Properties

Label 894.2.a
Level 894
Weight 2
Character orbit a
Rep. character \(\chi_{894}(1,\cdot)\)
Character field \(\Q\)
Dimension 25
Newform subspaces 12
Sturm bound 300
Trace bound 5

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Defining parameters

Level: \( N \) = \( 894 = 2 \cdot 3 \cdot 149 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 894.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 12 \)
Sturm bound: \(300\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(894))\).

Total New Old
Modular forms 154 25 129
Cusp forms 147 25 122
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.

\(2\)\(3\)\(149\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(2\)
\(+\)\(+\)\(-\)\(-\)\(4\)
\(+\)\(-\)\(+\)\(-\)\(6\)
\(+\)\(-\)\(-\)\(+\)\(1\)
\(-\)\(+\)\(+\)\(-\)\(4\)
\(-\)\(+\)\(-\)\(+\)\(2\)
\(-\)\(-\)\(+\)\(+\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(5\)
Plus space\(+\)\(6\)
Minus space\(-\)\(19\)

Trace form

\( 25q - q^{2} + q^{3} + 25q^{4} - 6q^{5} - q^{6} - 4q^{7} - q^{8} + 25q^{9} + O(q^{10}) \) \( 25q - q^{2} + q^{3} + 25q^{4} - 6q^{5} - q^{6} - 4q^{7} - q^{8} + 25q^{9} - 2q^{10} + 4q^{11} + q^{12} + 10q^{13} + 8q^{14} + 2q^{15} + 25q^{16} - 2q^{17} - q^{18} + 8q^{19} - 6q^{20} + 8q^{21} + 4q^{22} - q^{24} + 35q^{25} - 6q^{26} + q^{27} - 4q^{28} + 10q^{29} + 2q^{30} + 4q^{31} - q^{32} - 4q^{33} - 10q^{34} + 32q^{35} + 25q^{36} + 6q^{37} + 4q^{38} + 6q^{39} - 2q^{40} + 22q^{41} - 8q^{42} + 20q^{43} + 4q^{44} - 6q^{45} - 8q^{46} + 8q^{47} + q^{48} + 37q^{49} - 15q^{50} - 14q^{51} + 10q^{52} + 26q^{53} - q^{54} + 8q^{55} + 8q^{56} + 20q^{57} - 18q^{58} + 36q^{59} + 2q^{60} - 10q^{61} - 16q^{62} - 4q^{63} + 25q^{64} - 12q^{65} + 4q^{66} + 16q^{67} - 2q^{68} - 16q^{69} - 8q^{71} - q^{72} - 14q^{73} - 6q^{74} + 15q^{75} + 8q^{76} - 16q^{77} - 10q^{78} - 12q^{79} - 6q^{80} + 25q^{81} - 10q^{82} + 4q^{83} + 8q^{84} - 28q^{85} - 28q^{86} + 10q^{87} + 4q^{88} - 34q^{89} - 2q^{90} + 72q^{91} + 16q^{93} + 8q^{94} + 16q^{95} - q^{96} - 6q^{97} + 7q^{98} + 4q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(894))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 149
894.2.a.a \(1\) \(7.139\) \(\Q\) None \(-1\) \(-1\) \(-3\) \(0\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}+q^{4}-3q^{5}+q^{6}-q^{8}+\cdots\)
894.2.a.b \(1\) \(7.139\) \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
894.2.a.c \(1\) \(7.139\) \(\Q\) None \(-1\) \(1\) \(1\) \(-2\) \(+\) \(-\) \(-\) \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}-2q^{7}+\cdots\)
894.2.a.d \(1\) \(7.139\) \(\Q\) None \(-1\) \(1\) \(3\) \(2\) \(+\) \(-\) \(+\) \(q-q^{2}+q^{3}+q^{4}+3q^{5}-q^{6}+2q^{7}+\cdots\)
894.2.a.e \(1\) \(7.139\) \(\Q\) None \(1\) \(-1\) \(-3\) \(2\) \(-\) \(+\) \(-\) \(q+q^{2}-q^{3}+q^{4}-3q^{5}-q^{6}+2q^{7}+\cdots\)
894.2.a.f \(1\) \(7.139\) \(\Q\) None \(1\) \(-1\) \(0\) \(-4\) \(-\) \(+\) \(-\) \(q+q^{2}-q^{3}+q^{4}-q^{6}-4q^{7}+q^{8}+\cdots\)
894.2.a.g \(1\) \(7.139\) \(\Q\) None \(1\) \(1\) \(-3\) \(-4\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{3}+q^{4}-3q^{5}+q^{6}-4q^{7}+\cdots\)
894.2.a.h \(2\) \(7.139\) \(\Q(\sqrt{33}) \) None \(2\) \(2\) \(-1\) \(8\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{3}+q^{4}-\beta q^{5}+q^{6}+4q^{7}+\cdots\)
894.2.a.i \(3\) \(7.139\) \(\Q(\zeta_{18})^+\) None \(3\) \(3\) \(3\) \(-3\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{3}+q^{4}+(1-\beta _{1})q^{5}+q^{6}+\cdots\)
894.2.a.j \(4\) \(7.139\) 4.4.6809.1 None \(-4\) \(-4\) \(2\) \(-7\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}+(\beta _{1}-\beta _{2})q^{5}+q^{6}+\cdots\)
894.2.a.k \(4\) \(7.139\) 4.4.15529.1 None \(4\) \(-4\) \(0\) \(3\) \(-\) \(+\) \(+\) \(q+q^{2}-q^{3}+q^{4}+(\beta _{2}-\beta _{3})q^{5}-q^{6}+\cdots\)
894.2.a.l \(5\) \(7.139\) 5.5.5444689.1 None \(-5\) \(5\) \(-5\) \(1\) \(+\) \(-\) \(+\) \(q-q^{2}+q^{3}+q^{4}+(-1+\beta _{1}+\beta _{4})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(894))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(894)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(149))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(298))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(447))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{3} \))(\( ( 1 + T )^{4} \))(\( ( 1 - T )^{4} \))(\( ( 1 + T )^{5} \))
$3$ (\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{3} \))(\( ( 1 + T )^{4} \))(\( ( 1 + T )^{4} \))(\( ( 1 - T )^{5} \))
$5$ (\( 1 + 3 T + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 - T + 5 T^{2} \))(\( 1 - 3 T + 5 T^{2} \))(\( 1 + 3 T + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 + 3 T + 5 T^{2} \))(\( 1 + T + 2 T^{2} + 5 T^{3} + 25 T^{4} \))(\( 1 - 3 T + 15 T^{2} - 27 T^{3} + 75 T^{4} - 75 T^{5} + 125 T^{6} \))(\( 1 - 2 T + 9 T^{2} - 7 T^{3} + 39 T^{4} - 35 T^{5} + 225 T^{6} - 250 T^{7} + 625 T^{8} \))(\( 1 + 7 T^{2} + T^{3} + 49 T^{4} + 5 T^{5} + 175 T^{6} + 625 T^{8} \))(\( 1 + 5 T + 15 T^{2} + 29 T^{3} + 34 T^{4} + 24 T^{5} + 170 T^{6} + 725 T^{7} + 1875 T^{8} + 3125 T^{9} + 3125 T^{10} \))
$7$ (\( 1 + 7 T^{2} \))(\( 1 + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 - 2 T + 7 T^{2} \))(\( 1 - 2 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( ( 1 - 4 T + 7 T^{2} )^{2} \))(\( 1 + 3 T + 21 T^{2} + 41 T^{3} + 147 T^{4} + 147 T^{5} + 343 T^{6} \))(\( 1 + 7 T + 32 T^{2} + 118 T^{3} + 366 T^{4} + 826 T^{5} + 1568 T^{6} + 2401 T^{7} + 2401 T^{8} \))(\( 1 - 3 T + 18 T^{2} - 48 T^{3} + 152 T^{4} - 336 T^{5} + 882 T^{6} - 1029 T^{7} + 2401 T^{8} \))(\( 1 - T + 7 T^{2} + 15 T^{3} + 22 T^{4} + 180 T^{5} + 154 T^{6} + 735 T^{7} + 2401 T^{8} - 2401 T^{9} + 16807 T^{10} \))
$11$ (\( 1 - T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 + 3 T + 11 T^{2} \))(\( 1 + 3 T + 11 T^{2} \))(\( 1 + 5 T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 + T + 11 T^{2} \))(\( 1 + 3 T + 16 T^{2} + 33 T^{3} + 121 T^{4} \))(\( 1 - 6 T + 33 T^{2} - 108 T^{3} + 363 T^{4} - 726 T^{5} + 1331 T^{6} \))(\( 1 - 3 T + 22 T^{2} - 75 T^{3} + 250 T^{4} - 825 T^{5} + 2662 T^{6} - 3993 T^{7} + 14641 T^{8} \))(\( 1 - 9 T + 62 T^{2} - 297 T^{3} + 1114 T^{4} - 3267 T^{5} + 7502 T^{6} - 11979 T^{7} + 14641 T^{8} \))(\( 1 - 4 T + 27 T^{2} - 56 T^{3} + 238 T^{4} - 328 T^{5} + 2618 T^{6} - 6776 T^{7} + 35937 T^{8} - 58564 T^{9} + 161051 T^{10} \))
$13$ (\( 1 - 5 T + 13 T^{2} \))(\( 1 + T + 13 T^{2} \))(\( 1 + 5 T + 13 T^{2} \))(\( 1 - 5 T + 13 T^{2} \))(\( 1 - T + 13 T^{2} \))(\( 1 - T + 13 T^{2} \))(\( 1 + 5 T + 13 T^{2} \))(\( ( 1 - T + 13 T^{2} )^{2} \))(\( 1 - 3 T + 21 T^{2} - 95 T^{3} + 273 T^{4} - 507 T^{5} + 2197 T^{6} \))(\( 1 + 4 T + 17 T^{2} + 87 T^{3} + 417 T^{4} + 1131 T^{5} + 2873 T^{6} + 8788 T^{7} + 28561 T^{8} \))(\( 1 + 41 T^{2} - 9 T^{3} + 737 T^{4} - 117 T^{5} + 6929 T^{6} + 28561 T^{8} \))(\( 1 - 8 T + 64 T^{2} - 341 T^{3} + 1696 T^{4} - 6224 T^{5} + 22048 T^{6} - 57629 T^{7} + 140608 T^{8} - 228488 T^{9} + 371293 T^{10} \))
$17$ (\( 1 + 2 T + 17 T^{2} \))(\( 1 - T + 17 T^{2} \))(\( 1 + 4 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 3 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 5 T + 32 T^{2} + 85 T^{3} + 289 T^{4} \))(\( 1 + 39 T^{2} + 8 T^{3} + 663 T^{4} + 4913 T^{6} \))(\( 1 - 10 T + 80 T^{2} - 446 T^{3} + 2158 T^{4} - 7582 T^{5} + 23120 T^{6} - 49130 T^{7} + 83521 T^{8} \))(\( 1 + 32 T^{2} - 72 T^{3} + 478 T^{4} - 1224 T^{5} + 9248 T^{6} + 83521 T^{8} \))(\( 1 + T + 15 T^{2} + 88 T^{3} + 568 T^{4} + 286 T^{5} + 9656 T^{6} + 25432 T^{7} + 73695 T^{8} + 83521 T^{9} + 1419857 T^{10} \))
$19$ (\( 1 - T + 19 T^{2} \))(\( 1 - T + 19 T^{2} \))(\( 1 + 5 T + 19 T^{2} \))(\( 1 + T + 19 T^{2} \))(\( 1 + 5 T + 19 T^{2} \))(\( 1 - T + 19 T^{2} \))(\( 1 - T + 19 T^{2} \))(\( ( 1 - T + 19 T^{2} )^{2} \))(\( 1 - 6 T + 21 T^{2} - 76 T^{3} + 399 T^{4} - 2166 T^{5} + 6859 T^{6} \))(\( 1 + 5 T + 34 T^{2} + 153 T^{3} + 1010 T^{4} + 2907 T^{5} + 12274 T^{6} + 34295 T^{7} + 130321 T^{8} \))(\( 1 - T + 6 T^{2} - 37 T^{3} + 314 T^{4} - 703 T^{5} + 2166 T^{6} - 6859 T^{7} + 130321 T^{8} \))(\( 1 - 11 T + 61 T^{2} - 320 T^{3} + 1920 T^{4} - 9562 T^{5} + 36480 T^{6} - 115520 T^{7} + 418399 T^{8} - 1433531 T^{9} + 2476099 T^{10} \))
$23$ (\( 1 - 2 T + 23 T^{2} \))(\( 1 + T + 23 T^{2} \))(\( 1 - 2 T + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 + 3 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 + 5 T + 44 T^{2} + 115 T^{3} + 529 T^{4} \))(\( 1 - 3 T + 69 T^{2} - 135 T^{3} + 1587 T^{4} - 1587 T^{5} + 12167 T^{6} \))(\( 1 - 3 T + 54 T^{2} - 210 T^{3} + 1596 T^{4} - 4830 T^{5} + 28566 T^{6} - 36501 T^{7} + 279841 T^{8} \))(\( 1 - 13 T + 98 T^{2} - 550 T^{3} + 2704 T^{4} - 12650 T^{5} + 51842 T^{6} - 158171 T^{7} + 279841 T^{8} \))(\( 1 + 8 T + 82 T^{2} + 541 T^{3} + 3530 T^{4} + 16126 T^{5} + 81190 T^{6} + 286189 T^{7} + 997694 T^{8} + 2238728 T^{9} + 6436343 T^{10} \))
$29$ (\( 1 + 5 T + 29 T^{2} \))(\( 1 + 8 T + 29 T^{2} \))(\( 1 + T + 29 T^{2} \))(\( 1 + 3 T + 29 T^{2} \))(\( 1 + 5 T + 29 T^{2} \))(\( 1 + 8 T + 29 T^{2} \))(\( 1 + 5 T + 29 T^{2} \))(\( 1 - T + 50 T^{2} - 29 T^{3} + 841 T^{4} \))(\( 1 - 3 T + 15 T^{2} - 225 T^{3} + 435 T^{4} - 2523 T^{5} + 24389 T^{6} \))(\( 1 - 16 T + 175 T^{2} - 1235 T^{3} + 7563 T^{4} - 35815 T^{5} + 147175 T^{6} - 390224 T^{7} + 707281 T^{8} \))(\( 1 - 10 T + 117 T^{2} - 723 T^{3} + 4893 T^{4} - 20967 T^{5} + 98397 T^{6} - 243890 T^{7} + 707281 T^{8} \))(\( 1 - 15 T + 151 T^{2} - 1029 T^{3} + 6150 T^{4} - 31940 T^{5} + 178350 T^{6} - 865389 T^{7} + 3682739 T^{8} - 10609215 T^{9} + 20511149 T^{10} \))
$31$ (\( 1 + 6 T + 31 T^{2} \))(\( 1 - 6 T + 31 T^{2} \))(\( 1 + 2 T + 31 T^{2} \))(\( 1 + 10 T + 31 T^{2} \))(\( 1 - 2 T + 31 T^{2} \))(\( 1 + 10 T + 31 T^{2} \))(\( 1 + 2 T + 31 T^{2} \))(\( ( 1 + 2 T + 31 T^{2} )^{2} \))(\( 1 - 3 T + 33 T^{2} + 47 T^{3} + 1023 T^{4} - 2883 T^{5} + 29791 T^{6} \))(\( 1 + 3 T + 110 T^{2} + 268 T^{3} + 4920 T^{4} + 8308 T^{5} + 105710 T^{6} + 89373 T^{7} + 923521 T^{8} \))(\( 1 - 5 T - 10 T^{2} + 24 T^{3} + 1040 T^{4} + 744 T^{5} - 9610 T^{6} - 148955 T^{7} + 923521 T^{8} \))(\( 1 - 25 T + 385 T^{2} - 4071 T^{3} + 32866 T^{4} - 205664 T^{5} + 1018846 T^{6} - 3912231 T^{7} + 11469535 T^{8} - 23088025 T^{9} + 28629151 T^{10} \))
$37$ (\( 1 + 2 T + 37 T^{2} \))(\( 1 - 4 T + 37 T^{2} \))(\( 1 + 37 T^{2} \))(\( 1 - 8 T + 37 T^{2} \))(\( 1 + 4 T + 37 T^{2} \))(\( 1 + 4 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 + 6 T + 50 T^{2} + 222 T^{3} + 1369 T^{4} \))(\( 1 - 6 T + 75 T^{2} - 292 T^{3} + 2775 T^{4} - 8214 T^{5} + 50653 T^{6} \))(\( 1 + 8 T + 92 T^{2} + 536 T^{3} + 3894 T^{4} + 19832 T^{5} + 125948 T^{6} + 405224 T^{7} + 1874161 T^{8} \))(\( 1 - 6 T + 72 T^{2} - 194 T^{3} + 2622 T^{4} - 7178 T^{5} + 98568 T^{6} - 303918 T^{7} + 1874161 T^{8} \))(\( 1 - 4 T + 113 T^{2} - 304 T^{3} + 5842 T^{4} - 11864 T^{5} + 216154 T^{6} - 416176 T^{7} + 5723789 T^{8} - 7496644 T^{9} + 69343957 T^{10} \))
$41$ (\( 1 + 3 T + 41 T^{2} \))(\( 1 + 12 T + 41 T^{2} \))(\( 1 - 3 T + 41 T^{2} \))(\( 1 + 9 T + 41 T^{2} \))(\( 1 + 3 T + 41 T^{2} \))(\( 1 - 12 T + 41 T^{2} \))(\( 1 + 5 T + 41 T^{2} \))(\( 1 + 9 T + 94 T^{2} + 369 T^{3} + 1681 T^{4} \))(\( 1 - 3 T + 117 T^{2} - 229 T^{3} + 4797 T^{4} - 5043 T^{5} + 68921 T^{6} \))(\( 1 - 26 T + 383 T^{2} - 3861 T^{3} + 28685 T^{4} - 158301 T^{5} + 643823 T^{6} - 1791946 T^{7} + 2825761 T^{8} \))(\( 1 - 8 T + 177 T^{2} - 981 T^{3} + 11151 T^{4} - 40221 T^{5} + 297537 T^{6} - 551368 T^{7} + 2825761 T^{8} \))(\( 1 - 11 T + 133 T^{2} - 877 T^{3} + 6508 T^{4} - 34972 T^{5} + 266828 T^{6} - 1474237 T^{7} + 9166493 T^{8} - 31083371 T^{9} + 115856201 T^{10} \))
$43$ (\( 1 + 6 T + 43 T^{2} \))(\( 1 - 12 T + 43 T^{2} \))(\( 1 - 2 T + 43 T^{2} \))(\( 1 - 2 T + 43 T^{2} \))(\( 1 + 2 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 + 10 T + 43 T^{2} \))(\( 1 - 10 T + 78 T^{2} - 430 T^{3} + 1849 T^{4} \))(\( 1 + 9 T + 117 T^{2} + 703 T^{3} + 5031 T^{4} + 16641 T^{5} + 79507 T^{6} \))(\( 1 + 5 T + 126 T^{2} + 568 T^{3} + 7380 T^{4} + 24424 T^{5} + 232974 T^{6} + 397535 T^{7} + 3418801 T^{8} \))(\( 1 - 3 T + 162 T^{2} - 372 T^{3} + 10232 T^{4} - 15996 T^{5} + 299538 T^{6} - 238521 T^{7} + 3418801 T^{8} \))(\( 1 - 19 T + 331 T^{2} - 3519 T^{3} + 33598 T^{4} - 232388 T^{5} + 1444714 T^{6} - 6506631 T^{7} + 26316817 T^{8} - 64957219 T^{9} + 147008443 T^{10} \))
$47$ (\( 1 - 3 T + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 - 9 T + 47 T^{2} \))(\( 1 + 9 T + 47 T^{2} \))(\( 1 + 3 T + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 - 11 T + 47 T^{2} \))(\( 1 - 3 T + 88 T^{2} - 141 T^{3} + 2209 T^{4} \))(\( 1 + 57 T^{2} - 296 T^{3} + 2679 T^{4} + 103823 T^{6} \))(\( 1 - 5 T + 172 T^{2} - 637 T^{3} + 11710 T^{4} - 29939 T^{5} + 379948 T^{6} - 519115 T^{7} + 4879681 T^{8} \))(\( 1 - 3 T + 80 T^{2} - 531 T^{3} + 3750 T^{4} - 24957 T^{5} + 176720 T^{6} - 311469 T^{7} + 4879681 T^{8} \))(\( 1 + 2 T - 41 T^{2} - 56 T^{3} + 406 T^{4} - 11540 T^{5} + 19082 T^{6} - 123704 T^{7} - 4256743 T^{8} + 9759362 T^{9} + 229345007 T^{10} \))
$53$ (\( 1 + 6 T + 53 T^{2} \))(\( 1 + 53 T^{2} \))(\( 1 - 2 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 12 T + 53 T^{2} \))(\( 1 - 10 T + 53 T^{2} \))(\( 1 - 10 T + 98 T^{2} - 530 T^{3} + 2809 T^{4} \))(\( 1 + 21 T + 285 T^{2} + 2459 T^{3} + 15105 T^{4} + 58989 T^{5} + 148877 T^{6} \))(\( 1 - 9 T + 136 T^{2} - 1134 T^{3} + 8800 T^{4} - 60102 T^{5} + 382024 T^{6} - 1339893 T^{7} + 7890481 T^{8} \))(\( 1 - 19 T + 260 T^{2} - 2338 T^{3} + 19124 T^{4} - 123914 T^{5} + 730340 T^{6} - 2828663 T^{7} + 7890481 T^{8} \))(\( 1 + 9 T + 177 T^{2} + 1555 T^{3} + 15076 T^{4} + 116876 T^{5} + 799028 T^{6} + 4367995 T^{7} + 26351229 T^{8} + 71014329 T^{9} + 418195493 T^{10} \))
$59$ (\( 1 + 9 T + 59 T^{2} \))(\( 1 - 6 T + 59 T^{2} \))(\( 1 - 7 T + 59 T^{2} \))(\( 1 + 9 T + 59 T^{2} \))(\( 1 - 9 T + 59 T^{2} \))(\( 1 + 6 T + 59 T^{2} \))(\( 1 - T + 59 T^{2} \))(\( 1 + 13 T + 152 T^{2} + 767 T^{3} + 3481 T^{4} \))(\( 1 + 6 T + 33 T^{2} + 556 T^{3} + 1947 T^{4} + 20886 T^{5} + 205379 T^{6} \))(\( 1 - 17 T + 98 T^{2} + 807 T^{3} - 12670 T^{4} + 47613 T^{5} + 341138 T^{6} - 3491443 T^{7} + 12117361 T^{8} \))(\( 1 - 23 T + 402 T^{2} - 4503 T^{3} + 40834 T^{4} - 265677 T^{5} + 1399362 T^{6} - 4723717 T^{7} + 12117361 T^{8} \))(\( 1 - 16 T + 291 T^{2} - 3176 T^{3} + 34822 T^{4} - 265360 T^{5} + 2054498 T^{6} - 11055656 T^{7} + 59765289 T^{8} - 193877776 T^{9} + 714924299 T^{10} \))
$61$ (\( 1 - 10 T + 61 T^{2} \))(\( 1 + 14 T + 61 T^{2} \))(\( 1 + 12 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( ( 1 - 6 T + 61 T^{2} )^{2} \))(\( 1 + 12 T + 147 T^{2} + 1328 T^{3} + 8967 T^{4} + 44652 T^{5} + 226981 T^{6} \))(\( 1 - 10 T + 104 T^{2} - 598 T^{3} + 7182 T^{4} - 36478 T^{5} + 386984 T^{6} - 2269810 T^{7} + 13845841 T^{8} \))(\( 1 + 8 T + 160 T^{2} + 896 T^{3} + 12878 T^{4} + 54656 T^{5} + 595360 T^{6} + 1815848 T^{7} + 13845841 T^{8} \))(\( 1 + 225 T^{2} - 168 T^{3} + 23066 T^{4} - 20720 T^{5} + 1407026 T^{6} - 625128 T^{7} + 51070725 T^{8} + 844596301 T^{10} \))
$67$ (\( 1 - 8 T + 67 T^{2} \))(\( 1 + 13 T + 67 T^{2} \))(\( 1 - 12 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 - 11 T + 67 T^{2} \))(\( 1 + 8 T + 67 T^{2} \))(\( 1 - 3 T + 62 T^{2} - 201 T^{3} + 4489 T^{4} \))(\( 1 - 6 T + 177 T^{2} - 812 T^{3} + 11859 T^{4} - 26934 T^{5} + 300763 T^{6} \))(\( 1 - 10 T + 140 T^{2} - 738 T^{3} + 9846 T^{4} - 49446 T^{5} + 628460 T^{6} - 3007630 T^{7} + 20151121 T^{8} \))(\( 1 + 2 T + 244 T^{2} + 410 T^{3} + 23750 T^{4} + 27470 T^{5} + 1095316 T^{6} + 601526 T^{7} + 20151121 T^{8} \))(\( 1 + 3 T + 173 T^{2} + 508 T^{3} + 14768 T^{4} + 46738 T^{5} + 989456 T^{6} + 2280412 T^{7} + 52031999 T^{8} + 60453363 T^{9} + 1350125107 T^{10} \))
$71$ (\( 1 + 4 T + 71 T^{2} \))(\( 1 + 7 T + 71 T^{2} \))(\( 1 + 6 T + 71 T^{2} \))(\( 1 - 6 T + 71 T^{2} \))(\( 1 - 2 T + 71 T^{2} \))(\( 1 + 13 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 - 3 T + 70 T^{2} - 213 T^{3} + 5041 T^{4} \))(\( 1 + 9 T + 219 T^{2} + 1225 T^{3} + 15549 T^{4} + 45369 T^{5} + 357911 T^{6} \))(\( 1 - 17 T + 262 T^{2} - 2152 T^{3} + 21430 T^{4} - 152792 T^{5} + 1320742 T^{6} - 6084487 T^{7} + 25411681 T^{8} \))(\( 1 - 21 T + 328 T^{2} - 3914 T^{3} + 35956 T^{4} - 277894 T^{5} + 1653448 T^{6} - 7516131 T^{7} + 25411681 T^{8} \))(\( 1 + 18 T + 370 T^{2} + 4395 T^{3} + 52416 T^{4} + 446726 T^{5} + 3721536 T^{6} + 22155195 T^{7} + 132427070 T^{8} + 457410258 T^{9} + 1804229351 T^{10} \))
$73$ (\( 1 + T + 73 T^{2} \))(\( 1 + 7 T + 73 T^{2} \))(\( 1 - 3 T + 73 T^{2} \))(\( 1 - 11 T + 73 T^{2} \))(\( 1 + 13 T + 73 T^{2} \))(\( 1 - 17 T + 73 T^{2} \))(\( 1 + T + 73 T^{2} \))(\( 1 + 113 T^{2} + 5329 T^{4} \))(\( 1 + 9 T + 189 T^{2} + 1277 T^{3} + 13797 T^{4} + 47961 T^{5} + 389017 T^{6} \))(\( 1 + 2 T + 143 T^{2} - 413 T^{3} + 9487 T^{4} - 30149 T^{5} + 762047 T^{6} + 778034 T^{7} + 28398241 T^{8} \))(\( 1 + 30 T + 579 T^{2} + 7311 T^{3} + 72587 T^{4} + 533703 T^{5} + 3085491 T^{6} + 11670510 T^{7} + 28398241 T^{8} \))(\( 1 - 18 T + 350 T^{2} - 3961 T^{3} + 47072 T^{4} - 389392 T^{5} + 3436256 T^{6} - 21108169 T^{7} + 136155950 T^{8} - 511168338 T^{9} + 2073071593 T^{10} \))
$79$ (\( 1 + 17 T + 79 T^{2} \))(\( 1 + 11 T + 79 T^{2} \))(\( 1 + 5 T + 79 T^{2} \))(\( 1 - 17 T + 79 T^{2} \))(\( 1 - T + 79 T^{2} \))(\( 1 + 5 T + 79 T^{2} \))(\( 1 + 7 T + 79 T^{2} \))(\( ( 1 + 5 T + 79 T^{2} )^{2} \))(\( 1 - 18 T + 261 T^{2} - 2260 T^{3} + 20619 T^{4} - 112338 T^{5} + 493039 T^{6} \))(\( 1 - 15 T + 254 T^{2} - 2939 T^{3} + 26682 T^{4} - 232181 T^{5} + 1585214 T^{6} - 7395585 T^{7} + 38950081 T^{8} \))(\( 1 - T + 294 T^{2} - 253 T^{3} + 34010 T^{4} - 19987 T^{5} + 1834854 T^{6} - 493039 T^{7} + 38950081 T^{8} \))(\( 1 + 9 T + 357 T^{2} + 2460 T^{3} + 53780 T^{4} + 280342 T^{5} + 4248620 T^{6} + 15352860 T^{7} + 176014923 T^{8} + 350550729 T^{9} + 3077056399 T^{10} \))
$83$ (\( 1 - 9 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 - 9 T + 83 T^{2} \))(\( 1 + 15 T + 83 T^{2} \))(\( 1 + T + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( 1 - 15 T + 83 T^{2} \))(\( 1 - 3 T - 38 T^{2} - 249 T^{3} + 6889 T^{4} \))(\( 1 + 24 T + 297 T^{2} + 2768 T^{3} + 24651 T^{4} + 165336 T^{5} + 571787 T^{6} \))(\( 1 - T + 156 T^{2} + 663 T^{3} + 11030 T^{4} + 55029 T^{5} + 1074684 T^{6} - 571787 T^{7} + 47458321 T^{8} \))(\( 1 - 3 T + 64 T^{2} - 1259 T^{3} + 3694 T^{4} - 104497 T^{5} + 440896 T^{6} - 1715361 T^{7} + 47458321 T^{8} \))(\( ( 1 - 4 T + 83 T^{2} )^{5} \))
$89$ (\( 1 + T + 89 T^{2} \))(\( 1 + 10 T + 89 T^{2} \))(\( 1 + 7 T + 89 T^{2} \))(\( 1 + 3 T + 89 T^{2} \))(\( 1 - 15 T + 89 T^{2} \))(\( 1 + 18 T + 89 T^{2} \))(\( 1 - 9 T + 89 T^{2} \))(\( 1 - 7 T - 16 T^{2} - 623 T^{3} + 7921 T^{4} \))(\( 1 + 21 T + 375 T^{2} + 3897 T^{3} + 33375 T^{4} + 166341 T^{5} + 704969 T^{6} \))(\( 1 - 24 T + 475 T^{2} - 5907 T^{3} + 65545 T^{4} - 525723 T^{5} + 3762475 T^{6} - 16919256 T^{7} + 62742241 T^{8} \))(\( 1 + 30 T + 613 T^{2} + 8117 T^{3} + 88651 T^{4} + 722413 T^{5} + 4855573 T^{6} + 21149070 T^{7} + 62742241 T^{8} \))(\( 1 - T + 285 T^{2} - 819 T^{3} + 37406 T^{4} - 127040 T^{5} + 3329134 T^{6} - 6487299 T^{7} + 200916165 T^{8} - 62742241 T^{9} + 5584059449 T^{10} \))
$97$ (\( 1 + 18 T + 97 T^{2} \))(\( 1 - 18 T + 97 T^{2} \))(\( 1 + 8 T + 97 T^{2} \))(\( 1 + 16 T + 97 T^{2} \))(\( 1 - 12 T + 97 T^{2} \))(\( 1 - 6 T + 97 T^{2} \))(\( 1 - 6 T + 97 T^{2} \))(\( ( 1 + 2 T + 97 T^{2} )^{2} \))(\( 1 + 6 T + 219 T^{2} + 708 T^{3} + 21243 T^{4} + 56454 T^{5} + 912673 T^{6} \))(\( 1 - 24 T + 440 T^{2} - 5952 T^{3} + 67870 T^{4} - 577344 T^{5} + 4139960 T^{6} - 21904152 T^{7} + 88529281 T^{8} \))(\( 1 + 34 T + 796 T^{2} + 11950 T^{3} + 139286 T^{4} + 1159150 T^{5} + 7489564 T^{6} + 31030882 T^{7} + 88529281 T^{8} \))(\( 1 - 14 T + 257 T^{2} - 760 T^{3} + 6334 T^{4} + 131916 T^{5} + 614398 T^{6} - 7150840 T^{7} + 234556961 T^{8} - 1239409934 T^{9} + 8587340257 T^{10} \))
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