Properties

Label 670.2.a
Level 670
Weight 2
Character orbit a
Rep. character \(\chi_{670}(1,\cdot)\)
Character field \(\Q\)
Dimension 21
Newform subspaces 10
Sturm bound 204
Trace bound 3

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Defining parameters

Level: \( N \) = \( 670 = 2 \cdot 5 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 670.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 10 \)
Sturm bound: \(204\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(670))\).

Total New Old
Modular forms 106 21 85
Cusp forms 99 21 78
Eisenstein series 7 0 7

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

\(2\)\(5\)\(67\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(2\)
\(+\)\(+\)\(-\)\(-\)\(4\)
\(+\)\(-\)\(+\)\(-\)\(3\)
\(+\)\(-\)\(-\)\(+\)\(2\)
\(-\)\(+\)\(+\)\(-\)\(3\)
\(-\)\(+\)\(-\)\(+\)\(2\)
\(-\)\(-\)\(+\)\(+\)\(2\)
\(-\)\(-\)\(-\)\(-\)\(3\)
Plus space\(+\)\(8\)
Minus space\(-\)\(13\)

Trace form

\( 21q - q^{2} - 4q^{3} + 21q^{4} - q^{5} - 4q^{6} - 8q^{7} - q^{8} + 17q^{9} + O(q^{10}) \) \( 21q - q^{2} - 4q^{3} + 21q^{4} - q^{5} - 4q^{6} - 8q^{7} - q^{8} + 17q^{9} + q^{10} - 4q^{12} - 14q^{13} - 4q^{14} + 4q^{15} + 21q^{16} - 6q^{17} + 3q^{18} - 12q^{19} - q^{20} + 16q^{21} - 8q^{22} + 12q^{23} - 4q^{24} + 21q^{25} + 14q^{26} + 8q^{27} - 8q^{28} + 6q^{29} - 4q^{30} - 8q^{31} - q^{32} + 40q^{33} - 2q^{34} + 17q^{36} + 2q^{37} - 4q^{38} + q^{40} + 18q^{41} - 20q^{43} + 3q^{45} - 12q^{46} - 4q^{47} - 4q^{48} - 15q^{49} - q^{50} - 14q^{52} - 22q^{53} - 40q^{54} - 12q^{55} - 4q^{56} - 16q^{57} - 6q^{58} + 4q^{59} + 4q^{60} - 38q^{61} - 8q^{62} - 8q^{63} + 21q^{64} + 6q^{65} + 16q^{66} + q^{67} - 6q^{68} + 32q^{69} - 8q^{70} + 12q^{71} + 3q^{72} - 14q^{73} - 2q^{74} - 4q^{75} - 12q^{76} + 64q^{77} + 8q^{78} - 32q^{79} - q^{80} + 45q^{81} - 2q^{82} + 60q^{83} + 16q^{84} + 6q^{85} + 20q^{86} + 24q^{87} - 8q^{88} - 34q^{89} + 13q^{90} - 24q^{91} + 12q^{92} - 16q^{93} - 12q^{94} - 4q^{95} - 4q^{96} - 26q^{97} + 23q^{98} + 24q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(670))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 5 67
670.2.a.a \(1\) \(5.350\) \(\Q\) None \(-1\) \(-2\) \(1\) \(-1\) \(+\) \(-\) \(-\) \(q-q^{2}-2q^{3}+q^{4}+q^{5}+2q^{6}-q^{7}+\cdots\)
670.2.a.b \(1\) \(5.350\) \(\Q\) None \(-1\) \(0\) \(1\) \(1\) \(+\) \(-\) \(-\) \(q-q^{2}+q^{4}+q^{5}+q^{7}-q^{8}-3q^{9}+\cdots\)
670.2.a.c \(1\) \(5.350\) \(\Q\) None \(1\) \(-2\) \(-1\) \(1\) \(-\) \(+\) \(-\) \(q+q^{2}-2q^{3}+q^{4}-q^{5}-2q^{6}+q^{7}+\cdots\)
670.2.a.d \(1\) \(5.350\) \(\Q\) None \(1\) \(0\) \(-1\) \(-5\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{4}-q^{5}-5q^{7}+q^{8}-3q^{9}+\cdots\)
670.2.a.e \(2\) \(5.350\) \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(-2\) \(2\) \(+\) \(+\) \(+\) \(q-q^{2}+\beta q^{3}+q^{4}-q^{5}-\beta q^{6}+(1+\cdots)q^{7}+\cdots\)
670.2.a.f \(2\) \(5.350\) \(\Q(\sqrt{2}) \) None \(2\) \(-4\) \(2\) \(-6\) \(-\) \(-\) \(+\) \(q+q^{2}+(-2+\beta )q^{3}+q^{4}+q^{5}+(-2+\cdots)q^{6}+\cdots\)
670.2.a.g \(3\) \(5.350\) 3.3.148.1 None \(-3\) \(4\) \(3\) \(1\) \(+\) \(-\) \(+\) \(q-q^{2}+(1+\beta _{1})q^{3}+q^{4}+q^{5}+(-1+\cdots)q^{6}+\cdots\)
670.2.a.h \(3\) \(5.350\) 3.3.756.1 None \(3\) \(0\) \(-3\) \(3\) \(-\) \(+\) \(+\) \(q+q^{2}-\beta _{1}q^{3}+q^{4}-q^{5}-\beta _{1}q^{6}+\cdots\)
670.2.a.i \(3\) \(5.350\) 3.3.404.1 None \(3\) \(2\) \(3\) \(1\) \(-\) \(-\) \(-\) \(q+q^{2}+(1-\beta _{1})q^{3}+q^{4}+q^{5}+(1-\beta _{1}+\cdots)q^{6}+\cdots\)
670.2.a.j \(4\) \(5.350\) 4.4.15188.1 None \(-4\) \(-2\) \(-4\) \(-5\) \(+\) \(+\) \(-\) \(q-q^{2}+(-1-\beta _{2})q^{3}+q^{4}-q^{5}+(1+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(670))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(670)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(67))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(134))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(335))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{3} \))(\( ( 1 - T )^{3} \))(\( ( 1 - T )^{3} \))(\( ( 1 + T )^{4} \))
$3$ (\( 1 + 2 T + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 + 2 T + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 + 4 T^{2} + 9 T^{4} \))(\( 1 + 4 T + 8 T^{2} + 12 T^{3} + 9 T^{4} \))(\( 1 - 4 T + 11 T^{2} - 22 T^{3} + 33 T^{4} - 36 T^{5} + 27 T^{6} \))(\( 1 + 3 T^{2} + 2 T^{3} + 9 T^{4} + 27 T^{6} \))(\( 1 - 2 T + 5 T^{2} - 6 T^{3} + 15 T^{4} - 18 T^{5} + 27 T^{6} \))(\( 1 + 2 T + 4 T^{2} - 2 T^{4} + 36 T^{6} + 54 T^{7} + 81 T^{8} \))
$5$ (\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 + T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{3} \))(\( ( 1 + T )^{3} \))(\( ( 1 - T )^{3} \))(\( ( 1 + T )^{4} \))
$7$ (\( 1 + T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 + 5 T + 7 T^{2} \))(\( 1 - 2 T + 13 T^{2} - 14 T^{3} + 49 T^{4} \))(\( 1 + 6 T + 21 T^{2} + 42 T^{3} + 49 T^{4} \))(\( 1 - T + 18 T^{2} - 13 T^{3} + 126 T^{4} - 49 T^{5} + 343 T^{6} \))(\( 1 - 3 T + 18 T^{2} - 35 T^{3} + 126 T^{4} - 147 T^{5} + 343 T^{6} \))(\( 1 - T + 16 T^{2} - 15 T^{3} + 112 T^{4} - 49 T^{5} + 343 T^{6} \))(\( 1 + 5 T + 19 T^{2} + 30 T^{3} + 80 T^{4} + 210 T^{5} + 931 T^{6} + 1715 T^{7} + 2401 T^{8} \))
$11$ (\( 1 - 3 T + 11 T^{2} \))(\( 1 + 5 T + 11 T^{2} \))(\( 1 + 3 T + 11 T^{2} \))(\( 1 + 3 T + 11 T^{2} \))(\( ( 1 + 3 T + 11 T^{2} )^{2} \))(\( ( 1 + 5 T + 11 T^{2} )^{2} \))(\( ( 1 - T + 11 T^{2} )^{3} \))(\( ( 1 - 3 T + 11 T^{2} )^{3} \))(\( ( 1 - T + 11 T^{2} )^{3} \))(\( 1 - 9 T + 47 T^{2} - 212 T^{3} + 800 T^{4} - 2332 T^{5} + 5687 T^{6} - 11979 T^{7} + 14641 T^{8} \))
$13$ (\( 1 + 4 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 - 6 T + 13 T^{2} \))(\( 1 + 8 T^{2} + 169 T^{4} \))(\( 1 + 4 T + 12 T^{2} + 52 T^{3} + 169 T^{4} \))(\( 1 - 4 T + 23 T^{2} - 114 T^{3} + 299 T^{4} - 676 T^{5} + 2197 T^{6} \))(\( 1 + 27 T^{2} - 2 T^{3} + 351 T^{4} + 2197 T^{6} \))(\( 1 - 2 T + 33 T^{2} - 54 T^{3} + 429 T^{4} - 338 T^{5} + 2197 T^{6} \))(\( 1 + 12 T + 98 T^{2} + 526 T^{3} + 2214 T^{4} + 6838 T^{5} + 16562 T^{6} + 26364 T^{7} + 28561 T^{8} \))
$17$ (\( 1 + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 - 4 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 + 4 T + 36 T^{2} + 68 T^{3} + 289 T^{4} \))(\( 1 + 16 T^{2} + 289 T^{4} \))(\( 1 - 8 T + 57 T^{2} - 270 T^{3} + 969 T^{4} - 2312 T^{5} + 4913 T^{6} \))(\( 1 + 9 T^{2} - 98 T^{3} + 153 T^{4} + 4913 T^{6} \))(\( 1 + 2 T + 47 T^{2} + 62 T^{3} + 799 T^{4} + 578 T^{5} + 4913 T^{6} \))(\( 1 + 24 T^{2} - 98 T^{3} + 290 T^{4} - 1666 T^{5} + 6936 T^{6} + 83521 T^{8} \))
$19$ (\( 1 - 2 T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))(\( 1 + 4 T + 34 T^{2} + 76 T^{3} + 361 T^{4} \))(\( 1 + 4 T + 34 T^{2} + 76 T^{3} + 361 T^{4} \))(\( 1 + 8 T + 17 T^{2} + 323 T^{4} + 2888 T^{5} + 6859 T^{6} \))(\( 1 + 33 T^{2} - 16 T^{3} + 627 T^{4} + 6859 T^{6} \))(\( 1 + 25 T^{2} + 32 T^{3} + 475 T^{4} + 6859 T^{6} \))(\( 1 - 4 T - 4 T^{2} - 68 T^{3} + 790 T^{4} - 1292 T^{5} - 1444 T^{6} - 27436 T^{7} + 130321 T^{8} \))
$23$ (\( 1 + 23 T^{2} \))(\( 1 + 4 T + 23 T^{2} \))(\( 1 + 8 T + 23 T^{2} \))(\( 1 + 4 T + 23 T^{2} \))(\( ( 1 + 23 T^{2} )^{2} \))(\( 1 + 14 T^{2} + 529 T^{4} \))(\( 1 - 10 T + 89 T^{2} - 452 T^{3} + 2047 T^{4} - 5290 T^{5} + 12167 T^{6} \))(\( 1 - 6 T + 57 T^{2} - 252 T^{3} + 1311 T^{4} - 3174 T^{5} + 12167 T^{6} \))(\( ( 1 - 2 T + 23 T^{2} )^{3} \))(\( 1 - 6 T + 48 T^{2} - 230 T^{3} + 1118 T^{4} - 5290 T^{5} + 25392 T^{6} - 73002 T^{7} + 279841 T^{8} \))
$29$ (\( 1 + 6 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 + 10 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 + 4 T + 60 T^{2} + 116 T^{3} + 841 T^{4} \))(\( 1 + 8 T^{2} + 841 T^{4} \))(\( 1 + 2 T + 67 T^{2} + 66 T^{3} + 1943 T^{4} + 1682 T^{5} + 24389 T^{6} \))(\( 1 - 6 T + 87 T^{2} - 330 T^{3} + 2523 T^{4} - 5046 T^{5} + 24389 T^{6} \))(\( 1 - 4 T + 73 T^{2} - 178 T^{3} + 2117 T^{4} - 3364 T^{5} + 24389 T^{6} \))(\( 1 - 18 T + 206 T^{2} - 1600 T^{3} + 9854 T^{4} - 46400 T^{5} + 173246 T^{6} - 439002 T^{7} + 707281 T^{8} \))
$31$ (\( 1 - 2 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 + 10 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 + 4 T + 48 T^{2} + 124 T^{3} + 961 T^{4} \))(\( 1 - 8 T + 76 T^{2} - 248 T^{3} + 961 T^{4} \))(\( 1 + 89 T^{2} + 2 T^{3} + 2759 T^{4} + 29791 T^{6} \))(\( 1 + 45 T^{2} + 114 T^{3} + 1395 T^{4} + 29791 T^{6} \))(\( 1 + 2 T + 35 T^{2} + 142 T^{3} + 1085 T^{4} + 1922 T^{5} + 29791 T^{6} \))(\( 1 - 2 T + 82 T^{2} - 240 T^{3} + 3146 T^{4} - 7440 T^{5} + 78802 T^{6} - 59582 T^{7} + 923521 T^{8} \))
$37$ (\( 1 + 7 T + 37 T^{2} \))(\( 1 - 3 T + 37 T^{2} \))(\( 1 - T + 37 T^{2} \))(\( 1 - 7 T + 37 T^{2} \))(\( 1 + 2 T + 25 T^{2} + 74 T^{3} + 1369 T^{4} \))(\( 1 + 14 T + 121 T^{2} + 518 T^{3} + 1369 T^{4} \))(\( 1 + T + 46 T^{2} - 77 T^{3} + 1702 T^{4} + 1369 T^{5} + 50653 T^{6} \))(\( 1 - 3 T + 66 T^{2} - 289 T^{3} + 2442 T^{4} - 4107 T^{5} + 50653 T^{6} \))(\( 1 - 3 T + 48 T^{2} + 21 T^{3} + 1776 T^{4} - 4107 T^{5} + 50653 T^{6} \))(\( 1 - 9 T + 107 T^{2} - 670 T^{3} + 5406 T^{4} - 24790 T^{5} + 146483 T^{6} - 455877 T^{7} + 1874161 T^{8} \))
$41$ (\( 1 + 12 T + 41 T^{2} \))(\( 1 - 10 T + 41 T^{2} \))(\( 1 - 8 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( 1 + 12 T + 100 T^{2} + 492 T^{3} + 1681 T^{4} \))(\( 1 - 16 T^{2} + 1681 T^{4} \))(\( 1 - 12 T + 113 T^{2} - 646 T^{3} + 4633 T^{4} - 20172 T^{5} + 68921 T^{6} \))(\( 1 - 12 T + 129 T^{2} - 782 T^{3} + 5289 T^{4} - 20172 T^{5} + 68921 T^{6} \))(\( 1 + 10 T + 131 T^{2} + 738 T^{3} + 5371 T^{4} + 16810 T^{5} + 68921 T^{6} \))(\( 1 - 12 T + 124 T^{2} - 702 T^{3} + 4810 T^{4} - 28782 T^{5} + 208444 T^{6} - 827052 T^{7} + 2825761 T^{8} \))
$43$ (\( 1 + 4 T + 43 T^{2} \))(\( 1 + 6 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 2 T + 43 T^{2} \))(\( 1 - 12 T + 104 T^{2} - 516 T^{3} + 1849 T^{4} \))(\( 1 + 16 T + 132 T^{2} + 688 T^{3} + 1849 T^{4} \))(\( 1 + 6 T + 3 T^{2} - 290 T^{3} + 129 T^{4} + 11094 T^{5} + 79507 T^{6} \))(\( 1 - 6 T + 75 T^{2} - 394 T^{3} + 3225 T^{4} - 11094 T^{5} + 79507 T^{6} \))(\( 1 - 4 T + 109 T^{2} - 278 T^{3} + 4687 T^{4} - 7396 T^{5} + 79507 T^{6} \))(\( 1 + 16 T + 236 T^{2} + 2034 T^{3} + 16350 T^{4} + 87462 T^{5} + 436364 T^{6} + 1272112 T^{7} + 3418801 T^{8} \))
$47$ (\( 1 + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))(\( 1 - 2 T + 47 T^{2} \))(\( 1 + 8 T + 108 T^{2} + 376 T^{3} + 2209 T^{4} \))(\( 1 + 4 T + 80 T^{2} + 188 T^{3} + 2209 T^{4} \))(\( 1 - 8 T + 125 T^{2} - 622 T^{3} + 5875 T^{4} - 17672 T^{5} + 103823 T^{6} \))(\( 1 + 33 T^{2} + 54 T^{3} + 1551 T^{4} + 103823 T^{6} \))(\( 1 + 2 T + 47 T^{2} + 446 T^{3} + 2209 T^{4} + 4418 T^{5} + 103823 T^{6} \))(\( 1 - 10 T + 78 T^{2} - 612 T^{3} + 5978 T^{4} - 28764 T^{5} + 172302 T^{6} - 1038230 T^{7} + 4879681 T^{8} \))
$53$ (\( 1 + 6 T + 53 T^{2} \))(\( 1 + 12 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 4 T + 53 T^{2} \))(\( 1 + 4 T + 108 T^{2} + 212 T^{3} + 2809 T^{4} \))(\( 1 + 16 T + 168 T^{2} + 848 T^{3} + 2809 T^{4} \))(\( 1 - 12 T + 87 T^{2} - 354 T^{3} + 4611 T^{4} - 33708 T^{5} + 148877 T^{6} \))(\( 1 + 147 T^{2} - 2 T^{3} + 7791 T^{4} + 148877 T^{6} \))(\( 1 + 2 T + 101 T^{2} + 230 T^{3} + 5353 T^{4} + 5618 T^{5} + 148877 T^{6} \))(\( 1 - 8 T + 102 T^{2} - 810 T^{3} + 8870 T^{4} - 42930 T^{5} + 286518 T^{6} - 1191016 T^{7} + 7890481 T^{8} \))
$59$ (\( 1 + 12 T + 59 T^{2} \))(\( 1 - 14 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 - 6 T + 59 T^{2} \))(\( 1 + 12 T + 136 T^{2} + 708 T^{3} + 3481 T^{4} \))(\( 1 + 20 T^{2} + 3481 T^{4} \))(\( 1 + 4 T + 23 T^{2} + 482 T^{3} + 1357 T^{4} + 13924 T^{5} + 205379 T^{6} \))(\( 1 + 12 T + 183 T^{2} + 1214 T^{3} + 10797 T^{4} + 41772 T^{5} + 205379 T^{6} \))(\( 1 - 6 T + 145 T^{2} - 622 T^{3} + 8555 T^{4} - 20886 T^{5} + 205379 T^{6} \))(\( 1 - 18 T + 200 T^{2} - 1280 T^{3} + 8822 T^{4} - 75520 T^{5} + 696200 T^{6} - 3696822 T^{7} + 12117361 T^{8} \))
$61$ (\( 1 - 11 T + 61 T^{2} \))(\( 1 + 15 T + 61 T^{2} \))(\( 1 + 3 T + 61 T^{2} \))(\( 1 + 13 T + 61 T^{2} \))(\( 1 - 2 T - 39 T^{2} - 122 T^{3} + 3721 T^{4} \))(\( 1 - 10 T + 145 T^{2} - 610 T^{3} + 3721 T^{4} \))(\( 1 - 3 T + 86 T^{2} - 17 T^{3} + 5246 T^{4} - 11163 T^{5} + 226981 T^{6} \))(\( 1 + 3 T + 138 T^{2} + 433 T^{3} + 8418 T^{4} + 11163 T^{5} + 226981 T^{6} \))(\( 1 + 19 T + 296 T^{2} + 2519 T^{3} + 18056 T^{4} + 70699 T^{5} + 226981 T^{6} \))(\( 1 + 11 T + 139 T^{2} + 918 T^{3} + 9230 T^{4} + 55998 T^{5} + 517219 T^{6} + 2496791 T^{7} + 13845841 T^{8} \))
$67$ (\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{3} \))(\( ( 1 + T )^{3} \))(\( ( 1 - T )^{3} \))(\( ( 1 - T )^{4} \))
$71$ (\( 1 + 9 T + 71 T^{2} \))(\( 1 - 5 T + 71 T^{2} \))(\( 1 + 9 T + 71 T^{2} \))(\( 1 + 15 T + 71 T^{2} \))(\( 1 + 18 T + 173 T^{2} + 1278 T^{3} + 5041 T^{4} \))(\( 1 - 10 T + 149 T^{2} - 710 T^{3} + 5041 T^{4} \))(\( 1 - 13 T + 176 T^{2} - 1859 T^{3} + 12496 T^{4} - 65533 T^{5} + 357911 T^{6} \))(\( 1 - 9 T + 228 T^{2} - 1271 T^{3} + 16188 T^{4} - 45369 T^{5} + 357911 T^{6} \))(\( 1 + T + 118 T^{2} - 211 T^{3} + 8378 T^{4} + 5041 T^{5} + 357911 T^{6} \))(\( 1 - 27 T + 405 T^{2} - 4664 T^{3} + 44204 T^{4} - 331144 T^{5} + 2041605 T^{6} - 9663597 T^{7} + 25411681 T^{8} \))
$73$ (\( 1 + 10 T + 73 T^{2} \))(\( 1 + 4 T + 73 T^{2} \))(\( 1 - 14 T + 73 T^{2} \))(\( 1 + 8 T + 73 T^{2} \))(\( 1 - 8 T + 112 T^{2} - 584 T^{3} + 5329 T^{4} \))(\( 1 - 4 T + 100 T^{2} - 292 T^{3} + 5329 T^{4} \))(\( 1 - 16 T + 245 T^{2} - 1998 T^{3} + 17885 T^{4} - 85264 T^{5} + 389017 T^{6} \))(\( 1 + 24 T + 405 T^{2} + 3966 T^{3} + 29565 T^{4} + 127896 T^{5} + 389017 T^{6} \))(\( 1 + 14 T + 143 T^{2} + 1178 T^{3} + 10439 T^{4} + 74606 T^{5} + 389017 T^{6} \))(\( 1 - 4 T + 208 T^{2} - 458 T^{3} + 19258 T^{4} - 33434 T^{5} + 1108432 T^{6} - 1556068 T^{7} + 28398241 T^{8} \))
$79$ (\( 1 - 2 T + 79 T^{2} \))(\( 1 - 14 T + 79 T^{2} \))(\( 1 - 6 T + 79 T^{2} \))(\( 1 + 14 T + 79 T^{2} \))(\( ( 1 + 10 T + 79 T^{2} )^{2} \))(\( 1 - 4 T + 130 T^{2} - 316 T^{3} + 6241 T^{4} \))(\( 1 + 12 T + 141 T^{2} + 1816 T^{3} + 11139 T^{4} + 74892 T^{5} + 493039 T^{6} \))(\( 1 + 189 T^{2} + 16 T^{3} + 14931 T^{4} + 493039 T^{6} \))(\( 1 - 8 T + 189 T^{2} - 976 T^{3} + 14931 T^{4} - 49928 T^{5} + 493039 T^{6} \))(\( 1 + 20 T + 380 T^{2} + 4388 T^{3} + 47814 T^{4} + 346652 T^{5} + 2371580 T^{6} + 9860780 T^{7} + 38950081 T^{8} \))
$83$ (\( 1 + 3 T + 83 T^{2} \))(\( 1 + 7 T + 83 T^{2} \))(\( 1 - 5 T + 83 T^{2} \))(\( 1 - 9 T + 83 T^{2} \))(\( 1 - 18 T + 239 T^{2} - 1494 T^{3} + 6889 T^{4} \))(\( 1 - 2 T + 95 T^{2} - 166 T^{3} + 6889 T^{4} \))(\( 1 - 31 T + 536 T^{2} - 5831 T^{3} + 44488 T^{4} - 213559 T^{5} + 571787 T^{6} \))(\( 1 + 9 T + 192 T^{2} + 1465 T^{3} + 15936 T^{4} + 62001 T^{5} + 571787 T^{6} \))(\( 1 - 5 T + 240 T^{2} - 821 T^{3} + 19920 T^{4} - 34445 T^{5} + 571787 T^{6} \))(\( 1 - 9 T + 223 T^{2} - 2340 T^{3} + 23724 T^{4} - 194220 T^{5} + 1536247 T^{6} - 5146083 T^{7} + 47458321 T^{8} \))
$89$ (\( 1 + 3 T + 89 T^{2} \))(\( 1 + 15 T + 89 T^{2} \))(\( 1 - 13 T + 89 T^{2} \))(\( 1 + 15 T + 89 T^{2} \))(\( 1 + 6 T + 115 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( 1 + 14 T + 219 T^{2} + 1246 T^{3} + 7921 T^{4} \))(\( 1 - 11 T + 270 T^{2} - 1791 T^{3} + 24030 T^{4} - 87131 T^{5} + 704969 T^{6} \))(\( 1 + 21 T + 366 T^{2} + 3649 T^{3} + 32574 T^{4} + 166341 T^{5} + 704969 T^{6} \))(\( 1 + T + 182 T^{2} + 325 T^{3} + 16198 T^{4} + 7921 T^{5} + 704969 T^{6} \))(\( 1 - 17 T + 437 T^{2} - 4606 T^{3} + 61882 T^{4} - 409934 T^{5} + 3461477 T^{6} - 11984473 T^{7} + 62742241 T^{8} \))
$97$ (\( 1 - 5 T + 97 T^{2} \))(\( 1 - 13 T + 97 T^{2} \))(\( 1 - 19 T + 97 T^{2} \))(\( 1 - 3 T + 97 T^{2} \))(\( 1 + 22 T + 307 T^{2} + 2134 T^{3} + 9409 T^{4} \))(\( 1 + 18 T + 267 T^{2} + 1746 T^{3} + 9409 T^{4} \))(\( 1 - 9 T + 86 T^{2} + 283 T^{3} + 8342 T^{4} - 84681 T^{5} + 912673 T^{6} \))(\( 1 + 21 T + 414 T^{2} + 4233 T^{3} + 40158 T^{4} + 197589 T^{5} + 912673 T^{6} \))(\( 1 + 9 T + 286 T^{2} + 1645 T^{3} + 27742 T^{4} + 84681 T^{5} + 912673 T^{6} \))(\( 1 + 5 T + 277 T^{2} + 1126 T^{3} + 35698 T^{4} + 109222 T^{5} + 2606293 T^{6} + 4563365 T^{7} + 88529281 T^{8} \))
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