Properties

Label 575.2.a
Level 575
Weight 2
Character orbit a
Rep. character \(\chi_{575}(1,\cdot)\)
Character field \(\Q\)
Dimension 35
Newform subspaces 12
Sturm bound 120
Trace bound 3

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Defining parameters

Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 575.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 12 \)
Sturm bound: \(120\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(575))\).

Total New Old
Modular forms 66 35 31
Cusp forms 55 35 20
Eisenstein series 11 0 11

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

\(5\)\(23\)FrickeDim.
\(+\)\(+\)\(+\)\(4\)
\(+\)\(-\)\(-\)\(13\)
\(-\)\(+\)\(-\)\(12\)
\(-\)\(-\)\(+\)\(6\)
Plus space\(+\)\(10\)
Minus space\(-\)\(25\)

Trace form

\( 35q + 4q^{3} + 36q^{4} + 3q^{6} + 2q^{7} - 3q^{8} + 37q^{9} + O(q^{10}) \) \( 35q + 4q^{3} + 36q^{4} + 3q^{6} + 2q^{7} - 3q^{8} + 37q^{9} - 2q^{11} + 17q^{12} + 4q^{13} + 22q^{16} - 4q^{17} - q^{18} + 20q^{19} + 6q^{21} + 10q^{22} + 3q^{23} - 2q^{24} - 13q^{26} + 4q^{27} + 14q^{28} + 4q^{29} + 12q^{31} - 6q^{32} - 10q^{33} + 30q^{34} + 3q^{36} - 4q^{37} - 30q^{38} - 16q^{39} + 8q^{41} - 8q^{42} + 12q^{43} - 16q^{44} - 2q^{46} - 16q^{47} + 49q^{48} + 55q^{49} + 10q^{51} + 35q^{52} - 10q^{53} + 9q^{54} - 18q^{56} - 43q^{58} + 4q^{59} + 34q^{61} + 19q^{62} + 8q^{63} - 23q^{64} - 78q^{66} + 2q^{67} - 28q^{68} - 4q^{69} - 56q^{71} - 51q^{72} + 4q^{73} - 44q^{74} + 50q^{76} + 4q^{77} - 11q^{78} + 20q^{79} + 67q^{81} - 5q^{82} + 38q^{83} - 82q^{84} - 16q^{86} + 8q^{87} - 22q^{88} + 10q^{89} + 70q^{91} + 6q^{92} + 42q^{93} - 49q^{94} - 61q^{96} - 28q^{98} - 36q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(575))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 5 23
575.2.a.a \(1\) \(4.591\) \(\Q\) None \(-2\) \(-2\) \(0\) \(-1\) \(-\) \(-\) \(q-2q^{2}-2q^{3}+2q^{4}+4q^{6}-q^{7}+\cdots\)
575.2.a.b \(1\) \(4.591\) \(\Q\) None \(-2\) \(0\) \(0\) \(-1\) \(+\) \(+\) \(q-2q^{2}+2q^{4}-q^{7}-3q^{9}+2q^{11}+\cdots\)
575.2.a.c \(1\) \(4.591\) \(\Q\) None \(-1\) \(0\) \(0\) \(1\) \(-\) \(-\) \(q-q^{2}-q^{4}+q^{7}+3q^{8}-3q^{9}-q^{11}+\cdots\)
575.2.a.d \(1\) \(4.591\) \(\Q\) None \(1\) \(0\) \(0\) \(-1\) \(+\) \(+\) \(q+q^{2}-q^{4}-q^{7}-3q^{8}-3q^{9}-q^{11}+\cdots\)
575.2.a.e \(1\) \(4.591\) \(\Q\) None \(2\) \(2\) \(0\) \(1\) \(-\) \(+\) \(q+2q^{2}+2q^{3}+2q^{4}+4q^{6}+q^{7}+\cdots\)
575.2.a.f \(2\) \(4.591\) \(\Q(\sqrt{5}) \) None \(1\) \(0\) \(0\) \(-2\) \(+\) \(+\) \(q+\beta q^{2}+(1-2\beta )q^{3}+(-1+\beta )q^{4}+\cdots\)
575.2.a.g \(2\) \(4.591\) \(\Q(\sqrt{5}) \) None \(3\) \(2\) \(0\) \(2\) \(+\) \(-\) \(q+(1+\beta )q^{2}+q^{3}+3\beta q^{4}+(1+\beta )q^{6}+\cdots\)
575.2.a.h \(4\) \(4.591\) 4.4.15317.1 None \(-2\) \(2\) \(0\) \(3\) \(+\) \(-\) \(q-\beta _{1}q^{2}+(1+\beta _{2})q^{3}+(1+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
575.2.a.i \(4\) \(4.591\) 4.4.5744.1 None \(0\) \(-2\) \(0\) \(-6\) \(-\) \(-\) \(q-\beta _{1}q^{2}+(-1+\beta _{1}+\beta _{2})q^{3}+(\beta _{1}+\cdots)q^{4}+\cdots\)
575.2.a.j \(4\) \(4.591\) 4.4.5744.1 None \(0\) \(2\) \(0\) \(6\) \(-\) \(+\) \(q+\beta _{1}q^{2}+(1-\beta _{1}-\beta _{2})q^{3}+(\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
575.2.a.k \(7\) \(4.591\) \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-1\) \(0\) \(0\) \(3\) \(-\) \(+\) \(q-\beta _{1}q^{2}+\beta _{5}q^{3}+(2+\beta _{2})q^{4}+(1-\beta _{1}+\cdots)q^{6}+\cdots\)
575.2.a.l \(7\) \(4.591\) \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(1\) \(0\) \(0\) \(-3\) \(+\) \(-\) \(q+\beta _{1}q^{2}-\beta _{5}q^{3}+(2+\beta _{2})q^{4}+(1-\beta _{1}+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(575))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(575)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(115))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 2 T^{2} \))(\( 1 + 2 T + 2 T^{2} \))(\( 1 + T + 2 T^{2} \))(\( 1 - T + 2 T^{2} \))(\( 1 - 2 T + 2 T^{2} \))(\( 1 - T + 3 T^{2} - 2 T^{3} + 4 T^{4} \))(\( 1 - 3 T + 5 T^{2} - 6 T^{3} + 4 T^{4} \))(\( 1 + 2 T + 4 T^{2} + 7 T^{3} + 10 T^{4} + 14 T^{5} + 16 T^{6} + 16 T^{7} + 16 T^{8} \))(\( 1 + 3 T^{2} + 2 T^{3} + 5 T^{4} + 4 T^{5} + 12 T^{6} + 16 T^{8} \))(\( 1 + 3 T^{2} - 2 T^{3} + 5 T^{4} - 4 T^{5} + 12 T^{6} + 16 T^{8} \))(\( 1 + T + 2 T^{2} + 3 T^{3} + 7 T^{4} + 2 T^{5} + 9 T^{6} + 9 T^{7} + 18 T^{8} + 8 T^{9} + 56 T^{10} + 48 T^{11} + 64 T^{12} + 64 T^{13} + 128 T^{14} \))(\( 1 - T + 2 T^{2} - 3 T^{3} + 7 T^{4} - 2 T^{5} + 9 T^{6} - 9 T^{7} + 18 T^{8} - 8 T^{9} + 56 T^{10} - 48 T^{11} + 64 T^{12} - 64 T^{13} + 128 T^{14} \))
$3$ (\( 1 + 2 T + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 - 2 T + 3 T^{2} \))(\( 1 + T^{2} + 9 T^{4} \))(\( ( 1 - T + 3 T^{2} )^{2} \))(\( ( 1 - T + 2 T^{2} - 3 T^{3} + 9 T^{4} )^{2} \))(\( 1 + 2 T + 6 T^{2} + 12 T^{3} + 23 T^{4} + 36 T^{5} + 54 T^{6} + 54 T^{7} + 81 T^{8} \))(\( 1 - 2 T + 6 T^{2} - 12 T^{3} + 23 T^{4} - 36 T^{5} + 54 T^{6} - 54 T^{7} + 81 T^{8} \))(\( 1 + 3 T^{2} + 4 T^{4} - 8 T^{5} + 25 T^{6} - 28 T^{7} + 75 T^{8} - 72 T^{9} + 108 T^{10} + 729 T^{12} + 2187 T^{14} \))(\( 1 + 3 T^{2} + 4 T^{4} + 8 T^{5} + 25 T^{6} + 28 T^{7} + 75 T^{8} + 72 T^{9} + 108 T^{10} + 729 T^{12} + 2187 T^{14} \))
$5$ 1
$7$ (\( 1 + T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 + 2 T + 10 T^{2} + 14 T^{3} + 49 T^{4} \))(\( 1 - 2 T + 10 T^{2} - 14 T^{3} + 49 T^{4} \))(\( 1 - 3 T + 14 T^{2} - 11 T^{3} + 66 T^{4} - 77 T^{5} + 686 T^{6} - 1029 T^{7} + 2401 T^{8} \))(\( 1 + 6 T + 32 T^{2} + 114 T^{3} + 346 T^{4} + 798 T^{5} + 1568 T^{6} + 2058 T^{7} + 2401 T^{8} \))(\( 1 - 6 T + 32 T^{2} - 114 T^{3} + 346 T^{4} - 798 T^{5} + 1568 T^{6} - 2058 T^{7} + 2401 T^{8} \))(\( 1 - 3 T + 9 T^{2} - 14 T^{3} + 45 T^{4} + 35 T^{5} - 171 T^{6} + 1276 T^{7} - 1197 T^{8} + 1715 T^{9} + 15435 T^{10} - 33614 T^{11} + 151263 T^{12} - 352947 T^{13} + 823543 T^{14} \))(\( 1 + 3 T + 9 T^{2} + 14 T^{3} + 45 T^{4} - 35 T^{5} - 171 T^{6} - 1276 T^{7} - 1197 T^{8} - 1715 T^{9} + 15435 T^{10} + 33614 T^{11} + 151263 T^{12} + 352947 T^{13} + 823543 T^{14} \))
$11$ (\( 1 + 11 T^{2} \))(\( 1 - 2 T + 11 T^{2} \))(\( 1 + T + 11 T^{2} \))(\( 1 + T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 6 T + 26 T^{2} + 66 T^{3} + 121 T^{4} \))(\( 1 + 2 T + 18 T^{2} + 22 T^{3} + 121 T^{4} \))(\( 1 - 4 T + 28 T^{2} - 92 T^{3} + 406 T^{4} - 1012 T^{5} + 3388 T^{6} - 5324 T^{7} + 14641 T^{8} \))(\( 1 - 2 T + 28 T^{2} - 22 T^{3} + 346 T^{4} - 242 T^{5} + 3388 T^{6} - 2662 T^{7} + 14641 T^{8} \))(\( 1 - 2 T + 28 T^{2} - 22 T^{3} + 346 T^{4} - 242 T^{5} + 3388 T^{6} - 2662 T^{7} + 14641 T^{8} \))(\( 1 + T + 19 T^{2} - 18 T^{3} + 247 T^{4} - 153 T^{5} + 3637 T^{6} - 1148 T^{7} + 40007 T^{8} - 18513 T^{9} + 328757 T^{10} - 263538 T^{11} + 3059969 T^{12} + 1771561 T^{13} + 19487171 T^{14} \))(\( 1 + T + 19 T^{2} - 18 T^{3} + 247 T^{4} - 153 T^{5} + 3637 T^{6} - 1148 T^{7} + 40007 T^{8} - 18513 T^{9} + 328757 T^{10} - 263538 T^{11} + 3059969 T^{12} + 1771561 T^{13} + 19487171 T^{14} \))
$13$ (\( 1 + 2 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 - T + 13 T^{2} \))(\( 1 + T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( ( 1 + 3 T + 13 T^{2} )^{2} \))(\( 1 - 8 T + 37 T^{2} - 104 T^{3} + 169 T^{4} \))(\( 1 + 11 T^{2} + 160 T^{4} + 1859 T^{6} + 28561 T^{8} \))(\( 1 + 14 T + 118 T^{2} + 664 T^{3} + 2791 T^{4} + 8632 T^{5} + 19942 T^{6} + 30758 T^{7} + 28561 T^{8} \))(\( 1 - 14 T + 118 T^{2} - 664 T^{3} + 2791 T^{4} - 8632 T^{5} + 19942 T^{6} - 30758 T^{7} + 28561 T^{8} \))(\( 1 + 3 T + 45 T^{2} + 158 T^{3} + 1198 T^{4} + 4246 T^{5} + 21399 T^{6} + 68537 T^{7} + 278187 T^{8} + 717574 T^{9} + 2632006 T^{10} + 4512638 T^{11} + 16708185 T^{12} + 14480427 T^{13} + 62748517 T^{14} \))(\( 1 - 3 T + 45 T^{2} - 158 T^{3} + 1198 T^{4} - 4246 T^{5} + 21399 T^{6} - 68537 T^{7} + 278187 T^{8} - 717574 T^{9} + 2632006 T^{10} - 4512638 T^{11} + 16708185 T^{12} - 14480427 T^{13} + 62748517 T^{14} \))
$17$ (\( 1 - 5 T + 17 T^{2} \))(\( 1 + 3 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 5 T + 17 T^{2} \))(\( 1 + 6 T + 38 T^{2} + 102 T^{3} + 289 T^{4} \))(\( 1 - 4 T + 18 T^{2} - 68 T^{3} + 289 T^{4} \))(\( 1 - T + 50 T^{2} - 27 T^{3} + 1154 T^{4} - 459 T^{5} + 14450 T^{6} - 4913 T^{7} + 83521 T^{8} \))(\( 1 + 14 T + 126 T^{2} + 778 T^{3} + 3726 T^{4} + 13226 T^{5} + 36414 T^{6} + 68782 T^{7} + 83521 T^{8} \))(\( 1 - 14 T + 126 T^{2} - 778 T^{3} + 3726 T^{4} - 13226 T^{5} + 36414 T^{6} - 68782 T^{7} + 83521 T^{8} \))(\( 1 - 10 T + 83 T^{2} - 508 T^{3} + 3281 T^{4} - 17046 T^{5} + 82619 T^{6} - 338120 T^{7} + 1404523 T^{8} - 4926294 T^{9} + 16119553 T^{10} - 42428668 T^{11} + 117848131 T^{12} - 241375690 T^{13} + 410338673 T^{14} \))(\( 1 + 10 T + 83 T^{2} + 508 T^{3} + 3281 T^{4} + 17046 T^{5} + 82619 T^{6} + 338120 T^{7} + 1404523 T^{8} + 4926294 T^{9} + 16119553 T^{10} + 42428668 T^{11} + 117848131 T^{12} + 241375690 T^{13} + 410338673 T^{14} \))
$19$ (\( 1 - 8 T + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))(\( 1 + 5 T + 19 T^{2} \))(\( 1 + 5 T + 19 T^{2} \))(\( 1 - 8 T + 19 T^{2} \))(\( ( 1 + 2 T + 19 T^{2} )^{2} \))(\( 1 - 2 T - 6 T^{2} - 38 T^{3} + 361 T^{4} \))(\( 1 + 4 T + 60 T^{2} + 188 T^{3} + 1590 T^{4} + 3572 T^{5} + 21660 T^{6} + 27436 T^{7} + 130321 T^{8} \))(\( 1 + 4 T + 70 T^{2} + 200 T^{3} + 1918 T^{4} + 3800 T^{5} + 25270 T^{6} + 27436 T^{7} + 130321 T^{8} \))(\( 1 + 4 T + 70 T^{2} + 200 T^{3} + 1918 T^{4} + 3800 T^{5} + 25270 T^{6} + 27436 T^{7} + 130321 T^{8} \))(\( 1 - 15 T + 163 T^{2} - 1158 T^{3} + 6863 T^{4} - 31817 T^{5} + 140765 T^{6} - 580340 T^{7} + 2674535 T^{8} - 11485937 T^{9} + 47073317 T^{10} - 150911718 T^{11} + 403604137 T^{12} - 705688215 T^{13} + 893871739 T^{14} \))(\( 1 - 15 T + 163 T^{2} - 1158 T^{3} + 6863 T^{4} - 31817 T^{5} + 140765 T^{6} - 580340 T^{7} + 2674535 T^{8} - 11485937 T^{9} + 47073317 T^{10} - 150911718 T^{11} + 403604137 T^{12} - 705688215 T^{13} + 893871739 T^{14} \))
$23$ (\( 1 - T \))(\( 1 + T \))(\( 1 - T \))(\( 1 + T \))(\( 1 + T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{4} \))(\( ( 1 - T )^{4} \))(\( ( 1 + T )^{4} \))(\( ( 1 + T )^{7} \))(\( ( 1 - T )^{7} \))
$29$ (\( 1 + 5 T + 29 T^{2} \))(\( 1 - 7 T + 29 T^{2} \))(\( 1 + 5 T + 29 T^{2} \))(\( 1 + 5 T + 29 T^{2} \))(\( 1 + 5 T + 29 T^{2} \))(\( ( 1 + 3 T + 29 T^{2} )^{2} \))(\( 1 + 10 T + 63 T^{2} + 290 T^{3} + 841 T^{4} \))(\( 1 - 19 T + 233 T^{2} - 1922 T^{3} + 12034 T^{4} - 55738 T^{5} + 195953 T^{6} - 463391 T^{7} + 707281 T^{8} \))(\( 1 - 4 T + 94 T^{2} - 344 T^{3} + 3775 T^{4} - 9976 T^{5} + 79054 T^{6} - 97556 T^{7} + 707281 T^{8} \))(\( 1 - 4 T + 94 T^{2} - 344 T^{3} + 3775 T^{4} - 9976 T^{5} + 79054 T^{6} - 97556 T^{7} + 707281 T^{8} \))(\( 1 - 3 T + 81 T^{2} - 226 T^{3} + 2202 T^{4} - 6878 T^{5} + 11647 T^{6} - 168501 T^{7} + 337763 T^{8} - 5784398 T^{9} + 53704578 T^{10} - 159845506 T^{11} + 1661403069 T^{12} - 1784469963 T^{13} + 17249876309 T^{14} \))(\( 1 - 3 T + 81 T^{2} - 226 T^{3} + 2202 T^{4} - 6878 T^{5} + 11647 T^{6} - 168501 T^{7} + 337763 T^{8} - 5784398 T^{9} + 53704578 T^{10} - 159845506 T^{11} + 1661403069 T^{12} - 1784469963 T^{13} + 17249876309 T^{14} \))
$31$ (\( 1 + 5 T + 31 T^{2} \))(\( 1 + 5 T + 31 T^{2} \))(\( 1 + 2 T + 31 T^{2} \))(\( 1 + 2 T + 31 T^{2} \))(\( 1 + 5 T + 31 T^{2} \))(\( 1 + 17 T^{2} + 961 T^{4} \))(\( 1 - 4 T + 61 T^{2} - 124 T^{3} + 961 T^{4} \))(\( 1 + T + 23 T^{2} + 104 T^{3} + 1648 T^{4} + 3224 T^{5} + 22103 T^{6} + 29791 T^{7} + 923521 T^{8} \))(\( 1 + 50 T^{2} - 256 T^{3} + 1011 T^{4} - 7936 T^{5} + 48050 T^{6} + 923521 T^{8} \))(\( 1 + 50 T^{2} - 256 T^{3} + 1011 T^{4} - 7936 T^{5} + 48050 T^{6} + 923521 T^{8} \))(\( 1 - 14 T + 231 T^{2} - 2080 T^{3} + 20508 T^{4} - 140424 T^{5} + 1028881 T^{6} - 5567026 T^{7} + 31895311 T^{8} - 134947464 T^{9} + 610953828 T^{10} - 1920923680 T^{11} + 6613333881 T^{12} - 12425051534 T^{13} + 27512614111 T^{14} \))(\( 1 - 14 T + 231 T^{2} - 2080 T^{3} + 20508 T^{4} - 140424 T^{5} + 1028881 T^{6} - 5567026 T^{7} + 31895311 T^{8} - 134947464 T^{9} + 610953828 T^{10} - 1920923680 T^{11} + 6613333881 T^{12} - 12425051534 T^{13} + 27512614111 T^{14} \))
$37$ (\( 1 + 7 T + 37 T^{2} \))(\( 1 + 11 T + 37 T^{2} \))(\( 1 + 4 T + 37 T^{2} \))(\( 1 - 4 T + 37 T^{2} \))(\( 1 - 7 T + 37 T^{2} \))(\( 1 + 2 T + 70 T^{2} + 74 T^{3} + 1369 T^{4} \))(\( 1 - 6 T + 38 T^{2} - 222 T^{3} + 1369 T^{4} \))(\( 1 - 3 T + 32 T^{2} - 349 T^{3} + 1638 T^{4} - 12913 T^{5} + 43808 T^{6} - 151959 T^{7} + 1874161 T^{8} \))(\( 1 + 2 T + 76 T^{2} - 158 T^{3} + 2410 T^{4} - 5846 T^{5} + 104044 T^{6} + 101306 T^{7} + 1874161 T^{8} \))(\( 1 - 2 T + 76 T^{2} + 158 T^{3} + 2410 T^{4} + 5846 T^{5} + 104044 T^{6} - 101306 T^{7} + 1874161 T^{8} \))(\( 1 + 10 T + 143 T^{2} + 1100 T^{3} + 8905 T^{4} + 53670 T^{5} + 340191 T^{6} + 1982056 T^{7} + 12587067 T^{8} + 73474230 T^{9} + 451064965 T^{10} + 2061577100 T^{11} + 9916185851 T^{12} + 25657264090 T^{13} + 94931877133 T^{14} \))(\( 1 - 10 T + 143 T^{2} - 1100 T^{3} + 8905 T^{4} - 53670 T^{5} + 340191 T^{6} - 1982056 T^{7} + 12587067 T^{8} - 73474230 T^{9} + 451064965 T^{10} - 2061577100 T^{11} + 9916185851 T^{12} - 25657264090 T^{13} + 94931877133 T^{14} \))
$41$ (\( 1 + 7 T + 41 T^{2} \))(\( 1 - T + 41 T^{2} \))(\( 1 + 5 T + 41 T^{2} \))(\( 1 + 5 T + 41 T^{2} \))(\( 1 + 7 T + 41 T^{2} \))(\( 1 - 2 T + 63 T^{2} - 82 T^{3} + 1681 T^{4} \))(\( 1 + 6 T + 71 T^{2} + 246 T^{3} + 1681 T^{4} \))(\( 1 - 13 T + 209 T^{2} - 1602 T^{3} + 13682 T^{4} - 65682 T^{5} + 351329 T^{6} - 895973 T^{7} + 2825761 T^{8} \))(\( 1 + 8 T + 70 T^{2} + 616 T^{3} + 4863 T^{4} + 25256 T^{5} + 117670 T^{6} + 551368 T^{7} + 2825761 T^{8} \))(\( 1 + 8 T + 70 T^{2} + 616 T^{3} + 4863 T^{4} + 25256 T^{5} + 117670 T^{6} + 551368 T^{7} + 2825761 T^{8} \))(\( 1 - 19 T + 309 T^{2} - 3190 T^{3} + 30654 T^{4} - 228554 T^{5} + 1692151 T^{6} - 10621429 T^{7} + 69378191 T^{8} - 384199274 T^{9} + 2112704334 T^{10} - 9014177590 T^{11} + 35799566109 T^{12} - 90251980579 T^{13} + 194754273881 T^{14} \))(\( 1 - 19 T + 309 T^{2} - 3190 T^{3} + 30654 T^{4} - 228554 T^{5} + 1692151 T^{6} - 10621429 T^{7} + 69378191 T^{8} - 384199274 T^{9} + 2112704334 T^{10} - 9014177590 T^{11} + 35799566109 T^{12} - 90251980579 T^{13} + 194754273881 T^{14} \))
$43$ (\( 1 + 4 T + 43 T^{2} \))(\( 1 + 43 T^{2} \))(\( 1 + 9 T + 43 T^{2} \))(\( 1 - 9 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( ( 1 + 43 T^{2} )^{2} \))(\( 1 - 6 T + 50 T^{2} - 258 T^{3} + 1849 T^{4} \))(\( 1 - 6 T + 136 T^{2} - 758 T^{3} + 8126 T^{4} - 32594 T^{5} + 251464 T^{6} - 477042 T^{7} + 3418801 T^{8} \))(\( 1 + 4 T + 54 T^{2} + 80 T^{3} + 2910 T^{4} + 3440 T^{5} + 99846 T^{6} + 318028 T^{7} + 3418801 T^{8} \))(\( 1 - 4 T + 54 T^{2} - 80 T^{3} + 2910 T^{4} - 3440 T^{5} + 99846 T^{6} - 318028 T^{7} + 3418801 T^{8} \))(\( 1 - 5 T + 157 T^{2} - 826 T^{3} + 13117 T^{4} - 70451 T^{5} + 759257 T^{6} - 3813708 T^{7} + 32648051 T^{8} - 130263899 T^{9} + 1042893319 T^{10} - 2823929626 T^{11} + 23080325551 T^{12} - 31606815245 T^{13} + 271818611107 T^{14} \))(\( 1 + 5 T + 157 T^{2} + 826 T^{3} + 13117 T^{4} + 70451 T^{5} + 759257 T^{6} + 3813708 T^{7} + 32648051 T^{8} + 130263899 T^{9} + 1042893319 T^{10} + 2823929626 T^{11} + 23080325551 T^{12} + 31606815245 T^{13} + 271818611107 T^{14} \))
$47$ (\( 1 - 2 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 - 6 T + 47 T^{2} \))(\( 1 + 2 T + 47 T^{2} \))(\( 1 + 89 T^{2} + 2209 T^{4} \))(\( 1 + 10 T + 99 T^{2} + 470 T^{3} + 2209 T^{4} \))(\( 1 + 6 T + 105 T^{2} + 298 T^{3} + 5324 T^{4} + 14006 T^{5} + 231945 T^{6} + 622938 T^{7} + 4879681 T^{8} \))(\( 1 + 2 T + 50 T^{2} + 116 T^{3} + 4795 T^{4} + 5452 T^{5} + 110450 T^{6} + 207646 T^{7} + 4879681 T^{8} \))(\( 1 - 2 T + 50 T^{2} - 116 T^{3} + 4795 T^{4} - 5452 T^{5} + 110450 T^{6} - 207646 T^{7} + 4879681 T^{8} \))(\( 1 + 14 T + 311 T^{2} + 2944 T^{3} + 38680 T^{4} + 282800 T^{5} + 2790681 T^{6} + 16514822 T^{7} + 131162007 T^{8} + 624705200 T^{9} + 4015873640 T^{10} + 14365780864 T^{11} + 71326297177 T^{12} + 150909014606 T^{13} + 506623120463 T^{14} \))(\( 1 - 14 T + 311 T^{2} - 2944 T^{3} + 38680 T^{4} - 282800 T^{5} + 2790681 T^{6} - 16514822 T^{7} + 131162007 T^{8} - 624705200 T^{9} + 4015873640 T^{10} - 14365780864 T^{11} + 71326297177 T^{12} - 150909014606 T^{13} + 506623120463 T^{14} \))
$53$ (\( 1 - T + 53 T^{2} \))(\( 1 + 11 T + 53 T^{2} \))(\( 1 - 2 T + 53 T^{2} \))(\( 1 + 2 T + 53 T^{2} \))(\( 1 + T + 53 T^{2} \))(\( 1 - 8 T + 102 T^{2} - 424 T^{3} + 2809 T^{4} \))(\( ( 1 - 6 T + 53 T^{2} )^{2} \))(\( 1 + 19 T + 178 T^{2} + 929 T^{3} + 4474 T^{4} + 49237 T^{5} + 500002 T^{6} + 2828663 T^{7} + 7890481 T^{8} \))(\( 1 + 4 T + 108 T^{2} + 524 T^{3} + 8022 T^{4} + 27772 T^{5} + 303372 T^{6} + 595508 T^{7} + 7890481 T^{8} \))(\( 1 - 4 T + 108 T^{2} - 524 T^{3} + 8022 T^{4} - 27772 T^{5} + 303372 T^{6} - 595508 T^{7} + 7890481 T^{8} \))(\( 1 - 4 T + 247 T^{2} - 1232 T^{3} + 28177 T^{4} - 159580 T^{5} + 2044527 T^{6} - 11188576 T^{7} + 108359931 T^{8} - 448260220 T^{9} + 4194907229 T^{10} - 9721072592 T^{11} + 103294286771 T^{12} - 88657444516 T^{13} + 1174711139837 T^{14} \))(\( 1 + 4 T + 247 T^{2} + 1232 T^{3} + 28177 T^{4} + 159580 T^{5} + 2044527 T^{6} + 11188576 T^{7} + 108359931 T^{8} + 448260220 T^{9} + 4194907229 T^{10} + 9721072592 T^{11} + 103294286771 T^{12} + 88657444516 T^{13} + 1174711139837 T^{14} \))
$59$ (\( 1 - 3 T + 59 T^{2} \))(\( 1 + 13 T + 59 T^{2} \))(\( 1 - 8 T + 59 T^{2} \))(\( 1 - 8 T + 59 T^{2} \))(\( 1 - 3 T + 59 T^{2} \))(\( 1 - 4 T + 102 T^{2} - 236 T^{3} + 3481 T^{4} \))(\( 1 + 38 T^{2} + 3481 T^{4} \))(\( 1 - 23 T + 336 T^{2} - 3511 T^{3} + 29550 T^{4} - 207149 T^{5} + 1169616 T^{6} - 4723717 T^{7} + 12117361 T^{8} \))(\( 1 + 216 T^{2} + 16 T^{3} + 18542 T^{4} + 944 T^{5} + 751896 T^{6} + 12117361 T^{8} \))(\( 1 + 216 T^{2} + 16 T^{3} + 18542 T^{4} + 944 T^{5} + 751896 T^{6} + 12117361 T^{8} \))(\( 1 + 16 T + 322 T^{2} + 2460 T^{3} + 24083 T^{4} + 27416 T^{5} + 111413 T^{6} - 7169964 T^{7} + 6573367 T^{8} + 95435096 T^{9} + 4946142457 T^{10} + 29808708060 T^{11} + 230205624278 T^{12} + 674888538256 T^{13} + 2488651484819 T^{14} \))(\( 1 + 16 T + 322 T^{2} + 2460 T^{3} + 24083 T^{4} + 27416 T^{5} + 111413 T^{6} - 7169964 T^{7} + 6573367 T^{8} + 95435096 T^{9} + 4946142457 T^{10} + 29808708060 T^{11} + 230205624278 T^{12} + 674888538256 T^{13} + 2488651484819 T^{14} \))
$61$ (\( 1 + 6 T + 61 T^{2} \))(\( 1 + 8 T + 61 T^{2} \))(\( 1 + 8 T + 61 T^{2} \))(\( 1 + 8 T + 61 T^{2} \))(\( 1 + 6 T + 61 T^{2} \))(\( 1 - 4 T + 46 T^{2} - 244 T^{3} + 3721 T^{4} \))(\( 1 - 2 T - 2 T^{2} - 122 T^{3} + 3721 T^{4} \))(\( 1 + 188 T^{2} + 136 T^{3} + 15462 T^{4} + 8296 T^{5} + 699548 T^{6} + 13845841 T^{8} \))(\( 1 + 8 T + 142 T^{2} + 316 T^{3} + 7126 T^{4} + 19276 T^{5} + 528382 T^{6} + 1815848 T^{7} + 13845841 T^{8} \))(\( 1 + 8 T + 142 T^{2} + 316 T^{3} + 7126 T^{4} + 19276 T^{5} + 528382 T^{6} + 1815848 T^{7} + 13845841 T^{8} \))(\( 1 - 40 T + 971 T^{2} - 17008 T^{3} + 237341 T^{4} - 2736312 T^{5} + 26811983 T^{6} - 225335392 T^{7} + 1635530963 T^{8} - 10181816952 T^{9} + 53871897521 T^{10} - 235490063728 T^{11} + 820103008271 T^{12} - 2060814974440 T^{13} + 3142742836021 T^{14} \))(\( 1 - 40 T + 971 T^{2} - 17008 T^{3} + 237341 T^{4} - 2736312 T^{5} + 26811983 T^{6} - 225335392 T^{7} + 1635530963 T^{8} - 10181816952 T^{9} + 53871897521 T^{10} - 235490063728 T^{11} + 820103008271 T^{12} - 2060814974440 T^{13} + 3142742836021 T^{14} \))
$67$ (\( 1 + 13 T + 67 T^{2} \))(\( 1 + 5 T + 67 T^{2} \))(\( 1 - 8 T + 67 T^{2} \))(\( 1 + 8 T + 67 T^{2} \))(\( 1 - 13 T + 67 T^{2} \))(\( 1 - 10 T + 154 T^{2} - 670 T^{3} + 4489 T^{4} \))(\( 1 + 6 T + 98 T^{2} + 402 T^{3} + 4489 T^{4} \))(\( 1 - 3 T + 170 T^{2} - 391 T^{3} + 15834 T^{4} - 26197 T^{5} + 763130 T^{6} - 902289 T^{7} + 20151121 T^{8} \))(\( 1 - 2 T + 260 T^{2} - 390 T^{3} + 25866 T^{4} - 26130 T^{5} + 1167140 T^{6} - 601526 T^{7} + 20151121 T^{8} \))(\( 1 + 2 T + 260 T^{2} + 390 T^{3} + 25866 T^{4} + 26130 T^{5} + 1167140 T^{6} + 601526 T^{7} + 20151121 T^{8} \))(\( 1 + 4 T + 245 T^{2} + 680 T^{3} + 28669 T^{4} + 85884 T^{5} + 2488785 T^{6} + 7855600 T^{7} + 166748595 T^{8} + 385533276 T^{9} + 8622574447 T^{10} + 13702762280 T^{11} + 330780651215 T^{12} + 361833528676 T^{13} + 6060711605323 T^{14} \))(\( 1 - 4 T + 245 T^{2} - 680 T^{3} + 28669 T^{4} - 85884 T^{5} + 2488785 T^{6} - 7855600 T^{7} + 166748595 T^{8} - 385533276 T^{9} + 8622574447 T^{10} - 13702762280 T^{11} + 330780651215 T^{12} - 361833528676 T^{13} + 6060711605323 T^{14} \))
$71$ (\( 1 - 13 T + 71 T^{2} \))(\( 1 - 5 T + 71 T^{2} \))(\( 1 + 10 T + 71 T^{2} \))(\( 1 + 10 T + 71 T^{2} \))(\( 1 - 13 T + 71 T^{2} \))(\( 1 - 20 T + 237 T^{2} - 1420 T^{3} + 5041 T^{4} \))(\( 1 + 8 T + 153 T^{2} + 568 T^{3} + 5041 T^{4} \))(\( 1 + 3 T + 135 T^{2} + 104 T^{3} + 9080 T^{4} + 7384 T^{5} + 680535 T^{6} + 1073733 T^{7} + 25411681 T^{8} \))(\( 1 + 24 T + 350 T^{2} + 3544 T^{3} + 32183 T^{4} + 251624 T^{5} + 1764350 T^{6} + 8589864 T^{7} + 25411681 T^{8} \))(\( 1 + 24 T + 350 T^{2} + 3544 T^{3} + 32183 T^{4} + 251624 T^{5} + 1764350 T^{6} + 8589864 T^{7} + 25411681 T^{8} \))(\( 1 + 14 T + 479 T^{2} + 4960 T^{3} + 95992 T^{4} + 781136 T^{5} + 10923513 T^{6} + 70967270 T^{7} + 775569423 T^{8} + 3937706576 T^{9} + 34356592712 T^{10} + 126041937760 T^{11} + 864225859129 T^{12} + 1793403974894 T^{13} + 9095120158391 T^{14} \))(\( 1 + 14 T + 479 T^{2} + 4960 T^{3} + 95992 T^{4} + 781136 T^{5} + 10923513 T^{6} + 70967270 T^{7} + 775569423 T^{8} + 3937706576 T^{9} + 34356592712 T^{10} + 126041937760 T^{11} + 864225859129 T^{12} + 1793403974894 T^{13} + 9095120158391 T^{14} \))
$73$ (\( 1 + 8 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( 1 + 3 T + 73 T^{2} \))(\( 1 - 3 T + 73 T^{2} \))(\( 1 - 8 T + 73 T^{2} \))(\( 1 + 22 T + 247 T^{2} + 1606 T^{3} + 5329 T^{4} \))(\( 1 + 101 T^{2} + 5329 T^{4} \))(\( 1 - 32 T + 635 T^{2} - 8400 T^{3} + 83736 T^{4} - 613200 T^{5} + 3383915 T^{6} - 12448544 T^{7} + 28398241 T^{8} \))(\( 1 + 18 T + 270 T^{2} + 2088 T^{3} + 20423 T^{4} + 152424 T^{5} + 1438830 T^{6} + 7002306 T^{7} + 28398241 T^{8} \))(\( 1 - 18 T + 270 T^{2} - 2088 T^{3} + 20423 T^{4} - 152424 T^{5} + 1438830 T^{6} - 7002306 T^{7} + 28398241 T^{8} \))(\( 1 + 3 T + 325 T^{2} + 658 T^{3} + 54034 T^{4} + 90182 T^{5} + 5763275 T^{6} + 7883729 T^{7} + 420719075 T^{8} + 480579878 T^{9} + 21020144578 T^{10} + 18686042578 T^{11} + 673748267725 T^{12} + 454002678867 T^{13} + 11047398519097 T^{14} \))(\( 1 - 3 T + 325 T^{2} - 658 T^{3} + 54034 T^{4} - 90182 T^{5} + 5763275 T^{6} - 7883729 T^{7} + 420719075 T^{8} - 480579878 T^{9} + 21020144578 T^{10} - 18686042578 T^{11} + 673748267725 T^{12} - 454002678867 T^{13} + 11047398519097 T^{14} \))
$79$ (\( 1 + 14 T + 79 T^{2} \))(\( 1 + 12 T + 79 T^{2} \))(\( 1 + 3 T + 79 T^{2} \))(\( 1 + 3 T + 79 T^{2} \))(\( 1 + 14 T + 79 T^{2} \))(\( 1 + 4 T + 82 T^{2} + 316 T^{3} + 6241 T^{4} \))(\( 1 - 22 T + 274 T^{2} - 1738 T^{3} + 6241 T^{4} \))(\( 1 - 2 T + 176 T^{2} - 826 T^{3} + 15838 T^{4} - 65254 T^{5} + 1098416 T^{6} - 986078 T^{7} + 38950081 T^{8} \))(\( 1 - 24 T + 494 T^{2} - 6100 T^{3} + 65598 T^{4} - 481900 T^{5} + 3083054 T^{6} - 11832936 T^{7} + 38950081 T^{8} \))(\( 1 - 24 T + 494 T^{2} - 6100 T^{3} + 65598 T^{4} - 481900 T^{5} + 3083054 T^{6} - 11832936 T^{7} + 38950081 T^{8} \))(\( 1 + T + 321 T^{2} + 1538 T^{3} + 46557 T^{4} + 374223 T^{5} + 4556893 T^{6} + 40938716 T^{7} + 359994547 T^{8} + 2335525743 T^{9} + 22954416723 T^{10} + 59905224578 T^{11} + 987735104079 T^{12} + 243087455521 T^{13} + 19203908986159 T^{14} \))(\( 1 + T + 321 T^{2} + 1538 T^{3} + 46557 T^{4} + 374223 T^{5} + 4556893 T^{6} + 40938716 T^{7} + 359994547 T^{8} + 2335525743 T^{9} + 22954416723 T^{10} + 59905224578 T^{11} + 987735104079 T^{12} + 243087455521 T^{13} + 19203908986159 T^{14} \))
$83$ (\( 1 - 3 T + 83 T^{2} \))(\( 1 + 9 T + 83 T^{2} \))(\( 1 - 3 T + 83 T^{2} \))(\( 1 + 3 T + 83 T^{2} \))(\( 1 + 3 T + 83 T^{2} \))(\( 1 - 22 T + 282 T^{2} - 1826 T^{3} + 6889 T^{4} \))(\( 1 - 4 T + 150 T^{2} - 332 T^{3} + 6889 T^{4} \))(\( 1 - 21 T + 428 T^{2} - 5005 T^{3} + 56054 T^{4} - 415415 T^{5} + 2948492 T^{6} - 12007527 T^{7} + 47458321 T^{8} \))(\( 1 - 6 T + 62 T^{2} - 518 T^{3} + 11462 T^{4} - 42994 T^{5} + 427118 T^{6} - 3430722 T^{7} + 47458321 T^{8} \))(\( 1 + 6 T + 62 T^{2} + 518 T^{3} + 11462 T^{4} + 42994 T^{5} + 427118 T^{6} + 3430722 T^{7} + 47458321 T^{8} \))(\( 1 - 17 T + 541 T^{2} - 7850 T^{3} + 130949 T^{4} - 1547895 T^{5} + 17976145 T^{6} - 168233836 T^{7} + 1492020035 T^{8} - 10663448655 T^{9} + 74874935863 T^{10} - 372547819850 T^{11} + 2131020987863 T^{12} - 5557986347273 T^{13} + 27136050989627 T^{14} \))(\( 1 + 17 T + 541 T^{2} + 7850 T^{3} + 130949 T^{4} + 1547895 T^{5} + 17976145 T^{6} + 168233836 T^{7} + 1492020035 T^{8} + 10663448655 T^{9} + 74874935863 T^{10} + 372547819850 T^{11} + 2131020987863 T^{12} + 5557986347273 T^{13} + 27136050989627 T^{14} \))
$89$ (\( 1 + 14 T + 89 T^{2} \))(\( 1 - 4 T + 89 T^{2} \))(\( 1 - 10 T + 89 T^{2} \))(\( 1 - 10 T + 89 T^{2} \))(\( 1 + 14 T + 89 T^{2} \))(\( 1 + 12 T + 194 T^{2} + 1068 T^{3} + 7921 T^{4} \))(\( 1 - 10 T + 198 T^{2} - 890 T^{3} + 7921 T^{4} \))(\( 1 + 140 T^{2} - 1496 T^{3} + 6326 T^{4} - 133144 T^{5} + 1108940 T^{6} + 62742241 T^{8} \))(\( 1 + 8 T + 270 T^{2} + 1764 T^{3} + 34598 T^{4} + 156996 T^{5} + 2138670 T^{6} + 5639752 T^{7} + 62742241 T^{8} \))(\( 1 + 8 T + 270 T^{2} + 1764 T^{3} + 34598 T^{4} + 156996 T^{5} + 2138670 T^{6} + 5639752 T^{7} + 62742241 T^{8} \))(\( 1 - 16 T + 379 T^{2} - 4856 T^{3} + 79377 T^{4} - 825904 T^{5} + 10350019 T^{6} - 90162448 T^{7} + 921151691 T^{8} - 6541985584 T^{9} + 55958324313 T^{10} - 304676322296 T^{11} + 2116358531171 T^{12} - 7951700655376 T^{13} + 44231334895529 T^{14} \))(\( 1 - 16 T + 379 T^{2} - 4856 T^{3} + 79377 T^{4} - 825904 T^{5} + 10350019 T^{6} - 90162448 T^{7} + 921151691 T^{8} - 6541985584 T^{9} + 55958324313 T^{10} - 304676322296 T^{11} + 2116358531171 T^{12} - 7951700655376 T^{13} + 44231334895529 T^{14} \))
$97$ (\( 1 + 14 T + 97 T^{2} \))(\( 1 - 14 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( 1 - 14 T + 97 T^{2} \))(\( 1 + 22 T + 270 T^{2} + 2134 T^{3} + 9409 T^{4} \))(\( 1 + 10 T + 94 T^{2} + 970 T^{3} + 9409 T^{4} \))(\( 1 - 18 T + 460 T^{2} - 5038 T^{3} + 69350 T^{4} - 488686 T^{5} + 4328140 T^{6} - 16428114 T^{7} + 88529281 T^{8} \))(\( 1 + 34 T + 736 T^{2} + 10514 T^{3} + 119290 T^{4} + 1019858 T^{5} + 6925024 T^{6} + 31030882 T^{7} + 88529281 T^{8} \))(\( 1 - 34 T + 736 T^{2} - 10514 T^{3} + 119290 T^{4} - 1019858 T^{5} + 6925024 T^{6} - 31030882 T^{7} + 88529281 T^{8} \))(\( 1 + 24 T + 683 T^{2} + 11736 T^{3} + 196873 T^{4} + 2578984 T^{5} + 31528651 T^{6} + 323282256 T^{7} + 3058279147 T^{8} + 24265660456 T^{9} + 179680671529 T^{10} + 1038979641816 T^{11} + 5865153395531 T^{12} + 19991328118296 T^{13} + 80798284478113 T^{14} \))(\( 1 - 24 T + 683 T^{2} - 11736 T^{3} + 196873 T^{4} - 2578984 T^{5} + 31528651 T^{6} - 323282256 T^{7} + 3058279147 T^{8} - 24265660456 T^{9} + 179680671529 T^{10} - 1038979641816 T^{11} + 5865153395531 T^{12} - 19991328118296 T^{13} + 80798284478113 T^{14} \))
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