Properties

Label 1050.2.a
Level 1050
Weight 2
Character orbit a
Rep. character \(\chi_{1050}(1,\cdot)\)
Character field \(\Q\)
Dimension 18
Newform subspaces 18
Sturm bound 480
Trace bound 13

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Defining parameters

Level: \( N \) = \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1050.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 18 \)
Sturm bound: \(480\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(11\), \(13\), \(17\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1050))\).

Total New Old
Modular forms 264 18 246
Cusp forms 217 18 199
Eisenstein series 47 0 47

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

\(2\)\(3\)\(5\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(+\)\(1\)
\(+\)\(+\)\(+\)\(-\)\(-\)\(2\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(1\)
\(+\)\(+\)\(-\)\(-\)\(+\)\(1\)
\(+\)\(-\)\(+\)\(+\)\(-\)\(2\)
\(+\)\(-\)\(+\)\(-\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(+\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(-\)\(-\)\(1\)
\(-\)\(+\)\(+\)\(+\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(2\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(2\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(2\)
Plus space\(+\)\(4\)
Minus space\(-\)\(14\)

Trace form

\( 18q - 2q^{2} + 18q^{4} - 2q^{8} + 18q^{9} + O(q^{10}) \) \( 18q - 2q^{2} + 18q^{4} - 2q^{8} + 18q^{9} + 8q^{11} - 4q^{13} + 18q^{16} + 12q^{17} - 2q^{18} - 2q^{21} + 16q^{23} + 36q^{26} + 28q^{29} + 16q^{31} - 2q^{32} + 20q^{34} + 18q^{36} + 12q^{37} + 16q^{38} - 8q^{39} + 52q^{41} + 2q^{42} + 8q^{43} + 8q^{44} - 8q^{46} + 16q^{47} + 18q^{49} + 8q^{51} - 4q^{52} - 12q^{53} - 16q^{57} + 20q^{58} - 16q^{59} + 20q^{61} + 18q^{64} - 8q^{67} + 12q^{68} - 32q^{71} - 2q^{72} - 4q^{73} + 28q^{74} - 16q^{77} + 8q^{78} - 32q^{79} + 18q^{81} - 4q^{82} - 16q^{83} - 2q^{84} - 8q^{86} + 16q^{87} - 12q^{89} + 16q^{92} - 8q^{93} - 4q^{97} - 2q^{98} + 8q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1050))\) 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 7
1050.2.a.a \(1\) \(8.384\) \(\Q\) None \(-1\) \(-1\) \(0\) \(-1\) \(+\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{7}-q^{8}+\cdots\)
1050.2.a.b \(1\) \(8.384\) \(\Q\) None \(-1\) \(-1\) \(0\) \(-1\) \(+\) \(+\) \(-\) \(+\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{7}-q^{8}+\cdots\)
1050.2.a.c \(1\) \(8.384\) \(\Q\) None \(-1\) \(-1\) \(0\) \(1\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}+q^{6}+q^{7}-q^{8}+\cdots\)
1050.2.a.d \(1\) \(8.384\) \(\Q\) None \(-1\) \(-1\) \(0\) \(1\) \(+\) \(+\) \(-\) \(-\) \(q-q^{2}-q^{3}+q^{4}+q^{6}+q^{7}-q^{8}+\cdots\)
1050.2.a.e \(1\) \(8.384\) \(\Q\) None \(-1\) \(-1\) \(0\) \(1\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}+q^{6}+q^{7}-q^{8}+\cdots\)
1050.2.a.f \(1\) \(8.384\) \(\Q\) None \(-1\) \(1\) \(0\) \(-1\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{7}-q^{8}+\cdots\)
1050.2.a.g \(1\) \(8.384\) \(\Q\) None \(-1\) \(1\) \(0\) \(-1\) \(+\) \(-\) \(-\) \(+\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{7}-q^{8}+\cdots\)
1050.2.a.h \(1\) \(8.384\) \(\Q\) None \(-1\) \(1\) \(0\) \(-1\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{7}-q^{8}+\cdots\)
1050.2.a.i \(1\) \(8.384\) \(\Q\) None \(-1\) \(1\) \(0\) \(1\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}+q^{3}+q^{4}-q^{6}+q^{7}-q^{8}+\cdots\)
1050.2.a.j \(1\) \(8.384\) \(\Q\) None \(-1\) \(1\) \(0\) \(1\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}+q^{3}+q^{4}-q^{6}+q^{7}-q^{8}+\cdots\)
1050.2.a.k \(1\) \(8.384\) \(\Q\) None \(1\) \(-1\) \(0\) \(-1\) \(-\) \(+\) \(+\) \(+\) \(q+q^{2}-q^{3}+q^{4}-q^{6}-q^{7}+q^{8}+\cdots\)
1050.2.a.l \(1\) \(8.384\) \(\Q\) None \(1\) \(-1\) \(0\) \(-1\) \(-\) \(+\) \(+\) \(+\) \(q+q^{2}-q^{3}+q^{4}-q^{6}-q^{7}+q^{8}+\cdots\)
1050.2.a.m \(1\) \(8.384\) \(\Q\) None \(1\) \(-1\) \(0\) \(1\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{7}+q^{8}+\cdots\)
1050.2.a.n \(1\) \(8.384\) \(\Q\) None \(1\) \(-1\) \(0\) \(1\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{7}+q^{8}+\cdots\)
1050.2.a.o \(1\) \(8.384\) \(\Q\) None \(1\) \(1\) \(0\) \(-1\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{3}+q^{4}+q^{6}-q^{7}+q^{8}+\cdots\)
1050.2.a.p \(1\) \(8.384\) \(\Q\) None \(1\) \(1\) \(0\) \(-1\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{3}+q^{4}+q^{6}-q^{7}+q^{8}+\cdots\)
1050.2.a.q \(1\) \(8.384\) \(\Q\) None \(1\) \(1\) \(0\) \(1\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{7}+q^{8}+\cdots\)
1050.2.a.r \(1\) \(8.384\) \(\Q\) None \(1\) \(1\) \(0\) \(1\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{7}+q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1050))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1050)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(210))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(350))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(525))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))
$3$ (\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))
$5$ 1
$7$ (\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))
$11$ (\( 1 + 11 T^{2} \))(\( 1 - 2 T + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( 1 - 2 T + 11 T^{2} \))(\( 1 - 2 T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 - 4 T + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( 1 - 6 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - 6 T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 - 2 T + 11 T^{2} \))(\( 1 - 2 T + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( 1 - 2 T + 11 T^{2} \))
$13$ (\( 1 + 2 T + 13 T^{2} \))(\( 1 - T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + 6 T + 13 T^{2} \))(\( 1 + T + 13 T^{2} \))(\( 1 + 7 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + 6 T + 13 T^{2} \))(\( 1 + T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 - T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 - 7 T + 13 T^{2} \))(\( 1 - T + 13 T^{2} \))(\( 1 - 6 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + T + 13 T^{2} \))
$17$ (\( 1 - 6 T + 17 T^{2} \))(\( 1 + 3 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 + 4 T + 17 T^{2} \))(\( 1 - T + 17 T^{2} \))(\( 1 - 7 T + 17 T^{2} \))(\( 1 + 8 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 - 3 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 + 3 T + 17 T^{2} \))(\( 1 - 8 T + 17 T^{2} \))(\( 1 + 7 T + 17 T^{2} \))(\( 1 + T + 17 T^{2} \))(\( 1 - 4 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 - 3 T + 17 T^{2} \))
$19$ (\( 1 + 4 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 6 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 - 8 T + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 - 8 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))(\( 1 - 8 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 6 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 + 19 T^{2} \))
$23$ (\( 1 + 23 T^{2} \))(\( 1 - T + 23 T^{2} \))(\( 1 - 8 T + 23 T^{2} \))(\( 1 - 8 T + 23 T^{2} \))(\( 1 + 7 T + 23 T^{2} \))(\( 1 - 5 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 - 8 T + 23 T^{2} \))(\( 1 + 8 T + 23 T^{2} \))(\( 1 + 3 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 - 3 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 5 T + 23 T^{2} \))(\( 1 - 7 T + 23 T^{2} \))(\( 1 + 8 T + 23 T^{2} \))(\( 1 - 8 T + 23 T^{2} \))(\( 1 + T + 23 T^{2} \))
$29$ (\( 1 + 6 T + 29 T^{2} \))(\( 1 + 5 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 - T + 29 T^{2} \))(\( 1 - 9 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 - 3 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 - 3 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 - 9 T + 29 T^{2} \))(\( 1 - T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 - 10 T + 29 T^{2} \))(\( 1 + 5 T + 29 T^{2} \))
$31$ (\( 1 + 4 T + 31 T^{2} \))(\( 1 - 7 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 + 2 T + 31 T^{2} \))(\( 1 - 3 T + 31 T^{2} \))(\( 1 - T + 31 T^{2} \))(\( 1 - 6 T + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 - 5 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 - 5 T + 31 T^{2} \))(\( 1 - 6 T + 31 T^{2} \))(\( 1 - T + 31 T^{2} \))(\( 1 - 3 T + 31 T^{2} \))(\( 1 + 2 T + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( 1 - 7 T + 31 T^{2} \))
$37$ (\( 1 + 2 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 + 4 T + 37 T^{2} \))(\( 1 - 6 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 + 8 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 - 10 T + 37 T^{2} \))(\( 1 + 10 T + 37 T^{2} \))(\( 1 - 10 T + 37 T^{2} \))(\( 1 - 10 T + 37 T^{2} \))(\( 1 - 8 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 - 4 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))
$41$ (\( 1 - 6 T + 41 T^{2} \))(\( 1 - 7 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 + 3 T + 41 T^{2} \))(\( 1 - 11 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 - 9 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 - 9 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 - 11 T + 41 T^{2} \))(\( 1 + 3 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( 1 - 7 T + 41 T^{2} \))
$43$ (\( 1 + 8 T + 43 T^{2} \))(\( 1 - 11 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 - T + 43 T^{2} \))(\( 1 + 3 T + 43 T^{2} \))(\( 1 + 8 T + 43 T^{2} \))(\( 1 - 12 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 + T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))(\( 1 - 3 T + 43 T^{2} \))(\( 1 + T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 + 8 T + 43 T^{2} \))(\( 1 + 11 T + 43 T^{2} \))
$47$ (\( 1 - 12 T + 47 T^{2} \))(\( 1 + 8 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 8 T + 47 T^{2} \))(\( 1 - 12 T + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 - 4 T + 47 T^{2} \))(\( 1 - 4 T + 47 T^{2} \))(\( 1 + 12 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))
$53$ (\( 1 + 6 T + 53 T^{2} \))(\( 1 - T + 53 T^{2} \))(\( 1 - 10 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 11 T + 53 T^{2} \))(\( 1 + 3 T + 53 T^{2} \))(\( 1 - 2 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 9 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 + 9 T + 53 T^{2} \))(\( 1 + 2 T + 53 T^{2} \))(\( 1 - 3 T + 53 T^{2} \))(\( 1 - 11 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 + 10 T + 53 T^{2} \))(\( 1 + T + 53 T^{2} \))
$59$ (\( 1 + 12 T + 59 T^{2} \))(\( 1 + 5 T + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 + 8 T + 59 T^{2} \))(\( 1 + 3 T + 59 T^{2} \))(\( 1 - 7 T + 59 T^{2} \))(\( 1 + 8 T + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( 1 - 9 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 - 9 T + 59 T^{2} \))(\( 1 + 8 T + 59 T^{2} \))(\( 1 - 7 T + 59 T^{2} \))(\( 1 + 3 T + 59 T^{2} \))(\( 1 + 8 T + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( 1 + 5 T + 59 T^{2} \))
$61$ (\( 1 - 2 T + 61 T^{2} \))(\( 1 + 3 T + 61 T^{2} \))(\( 1 - 14 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 - 5 T + 61 T^{2} \))(\( 1 + 5 T + 61 T^{2} \))(\( 1 - 10 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( 1 - 6 T + 61 T^{2} \))(\( 1 - 11 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 - 11 T + 61 T^{2} \))(\( 1 - 10 T + 61 T^{2} \))(\( 1 + 5 T + 61 T^{2} \))(\( 1 - 5 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 + 6 T + 61 T^{2} \))(\( 1 + 3 T + 61 T^{2} \))
$67$ (\( 1 + 8 T + 67 T^{2} \))(\( 1 - 12 T + 67 T^{2} \))(\( 1 - 12 T + 67 T^{2} \))(\( 1 + 8 T + 67 T^{2} \))(\( 1 - 12 T + 67 T^{2} \))(\( 1 - 12 T + 67 T^{2} \))(\( 1 - 12 T + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))(\( 1 - 8 T + 67 T^{2} \))(\( 1 + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))
$71$ (\( 1 + 71 T^{2} \))(\( 1 - 12 T + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 + 6 T + 71 T^{2} \))(\( 1 - 4 T + 71 T^{2} \))(\( 1 + 4 T + 71 T^{2} \))(\( 1 + 14 T + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 + 12 T + 71 T^{2} \))(\( 1 - 12 T + 71 T^{2} \))(\( 1 + 12 T + 71 T^{2} \))(\( 1 + 14 T + 71 T^{2} \))(\( 1 + 4 T + 71 T^{2} \))(\( 1 - 4 T + 71 T^{2} \))(\( 1 + 6 T + 71 T^{2} \))(\( 1 + 12 T + 71 T^{2} \))(\( 1 - 12 T + 71 T^{2} \))
$73$ (\( 1 + 14 T + 73 T^{2} \))(\( 1 - 6 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 - 14 T + 73 T^{2} \))(\( 1 - 14 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 - 14 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 + 14 T + 73 T^{2} \))(\( 1 + 14 T + 73 T^{2} \))(\( 1 - 6 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))
$79$ (\( 1 + 16 T + 79 T^{2} \))(\( 1 - 10 T + 79 T^{2} \))(\( 1 - 16 T + 79 T^{2} \))(\( 1 + 12 T + 79 T^{2} \))(\( 1 + 2 T + 79 T^{2} \))(\( 1 + 6 T + 79 T^{2} \))(\( 1 - 4 T + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 + 10 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 + 10 T + 79 T^{2} \))(\( 1 - 4 T + 79 T^{2} \))(\( 1 + 6 T + 79 T^{2} \))(\( 1 + 2 T + 79 T^{2} \))(\( 1 + 12 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 - 10 T + 79 T^{2} \))
$83$ (\( 1 + 12 T + 83 T^{2} \))(\( 1 - 11 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 - 8 T + 83 T^{2} \))(\( 1 - 3 T + 83 T^{2} \))(\( 1 - 9 T + 83 T^{2} \))(\( 1 + 16 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 - 4 T + 83 T^{2} \))(\( 1 - 9 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 + 9 T + 83 T^{2} \))(\( 1 - 16 T + 83 T^{2} \))(\( 1 + 9 T + 83 T^{2} \))(\( 1 + 3 T + 83 T^{2} \))(\( 1 + 8 T + 83 T^{2} \))(\( 1 - 4 T + 83 T^{2} \))(\( 1 + 11 T + 83 T^{2} \))
$89$ (\( 1 - 6 T + 89 T^{2} \))(\( 1 + 10 T + 89 T^{2} \))(\( 1 - 10 T + 89 T^{2} \))(\( 1 + 10 T + 89 T^{2} \))(\( 1 - 10 T + 89 T^{2} \))(\( 1 + 10 T + 89 T^{2} \))(\( 1 - 10 T + 89 T^{2} \))(\( 1 - 2 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 - 10 T + 89 T^{2} \))(\( 1 + 10 T + 89 T^{2} \))(\( 1 - 10 T + 89 T^{2} \))(\( 1 + 10 T + 89 T^{2} \))(\( 1 - 14 T + 89 T^{2} \))(\( 1 + 10 T + 89 T^{2} \))
$97$ (\( 1 + 14 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 - 10 T + 97 T^{2} \))(\( 1 - 10 T + 97 T^{2} \))(\( 1 + 10 T + 97 T^{2} \))(\( 1 + 10 T + 97 T^{2} \))(\( 1 + 10 T + 97 T^{2} \))(\( 1 - 14 T + 97 T^{2} \))(\( 1 - 14 T + 97 T^{2} \))(\( 1 - 10 T + 97 T^{2} \))(\( 1 + 14 T + 97 T^{2} \))(\( 1 - 10 T + 97 T^{2} \))(\( 1 - 10 T + 97 T^{2} \))(\( 1 + 10 T + 97 T^{2} \))(\( 1 + 10 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))
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