73 T^{2} \))(\( 1 - 7 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 + 127 T^{2} + 5329 T^{4} \))(\( 1 + 127 T^{2} + 5329 T^{4} \))
$79$ (\( 1 - 8 T + 79 T^{2} \))(\( 1 - 9 T + 79 T^{2} \))(\( 1 + T + 79 T^{2} \))(\( 1 + 13 T + 79 T^{2} \))(\( 1 + 10 T + 79 T^{2} \))(\( 1 + T + 79 T^{2} \))(\( 1 + T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 + 10 T + 79 T^{2} \))(\( 1 - 9 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 - 9 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 + 10 T + 79 T^{2} \))(\( 1 + 13 T + 79 T^{2} \))(\( 1 + T + 79 T^{2} \))(\( 1 + T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 + T + 79 T^{2} \))(\( 1 + 10 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 - 9 T + 79 T^{2} \))(\( ( 1 + 79 T^{2} )^{2} \))(\( ( 1 + 79 T^{2} )^{2} \))
$83$ (\( 1 - 12 T + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( 1 + 18 T + 83 T^{2} \))(\( 1 + 6 T + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))(\( 1 + 3 T + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 - 4 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))(\( 1 - 18 T + 83 T^{2} \))(\( 1 + 6 T + 83 T^{2} \))(\( 1 - 3 T + 83 T^{2} \))(\( 1 + 6 T + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 - 4 T + 83 T^{2} \))(\( ( 1 - 5 T + 83 T^{2} )^{2} \))(\( ( 1 + 5 T + 83 T^{2} )^{2} \))
$89$ (\( 1 + 12 T + 89 T^{2} \))(\( 1 - 14 T + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 + 12 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 - 18 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 14 T + 89 T^{2} \))(\( 1 - 12 T + 89 T^{2} \))(\( 1 + 14 T + 89 T^{2} \))(\( 1 - 12 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 - 12 T + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 + 18 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 + 12 T + 89 T^{2} \))(\( 1 - 14 T + 89 T^{2} \))(\( 1 + 102 T^{2} + 7921 T^{4} \))(\( 1 + 102 T^{2} + 7921 T^{4} \))
$97$ (\( 1 + 19 T + 97 T^{2} \))(\( 1 - 10 T + 97 T^{2} \))(\( 1 + 14 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 - 7 T + 97 T^{2} \))(\( 1 - 5 T + 97 T^{2} \))(\( 1 + 5 T + 97 T^{2} \))(\( 1 - T + 97 T^{2} \))(\( 1 + 7 T + 97 T^{2} \))(\( 1 + 10 T + 97 T^{2} \))(\( 1 - 19 T + 97 T^{2} \))(\( 1 - 10 T + 97 T^{2} \))(\( 1 + 19 T + 97 T^{2} \))(\( 1 - 7 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 + 14 T + 97 T^{2} \))(\( 1 - 5 T + 97 T^{2} \))(\( 1 - T + 97 T^{2} \))(\( 1 + 5 T + 97 T^{2} \))(\( 1 + 7 T + 97 T^{2} \))(\( 1 - 19 T + 97 T^{2} \))(\( 1 + 10 T + 97 T^{2} \))(\( 1 + 175 T^{2} + 9409 T^{4} \))(\( 1 + 175 T^{2} + 9409 T^{4} \))
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