Properties

Label 574.2.a
Level 574
Weight 2
Character orbit a
Rep. character \(\chi_{574}(1,\cdot)\)
Character field \(\Q\)
Dimension 19
Newform subspaces 13
Sturm bound 168
Trace bound 5

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Defining parameters

Level: \( N \) = \( 574 = 2 \cdot 7 \cdot 41 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 574.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 13 \)
Sturm bound: \(168\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\), \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(574))\).

Total New Old
Modular forms 88 19 69
Cusp forms 81 19 62
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.

\(2\)\(7\)\(41\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(2\)
\(+\)\(+\)\(-\)\(-\)\(4\)
\(+\)\(-\)\(+\)\(-\)\(3\)
\(+\)\(-\)\(-\)\(+\)\(1\)
\(-\)\(+\)\(+\)\(-\)\(3\)
\(-\)\(+\)\(-\)\(+\)\(1\)
\(-\)\(-\)\(+\)\(+\)\(2\)
\(-\)\(-\)\(-\)\(-\)\(3\)
Plus space\(+\)\(6\)
Minus space\(-\)\(13\)

Trace form

\( 19q - q^{2} + 4q^{3} + 19q^{4} + 2q^{5} - 4q^{6} - q^{7} - q^{8} + 23q^{9} + O(q^{10}) \) \( 19q - q^{2} + 4q^{3} + 19q^{4} + 2q^{5} - 4q^{6} - q^{7} - q^{8} + 23q^{9} - 6q^{10} + 4q^{12} - 14q^{13} + 3q^{14} + 8q^{15} + 19q^{16} - 10q^{17} - 5q^{18} - 12q^{19} + 2q^{20} + 4q^{21} - 8q^{22} + 8q^{23} - 4q^{24} + 13q^{25} - 6q^{26} - 8q^{27} - q^{28} - 2q^{29} + 16q^{30} + 24q^{31} - q^{32} - 8q^{33} - 2q^{34} - 2q^{35} + 23q^{36} + 2q^{37} + 12q^{38} - 16q^{39} - 6q^{40} - q^{41} - 4q^{42} - 36q^{43} + 26q^{45} - 8q^{46} + 16q^{47} + 4q^{48} + 19q^{49} - 23q^{50} - 8q^{51} - 14q^{52} - 10q^{53} + 8q^{54} + 24q^{55} + 3q^{56} + 24q^{57} - 18q^{58} + 4q^{59} + 8q^{60} - 6q^{61} - 5q^{63} + 19q^{64} + 68q^{65} + 16q^{67} - 10q^{68} + 8q^{69} - 2q^{70} + 32q^{71} - 5q^{72} - 34q^{73} + 10q^{74} + 4q^{75} - 12q^{76} + 4q^{77} - 16q^{79} + 2q^{80} + 35q^{81} - q^{82} - 20q^{83} + 4q^{84} - 4q^{85} - 12q^{86} - 32q^{87} - 8q^{88} + 86q^{89} - 46q^{90} + 10q^{91} + 8q^{92} - 40q^{93} + 8q^{94} - 40q^{95} - 4q^{96} - 18q^{97} - q^{98} - 24q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(574))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 7 41
574.2.a.a \(1\) \(4.583\) \(\Q\) None \(-1\) \(-2\) \(4\) \(1\) \(+\) \(-\) \(+\) \(q-q^{2}-2q^{3}+q^{4}+4q^{5}+2q^{6}+q^{7}+\cdots\)
574.2.a.b \(1\) \(4.583\) \(\Q\) None \(-1\) \(-1\) \(1\) \(-1\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-q^{7}+\cdots\)
574.2.a.c \(1\) \(4.583\) \(\Q\) None \(-1\) \(1\) \(-3\) \(1\) \(+\) \(-\) \(-\) \(q-q^{2}+q^{3}+q^{4}-3q^{5}-q^{6}+q^{7}+\cdots\)
574.2.a.d \(1\) \(4.583\) \(\Q\) None \(-1\) \(2\) \(-2\) \(-1\) \(+\) \(+\) \(+\) \(q-q^{2}+2q^{3}+q^{4}-2q^{5}-2q^{6}-q^{7}+\cdots\)
574.2.a.e \(1\) \(4.583\) \(\Q\) None \(-1\) \(2\) \(2\) \(1\) \(+\) \(-\) \(+\) \(q-q^{2}+2q^{3}+q^{4}+2q^{5}-2q^{6}+q^{7}+\cdots\)
574.2.a.f \(1\) \(4.583\) \(\Q\) None \(-1\) \(3\) \(-1\) \(1\) \(+\) \(-\) \(+\) \(q-q^{2}+3q^{3}+q^{4}-q^{5}-3q^{6}+q^{7}+\cdots\)
574.2.a.g \(1\) \(4.583\) \(\Q\) None \(1\) \(-3\) \(-1\) \(1\) \(-\) \(-\) \(+\) \(q+q^{2}-3q^{3}+q^{4}-q^{5}-3q^{6}+q^{7}+\cdots\)
574.2.a.h \(1\) \(4.583\) \(\Q\) None \(1\) \(-1\) \(-1\) \(-1\) \(-\) \(+\) \(-\) \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}-q^{7}+\cdots\)
574.2.a.i \(1\) \(4.583\) \(\Q\) None \(1\) \(-1\) \(1\) \(1\) \(-\) \(-\) \(-\) \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+q^{7}+\cdots\)
574.2.a.j \(1\) \(4.583\) \(\Q\) None \(1\) \(0\) \(-4\) \(1\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{4}-4q^{5}+q^{7}+q^{8}-3q^{9}+\cdots\)
574.2.a.k \(2\) \(4.583\) \(\Q(\sqrt{3}) \) None \(2\) \(4\) \(2\) \(2\) \(-\) \(-\) \(-\) \(q+q^{2}+2q^{3}+q^{4}+(1+\beta )q^{5}+2q^{6}+\cdots\)
574.2.a.l \(3\) \(4.583\) 3.3.568.1 None \(3\) \(1\) \(1\) \(-3\) \(-\) \(+\) \(+\) \(q+q^{2}-\beta _{2}q^{3}+q^{4}+\beta _{1}q^{5}-\beta _{2}q^{6}+\cdots\)
574.2.a.m \(4\) \(4.583\) 4.4.11348.1 None \(-4\) \(-1\) \(3\) \(-4\) \(+\) \(+\) \(-\) \(q-q^{2}-\beta _{2}q^{3}+q^{4}+(1-\beta _{2}+\beta _{3})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(574))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(574)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(41))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(82))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(287))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{3} \))(\( ( 1 + T )^{4} \))
$3$ (\( 1 + 2 T + 3 T^{2} \))(\( 1 + T + 3 T^{2} \))(\( 1 - T + 3 T^{2} \))(\( 1 - 2 T + 3 T^{2} \))(\( 1 - 2 T + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))(\( 1 + 3 T + 3 T^{2} \))(\( 1 + T + 3 T^{2} \))(\( 1 + T + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( ( 1 - 2 T + 3 T^{2} )^{2} \))(\( 1 - T + T^{2} - 2 T^{3} + 3 T^{4} - 9 T^{5} + 27 T^{6} \))(\( 1 + T + 2 T^{2} + 5 T^{3} + 2 T^{4} + 15 T^{5} + 18 T^{6} + 27 T^{7} + 81 T^{8} \))
$5$ (\( 1 - 4 T + 5 T^{2} \))(\( 1 - T + 5 T^{2} \))(\( 1 + 3 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 + T + 5 T^{2} \))(\( 1 + T + 5 T^{2} \))(\( 1 + T + 5 T^{2} \))(\( 1 - T + 5 T^{2} \))(\( 1 + 4 T + 5 T^{2} \))(\( 1 - 2 T + 8 T^{2} - 10 T^{3} + 25 T^{4} \))(\( 1 - T + 9 T^{2} - 12 T^{3} + 45 T^{4} - 25 T^{5} + 125 T^{6} \))(\( 1 - 3 T + 8 T^{2} - 5 T^{3} + 2 T^{4} - 25 T^{5} + 200 T^{6} - 375 T^{7} + 625 T^{8} \))
$7$ (\( 1 - T \))(\( 1 + T \))(\( 1 - T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{3} \))(\( ( 1 + T )^{4} \))
$11$ (\( 1 - 4 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 6 T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 - 4 T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 + 6 T + 11 T^{2} \))(\( 1 - 2 T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 + 2 T + 20 T^{2} + 22 T^{3} + 121 T^{4} \))(\( 1 - 6 T + 35 T^{2} - 112 T^{3} + 385 T^{4} - 726 T^{5} + 1331 T^{6} \))(\( 1 - 4 T + 36 T^{2} - 104 T^{3} + 534 T^{4} - 1144 T^{5} + 4356 T^{6} - 5324 T^{7} + 14641 T^{8} \))
$13$ (\( 1 - 4 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 + 6 T + 13 T^{2} \))(\( 1 + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 + 6 T + 13 T^{2} \))(\( 1 + 4 T + 18 T^{2} + 52 T^{3} + 169 T^{4} \))(\( ( 1 + 13 T^{2} )^{3} \))(\( 1 + 6 T + 44 T^{2} + 170 T^{3} + 774 T^{4} + 2210 T^{5} + 7436 T^{6} + 13182 T^{7} + 28561 T^{8} \))
$17$ (\( 1 + 2 T + 17 T^{2} \))(\( 1 + 5 T + 17 T^{2} \))(\( 1 + 3 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 - 3 T + 17 T^{2} \))(\( 1 + 3 T + 17 T^{2} \))(\( 1 - 7 T + 17 T^{2} \))(\( 1 - 3 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( ( 1 + 17 T^{2} )^{2} \))(\( 1 + 7 T + 59 T^{2} + 230 T^{3} + 1003 T^{4} + 2023 T^{5} + 4913 T^{6} \))(\( 1 + T + 10 T^{2} + 23 T^{3} - 86 T^{4} + 391 T^{5} + 2890 T^{6} + 4913 T^{7} + 83521 T^{8} \))
$19$ (\( 1 + 6 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 + 6 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 8 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 26 T^{2} + 361 T^{4} \))(\( 1 - 4 T + 29 T^{2} - 120 T^{3} + 551 T^{4} - 1444 T^{5} + 6859 T^{6} \))(\( 1 - 2 T + 56 T^{2} - 106 T^{3} + 1438 T^{4} - 2014 T^{5} + 20216 T^{6} - 13718 T^{7} + 130321 T^{8} \))
$23$ (\( 1 + 8 T + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 - 8 T + 23 T^{2} \))(\( 1 - 2 T + 23 T^{2} \))(\( 1 + 4 T + 23 T^{2} \))(\( 1 + 8 T + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 - 8 T + 23 T^{2} \))(\( ( 1 + 2 T + 23 T^{2} )^{2} \))(\( 1 - 4 T + 49 T^{2} - 120 T^{3} + 1127 T^{4} - 2116 T^{5} + 12167 T^{6} \))(\( 1 - 6 T + 84 T^{2} - 350 T^{3} + 2774 T^{4} - 8050 T^{5} + 44436 T^{6} - 73002 T^{7} + 279841 T^{8} \))
$29$ (\( 1 - 6 T + 29 T^{2} \))(\( 1 + 9 T + 29 T^{2} \))(\( 1 + 3 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 + 4 T + 29 T^{2} \))(\( 1 - T + 29 T^{2} \))(\( 1 + 5 T + 29 T^{2} \))(\( 1 - T + 29 T^{2} \))(\( 1 + 5 T + 29 T^{2} \))(\( 1 + 8 T + 29 T^{2} \))(\( 1 - 2 T - 16 T^{2} - 58 T^{3} + 841 T^{4} \))(\( 1 - 5 T + 89 T^{2} - 280 T^{3} + 2581 T^{4} - 4205 T^{5} + 24389 T^{6} \))(\( 1 - 17 T + 178 T^{2} - 1231 T^{3} + 7294 T^{4} - 35699 T^{5} + 149698 T^{6} - 414613 T^{7} + 707281 T^{8} \))
$31$ (\( 1 - 4 T + 31 T^{2} \))(\( 1 - 3 T + 31 T^{2} \))(\( 1 + T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( 1 - 9 T + 31 T^{2} \))(\( 1 + 3 T + 31 T^{2} \))(\( 1 - 5 T + 31 T^{2} \))(\( 1 - 7 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 + 4 T + 54 T^{2} + 124 T^{3} + 961 T^{4} \))(\( 1 - 7 T + 101 T^{2} - 426 T^{3} + 3131 T^{4} - 6727 T^{5} + 29791 T^{6} \))(\( 1 - 5 T + 68 T^{2} - 289 T^{3} + 2326 T^{4} - 8959 T^{5} + 65348 T^{6} - 148955 T^{7} + 923521 T^{8} \))
$37$ (\( 1 - 2 T + 37 T^{2} \))(\( 1 + 8 T + 37 T^{2} \))(\( 1 + 4 T + 37 T^{2} \))(\( 1 - 10 T + 37 T^{2} \))(\( 1 - 10 T + 37 T^{2} \))(\( 1 + 8 T + 37 T^{2} \))(\( 1 - 10 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 + 4 T + 30 T^{2} + 148 T^{3} + 1369 T^{4} \))(\( ( 1 - 2 T + 37 T^{2} )^{3} \))(\( 1 + 6 T + 120 T^{2} + 546 T^{3} + 6174 T^{4} + 20202 T^{5} + 164280 T^{6} + 303918 T^{7} + 1874161 T^{8} \))
$41$ (\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{3} \))(\( ( 1 - T )^{4} \))
$43$ (\( 1 + 43 T^{2} \))(\( 1 - 7 T + 43 T^{2} \))(\( 1 + T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 + 5 T + 43 T^{2} \))(\( 1 + 5 T + 43 T^{2} \))(\( 1 + 5 T + 43 T^{2} \))(\( 1 + T + 43 T^{2} \))(\( 1 + 8 T + 43 T^{2} \))(\( 1 - 4 T - 18 T^{2} - 172 T^{3} + 1849 T^{4} \))(\( 1 + 9 T + 81 T^{2} + 542 T^{3} + 3483 T^{4} + 16641 T^{5} + 79507 T^{6} \))(\( 1 + 13 T + 220 T^{2} + 1709 T^{3} + 15190 T^{4} + 73487 T^{5} + 406780 T^{6} + 1033591 T^{7} + 3418801 T^{8} \))
$47$ (\( 1 - 4 T + 47 T^{2} \))(\( 1 + 10 T + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 - 6 T + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 + 2 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 - 8 T + 98 T^{2} - 376 T^{3} + 2209 T^{4} \))(\( ( 1 - 2 T + 47 T^{2} )^{3} \))(\( 1 - 6 T + 108 T^{2} - 190 T^{3} + 4550 T^{4} - 8930 T^{5} + 238572 T^{6} - 622938 T^{7} + 4879681 T^{8} \))
$53$ (\( 1 - 2 T + 53 T^{2} \))(\( 1 + 13 T + 53 T^{2} \))(\( 1 - 9 T + 53 T^{2} \))(\( 1 + 4 T + 53 T^{2} \))(\( 1 + 8 T + 53 T^{2} \))(\( 1 + 3 T + 53 T^{2} \))(\( 1 + 9 T + 53 T^{2} \))(\( 1 + 3 T + 53 T^{2} \))(\( 1 + T + 53 T^{2} \))(\( 1 + 53 T^{2} \))(\( 1 + 2 T + 32 T^{2} + 106 T^{3} + 2809 T^{4} \))(\( 1 + 7 T + 113 T^{2} + 632 T^{3} + 5989 T^{4} + 19663 T^{5} + 148877 T^{6} \))(\( 1 - 29 T + 362 T^{2} - 2619 T^{3} + 16622 T^{4} - 138807 T^{5} + 1016858 T^{6} - 4317433 T^{7} + 7890481 T^{8} \))
$59$ (\( 1 - 14 T + 59 T^{2} \))(\( 1 + 14 T + 59 T^{2} \))(\( 1 + 6 T + 59 T^{2} \))(\( 1 + 8 T + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 - 14 T + 59 T^{2} \))(\( 1 + 10 T + 59 T^{2} \))(\( 1 + 10 T + 59 T^{2} \))(\( 1 - 10 T + 59 T^{2} \))(\( 1 - 2 T + 59 T^{2} \))(\( 1 - 2 T - 28 T^{2} - 118 T^{3} + 3481 T^{4} \))(\( 1 + 2 T + 127 T^{2} + 336 T^{3} + 7493 T^{4} + 6962 T^{5} + 205379 T^{6} \))(\( 1 - 16 T + 264 T^{2} - 2444 T^{3} + 23526 T^{4} - 144196 T^{5} + 918984 T^{6} - 3286064 T^{7} + 12117361 T^{8} \))
$61$ (\( 1 - 4 T + 61 T^{2} \))(\( 1 + 5 T + 61 T^{2} \))(\( 1 + T + 61 T^{2} \))(\( 1 - 10 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 + 11 T + 61 T^{2} \))(\( 1 - 13 T + 61 T^{2} \))(\( 1 + 3 T + 61 T^{2} \))(\( 1 + 13 T + 61 T^{2} \))(\( 1 + 8 T + 61 T^{2} \))(\( 1 - 10 T + 144 T^{2} - 610 T^{3} + 3721 T^{4} \))(\( 1 + 13 T + 209 T^{2} + 1596 T^{3} + 12749 T^{4} + 48373 T^{5} + 226981 T^{6} \))(\( 1 - 21 T + 276 T^{2} - 2979 T^{3} + 26922 T^{4} - 181719 T^{5} + 1026996 T^{6} - 4766601 T^{7} + 13845841 T^{8} \))
$67$ (\( 1 + 8 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 - 8 T + 67 T^{2} \))(\( 1 + 14 T + 67 T^{2} \))(\( 1 + 2 T + 67 T^{2} \))(\( 1 + 8 T + 67 T^{2} \))(\( 1 + 2 T + 67 T^{2} \))(\( 1 - 14 T + 67 T^{2} \))(\( 1 + 2 T + 67 T^{2} \))(\( 1 - 10 T + 67 T^{2} \))(\( 1 + 10 T + 132 T^{2} + 670 T^{3} + 4489 T^{4} \))(\( 1 - 22 T + 267 T^{2} - 2368 T^{3} + 17889 T^{4} - 98758 T^{5} + 300763 T^{6} \))(\( 1 - 4 T + 172 T^{2} - 384 T^{3} + 14822 T^{4} - 25728 T^{5} + 772108 T^{6} - 1203052 T^{7} + 20151121 T^{8} \))
$71$ (\( 1 - 8 T + 71 T^{2} \))(\( 1 + 15 T + 71 T^{2} \))(\( 1 - 3 T + 71 T^{2} \))(\( 1 + 12 T + 71 T^{2} \))(\( 1 - 12 T + 71 T^{2} \))(\( 1 - 3 T + 71 T^{2} \))(\( 1 - 9 T + 71 T^{2} \))(\( 1 - 3 T + 71 T^{2} \))(\( 1 + 3 T + 71 T^{2} \))(\( 1 + 12 T + 71 T^{2} \))(\( 1 - 12 T + 166 T^{2} - 852 T^{3} + 5041 T^{4} \))(\( 1 - 7 T + 109 T^{2} - 666 T^{3} + 7739 T^{4} - 35287 T^{5} + 357911 T^{6} \))(\( 1 - 17 T + 84 T^{2} + 1019 T^{3} - 16170 T^{4} + 72349 T^{5} + 423444 T^{6} - 6084487 T^{7} + 25411681 T^{8} \))
$73$ (\( 1 + 14 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 + 14 T + 73 T^{2} \))(\( 1 - 4 T + 73 T^{2} \))(\( 1 - 8 T + 73 T^{2} \))(\( 1 - 4 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 + 4 T + 102 T^{2} + 292 T^{3} + 5329 T^{4} \))(\( 1 + 28 T + 447 T^{2} + 4600 T^{3} + 32631 T^{4} + 149212 T^{5} + 389017 T^{6} \))(\( 1 - 12 T + 116 T^{2} - 276 T^{3} + 3382 T^{4} - 20148 T^{5} + 618164 T^{6} - 4668204 T^{7} + 28398241 T^{8} \))
$79$ (\( 1 + 8 T + 79 T^{2} \))(\( 1 + 3 T + 79 T^{2} \))(\( 1 + T + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 - 7 T + 79 T^{2} \))(\( 1 + 11 T + 79 T^{2} \))(\( 1 - 7 T + 79 T^{2} \))(\( 1 + 15 T + 79 T^{2} \))(\( 1 - 16 T + 79 T^{2} \))(\( ( 1 + 79 T^{2} )^{2} \))(\( 1 - 19 T + 333 T^{2} - 3130 T^{3} + 26307 T^{4} - 118579 T^{5} + 493039 T^{6} \))(\( 1 + 27 T + 548 T^{2} + 7039 T^{3} + 74358 T^{4} + 556081 T^{5} + 3420068 T^{6} + 13312053 T^{7} + 38950081 T^{8} \))
$83$ (\( 1 - 6 T + 83 T^{2} \))(\( 1 - 16 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 - 4 T + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 - 16 T + 83 T^{2} \))(\( 1 + 14 T + 83 T^{2} \))(\( 1 + 2 T + 83 T^{2} \))(\( 1 + 6 T + 83 T^{2} \))(\( 1 + 2 T + 83 T^{2} \))(\( 1 - 6 T + 172 T^{2} - 498 T^{3} + 6889 T^{4} \))(\( 1 + 14 T + 295 T^{2} + 2304 T^{3} + 24485 T^{4} + 96446 T^{5} + 571787 T^{6} \))(\( 1 + 18 T + 412 T^{2} + 4590 T^{3} + 54630 T^{4} + 380970 T^{5} + 2838268 T^{6} + 10292166 T^{7} + 47458321 T^{8} \))
$89$ (\( 1 - 10 T + 89 T^{2} \))(\( 1 - 13 T + 89 T^{2} \))(\( 1 - 3 T + 89 T^{2} \))(\( 1 + 14 T + 89 T^{2} \))(\( 1 - 18 T + 89 T^{2} \))(\( 1 - 5 T + 89 T^{2} \))(\( 1 + T + 89 T^{2} \))(\( 1 - 5 T + 89 T^{2} \))(\( 1 + 15 T + 89 T^{2} \))(\( 1 + 10 T + 89 T^{2} \))(\( 1 - 12 T + 166 T^{2} - 1068 T^{3} + 7921 T^{4} \))(\( 1 - 23 T + 419 T^{2} - 4330 T^{3} + 37291 T^{4} - 182183 T^{5} + 704969 T^{6} \))(\( 1 - 37 T + 814 T^{2} - 11891 T^{3} + 130434 T^{4} - 1058299 T^{5} + 6447694 T^{6} - 26083853 T^{7} + 62742241 T^{8} \))
$97$ (\( 1 - 6 T + 97 T^{2} \))(\( 1 - 17 T + 97 T^{2} \))(\( 1 + T + 97 T^{2} \))(\( 1 + 10 T + 97 T^{2} \))(\( 1 + 10 T + 97 T^{2} \))(\( 1 - T + 97 T^{2} \))(\( 1 - 7 T + 97 T^{2} \))(\( 1 - 5 T + 97 T^{2} \))(\( 1 + 7 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( ( 1 - 2 T + 97 T^{2} )^{2} \))(\( 1 + 25 T + 475 T^{2} + 5254 T^{3} + 46075 T^{4} + 235225 T^{5} + 912673 T^{6} \))(\( 1 + 3 T + 290 T^{2} + 621 T^{3} + 37786 T^{4} + 60237 T^{5} + 2728610 T^{6} + 2738019 T^{7} + 88529281 T^{8} \))
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