Properties

Label 618.2.a
Level 618
Weight 2
Character orbit a
Rep. character \(\chi_{618}(1,\cdot)\)
Character field \(\Q\)
Dimension 17
Newform subspaces 11
Sturm bound 208
Trace bound 5

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Defining parameters

Level: \( N \) \(=\) \( 618 = 2 \cdot 3 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 618.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 11 \)
Sturm bound: \(208\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(618))\).

Total New Old
Modular forms 108 17 91
Cusp forms 101 17 84
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.

\(2\)\(3\)\(103\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(2\)
\(+\)\(+\)\(-\)\(-\)\(2\)
\(+\)\(-\)\(+\)\(-\)\(2\)
\(+\)\(-\)\(-\)\(+\)\(2\)
\(-\)\(+\)\(+\)\(-\)\(4\)
\(-\)\(+\)\(-\)\(+\)\(1\)
\(-\)\(-\)\(+\)\(+\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(3\)
Plus space\(+\)\(6\)
Minus space\(-\)\(11\)

Trace form

\( 17q + q^{2} - q^{3} + 17q^{4} + 6q^{5} - q^{6} + q^{8} + 17q^{9} + O(q^{10}) \) \( 17q + q^{2} - q^{3} + 17q^{4} + 6q^{5} - q^{6} + q^{8} + 17q^{9} - 2q^{10} - 4q^{11} - q^{12} - 6q^{13} - 6q^{15} + 17q^{16} + 2q^{17} + q^{18} - 16q^{19} + 6q^{20} - 8q^{21} - 4q^{22} - q^{24} + 19q^{25} - 2q^{26} - q^{27} + 6q^{29} - 2q^{30} - 24q^{31} + q^{32} - 8q^{33} + 18q^{34} + 24q^{35} + 17q^{36} - 2q^{37} + 4q^{38} - 6q^{39} - 2q^{40} + 2q^{41} + 4q^{43} - 4q^{44} + 6q^{45} - 8q^{46} + 8q^{47} - q^{48} + 9q^{49} + 31q^{50} - 2q^{51} - 6q^{52} - 2q^{53} - q^{54} + 24q^{55} - 4q^{57} + 2q^{58} - 20q^{59} - 6q^{60} + 34q^{61} + 24q^{62} + 17q^{64} + 20q^{65} + 4q^{66} - 20q^{67} + 2q^{68} + 8q^{69} + 8q^{70} - 32q^{71} + q^{72} - 54q^{73} - 2q^{74} + q^{75} - 16q^{76} - 16q^{77} - 14q^{78} - 16q^{79} + 6q^{80} + 17q^{81} + 10q^{82} - 28q^{83} - 8q^{84} - 28q^{85} - 4q^{86} + 2q^{87} - 4q^{88} - 30q^{89} - 2q^{90} + 24q^{91} - 28q^{93} - 16q^{94} - 32q^{95} - q^{96} + 10q^{97} + 9q^{98} - 4q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(618))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 103
618.2.a.a \(1\) \(4.935\) \(\Q\) None \(-1\) \(-1\) \(-1\) \(-2\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}-2q^{7}+\cdots\)
618.2.a.b \(1\) \(4.935\) \(\Q\) None \(-1\) \(-1\) \(2\) \(-2\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}+q^{4}+2q^{5}+q^{6}-2q^{7}+\cdots\)
618.2.a.c \(1\) \(4.935\) \(\Q\) None \(-1\) \(1\) \(-3\) \(2\) \(+\) \(-\) \(-\) \(q-q^{2}+q^{3}+q^{4}-3q^{5}-q^{6}+2q^{7}+\cdots\)
618.2.a.d \(1\) \(4.935\) \(\Q\) None \(-1\) \(1\) \(0\) \(-4\) \(+\) \(-\) \(-\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-4q^{7}-q^{8}+\cdots\)
618.2.a.e \(1\) \(4.935\) \(\Q\) None \(1\) \(-1\) \(-2\) \(-2\) \(-\) \(+\) \(-\) \(q+q^{2}-q^{3}+q^{4}-2q^{5}-q^{6}-2q^{7}+\cdots\)
618.2.a.f \(1\) \(4.935\) \(\Q\) None \(1\) \(1\) \(-4\) \(-4\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{3}+q^{4}-4q^{5}+q^{6}-4q^{7}+\cdots\)
618.2.a.g \(1\) \(4.935\) \(\Q\) None \(1\) \(1\) \(3\) \(-2\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{3}+q^{4}+3q^{5}+q^{6}-2q^{7}+\cdots\)
618.2.a.h \(2\) \(4.935\) \(\Q(\sqrt{3}) \) None \(-2\) \(-2\) \(2\) \(6\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}+(1+\beta )q^{5}+q^{6}+\cdots\)
618.2.a.i \(2\) \(4.935\) \(\Q(\sqrt{2}) \) None \(-2\) \(2\) \(4\) \(0\) \(+\) \(-\) \(+\) \(q-q^{2}+q^{3}+q^{4}+(2+\beta )q^{5}-q^{6}+\cdots\)
618.2.a.j \(2\) \(4.935\) \(\Q(\sqrt{2}) \) None \(2\) \(2\) \(0\) \(4\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{3}+q^{4}+\beta q^{5}+q^{6}+(2+\beta )q^{7}+\cdots\)
618.2.a.k \(4\) \(4.935\) 4.4.54332.1 None \(4\) \(-4\) \(5\) \(4\) \(-\) \(+\) \(+\) \(q+q^{2}-q^{3}+q^{4}+(1+\beta _{1})q^{5}-q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(618))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(618)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(103))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(206))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(309))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{4} \))
$3$ (\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{4} \))
$5$ (\( 1 + T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 + 3 T + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 4 T + 5 T^{2} \))(\( 1 - 3 T + 5 T^{2} \))(\( 1 - 2 T + 8 T^{2} - 10 T^{3} + 25 T^{4} \))(\( 1 - 4 T + 12 T^{2} - 20 T^{3} + 25 T^{4} \))(\( 1 + 8 T^{2} + 25 T^{4} \))(\( 1 - 5 T + 14 T^{2} - 21 T^{3} + 38 T^{4} - 105 T^{5} + 350 T^{6} - 625 T^{7} + 625 T^{8} \))
$7$ (\( 1 + 2 T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 - 2 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 - 6 T + 20 T^{2} - 42 T^{3} + 49 T^{4} \))(\( 1 + 12 T^{2} + 49 T^{4} \))(\( 1 - 4 T + 16 T^{2} - 28 T^{3} + 49 T^{4} \))(\( 1 - 4 T + 18 T^{2} - 56 T^{3} + 186 T^{4} - 392 T^{5} + 882 T^{6} - 1372 T^{7} + 2401 T^{8} \))
$11$ (\( 1 - 6 T + 11 T^{2} \))(\( 1 + 3 T + 11 T^{2} \))(\( 1 + 6 T + 11 T^{2} \))(\( 1 + 3 T + 11 T^{2} \))(\( 1 - T + 11 T^{2} \))(\( 1 + 3 T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 + 19 T^{2} + 121 T^{4} \))(\( 1 - 6 T + 29 T^{2} - 66 T^{3} + 121 T^{4} \))(\( 1 - 2 T + 21 T^{2} - 22 T^{3} + 121 T^{4} \))(\( 1 + 2 T + 21 T^{2} + 76 T^{3} + 228 T^{4} + 836 T^{5} + 2541 T^{6} + 2662 T^{7} + 14641 T^{8} \))
$13$ (\( 1 + T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 + T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 + 6 T + 13 T^{2} \))(\( 1 - 3 T + 13 T^{2} \))(\( 1 - 2 T + 24 T^{2} - 26 T^{3} + 169 T^{4} \))(\( 1 + 24 T^{2} + 169 T^{4} \))(\( 1 + 4 T + 12 T^{2} + 52 T^{3} + 169 T^{4} \))(\( 1 - 7 T + 38 T^{2} - 111 T^{3} + 414 T^{4} - 1443 T^{5} + 6422 T^{6} - 15379 T^{7} + 28561 T^{8} \))
$17$ (\( 1 + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 + 4 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 2 T + 32 T^{2} + 34 T^{3} + 289 T^{4} \))(\( 1 - 16 T^{2} + 289 T^{4} \))(\( 1 - 4 T + 20 T^{2} - 68 T^{3} + 289 T^{4} \))(\( 1 - 8 T + 42 T^{2} - 172 T^{3} + 594 T^{4} - 2924 T^{5} + 12138 T^{6} - 39304 T^{7} + 83521 T^{8} \))
$19$ (\( 1 + 8 T + 19 T^{2} \))(\( 1 - T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 + T + 19 T^{2} \))(\( 1 + 3 T + 19 T^{2} \))(\( 1 - 3 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 - 4 T + 39 T^{2} - 76 T^{3} + 361 T^{4} \))(\( 1 + 2 T + 21 T^{2} + 38 T^{3} + 361 T^{4} \))(\( 1 + 6 T + 45 T^{2} + 114 T^{3} + 361 T^{4} \))(\( 1 + 21 T^{2} + 828 T^{4} + 7581 T^{6} + 130321 T^{8} \))
$23$ (\( 1 - 3 T + 23 T^{2} \))(\( 1 + 3 T + 23 T^{2} \))(\( 1 - 3 T + 23 T^{2} \))(\( 1 + 9 T + 23 T^{2} \))(\( 1 + 5 T + 23 T^{2} \))(\( 1 - T + 23 T^{2} \))(\( 1 + 3 T + 23 T^{2} \))(\( 1 + 19 T^{2} + 529 T^{4} \))(\( 1 - 10 T + 69 T^{2} - 230 T^{3} + 529 T^{4} \))(\( 1 - 2 T + 45 T^{2} - 46 T^{3} + 529 T^{4} \))(\( 1 - T + 77 T^{2} - 38 T^{3} + 2476 T^{4} - 874 T^{5} + 40733 T^{6} - 12167 T^{7} + 279841 T^{8} \))
$29$ (\( 1 + 6 T + 29 T^{2} \))(\( 1 - 9 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 - 9 T + 29 T^{2} \))(\( 1 - 3 T + 29 T^{2} \))(\( 1 + 5 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 + 6 T + 19 T^{2} + 174 T^{3} + 841 T^{4} \))(\( ( 1 - T + 29 T^{2} )^{2} \))(\( 1 - 6 T + 35 T^{2} - 174 T^{3} + 841 T^{4} \))(\( ( 1 - T + 20 T^{2} - 29 T^{3} + 841 T^{4} )^{2} \))
$31$ (\( 1 + 5 T + 31 T^{2} \))(\( 1 + 5 T + 31 T^{2} \))(\( 1 + 7 T + 31 T^{2} \))(\( 1 + T + 31 T^{2} \))(\( 1 + 3 T + 31 T^{2} \))(\( 1 + 3 T + 31 T^{2} \))(\( 1 + 3 T + 31 T^{2} \))(\( 1 - 4 T - 9 T^{2} - 124 T^{3} + 961 T^{4} \))(\( 1 + 10 T + 69 T^{2} + 310 T^{3} + 961 T^{4} \))(\( 1 + 2 T + 45 T^{2} + 62 T^{3} + 961 T^{4} \))(\( 1 - 11 T + 53 T^{2} + 182 T^{3} - 2204 T^{4} + 5642 T^{5} + 50933 T^{6} - 327701 T^{7} + 923521 T^{8} \))
$37$ (\( 1 - 6 T + 37 T^{2} \))(\( 1 + 9 T + 37 T^{2} \))(\( 1 + 10 T + 37 T^{2} \))(\( 1 - 5 T + 37 T^{2} \))(\( 1 + 11 T + 37 T^{2} \))(\( 1 - 11 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( ( 1 - 5 T + 37 T^{2} )^{2} \))(\( 1 + 2 T + 67 T^{2} + 74 T^{3} + 1369 T^{4} \))(\( 1 + 6 T + 51 T^{2} + 222 T^{3} + 1369 T^{4} \))(\( 1 - 2 T + 77 T^{2} - 14 T^{3} + 3372 T^{4} - 518 T^{5} + 105413 T^{6} - 101306 T^{7} + 1874161 T^{8} \))
$41$ (\( 1 - 4 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 + 12 T + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 + 2 T + 56 T^{2} + 82 T^{3} + 1681 T^{4} \))(\( 1 + 4 T + 68 T^{2} + 164 T^{3} + 1681 T^{4} \))(\( 1 - 16 T^{2} + 1681 T^{4} \))(\( 1 - 12 T + 202 T^{2} - 1488 T^{3} + 13186 T^{4} - 61008 T^{5} + 339562 T^{6} - 827052 T^{7} + 2825761 T^{8} \))
$43$ (\( 1 + 9 T + 43 T^{2} \))(\( 1 + 6 T + 43 T^{2} \))(\( 1 - 5 T + 43 T^{2} \))(\( 1 - 2 T + 43 T^{2} \))(\( 1 - 6 T + 43 T^{2} \))(\( 1 - 6 T + 43 T^{2} \))(\( 1 + 3 T + 43 T^{2} \))(\( 1 - 16 T + 138 T^{2} - 688 T^{3} + 1849 T^{4} \))(\( 1 + 4 T + 18 T^{2} + 172 T^{3} + 1849 T^{4} \))(\( 1 + 4 T + 82 T^{2} + 172 T^{3} + 1849 T^{4} \))(\( 1 + 5 T + 140 T^{2} + 441 T^{3} + 8070 T^{4} + 18963 T^{5} + 258860 T^{6} + 397535 T^{7} + 3418801 T^{8} \))
$47$ (\( 1 + 10 T + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))(\( 1 - 6 T + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 10 T + 47 T^{2} \))(\( 1 + 10 T + 47 T^{2} \))(\( 1 - 14 T + 140 T^{2} - 658 T^{3} + 2209 T^{4} \))(\( 1 - 12 T + 80 T^{2} - 564 T^{3} + 2209 T^{4} \))(\( 1 - 16 T + 156 T^{2} - 752 T^{3} + 2209 T^{4} \))(\( 1 + 138 T^{2} + 68 T^{3} + 8618 T^{4} + 3196 T^{5} + 304842 T^{6} + 4879681 T^{8} \))
$53$ (\( 1 + 6 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 + 12 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 + 4 T + 53 T^{2} \))(\( 1 - 10 T + 53 T^{2} \))(\( 1 - 6 T + 40 T^{2} - 318 T^{3} + 2809 T^{4} \))(\( 1 - 4 T + 92 T^{2} - 212 T^{3} + 2809 T^{4} \))(\( 1 + 16 T + 168 T^{2} + 848 T^{3} + 2809 T^{4} \))(\( 1 + 2 T + 82 T^{2} + 238 T^{3} + 3178 T^{4} + 12614 T^{5} + 230338 T^{6} + 297754 T^{7} + 7890481 T^{8} \))
$59$ (\( 1 + 5 T + 59 T^{2} \))(\( 1 + 2 T + 59 T^{2} \))(\( 1 - 3 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 - 6 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 - 9 T + 59 T^{2} \))(\( 1 - 6 T + 100 T^{2} - 354 T^{3} + 3481 T^{4} \))(\( 1 - 8 T + 132 T^{2} - 472 T^{3} + 3481 T^{4} \))(\( 1 + 4 T + 120 T^{2} + 236 T^{3} + 3481 T^{4} \))(\( 1 + 29 T + 536 T^{2} + 6411 T^{3} + 58070 T^{4} + 378249 T^{5} + 1865816 T^{6} + 5955991 T^{7} + 12117361 T^{8} \))
$61$ (\( 1 - 15 T + 61 T^{2} \))(\( 1 + 12 T + 61 T^{2} \))(\( 1 + T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 - 4 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 + 5 T + 61 T^{2} \))(\( 1 - 8 T + 90 T^{2} - 488 T^{3} + 3721 T^{4} \))(\( ( 1 + 4 T + 61 T^{2} )^{2} \))(\( 1 - 8 T + 106 T^{2} - 488 T^{3} + 3721 T^{4} \))(\( 1 - 9 T + 158 T^{2} - 1103 T^{3} + 14010 T^{4} - 67283 T^{5} + 587918 T^{6} - 2042829 T^{7} + 13845841 T^{8} \))
$67$ (\( 1 + 15 T + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))(\( 1 + 13 T + 67 T^{2} \))(\( 1 - 14 T + 67 T^{2} \))(\( 1 - 8 T + 67 T^{2} \))(\( 1 + 2 T + 67 T^{2} \))(\( 1 + 5 T + 67 T^{2} \))(\( 1 - 14 T + 180 T^{2} - 938 T^{3} + 4489 T^{4} \))(\( 1 - 4 T + 136 T^{2} - 268 T^{3} + 4489 T^{4} \))(\( 1 + 8 T + 100 T^{2} + 536 T^{3} + 4489 T^{4} \))(\( 1 + 5 T + 32 T^{2} - 389 T^{3} - 6458 T^{4} - 26063 T^{5} + 143648 T^{6} + 1503815 T^{7} + 20151121 T^{8} \))
$71$ (\( 1 + 71 T^{2} \))(\( 1 - 6 T + 71 T^{2} \))(\( 1 + 12 T + 71 T^{2} \))(\( 1 + 12 T + 71 T^{2} \))(\( 1 - 2 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 + 6 T + 4 T^{2} + 426 T^{3} + 5041 T^{4} \))(\( 1 - 8 T + 140 T^{2} - 568 T^{3} + 5041 T^{4} \))(\( 1 + 4 T + 144 T^{2} + 284 T^{3} + 5041 T^{4} \))(\( 1 + 22 T + 374 T^{2} + 4022 T^{3} + 39250 T^{4} + 285562 T^{5} + 1885334 T^{6} + 7874042 T^{7} + 25411681 T^{8} \))
$73$ (\( 1 + 16 T + 73 T^{2} \))(\( 1 - 14 T + 73 T^{2} \))(\( 1 + 16 T + 73 T^{2} \))(\( 1 + 16 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 + 4 T + 73 T^{2} \))(\( 1 + 4 T + 73 T^{2} \))(\( 1 + 10 T + 96 T^{2} + 730 T^{3} + 5329 T^{4} \))(\( 1 - 12 T + 164 T^{2} - 876 T^{3} + 5329 T^{4} \))(\( 1 + 144 T^{2} + 5329 T^{4} \))(\( 1 + 4 T + 154 T^{2} + 592 T^{3} + 15490 T^{4} + 43216 T^{5} + 820666 T^{6} + 1556068 T^{7} + 28398241 T^{8} \))
$79$ (\( 1 - 6 T + 79 T^{2} \))(\( 1 + 6 T + 79 T^{2} \))(\( 1 - 14 T + 79 T^{2} \))(\( 1 - 2 T + 79 T^{2} \))(\( 1 - 10 T + 79 T^{2} \))(\( 1 + 14 T + 79 T^{2} \))(\( 1 + 2 T + 79 T^{2} \))(\( 1 + 146 T^{2} + 6241 T^{4} \))(\( 1 - 4 T - 38 T^{2} - 316 T^{3} + 6241 T^{4} \))(\( 1 + 20 T + 250 T^{2} + 1580 T^{3} + 6241 T^{4} \))(\( 1 + 10 T + 224 T^{2} + 2346 T^{3} + 23166 T^{4} + 185334 T^{5} + 1397984 T^{6} + 4930390 T^{7} + 38950081 T^{8} \))
$83$ (\( 1 + 9 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 + 9 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 + 3 T + 83 T^{2} \))(\( ( 1 + 4 T + 83 T^{2} )^{2} \))(\( ( 1 - 12 T + 83 T^{2} )^{2} \))(\( 1 + 16 T + 198 T^{2} + 1328 T^{3} + 6889 T^{4} \))(\( 1 + 7 T + 312 T^{2} + 1679 T^{3} + 38142 T^{4} + 139357 T^{5} + 2149368 T^{6} + 4002509 T^{7} + 47458321 T^{8} \))
$89$ (\( 1 - 2 T + 89 T^{2} \))(\( 1 + 7 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 - 3 T + 89 T^{2} \))(\( 1 - T + 89 T^{2} \))(\( 1 + 9 T + 89 T^{2} \))(\( 1 - 10 T + 89 T^{2} \))(\( 1 + 14 T + 179 T^{2} + 1246 T^{3} + 7921 T^{4} \))(\( ( 1 + T + 89 T^{2} )^{2} \))(\( 1 - 14 T + 155 T^{2} - 1246 T^{3} + 7921 T^{4} \))(\( 1 + 34 T + 717 T^{2} + 10166 T^{3} + 111108 T^{4} + 904774 T^{5} + 5679357 T^{6} + 23968946 T^{7} + 62742241 T^{8} \))
$97$ (\( 1 - T + 97 T^{2} \))(\( 1 - 7 T + 97 T^{2} \))(\( 1 - 17 T + 97 T^{2} \))(\( 1 + T + 97 T^{2} \))(\( 1 + 17 T + 97 T^{2} \))(\( 1 + 17 T + 97 T^{2} \))(\( 1 + 7 T + 97 T^{2} \))(\( 1 - 10 T + 171 T^{2} - 970 T^{3} + 9409 T^{4} \))(\( 1 - 2 T + 123 T^{2} - 194 T^{3} + 9409 T^{4} \))(\( 1 - 2 T + 187 T^{2} - 194 T^{3} + 9409 T^{4} \))(\( 1 - 13 T + 413 T^{2} - 3622 T^{3} + 61002 T^{4} - 351334 T^{5} + 3885917 T^{6} - 11864749 T^{7} + 88529281 T^{8} \))
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