Properties

Label 315.2.j
Level 315
Weight 2
Character orbit j
Rep. character \(\chi_{315}(46,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 28
Newform subspaces 7
Sturm bound 96
Trace bound 2

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Defining parameters

Level: \( N \) = \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 315.j (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 7 \)
Sturm bound: \(96\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(315, [\chi])\).

Total New Old
Modular forms 112 28 84
Cusp forms 80 28 52
Eisenstein series 32 0 32

Trace form

\( 28q - 2q^{2} - 18q^{4} + 2q^{5} + 6q^{7} + 12q^{8} + O(q^{10}) \) \( 28q - 2q^{2} - 18q^{4} + 2q^{5} + 6q^{7} + 12q^{8} - 2q^{10} + 16q^{13} + 8q^{14} - 22q^{16} + 12q^{17} - 4q^{19} - 20q^{20} + 24q^{22} - 10q^{23} - 14q^{25} - 4q^{26} - 14q^{28} + 12q^{29} + 8q^{31} - 14q^{32} - 40q^{34} - 4q^{35} - 12q^{37} - 24q^{38} - 6q^{40} + 12q^{41} - 36q^{43} - 16q^{44} + 22q^{46} + 12q^{47} - 10q^{49} + 4q^{50} - 28q^{52} + 8q^{53} + 24q^{55} + 18q^{56} + 10q^{58} - 12q^{59} + 46q^{61} + 116q^{64} + 8q^{65} + 22q^{67} + 8q^{68} + 14q^{70} - 16q^{71} - 32q^{73} + 4q^{74} - 48q^{76} - 72q^{77} - 4q^{79} + 22q^{80} + 2q^{82} - 12q^{83} + 8q^{85} - 14q^{86} - 76q^{88} - 22q^{89} - 24q^{91} + 76q^{92} + 36q^{94} - 24q^{97} + 14q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(315, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
315.2.j.a \(2\) \(2.515\) \(\Q(\sqrt{-3}) \) None \(-2\) \(0\) \(1\) \(-1\) \(q-2\zeta_{6}q^{2}+(-2+2\zeta_{6})q^{4}+\zeta_{6}q^{5}+\cdots\)
315.2.j.b \(2\) \(2.515\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-1\) \(5\) \(q+(2-2\zeta_{6})q^{4}-\zeta_{6}q^{5}+(2+\zeta_{6})q^{7}+\cdots\)
315.2.j.c \(4\) \(2.515\) \(\Q(\zeta_{12})\) None \(-2\) \(0\) \(2\) \(0\) \(q+(\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+(-2+2\zeta_{12}+\cdots)q^{4}+\cdots\)
315.2.j.d \(4\) \(2.515\) \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(0\) \(-2\) \(-2\) \(q+\beta _{1}q^{2}+(-1-\beta _{2})q^{5}+(-1-\beta _{1}+\cdots)q^{7}+\cdots\)
315.2.j.e \(4\) \(2.515\) \(\Q(\sqrt{2}, \sqrt{-3})\) None \(2\) \(0\) \(2\) \(2\) \(q+(1+\beta _{1}+\beta _{2})q^{2}+(2\beta _{1}+\beta _{2}+2\beta _{3})q^{4}+\cdots\)
315.2.j.f \(6\) \(2.515\) 6.0.4406832.1 None \(-2\) \(0\) \(-3\) \(1\) \(q+(-\beta _{1}+\beta _{2}-\beta _{4})q^{2}+(-2+\beta _{1}+\cdots)q^{4}+\cdots\)
315.2.j.g \(6\) \(2.515\) 6.0.4406832.1 None \(2\) \(0\) \(3\) \(1\) \(q+(\beta _{1}-\beta _{2}+\beta _{4})q^{2}+(-2+\beta _{1}+\beta _{3}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(315, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(315, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} \))(\( 1 - 2 T^{2} + 4 T^{4} \))(\( ( 1 + 2 T + 2 T^{2} )^{2}( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} ) \))(\( ( 1 - 2 T^{2} )^{2}( 1 + 2 T^{2} + 4 T^{4} ) \))(\( 1 - 2 T + T^{2} + 2 T^{3} - 3 T^{4} + 4 T^{5} + 4 T^{6} - 16 T^{7} + 16 T^{8} \))(\( 1 + 2 T + 2 T^{2} - 4 T^{4} - 4 T^{5} - 4 T^{6} - 8 T^{7} - 16 T^{8} + 32 T^{10} + 64 T^{11} + 64 T^{12} \))(\( 1 - 2 T + 2 T^{2} - 4 T^{4} + 4 T^{5} - 4 T^{6} + 8 T^{7} - 16 T^{8} + 32 T^{10} - 64 T^{11} + 64 T^{12} \))
$3$ 1
$5$ (\( 1 - T + T^{2} \))(\( 1 + T + T^{2} \))(\( ( 1 - T + T^{2} )^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{2} \))(\( ( 1 + T + T^{2} )^{3} \))(\( ( 1 - T + T^{2} )^{3} \))
$7$ (\( 1 + T + 7 T^{2} \))(\( 1 - 5 T + 7 T^{2} \))(\( 1 + 11 T^{2} + 49 T^{4} \))(\( 1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4} \))(\( 1 - 2 T - 3 T^{2} - 14 T^{3} + 49 T^{4} \))(\( 1 - T + 2 T^{2} + 23 T^{3} + 14 T^{4} - 49 T^{5} + 343 T^{6} \))(\( 1 - T + 2 T^{2} + 23 T^{3} + 14 T^{4} - 49 T^{5} + 343 T^{6} \))
$11$ (\( 1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 + 2 T - 16 T^{2} - 4 T^{3} + 235 T^{4} - 44 T^{5} - 1936 T^{6} + 2662 T^{7} + 14641 T^{8} \))(\( 1 - 4 T - 8 T^{2} - 8 T^{3} + 279 T^{4} - 88 T^{5} - 968 T^{6} - 5324 T^{7} + 14641 T^{8} \))(\( 1 - 4 T - 2 T^{2} + 16 T^{3} + 27 T^{4} + 176 T^{5} - 242 T^{6} - 5324 T^{7} + 14641 T^{8} \))(\( 1 + 10 T + 41 T^{2} + 138 T^{3} + 572 T^{4} + 1564 T^{5} + 3347 T^{6} + 17204 T^{7} + 69212 T^{8} + 183678 T^{9} + 600281 T^{10} + 1610510 T^{11} + 1771561 T^{12} \))(\( 1 - 10 T + 41 T^{2} - 138 T^{3} + 572 T^{4} - 1564 T^{5} + 3347 T^{6} - 17204 T^{7} + 69212 T^{8} - 183678 T^{9} + 600281 T^{10} - 1610510 T^{11} + 1771561 T^{12} \))
$13$ (\( ( 1 + 3 T + 13 T^{2} )^{2} \))(\( ( 1 + T + 13 T^{2} )^{2} \))(\( ( 1 - 8 T + 39 T^{2} - 104 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 + 6 T + 33 T^{2} + 78 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 + 4 T + 22 T^{2} + 52 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 - 7 T + 48 T^{2} - 171 T^{3} + 624 T^{4} - 1183 T^{5} + 2197 T^{6} )^{2} \))(\( ( 1 - 7 T + 48 T^{2} - 171 T^{3} + 624 T^{4} - 1183 T^{5} + 2197 T^{6} )^{2} \))
$17$ (\( 1 + 4 T - T^{2} + 68 T^{3} + 289 T^{4} \))(\( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} \))(\( 1 - 10 T + 44 T^{2} - 220 T^{3} + 1147 T^{4} - 3740 T^{5} + 12716 T^{6} - 49130 T^{7} + 83521 T^{8} \))(\( 1 + 4 T - 4 T^{2} - 56 T^{3} - 161 T^{4} - 952 T^{5} - 1156 T^{6} + 19652 T^{7} + 83521 T^{8} \))(\( 1 - 4 T - 14 T^{2} + 16 T^{3} + 339 T^{4} + 272 T^{5} - 4046 T^{6} - 19652 T^{7} + 83521 T^{8} \))(\( 1 - 2 T - 43 T^{2} + 30 T^{3} + 1286 T^{4} - 296 T^{5} - 25039 T^{6} - 5032 T^{7} + 371654 T^{8} + 147390 T^{9} - 3591403 T^{10} - 2839714 T^{11} + 24137569 T^{12} \))(\( 1 + 2 T - 43 T^{2} - 30 T^{3} + 1286 T^{4} + 296 T^{5} - 25039 T^{6} + 5032 T^{7} + 371654 T^{8} - 147390 T^{9} - 3591403 T^{10} + 2839714 T^{11} + 24137569 T^{12} \))
$19$ (\( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( 1 + 5 T + 6 T^{2} + 95 T^{3} + 361 T^{4} \))(\( 1 + 2 T - 23 T^{2} - 22 T^{3} + 292 T^{4} - 418 T^{5} - 8303 T^{6} + 13718 T^{7} + 130321 T^{8} \))(\( 1 - 2 T - 3 T^{2} + 62 T^{3} - 388 T^{4} + 1178 T^{5} - 1083 T^{6} - 13718 T^{7} + 130321 T^{8} \))(\( 1 - 30 T^{2} + 539 T^{4} - 10830 T^{6} + 130321 T^{8} \))(\( 1 - T - 15 T^{2} + 146 T^{3} - 129 T^{4} - 1049 T^{5} + 15054 T^{6} - 19931 T^{7} - 46569 T^{8} + 1001414 T^{9} - 1954815 T^{10} - 2476099 T^{11} + 47045881 T^{12} \))(\( 1 - T - 15 T^{2} + 146 T^{3} - 129 T^{4} - 1049 T^{5} + 15054 T^{6} - 19931 T^{7} - 46569 T^{8} + 1001414 T^{9} - 1954815 T^{10} - 2476099 T^{11} + 47045881 T^{12} \))
$23$ (\( 1 + 4 T - 7 T^{2} + 92 T^{3} + 529 T^{4} \))(\( 1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4} \))(\( 1 + 6 T - 16 T^{2} + 36 T^{3} + 1347 T^{4} + 828 T^{5} - 8464 T^{6} + 73002 T^{7} + 279841 T^{8} \))(\( 1 + 4 T - 16 T^{2} - 56 T^{3} + 127 T^{4} - 1288 T^{5} - 8464 T^{6} + 48668 T^{7} + 279841 T^{8} \))(\( 1 + 2 T - 41 T^{2} - 2 T^{3} + 1404 T^{4} - 46 T^{5} - 21689 T^{6} + 24334 T^{7} + 279841 T^{8} \))(\( 1 + 10 T + 47 T^{2} + 126 T^{3} - 430 T^{4} - 8384 T^{5} - 52369 T^{6} - 192832 T^{7} - 227470 T^{8} + 1533042 T^{9} + 13152527 T^{10} + 64363430 T^{11} + 148035889 T^{12} \))(\( 1 - 10 T + 47 T^{2} - 126 T^{3} - 430 T^{4} + 8384 T^{5} - 52369 T^{6} + 192832 T^{7} - 227470 T^{8} - 1533042 T^{9} + 13152527 T^{10} - 64363430 T^{11} + 148035889 T^{12} \))
$29$ (\( ( 1 - 8 T + 29 T^{2} )^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 2 T + 32 T^{2} + 58 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 + 8 T + 56 T^{2} + 232 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 - T + 29 T^{2} )^{4} \))(\( ( 1 - 4 T + 73 T^{2} - 178 T^{3} + 2117 T^{4} - 3364 T^{5} + 24389 T^{6} )^{2} \))(\( ( 1 + 4 T + 73 T^{2} + 178 T^{3} + 2117 T^{4} + 3364 T^{5} + 24389 T^{6} )^{2} \))
$31$ (\( 1 + T - 30 T^{2} + 31 T^{3} + 961 T^{4} \))(\( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \))(\( 1 + 6 T - 23 T^{2} - 18 T^{3} + 1404 T^{4} - 558 T^{5} - 22103 T^{6} + 178746 T^{7} + 923521 T^{8} \))(\( 1 - 6 T - 27 T^{2} - 6 T^{3} + 1892 T^{4} - 186 T^{5} - 25947 T^{6} - 178746 T^{7} + 923521 T^{8} \))(\( ( 1 - 6 T + 5 T^{2} - 186 T^{3} + 961 T^{4} )^{2} \))(\( 1 - T - 23 T^{2} + 298 T^{3} - 329 T^{4} - 3337 T^{5} + 66686 T^{6} - 103447 T^{7} - 316169 T^{8} + 8877718 T^{9} - 21240983 T^{10} - 28629151 T^{11} + 887503681 T^{12} \))(\( 1 - T - 23 T^{2} + 298 T^{3} - 329 T^{4} - 3337 T^{5} + 66686 T^{6} - 103447 T^{7} - 316169 T^{8} + 8877718 T^{9} - 21240983 T^{10} - 28629151 T^{11} + 887503681 T^{12} \))
$37$ (\( 1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4} \))(\( 1 - 7 T + 12 T^{2} - 259 T^{3} + 1369 T^{4} \))(\( 1 + 4 T - 35 T^{2} - 92 T^{3} + 640 T^{4} - 3404 T^{5} - 47915 T^{6} + 202612 T^{7} + 1874161 T^{8} \))(\( 1 - 14 T + 75 T^{2} - 658 T^{3} + 6020 T^{4} - 24346 T^{5} + 102675 T^{6} - 709142 T^{7} + 1874161 T^{8} \))(\( ( 1 - 37 T^{2} + 1369 T^{4} )^{2} \))(\( 1 + 11 T - 5 T^{2} - 224 T^{3} + 2359 T^{4} + 13925 T^{5} - 3730 T^{6} + 515225 T^{7} + 3229471 T^{8} - 11346272 T^{9} - 9370805 T^{10} + 762783527 T^{11} + 2565726409 T^{12} \))(\( 1 + 11 T - 5 T^{2} - 224 T^{3} + 2359 T^{4} + 13925 T^{5} - 3730 T^{6} + 515225 T^{7} + 3229471 T^{8} - 11346272 T^{9} - 9370805 T^{10} + 762783527 T^{11} + 2565726409 T^{12} \))
$41$ (\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 12 T + 41 T^{2} )^{2} \))(\( ( 1 + 2 T + 80 T^{2} + 82 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 - 4 T + 68 T^{2} - 164 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 - 10 T + 99 T^{2} - 410 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 - 2 T + 65 T^{2} - 182 T^{3} + 2665 T^{4} - 3362 T^{5} + 68921 T^{6} )^{2} \))(\( ( 1 + 2 T + 65 T^{2} + 182 T^{3} + 2665 T^{4} + 3362 T^{5} + 68921 T^{6} )^{2} \))
$43$ (\( ( 1 - T + 43 T^{2} )^{2} \))(\( ( 1 + T + 43 T^{2} )^{2} \))(\( ( 1 + 4 T + 63 T^{2} + 172 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 + 18 T + 165 T^{2} + 774 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 - 10 T + 109 T^{2} - 430 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 + 3 T + 88 T^{2} + 209 T^{3} + 3784 T^{4} + 5547 T^{5} + 79507 T^{6} )^{2} \))(\( ( 1 + 3 T + 88 T^{2} + 209 T^{3} + 3784 T^{4} + 5547 T^{5} + 79507 T^{6} )^{2} \))
$47$ (\( 1 - 2 T - 43 T^{2} - 94 T^{3} + 2209 T^{4} \))(\( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} \))(\( ( 1 - 2 T - 43 T^{2} - 94 T^{3} + 2209 T^{4} )^{2} \))(\( 1 + 4 T + 46 T^{2} - 496 T^{3} - 2061 T^{4} - 23312 T^{5} + 101614 T^{6} + 415292 T^{7} + 4879681 T^{8} \))(\( ( 1 - 2 T - 43 T^{2} - 94 T^{3} + 2209 T^{4} )^{2} \))(\( 1 - 2 T - 69 T^{2} - 106 T^{3} + 1858 T^{4} + 7846 T^{5} - 51253 T^{6} + 368762 T^{7} + 4104322 T^{8} - 11005238 T^{9} - 336697989 T^{10} - 458690014 T^{11} + 10779215329 T^{12} \))(\( 1 + 2 T - 69 T^{2} + 106 T^{3} + 1858 T^{4} - 7846 T^{5} - 51253 T^{6} - 368762 T^{7} + 4104322 T^{8} + 11005238 T^{9} - 336697989 T^{10} + 458690014 T^{11} + 10779215329 T^{12} \))
$53$ (\( 1 - 4 T - 37 T^{2} - 212 T^{3} + 2809 T^{4} \))(\( 1 - 53 T^{2} + 2809 T^{4} \))(\( 1 - 4 T + 14 T^{2} + 416 T^{3} - 3653 T^{4} + 22048 T^{5} + 39326 T^{6} - 595508 T^{7} + 7890481 T^{8} \))(\( 1 - 8 T - 50 T^{2} - 64 T^{3} + 6795 T^{4} - 3392 T^{5} - 140450 T^{6} - 1191016 T^{7} + 7890481 T^{8} \))(\( 1 + 8 T - 50 T^{2} + 64 T^{3} + 6795 T^{4} + 3392 T^{5} - 140450 T^{6} + 1191016 T^{7} + 7890481 T^{8} \))(\( 1 + 4 T - 127 T^{2} - 180 T^{3} + 11366 T^{4} + 4372 T^{5} - 689611 T^{6} + 231716 T^{7} + 31927094 T^{8} - 26797860 T^{9} - 1002091087 T^{10} + 1672781972 T^{11} + 22164361129 T^{12} \))(\( 1 - 4 T - 127 T^{2} + 180 T^{3} + 11366 T^{4} - 4372 T^{5} - 689611 T^{6} - 231716 T^{7} + 31927094 T^{8} + 26797860 T^{9} - 1002091087 T^{10} - 1672781972 T^{11} + 22164361129 T^{12} \))
$59$ (\( 1 + 8 T + 5 T^{2} + 472 T^{3} + 3481 T^{4} \))(\( 1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4} \))(\( 1 - 10 T - 16 T^{2} + 20 T^{3} + 4075 T^{4} + 1180 T^{5} - 55696 T^{6} - 2053790 T^{7} + 12117361 T^{8} \))(\( 1 - 116 T^{2} + 9975 T^{4} - 403796 T^{6} + 12117361 T^{8} \))(\( 1 + 8 T + 2 T^{2} - 448 T^{3} - 3413 T^{4} - 26432 T^{5} + 6962 T^{6} + 1643032 T^{7} + 12117361 T^{8} \))(\( 1 + 4 T + 9 T^{2} - 472 T^{3} - 1364 T^{4} + 12370 T^{5} + 414671 T^{6} + 729830 T^{7} - 4748084 T^{8} - 96938888 T^{9} + 109056249 T^{10} + 2859697196 T^{11} + 42180533641 T^{12} \))(\( 1 - 4 T + 9 T^{2} + 472 T^{3} - 1364 T^{4} - 12370 T^{5} + 414671 T^{6} - 729830 T^{7} - 4748084 T^{8} + 96938888 T^{9} + 109056249 T^{10} - 2859697196 T^{11} + 42180533641 T^{12} \))
$61$ (\( ( 1 - 13 T + 61 T^{2} )( 1 - T + 61 T^{2} ) \))(\( 1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4} \))(\( ( 1 + 4 T - 45 T^{2} + 244 T^{3} + 3721 T^{4} )^{2} \))(\( 1 + 8 T - 2 T^{2} - 448 T^{3} - 3269 T^{4} - 27328 T^{5} - 7442 T^{6} + 1815848 T^{7} + 13845841 T^{8} \))(\( 1 - 6 T - 23 T^{2} + 378 T^{3} - 2436 T^{4} + 23058 T^{5} - 85583 T^{6} - 1361886 T^{7} + 13845841 T^{8} \))(\( 1 - 22 T + 157 T^{2} - 1322 T^{3} + 22390 T^{4} - 164266 T^{5} + 750425 T^{6} - 10020226 T^{7} + 83313190 T^{8} - 300068882 T^{9} + 2173797037 T^{10} - 18581118622 T^{11} + 51520374361 T^{12} \))(\( 1 - 22 T + 157 T^{2} - 1322 T^{3} + 22390 T^{4} - 164266 T^{5} + 750425 T^{6} - 10020226 T^{7} + 83313190 T^{8} - 300068882 T^{9} + 2173797037 T^{10} - 18581118622 T^{11} + 51520374361 T^{12} \))
$67$ (\( 1 + 7 T - 18 T^{2} + 469 T^{3} + 4489 T^{4} \))(\( 1 - 7 T - 18 T^{2} - 469 T^{3} + 4489 T^{4} \))(\( 1 - 12 T + 49 T^{2} + 468 T^{3} - 5112 T^{4} + 31356 T^{5} + 219961 T^{6} - 3609156 T^{7} + 20151121 T^{8} \))(\( 1 - 2 T + 31 T^{2} + 322 T^{3} - 4028 T^{4} + 21574 T^{5} + 139159 T^{6} - 601526 T^{7} + 20151121 T^{8} \))(\( 1 + 22 T + 231 T^{2} + 2618 T^{3} + 27092 T^{4} + 175406 T^{5} + 1036959 T^{6} + 6616786 T^{7} + 20151121 T^{8} \))(\( 1 - 15 T - 13 T^{2} + 148 T^{3} + 12813 T^{4} - 31397 T^{5} - 747766 T^{6} - 2103599 T^{7} + 57517557 T^{8} + 44512924 T^{9} - 261964573 T^{10} - 20251876605 T^{11} + 90458382169 T^{12} \))(\( 1 - 15 T - 13 T^{2} + 148 T^{3} + 12813 T^{4} - 31397 T^{5} - 747766 T^{6} - 2103599 T^{7} + 57517557 T^{8} + 44512924 T^{9} - 261964573 T^{10} - 20251876605 T^{11} + 90458382169 T^{12} \))
$71$ (\( ( 1 + 6 T + 71 T^{2} )^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 2 T + 116 T^{2} + 142 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 - 4 T + 144 T^{2} - 284 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 - 8 T + 86 T^{2} - 568 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 - 16 T + 187 T^{2} - 1390 T^{3} + 13277 T^{4} - 80656 T^{5} + 357911 T^{6} )^{2} \))(\( ( 1 + 16 T + 187 T^{2} + 1390 T^{3} + 13277 T^{4} + 80656 T^{5} + 357911 T^{6} )^{2} \))
$73$ (\( 1 + T - 72 T^{2} + 73 T^{3} + 5329 T^{4} \))(\( 1 + 11 T + 48 T^{2} + 803 T^{3} + 5329 T^{4} \))(\( 1 + 8 T - 23 T^{2} - 472 T^{3} - 2432 T^{4} - 34456 T^{5} - 122567 T^{6} + 3112136 T^{7} + 28398241 T^{8} \))(\( 1 - 10 T - 21 T^{2} + 250 T^{3} + 2012 T^{4} + 18250 T^{5} - 111909 T^{6} - 3890170 T^{7} + 28398241 T^{8} \))(\( 1 + 4 T - 126 T^{2} - 16 T^{3} + 13667 T^{4} - 1168 T^{5} - 671454 T^{6} + 1556068 T^{7} + 28398241 T^{8} \))(\( 1 + 9 T - 49 T^{2} - 1532 T^{3} - 4749 T^{4} + 52771 T^{5} + 937886 T^{6} + 3852283 T^{7} - 25307421 T^{8} - 595974044 T^{9} - 1391513809 T^{10} + 18657644337 T^{11} + 151334226289 T^{12} \))(\( 1 + 9 T - 49 T^{2} - 1532 T^{3} - 4749 T^{4} + 52771 T^{5} + 937886 T^{6} + 3852283 T^{7} - 25307421 T^{8} - 595974044 T^{9} - 1391513809 T^{10} + 18657644337 T^{11} + 151334226289 T^{12} \))
$79$ (\( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \))(\( ( 1 - 17 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} ) \))(\( 1 + 6 T - 23 T^{2} - 594 T^{3} - 5604 T^{4} - 46926 T^{5} - 143543 T^{6} + 2958234 T^{7} + 38950081 T^{8} \))(\( 1 + 2 T - 123 T^{2} - 62 T^{3} + 9572 T^{4} - 4898 T^{5} - 767643 T^{6} + 986078 T^{7} + 38950081 T^{8} \))(\( 1 + 24 T + 282 T^{2} + 3264 T^{3} + 34691 T^{4} + 257856 T^{5} + 1759962 T^{6} + 11832936 T^{7} + 38950081 T^{8} \))(\( 1 - 7 T - 135 T^{2} + 1070 T^{3} + 11067 T^{4} - 57119 T^{5} - 604314 T^{6} - 4512401 T^{7} + 69069147 T^{8} + 527551730 T^{9} - 5258260935 T^{10} - 21539394793 T^{11} + 243087455521 T^{12} \))(\( 1 - 7 T - 135 T^{2} + 1070 T^{3} + 11067 T^{4} - 57119 T^{5} - 604314 T^{6} - 4512401 T^{7} + 69069147 T^{8} + 527551730 T^{9} - 5258260935 T^{10} - 21539394793 T^{11} + 243087455521 T^{12} \))
$83$ (\( ( 1 + 2 T + 83 T^{2} )^{2} \))(\( ( 1 - 12 T + 83 T^{2} )^{2} \))(\( ( 1 + 6 T + 28 T^{2} + 498 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 + 8 T + 180 T^{2} + 664 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 + 2 T + 5 T^{2} + 166 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 - 14 T + 289 T^{2} - 2342 T^{3} + 23987 T^{4} - 96446 T^{5} + 571787 T^{6} )^{2} \))(\( ( 1 + 14 T + 289 T^{2} + 2342 T^{3} + 23987 T^{4} + 96446 T^{5} + 571787 T^{6} )^{2} \))
$89$ (\( 1 + 12 T + 55 T^{2} + 1068 T^{3} + 7921 T^{4} \))(\( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 - 6 T - 4 T^{2} + 828 T^{3} - 9525 T^{4} + 73692 T^{5} - 31684 T^{6} - 4229814 T^{7} + 62742241 T^{8} \))(\( 1 + 16 T + 32 T^{2} + 736 T^{3} + 19471 T^{4} + 65504 T^{5} + 253472 T^{6} + 11279504 T^{7} + 62742241 T^{8} \))(\( 1 + 6 T - 119 T^{2} - 138 T^{3} + 12900 T^{4} - 12282 T^{5} - 942599 T^{6} + 4229814 T^{7} + 62742241 T^{8} \))(\( 1 - 30 T + 371 T^{2} - 4122 T^{3} + 56540 T^{4} - 520116 T^{5} + 3879065 T^{6} - 46290324 T^{7} + 447853340 T^{8} - 2905882218 T^{9} + 23277371411 T^{10} - 167521783470 T^{11} + 496981290961 T^{12} \))(\( 1 + 30 T + 371 T^{2} + 4122 T^{3} + 56540 T^{4} + 520116 T^{5} + 3879065 T^{6} + 46290324 T^{7} + 447853340 T^{8} + 2905882218 T^{9} + 23277371411 T^{10} + 167521783470 T^{11} + 496981290961 T^{12} \))
$97$ (\( ( 1 + 6 T + 97 T^{2} )^{2} \))(\( ( 1 + 10 T + 97 T^{2} )^{2} \))(\( ( 1 - 16 T + 210 T^{2} - 1552 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 + 16 T + 250 T^{2} + 1552 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 12 T + 198 T^{2} - 1164 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 + 4 T + 211 T^{2} + 564 T^{3} + 20467 T^{4} + 37636 T^{5} + 912673 T^{6} )^{2} \))(\( ( 1 + 4 T + 211 T^{2} + 564 T^{3} + 20467 T^{4} + 37636 T^{5} + 912673 T^{6} )^{2} \))
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