Properties

Label 525.2.a
Level 525
Weight 2
Character orbit a
Rep. character \(\chi_{525}(1,\cdot)\)
Character field \(\Q\)
Dimension 20
Newform subspaces 11
Sturm bound 160
Trace bound 11

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Defining parameters

Level: \( N \) = \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 525.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 11 \)
Sturm bound: \(160\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(2\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(525))\).

Total New Old
Modular forms 92 20 72
Cusp forms 69 20 49
Eisenstein series 23 0 23

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

\(3\)\(5\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(3\)
\(+\)\(+\)\(-\)\(-\)\(1\)
\(+\)\(-\)\(+\)\(-\)\(3\)
\(+\)\(-\)\(-\)\(+\)\(3\)
\(-\)\(+\)\(+\)\(-\)\(4\)
\(-\)\(-\)\(+\)\(+\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(5\)
Plus space\(+\)\(7\)
Minus space\(-\)\(13\)

Trace form

\( 20q + 24q^{4} + 4q^{6} - 2q^{7} + 20q^{9} + O(q^{10}) \) \( 20q + 24q^{4} + 4q^{6} - 2q^{7} + 20q^{9} - 8q^{11} + 8q^{12} + 8q^{13} - 2q^{14} + 40q^{16} + 8q^{17} + 8q^{19} + 2q^{21} + 24q^{22} - 16q^{23} + 12q^{24} - 16q^{26} - 6q^{28} - 32q^{29} - 32q^{34} + 24q^{36} - 8q^{37} - 8q^{38} + 16q^{39} - 32q^{41} - 2q^{42} - 44q^{44} - 28q^{46} - 16q^{47} + 20q^{49} + 8q^{51} - 8q^{52} + 4q^{54} - 30q^{56} + 8q^{57} - 8q^{59} - 8q^{61} - 24q^{62} - 2q^{63} + 28q^{64} - 8q^{66} + 8q^{68} + 16q^{69} + 8q^{71} + 24q^{73} - 68q^{74} + 40q^{76} - 16q^{78} + 32q^{79} + 20q^{81} + 8q^{82} + 32q^{83} + 6q^{84} + 52q^{86} + 8q^{88} + 8q^{89} + 4q^{91} - 16q^{92} + 8q^{93} + 16q^{94} - 52q^{96} - 8q^{97} - 8q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(525))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 5 7
525.2.a.a \(1\) \(4.192\) \(\Q\) None \(-1\) \(-1\) \(0\) \(-1\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}-q^{4}+q^{6}-q^{7}+3q^{8}+\cdots\)
525.2.a.b \(1\) \(4.192\) \(\Q\) None \(-1\) \(1\) \(0\) \(-1\) \(-\) \(-\) \(+\) \(q-q^{2}+q^{3}-q^{4}-q^{6}-q^{7}+3q^{8}+\cdots\)
525.2.a.c \(1\) \(4.192\) \(\Q\) None \(1\) \(-1\) \(0\) \(1\) \(+\) \(-\) \(-\) \(q+q^{2}-q^{3}-q^{4}-q^{6}+q^{7}-3q^{8}+\cdots\)
525.2.a.d \(1\) \(4.192\) \(\Q\) None \(1\) \(-1\) \(0\) \(1\) \(+\) \(+\) \(-\) \(q+q^{2}-q^{3}-q^{4}-q^{6}+q^{7}-3q^{8}+\cdots\)
525.2.a.e \(2\) \(4.192\) \(\Q(\sqrt{5}) \) None \(-3\) \(-2\) \(0\) \(2\) \(+\) \(-\) \(-\) \(q+(-1-\beta )q^{2}-q^{3}+3\beta q^{4}+(1+\beta )q^{6}+\cdots\)
525.2.a.f \(2\) \(4.192\) \(\Q(\sqrt{13}) \) None \(-1\) \(-2\) \(0\) \(-2\) \(+\) \(+\) \(+\) \(q-\beta q^{2}-q^{3}+(1+\beta )q^{4}+\beta q^{6}-q^{7}+\cdots\)
525.2.a.g \(2\) \(4.192\) \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(0\) \(-2\) \(-\) \(+\) \(+\) \(q-\beta q^{2}+q^{3}+3q^{4}-\beta q^{6}-q^{7}-\beta q^{8}+\cdots\)
525.2.a.h \(2\) \(4.192\) \(\Q(\sqrt{13}) \) None \(1\) \(2\) \(0\) \(2\) \(-\) \(-\) \(-\) \(q+\beta q^{2}+q^{3}+(1+\beta )q^{4}+\beta q^{6}+q^{7}+\cdots\)
525.2.a.i \(2\) \(4.192\) \(\Q(\sqrt{5}) \) None \(3\) \(2\) \(0\) \(-2\) \(-\) \(+\) \(+\) \(q+(1+\beta )q^{2}+q^{3}+3\beta q^{4}+(1+\beta )q^{6}+\cdots\)
525.2.a.j \(3\) \(4.192\) 3.3.148.1 None \(-1\) \(3\) \(0\) \(3\) \(-\) \(-\) \(-\) \(q-\beta _{1}q^{2}+q^{3}+(1+\beta _{1}+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)
525.2.a.k \(3\) \(4.192\) 3.3.148.1 None \(1\) \(-3\) \(0\) \(-3\) \(+\) \(-\) \(+\) \(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{1}+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(525))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(525)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T + 2 T^{2} \))(\( 1 + T + 2 T^{2} \))(\( 1 - T + 2 T^{2} \))(\( 1 - T + 2 T^{2} \))(\( 1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4} \))(\( 1 + T + T^{2} + 2 T^{3} + 4 T^{4} \))(\( 1 - T^{2} + 4 T^{4} \))(\( 1 - T + T^{2} - 2 T^{3} + 4 T^{4} \))(\( 1 - 3 T + 5 T^{2} - 6 T^{3} + 4 T^{4} \))(\( 1 + T + T^{2} + 3 T^{3} + 2 T^{4} + 4 T^{5} + 8 T^{6} \))(\( 1 - T + T^{2} - 3 T^{3} + 2 T^{4} - 4 T^{5} + 8 T^{6} \))
$3$ (\( 1 + T \))(\( 1 - T \))(\( 1 + T \))(\( 1 + T \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{3} \))(\( ( 1 + T )^{3} \))
$5$ 1
$7$ (\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{3} \))(\( ( 1 + T )^{3} \))
$11$ (\( 1 + 11 T^{2} \))(\( 1 + 6 T + 11 T^{2} \))(\( 1 + 6 T + 11 T^{2} \))(\( 1 - 4 T + 11 T^{2} \))(\( 1 + 2 T + 3 T^{2} + 22 T^{3} + 121 T^{4} \))(\( ( 1 + 3 T + 11 T^{2} )^{2} \))(\( 1 - 4 T + 6 T^{2} - 44 T^{3} + 121 T^{4} \))(\( ( 1 + 3 T + 11 T^{2} )^{2} \))(\( 1 + 2 T + 3 T^{2} + 22 T^{3} + 121 T^{4} \))(\( ( 1 - 2 T + 11 T^{2} )^{3} \))(\( ( 1 - 2 T + 11 T^{2} )^{3} \))
$13$ (\( 1 - 6 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + 6 T + 30 T^{2} + 78 T^{3} + 169 T^{4} \))(\( 1 + 2 T + 14 T^{2} + 26 T^{3} + 169 T^{4} \))(\( 1 + 6 T^{2} + 169 T^{4} \))(\( 1 - 2 T + 14 T^{2} - 26 T^{3} + 169 T^{4} \))(\( 1 - 6 T + 30 T^{2} - 78 T^{3} + 169 T^{4} \))(\( 1 - 6 T + 35 T^{2} - 148 T^{3} + 455 T^{4} - 1014 T^{5} + 2197 T^{6} \))(\( 1 + 6 T + 35 T^{2} + 148 T^{3} + 455 T^{4} + 1014 T^{5} + 2197 T^{6} \))
$17$ (\( 1 + 2 T + 17 T^{2} \))(\( 1 - 4 T + 17 T^{2} \))(\( 1 + 4 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 - 2 T - 10 T^{2} - 34 T^{3} + 289 T^{4} \))(\( 1 + 2 T + 22 T^{2} + 34 T^{3} + 289 T^{4} \))(\( ( 1 - 2 T + 17 T^{2} )^{2} \))(\( 1 - 2 T + 22 T^{2} - 34 T^{3} + 289 T^{4} \))(\( 1 + 2 T - 10 T^{2} + 34 T^{3} + 289 T^{4} \))(\( 1 + 35 T^{2} + 16 T^{3} + 595 T^{4} + 4913 T^{6} \))(\( 1 + 35 T^{2} - 16 T^{3} + 595 T^{4} + 4913 T^{6} \))
$19$ (\( 1 + 8 T + 19 T^{2} \))(\( 1 + 6 T + 19 T^{2} \))(\( 1 + 6 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 2 T + 34 T^{2} + 38 T^{3} + 361 T^{4} \))(\( 1 - 6 T + 34 T^{2} - 114 T^{3} + 361 T^{4} \))(\( 1 - 4 T + 22 T^{2} - 76 T^{3} + 361 T^{4} \))(\( 1 - 6 T + 34 T^{2} - 114 T^{3} + 361 T^{4} \))(\( 1 + 2 T + 34 T^{2} + 38 T^{3} + 361 T^{4} \))(\( 1 - 6 T + 53 T^{2} - 188 T^{3} + 1007 T^{4} - 2166 T^{5} + 6859 T^{6} \))(\( 1 - 6 T + 53 T^{2} - 188 T^{3} + 1007 T^{4} - 2166 T^{5} + 6859 T^{6} \))
$23$ (\( 1 + 8 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( ( 1 + 5 T + 23 T^{2} )^{2} \))(\( 1 + 2 T - 5 T^{2} + 46 T^{3} + 529 T^{4} \))(\( ( 1 + 4 T + 23 T^{2} )^{2} \))(\( 1 - 2 T - 5 T^{2} - 46 T^{3} + 529 T^{4} \))(\( ( 1 - 5 T + 23 T^{2} )^{2} \))(\( 1 + 4 T + 61 T^{2} + 168 T^{3} + 1403 T^{4} + 2116 T^{5} + 12167 T^{6} \))(\( 1 - 4 T + 61 T^{2} - 168 T^{3} + 1403 T^{4} - 2116 T^{5} + 12167 T^{6} \))
$29$ (\( 1 + 2 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 + 4 T + 17 T^{2} + 116 T^{3} + 841 T^{4} \))(\( 1 + 8 T + 61 T^{2} + 232 T^{3} + 841 T^{4} \))(\( ( 1 + 2 T + 29 T^{2} )^{2} \))(\( 1 + 8 T + 61 T^{2} + 232 T^{3} + 841 T^{4} \))(\( 1 + 4 T + 17 T^{2} + 116 T^{3} + 841 T^{4} \))(\( 1 - 2 T + 35 T^{2} - 76 T^{3} + 1015 T^{4} - 1682 T^{5} + 24389 T^{6} \))(\( 1 - 2 T + 35 T^{2} - 76 T^{3} + 1015 T^{4} - 1682 T^{5} + 24389 T^{6} \))
$31$ (\( 1 - 4 T + 31 T^{2} \))(\( 1 + 10 T + 31 T^{2} \))(\( 1 + 10 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 + 42 T^{2} + 961 T^{4} \))(\( 1 + 10 T^{2} + 961 T^{4} \))(\( 1 - 12 T + 78 T^{2} - 372 T^{3} + 961 T^{4} \))(\( 1 + 10 T^{2} + 961 T^{4} \))(\( 1 + 42 T^{2} + 961 T^{4} \))(\( 1 - 2 T + 41 T^{2} + 60 T^{3} + 1271 T^{4} - 1922 T^{5} + 29791 T^{6} \))(\( 1 - 2 T + 41 T^{2} + 60 T^{3} + 1271 T^{4} - 1922 T^{5} + 29791 T^{6} \))
$37$ (\( 1 - 2 T + 37 T^{2} \))(\( 1 + 4 T + 37 T^{2} \))(\( 1 - 4 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 + 2 T + 55 T^{2} + 74 T^{3} + 1369 T^{4} \))(\( 1 + 6 T + 31 T^{2} + 222 T^{3} + 1369 T^{4} \))(\( 1 + 4 T - 2 T^{2} + 148 T^{3} + 1369 T^{4} \))(\( 1 - 6 T + 31 T^{2} - 222 T^{3} + 1369 T^{4} \))(\( 1 - 2 T + 55 T^{2} - 74 T^{3} + 1369 T^{4} \))(\( 1 - 4 T + 31 T^{2} - 232 T^{3} + 1147 T^{4} - 5476 T^{5} + 50653 T^{6} \))(\( 1 + 4 T + 31 T^{2} + 232 T^{3} + 1147 T^{4} + 5476 T^{5} + 50653 T^{6} \))
$41$ (\( 1 + 6 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( ( 1 + 8 T + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 2 T + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 8 T + 41 T^{2} )^{2} \))(\( 1 - 2 T + 63 T^{2} + 36 T^{3} + 2583 T^{4} - 3362 T^{5} + 68921 T^{6} \))(\( 1 - 2 T + 63 T^{2} + 36 T^{3} + 2583 T^{4} - 3362 T^{5} + 68921 T^{6} \))
$43$ (\( 1 + 4 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 + 12 T + 117 T^{2} + 516 T^{3} + 1849 T^{4} \))(\( 1 + 12 T + 109 T^{2} + 516 T^{3} + 1849 T^{4} \))(\( 1 + 6 T^{2} + 1849 T^{4} \))(\( 1 - 12 T + 109 T^{2} - 516 T^{3} + 1849 T^{4} \))(\( 1 - 12 T + 117 T^{2} - 516 T^{3} + 1849 T^{4} \))(\( 1 + 4 T - 15 T^{2} - 488 T^{3} - 645 T^{4} + 7396 T^{5} + 79507 T^{6} \))(\( 1 - 4 T - 15 T^{2} + 488 T^{3} - 645 T^{4} - 7396 T^{5} + 79507 T^{6} \))
$47$ (\( 1 + 8 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 10 T + 114 T^{2} + 470 T^{3} + 2209 T^{4} \))(\( 1 + 14 T + 130 T^{2} + 658 T^{3} + 2209 T^{4} \))(\( 1 + 8 T + 30 T^{2} + 376 T^{3} + 2209 T^{4} \))(\( 1 - 14 T + 130 T^{2} - 658 T^{3} + 2209 T^{4} \))(\( 1 - 10 T + 114 T^{2} - 470 T^{3} + 2209 T^{4} \))(\( 1 + 8 T + 109 T^{2} + 624 T^{3} + 5123 T^{4} + 17672 T^{5} + 103823 T^{6} \))(\( 1 - 8 T + 109 T^{2} - 624 T^{3} + 5123 T^{4} - 17672 T^{5} + 103823 T^{6} \))
$53$ (\( 1 + 10 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 8 T + 102 T^{2} - 424 T^{3} + 2809 T^{4} \))(\( 1 - 8 T + 70 T^{2} - 424 T^{3} + 2809 T^{4} \))(\( 1 - 16 T + 150 T^{2} - 848 T^{3} + 2809 T^{4} \))(\( 1 + 8 T + 70 T^{2} + 424 T^{3} + 2809 T^{4} \))(\( 1 + 8 T + 102 T^{2} + 424 T^{3} + 2809 T^{4} \))(\( 1 + 14 T + 171 T^{2} + 1188 T^{3} + 9063 T^{4} + 39326 T^{5} + 148877 T^{6} \))(\( 1 - 14 T + 171 T^{2} - 1188 T^{3} + 9063 T^{4} - 39326 T^{5} + 148877 T^{6} \))
$59$ (\( 1 - 4 T + 59 T^{2} \))(\( 1 + 8 T + 59 T^{2} \))(\( 1 + 8 T + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 + 6 T + 122 T^{2} + 354 T^{3} + 3481 T^{4} \))(\( 1 + 14 T + 154 T^{2} + 826 T^{3} + 3481 T^{4} \))(\( 1 + 38 T^{2} + 3481 T^{4} \))(\( 1 + 14 T + 154 T^{2} + 826 T^{3} + 3481 T^{4} \))(\( 1 + 6 T + 122 T^{2} + 354 T^{3} + 3481 T^{4} \))(\( 1 - 16 T + 113 T^{2} - 608 T^{3} + 6667 T^{4} - 55696 T^{5} + 205379 T^{6} \))(\( 1 - 16 T + 113 T^{2} - 608 T^{3} + 6667 T^{4} - 55696 T^{5} + 205379 T^{6} \))
$61$ (\( 1 + 2 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( 1 + 4 T - 54 T^{2} + 244 T^{3} + 3721 T^{4} \))(\( 1 - 12 T + 106 T^{2} - 732 T^{3} + 3721 T^{4} \))(\( ( 1 + 2 T + 61 T^{2} )^{2} \))(\( 1 - 12 T + 106 T^{2} - 732 T^{3} + 3721 T^{4} \))(\( 1 + 4 T - 54 T^{2} + 244 T^{3} + 3721 T^{4} \))(\( 1 + 6 T + 131 T^{2} + 484 T^{3} + 7991 T^{4} + 22326 T^{5} + 226981 T^{6} \))(\( 1 + 6 T + 131 T^{2} + 484 T^{3} + 7991 T^{4} + 22326 T^{5} + 226981 T^{6} \))
$67$ (\( 1 + 4 T + 67 T^{2} \))(\( 1 + 16 T + 67 T^{2} \))(\( 1 - 16 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 + 8 T + 105 T^{2} + 536 T^{3} + 4489 T^{4} \))(\( 1 + 24 T + 265 T^{2} + 1608 T^{3} + 4489 T^{4} \))(\( ( 1 - 4 T + 67 T^{2} )^{2} \))(\( 1 - 24 T + 265 T^{2} - 1608 T^{3} + 4489 T^{4} \))(\( 1 - 8 T + 105 T^{2} - 536 T^{3} + 4489 T^{4} \))(\( 1 + 8 T + 169 T^{2} + 944 T^{3} + 11323 T^{4} + 35912 T^{5} + 300763 T^{6} \))(\( 1 - 8 T + 169 T^{2} - 944 T^{3} + 11323 T^{4} - 35912 T^{5} + 300763 T^{6} \))
$71$ (\( 1 + 12 T + 71 T^{2} \))(\( 1 - 10 T + 71 T^{2} \))(\( 1 - 10 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 10 T + 147 T^{2} + 710 T^{3} + 5041 T^{4} \))(\( ( 1 + 3 T + 71 T^{2} )^{2} \))(\( 1 - 20 T + 222 T^{2} - 1420 T^{3} + 5041 T^{4} \))(\( ( 1 + 3 T + 71 T^{2} )^{2} \))(\( 1 + 10 T + 147 T^{2} + 710 T^{3} + 5041 T^{4} \))(\( ( 1 - 2 T + 71 T^{2} )^{3} \))(\( ( 1 - 2 T + 71 T^{2} )^{3} \))
$73$ (\( 1 - 2 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( 1 - 6 T + 73 T^{2} \))(\( 1 - 6 T + 73 T^{2} \))(\( 1 - 2 T + 142 T^{2} - 146 T^{3} + 5329 T^{4} \))(\( 1 - 6 T + 142 T^{2} - 438 T^{3} + 5329 T^{4} \))(\( 1 - 16 T + 190 T^{2} - 1168 T^{3} + 5329 T^{4} \))(\( 1 + 6 T + 142 T^{2} + 438 T^{3} + 5329 T^{4} \))(\( 1 + 2 T + 142 T^{2} + 146 T^{3} + 5329 T^{4} \))(\( 1 - 18 T + 311 T^{2} - 2732 T^{3} + 22703 T^{4} - 95922 T^{5} + 389017 T^{6} \))(\( 1 + 18 T + 311 T^{2} + 2732 T^{3} + 22703 T^{4} + 95922 T^{5} + 389017 T^{6} \))
$79$ (\( 1 - 8 T + 79 T^{2} \))(\( 1 - 4 T + 79 T^{2} \))(\( 1 - 4 T + 79 T^{2} \))(\( 1 + 16 T + 79 T^{2} \))(\( 1 - 8 T + 169 T^{2} - 632 T^{3} + 6241 T^{4} \))(\( 1 + 8 T + 57 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( 1 - 8 T + 94 T^{2} - 632 T^{3} + 6241 T^{4} \))(\( 1 + 8 T + 57 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( 1 - 8 T + 169 T^{2} - 632 T^{3} + 6241 T^{4} \))(\( 1 - 12 T + 221 T^{2} - 1576 T^{3} + 17459 T^{4} - 74892 T^{5} + 493039 T^{6} \))(\( 1 - 12 T + 221 T^{2} - 1576 T^{3} + 17459 T^{4} - 74892 T^{5} + 493039 T^{6} \))
$83$ (\( 1 - 4 T + 83 T^{2} \))(\( 1 - 8 T + 83 T^{2} \))(\( 1 + 8 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 + 16 T + 210 T^{2} + 1328 T^{3} + 6889 T^{4} \))(\( 1 + 8 T + 130 T^{2} + 664 T^{3} + 6889 T^{4} \))(\( 1 - 16 T + 150 T^{2} - 1328 T^{3} + 6889 T^{4} \))(\( 1 - 8 T + 130 T^{2} - 664 T^{3} + 6889 T^{4} \))(\( 1 - 16 T + 210 T^{2} - 1328 T^{3} + 6889 T^{4} \))(\( 1 + 8 T + 185 T^{2} + 1072 T^{3} + 15355 T^{4} + 55112 T^{5} + 571787 T^{6} \))(\( 1 - 8 T + 185 T^{2} - 1072 T^{3} + 15355 T^{4} - 55112 T^{5} + 571787 T^{6} \))
$89$ (\( 1 + 6 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 14 T + 89 T^{2} \))(\( 1 - 2 T + 134 T^{2} - 178 T^{3} + 7921 T^{4} \))(\( 1 + 6 T + 70 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( ( 1 + 2 T + 89 T^{2} )^{2} \))(\( 1 + 6 T + 70 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( 1 - 2 T + 134 T^{2} - 178 T^{3} + 7921 T^{4} \))(\( 1 - 14 T + 319 T^{2} - 2532 T^{3} + 28391 T^{4} - 110894 T^{5} + 704969 T^{6} \))(\( 1 - 14 T + 319 T^{2} - 2532 T^{3} + 28391 T^{4} - 110894 T^{5} + 704969 T^{6} \))
$97$ (\( 1 - 18 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( 1 + 18 T + 97 T^{2} \))(\( 1 + 16 T + 238 T^{2} + 1552 T^{3} + 9409 T^{4} \))(\( 1 - 16 T + 206 T^{2} - 1552 T^{3} + 9409 T^{4} \))(\( 1 + 8 T + 190 T^{2} + 776 T^{3} + 9409 T^{4} \))(\( 1 + 16 T + 206 T^{2} + 1552 T^{3} + 9409 T^{4} \))(\( 1 - 16 T + 238 T^{2} - 1552 T^{3} + 9409 T^{4} \))(\( 1 - 22 T + 255 T^{2} - 2404 T^{3} + 24735 T^{4} - 206998 T^{5} + 912673 T^{6} \))(\( 1 + 22 T + 255 T^{2} + 2404 T^{3} + 24735 T^{4} + 206998 T^{5} + 912673 T^{6} \))
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