Properties

Label 1056.2.a
Level $1056$
Weight $2$
Character orbit 1056.a
Rep. character $\chi_{1056}(1,\cdot)$
Character field $\Q$
Dimension $20$
Newform subspaces $14$
Sturm bound $384$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 1056 = 2^{5} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1056.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 14 \)
Sturm bound: \(384\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(7\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1056))\).

Total New Old
Modular forms 208 20 188
Cusp forms 177 20 157
Eisenstein series 31 0 31

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

\(2\)\(3\)\(11\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(2\)
\(+\)\(+\)\(-\)\(-\)\(3\)
\(+\)\(-\)\(+\)\(-\)\(4\)
\(+\)\(-\)\(-\)\(+\)\(1\)
\(-\)\(+\)\(+\)\(-\)\(3\)
\(-\)\(+\)\(-\)\(+\)\(2\)
\(-\)\(-\)\(+\)\(+\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(4\)
Plus space\(+\)\(6\)
Minus space\(-\)\(14\)

Trace form

\( 20q - 8q^{5} + 20q^{9} + O(q^{10}) \) \( 20q - 8q^{5} + 20q^{9} + 8q^{13} + 24q^{17} + 16q^{21} + 44q^{25} - 8q^{29} + 8q^{37} + 24q^{41} - 8q^{45} + 36q^{49} - 8q^{53} + 16q^{57} + 8q^{61} + 48q^{65} + 40q^{73} + 20q^{81} - 16q^{85} + 8q^{89} + 16q^{93} + 24q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1056))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 11
1056.2.a.a \(1\) \(8.432\) \(\Q\) None \(0\) \(-1\) \(-4\) \(2\) \(-\) \(+\) \(-\) \(q-q^{3}-4q^{5}+2q^{7}+q^{9}+q^{11}+4q^{13}+\cdots\)
1056.2.a.b \(1\) \(8.432\) \(\Q\) None \(0\) \(-1\) \(-2\) \(0\) \(+\) \(+\) \(-\) \(q-q^{3}-2q^{5}+q^{9}+q^{11}-6q^{13}+\cdots\)
1056.2.a.c \(1\) \(8.432\) \(\Q\) None \(0\) \(-1\) \(-2\) \(2\) \(+\) \(+\) \(+\) \(q-q^{3}-2q^{5}+2q^{7}+q^{9}-q^{11}-2q^{13}+\cdots\)
1056.2.a.d \(1\) \(8.432\) \(\Q\) None \(0\) \(-1\) \(0\) \(-2\) \(+\) \(+\) \(+\) \(q-q^{3}-2q^{7}+q^{9}-q^{11}+4q^{13}+\cdots\)
1056.2.a.e \(1\) \(8.432\) \(\Q\) None \(0\) \(-1\) \(2\) \(-4\) \(-\) \(+\) \(-\) \(q-q^{3}+2q^{5}-4q^{7}+q^{9}+q^{11}-2q^{13}+\cdots\)
1056.2.a.f \(1\) \(8.432\) \(\Q\) None \(0\) \(1\) \(-4\) \(-2\) \(+\) \(-\) \(+\) \(q+q^{3}-4q^{5}-2q^{7}+q^{9}-q^{11}+4q^{13}+\cdots\)
1056.2.a.g \(1\) \(8.432\) \(\Q\) None \(0\) \(1\) \(-2\) \(-2\) \(+\) \(-\) \(-\) \(q+q^{3}-2q^{5}-2q^{7}+q^{9}+q^{11}-2q^{13}+\cdots\)
1056.2.a.h \(1\) \(8.432\) \(\Q\) None \(0\) \(1\) \(-2\) \(0\) \(-\) \(-\) \(+\) \(q+q^{3}-2q^{5}+q^{9}-q^{11}-6q^{13}+\cdots\)
1056.2.a.i \(1\) \(8.432\) \(\Q\) None \(0\) \(1\) \(0\) \(2\) \(-\) \(-\) \(-\) \(q+q^{3}+2q^{7}+q^{9}+q^{11}+4q^{13}+\cdots\)
1056.2.a.j \(1\) \(8.432\) \(\Q\) None \(0\) \(1\) \(2\) \(4\) \(+\) \(-\) \(+\) \(q+q^{3}+2q^{5}+4q^{7}+q^{9}-q^{11}-2q^{13}+\cdots\)
1056.2.a.k \(2\) \(8.432\) \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(2\) \(-2\) \(+\) \(+\) \(-\) \(q-q^{3}+(1+\beta )q^{5}+(-1+\beta )q^{7}+q^{9}+\cdots\)
1056.2.a.l \(2\) \(8.432\) \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(2\) \(2\) \(+\) \(-\) \(+\) \(q+q^{3}+(1+\beta )q^{5}+(1-\beta )q^{7}+q^{9}+\cdots\)
1056.2.a.m \(3\) \(8.432\) 3.3.229.1 None \(0\) \(-3\) \(0\) \(-4\) \(-\) \(+\) \(+\) \(q-q^{3}-\beta _{1}q^{5}+(-1-\beta _{2})q^{7}+q^{9}+\cdots\)
1056.2.a.n \(3\) \(8.432\) 3.3.229.1 None \(0\) \(3\) \(0\) \(4\) \(-\) \(-\) \(-\) \(q+q^{3}-\beta _{1}q^{5}+(1+\beta _{2})q^{7}+q^{9}+q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1056))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1056)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(66))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(88))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(132))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(176))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(264))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(352))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(528))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{3} \))(\( ( 1 - T )^{3} \))
$5$ (\( 1 + 4 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 + 4 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 - 2 T + 6 T^{2} - 10 T^{3} + 25 T^{4} \))(\( 1 - 2 T + 6 T^{2} - 10 T^{3} + 25 T^{4} \))(\( 1 - T^{2} + 8 T^{3} - 5 T^{4} + 125 T^{6} \))(\( 1 - T^{2} + 8 T^{3} - 5 T^{4} + 125 T^{6} \))
$7$ (\( 1 - 2 T + 7 T^{2} \))(\( 1 + 7 T^{2} \))(\( 1 - 2 T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 + 7 T^{2} \))(\( 1 - 2 T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 + 2 T + 10 T^{2} + 14 T^{3} + 49 T^{4} \))(\( 1 - 2 T + 10 T^{2} - 14 T^{3} + 49 T^{4} \))(\( 1 + 4 T + 5 T^{2} + 35 T^{4} + 196 T^{5} + 343 T^{6} \))(\( 1 - 4 T + 5 T^{2} + 35 T^{4} - 196 T^{5} + 343 T^{6} \))
$11$ (\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 + T \))(\( 1 - T \))(\( 1 + T \))(\( 1 - T \))(\( 1 + T \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{3} \))(\( ( 1 - T )^{3} \))
$13$ (\( 1 - 4 T + 13 T^{2} \))(\( 1 + 6 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 + 6 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 - 6 T + 30 T^{2} - 78 T^{3} + 169 T^{4} \))(\( 1 - 6 T + 30 T^{2} - 78 T^{3} + 169 T^{4} \))(\( 1 + 23 T^{2} + 8 T^{3} + 299 T^{4} + 2197 T^{6} \))(\( 1 + 23 T^{2} + 8 T^{3} + 299 T^{4} + 2197 T^{6} \))
$17$ (\( 1 - 2 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 14 T^{2} + 289 T^{4} \))(\( 1 + 14 T^{2} + 289 T^{4} \))(\( 1 - 8 T + 47 T^{2} - 208 T^{3} + 799 T^{4} - 2312 T^{5} + 4913 T^{6} \))(\( 1 - 8 T + 47 T^{2} - 208 T^{3} + 799 T^{4} - 2312 T^{5} + 4913 T^{6} \))
$19$ (\( 1 + 4 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 4 T + 22 T^{2} + 76 T^{3} + 361 T^{4} \))(\( 1 - 4 T + 22 T^{2} - 76 T^{3} + 361 T^{4} \))(\( 1 + 2 T + 33 T^{2} + 44 T^{3} + 627 T^{4} + 722 T^{5} + 6859 T^{6} \))(\( 1 - 2 T + 33 T^{2} - 44 T^{3} + 627 T^{4} - 722 T^{5} + 6859 T^{6} \))
$23$ (\( 1 + 6 T + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 2 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 4 T + 23 T^{2} \))(\( 1 - 2 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 2 T + 2 T^{2} + 46 T^{3} + 529 T^{4} \))(\( 1 - 2 T + 2 T^{2} - 46 T^{3} + 529 T^{4} \))(\( 1 - 6 T + 65 T^{2} - 244 T^{3} + 1495 T^{4} - 3174 T^{5} + 12167 T^{6} \))(\( 1 + 6 T + 65 T^{2} + 244 T^{3} + 1495 T^{4} + 3174 T^{5} + 12167 T^{6} \))
$29$ (\( 1 + 6 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 + 8 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 8 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 - 8 T + 54 T^{2} - 232 T^{3} + 841 T^{4} \))(\( 1 - 8 T + 54 T^{2} - 232 T^{3} + 841 T^{4} \))(\( 1 - 8 T + 83 T^{2} - 400 T^{3} + 2407 T^{4} - 6728 T^{5} + 24389 T^{6} \))(\( 1 - 8 T + 83 T^{2} - 400 T^{3} + 2407 T^{4} - 6728 T^{5} + 24389 T^{6} \))
$31$ (\( 1 + 4 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( ( 1 + 31 T^{2} )^{2} \))(\( ( 1 + 31 T^{2} )^{2} \))(\( 1 + 4 T - 3 T^{2} - 8 T^{3} - 93 T^{4} + 3844 T^{5} + 29791 T^{6} \))(\( 1 - 4 T - 3 T^{2} + 8 T^{3} - 93 T^{4} - 3844 T^{5} + 29791 T^{6} \))
$37$ (\( 1 + 2 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 - 6 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 - 6 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( ( 1 - 6 T + 37 T^{2} )^{2} \))(\( ( 1 - 6 T + 37 T^{2} )^{2} \))(\( 1 + 2 T + 11 T^{2} + 204 T^{3} + 407 T^{4} + 2738 T^{5} + 50653 T^{6} \))(\( 1 + 2 T + 11 T^{2} + 204 T^{3} + 407 T^{4} + 2738 T^{5} + 50653 T^{6} \))
$41$ (\( 1 - 6 T + 41 T^{2} \))(\( 1 - 10 T + 41 T^{2} \))(\( 1 + 4 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 + 4 T + 41 T^{2} \))(\( 1 - 10 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 - 4 T + 6 T^{2} - 164 T^{3} + 1681 T^{4} \))(\( 1 - 4 T + 6 T^{2} - 164 T^{3} + 1681 T^{4} \))(\( 1 - 8 T + 95 T^{2} - 448 T^{3} + 3895 T^{4} - 13448 T^{5} + 68921 T^{6} \))(\( 1 - 8 T + 95 T^{2} - 448 T^{3} + 3895 T^{4} - 13448 T^{5} + 68921 T^{6} \))
$43$ (\( 1 + 8 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 6 T + 43 T^{2} \))(\( 1 + 12 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))(\( 1 + 6 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 - 12 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 - 8 T + 22 T^{2} - 344 T^{3} + 1849 T^{4} \))(\( 1 + 8 T + 22 T^{2} + 344 T^{3} + 1849 T^{4} \))(\( 1 + 10 T + 113 T^{2} + 828 T^{3} + 4859 T^{4} + 18490 T^{5} + 79507 T^{6} \))(\( 1 - 10 T + 113 T^{2} - 828 T^{3} + 4859 T^{4} - 18490 T^{5} + 79507 T^{6} \))
$47$ (\( 1 + 10 T + 47 T^{2} \))(\( 1 - 4 T + 47 T^{2} \))(\( 1 + 8 T + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 - 10 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))(\( 1 - 6 T + 47 T^{2} \))(\( 1 + 8 T + 47 T^{2} \))(\( 1 - 2 T + 50 T^{2} - 94 T^{3} + 2209 T^{4} \))(\( 1 + 2 T + 50 T^{2} + 94 T^{3} + 2209 T^{4} \))(\( 1 - 18 T + 233 T^{2} - 1820 T^{3} + 10951 T^{4} - 39762 T^{5} + 103823 T^{6} \))(\( 1 + 18 T + 233 T^{2} + 1820 T^{3} + 10951 T^{4} + 39762 T^{5} + 103823 T^{6} \))
$53$ (\( 1 + 12 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 12 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 18 T + 182 T^{2} - 954 T^{3} + 2809 T^{4} \))(\( 1 - 18 T + 182 T^{2} - 954 T^{3} + 2809 T^{4} \))(\( 1 + 4 T + 39 T^{2} + 192 T^{3} + 2067 T^{4} + 11236 T^{5} + 148877 T^{6} \))(\( 1 + 4 T + 39 T^{2} + 192 T^{3} + 2067 T^{4} + 11236 T^{5} + 148877 T^{6} \))
$59$ (\( 1 + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 - 20 T + 198 T^{2} - 1180 T^{3} + 3481 T^{4} \))(\( 1 + 20 T + 198 T^{2} + 1180 T^{3} + 3481 T^{4} \))(\( 1 + 113 T^{2} + 64 T^{3} + 6667 T^{4} + 205379 T^{6} \))(\( 1 + 113 T^{2} - 64 T^{3} + 6667 T^{4} + 205379 T^{6} \))
$61$ (\( 1 - 8 T + 61 T^{2} \))(\( 1 + 6 T + 61 T^{2} \))(\( 1 + 6 T + 61 T^{2} \))(\( 1 + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 + 6 T + 61 T^{2} \))(\( 1 + 6 T + 61 T^{2} \))(\( 1 + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 - 10 T + 102 T^{2} - 610 T^{3} + 3721 T^{4} \))(\( 1 - 10 T + 102 T^{2} - 610 T^{3} + 3721 T^{4} \))(\( 1 - 8 T + 127 T^{2} - 584 T^{3} + 7747 T^{4} - 29768 T^{5} + 226981 T^{6} \))(\( 1 - 8 T + 127 T^{2} - 584 T^{3} + 7747 T^{4} - 29768 T^{5} + 226981 T^{6} \))
$67$ (\( 1 + 4 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( ( 1 + 8 T + 67 T^{2} )^{2} \))(\( ( 1 - 8 T + 67 T^{2} )^{2} \))(\( 1 - 4 T + 9 T^{2} + 488 T^{3} + 603 T^{4} - 17956 T^{5} + 300763 T^{6} \))(\( 1 + 4 T + 9 T^{2} - 488 T^{3} + 603 T^{4} + 17956 T^{5} + 300763 T^{6} \))
$71$ (\( 1 + 6 T + 71 T^{2} \))(\( 1 - 4 T + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 + 10 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 - 6 T + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 + 4 T + 71 T^{2} \))(\( 1 - 10 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 2 T + 98 T^{2} + 142 T^{3} + 5041 T^{4} \))(\( 1 - 2 T + 98 T^{2} - 142 T^{3} + 5041 T^{4} \))(\( 1 - 30 T + 497 T^{2} - 5092 T^{3} + 35287 T^{4} - 151230 T^{5} + 357911 T^{6} \))(\( 1 + 30 T + 497 T^{2} + 5092 T^{3} + 35287 T^{4} + 151230 T^{5} + 357911 T^{6} \))
$73$ (\( 1 - 10 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( ( 1 + 6 T + 73 T^{2} )^{2} \))(\( ( 1 + 6 T + 73 T^{2} )^{2} \))(\( 1 - 10 T + 55 T^{2} - 76 T^{3} + 4015 T^{4} - 53290 T^{5} + 389017 T^{6} \))(\( 1 - 10 T + 55 T^{2} - 76 T^{3} + 4015 T^{4} - 53290 T^{5} + 389017 T^{6} \))
$79$ (\( 1 - 6 T + 79 T^{2} \))(\( 1 - 16 T + 79 T^{2} \))(\( 1 + 14 T + 79 T^{2} \))(\( 1 - 2 T + 79 T^{2} \))(\( 1 + 12 T + 79 T^{2} \))(\( 1 + 6 T + 79 T^{2} \))(\( 1 - 14 T + 79 T^{2} \))(\( 1 + 16 T + 79 T^{2} \))(\( 1 + 2 T + 79 T^{2} \))(\( 1 - 12 T + 79 T^{2} \))(\( 1 - 10 T + 138 T^{2} - 790 T^{3} + 6241 T^{4} \))(\( 1 + 10 T + 138 T^{2} + 790 T^{3} + 6241 T^{4} \))(\( 1 + 16 T + 261 T^{2} + 2536 T^{3} + 20619 T^{4} + 99856 T^{5} + 493039 T^{6} \))(\( 1 - 16 T + 261 T^{2} - 2536 T^{3} + 20619 T^{4} - 99856 T^{5} + 493039 T^{6} \))
$83$ (\( 1 - 12 T + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( 1 + 8 T + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 - 8 T + 83 T^{2} \))(\( 1 - 4 T + 83 T^{2} \))(\( 1 - 4 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 - 8 T + 102 T^{2} - 664 T^{3} + 6889 T^{4} \))(\( 1 + 8 T + 102 T^{2} + 664 T^{3} + 6889 T^{4} \))(\( ( 1 - 8 T + 83 T^{2} )^{3} \))(\( ( 1 + 8 T + 83 T^{2} )^{3} \))
$89$ (\( 1 + 14 T + 89 T^{2} \))(\( 1 - 10 T + 89 T^{2} \))(\( 1 - 2 T + 89 T^{2} \))(\( 1 + 14 T + 89 T^{2} \))(\( 1 - 10 T + 89 T^{2} \))(\( 1 + 14 T + 89 T^{2} \))(\( 1 - 2 T + 89 T^{2} \))(\( 1 - 10 T + 89 T^{2} \))(\( 1 + 14 T + 89 T^{2} \))(\( 1 - 10 T + 89 T^{2} \))(\( ( 1 - 2 T + 89 T^{2} )^{2} \))(\( ( 1 - 2 T + 89 T^{2} )^{2} \))(\( ( 1 - 2 T + 89 T^{2} )^{3} \))(\( ( 1 - 2 T + 89 T^{2} )^{3} \))
$97$ (\( 1 + 18 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( 1 - 14 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 - 18 T + 97 T^{2} \))(\( 1 + 18 T + 97 T^{2} \))(\( 1 - 14 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 - 18 T + 97 T^{2} \))(\( 1 + 8 T + 190 T^{2} + 776 T^{3} + 9409 T^{4} \))(\( 1 + 8 T + 190 T^{2} + 776 T^{3} + 9409 T^{4} \))(\( 1 - 6 T + 239 T^{2} - 980 T^{3} + 23183 T^{4} - 56454 T^{5} + 912673 T^{6} \))(\( 1 - 6 T + 239 T^{2} - 980 T^{3} + 23183 T^{4} - 56454 T^{5} + 912673 T^{6} \))
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