Properties

Label 1340.2.a
Level 1340
Weight 2
Character orbit a
Rep. character \(\chi_{1340}(1,\cdot)\)
Character field \(\Q\)
Dimension 22
Newform subspaces 9
Sturm bound 408
Trace bound 7

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Defining parameters

Level: \( N \) = \( 1340 = 2^{2} \cdot 5 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1340.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 9 \)
Sturm bound: \(408\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1340))\).

Total New Old
Modular forms 210 22 188
Cusp forms 199 22 177
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.

\(2\)\(5\)\(67\)FrickeDim.
\(-\)\(+\)\(+\)\(-\)\(5\)
\(-\)\(+\)\(-\)\(+\)\(6\)
\(-\)\(-\)\(+\)\(+\)\(5\)
\(-\)\(-\)\(-\)\(-\)\(6\)
Plus space\(+\)\(11\)
Minus space\(-\)\(11\)

Trace form

\( 22q + 18q^{9} + O(q^{10}) \) \( 22q + 18q^{9} - 4q^{15} + 6q^{17} + 2q^{19} - 6q^{23} + 22q^{25} - 24q^{27} - 18q^{29} - 8q^{33} + 8q^{35} + 10q^{37} - 16q^{39} - 12q^{41} - 12q^{43} - 8q^{45} - 10q^{47} + 42q^{49} - 24q^{51} + 20q^{53} - 32q^{57} - 26q^{59} - 12q^{61} + 8q^{65} + 2q^{67} + 8q^{69} + 36q^{71} - 6q^{73} - 8q^{77} - 24q^{79} + 46q^{81} - 24q^{83} - 12q^{85} + 12q^{87} + 14q^{89} + 16q^{91} + 28q^{93} - 8q^{95} + 24q^{97} - 12q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1340))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 5 67
1340.2.a.a \(1\) \(10.700\) \(\Q\) None \(0\) \(-2\) \(-1\) \(2\) \(-\) \(+\) \(+\) \(q-2q^{3}-q^{5}+2q^{7}+q^{9}+2q^{13}+\cdots\)
1340.2.a.b \(1\) \(10.700\) \(\Q\) None \(0\) \(1\) \(-1\) \(1\) \(-\) \(+\) \(-\) \(q+q^{3}-q^{5}+q^{7}-2q^{9}-6q^{11}+2q^{13}+\cdots\)
1340.2.a.c \(1\) \(10.700\) \(\Q\) None \(0\) \(1\) \(-1\) \(5\) \(-\) \(+\) \(+\) \(q+q^{3}-q^{5}+5q^{7}-2q^{9}+2q^{13}+\cdots\)
1340.2.a.d \(2\) \(10.700\) \(\Q(\sqrt{17}) \) None \(0\) \(-1\) \(2\) \(-3\) \(-\) \(-\) \(+\) \(q-\beta q^{3}+q^{5}+(-2+\beta )q^{7}+(1+\beta )q^{9}+\cdots\)
1340.2.a.e \(2\) \(10.700\) \(\Q(\sqrt{3}) \) None \(0\) \(2\) \(2\) \(4\) \(-\) \(-\) \(-\) \(q+(1+\beta )q^{3}+q^{5}+2q^{7}+(1+2\beta )q^{9}+\cdots\)
1340.2.a.f \(3\) \(10.700\) \(\Q(\zeta_{14})^+\) None \(0\) \(-2\) \(3\) \(-2\) \(-\) \(-\) \(+\) \(q+(-1-\beta _{2})q^{3}+q^{5}+(1-3\beta _{1}+2\beta _{2})q^{7}+\cdots\)
1340.2.a.g \(3\) \(10.700\) 3.3.257.1 None \(0\) \(4\) \(-3\) \(-2\) \(-\) \(+\) \(+\) \(q+(1+\beta _{1})q^{3}-q^{5}+(-1+\beta _{1}+\beta _{2})q^{7}+\cdots\)
1340.2.a.h \(4\) \(10.700\) 4.4.60513.1 None \(0\) \(-1\) \(4\) \(5\) \(-\) \(-\) \(-\) \(q-\beta _{1}q^{3}+q^{5}+(1+\beta _{3})q^{7}+(1+\beta _{1}+\cdots)q^{9}+\cdots\)
1340.2.a.i \(5\) \(10.700\) 5.5.1465016.1 None \(0\) \(-2\) \(-5\) \(-10\) \(-\) \(+\) \(-\) \(q-\beta _{1}q^{3}-q^{5}+(-2+\beta _{2})q^{7}+(2+\beta _{1}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1340))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1340)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(67))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(134))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(268))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(335))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(670))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 2 T + 3 T^{2} \))(\( 1 - T + 3 T^{2} \))(\( 1 - T + 3 T^{2} \))(\( 1 + T + 2 T^{2} + 3 T^{3} + 9 T^{4} \))(\( 1 - 2 T + 4 T^{2} - 6 T^{3} + 9 T^{4} \))(\( 1 + 2 T + 8 T^{2} + 11 T^{3} + 24 T^{4} + 18 T^{5} + 27 T^{6} \))(\( 1 - 4 T + 10 T^{2} - 19 T^{3} + 30 T^{4} - 36 T^{5} + 27 T^{6} \))(\( 1 + T + 5 T^{2} + 9 T^{3} + 15 T^{4} + 27 T^{5} + 45 T^{6} + 27 T^{7} + 81 T^{8} \))(\( 1 + 2 T + 4 T^{2} + 9 T^{3} + 13 T^{4} + 12 T^{5} + 39 T^{6} + 81 T^{7} + 108 T^{8} + 162 T^{9} + 243 T^{10} \))
$5$ (\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{3} \))(\( ( 1 + T )^{3} \))(\( ( 1 - T )^{4} \))(\( ( 1 + T )^{5} \))
$7$ (\( 1 - 2 T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 - 5 T + 7 T^{2} \))(\( 1 + 3 T + 12 T^{2} + 21 T^{3} + 49 T^{4} \))(\( ( 1 - 2 T + 7 T^{2} )^{2} \))(\( 1 + 2 T + 6 T^{2} - T^{3} + 42 T^{4} + 98 T^{5} + 343 T^{6} \))(\( 1 + 2 T + 12 T^{2} + 9 T^{3} + 84 T^{4} + 98 T^{5} + 343 T^{6} \))(\( 1 - 5 T + 21 T^{2} - 59 T^{3} + 179 T^{4} - 413 T^{5} + 1029 T^{6} - 1715 T^{7} + 2401 T^{8} \))(\( 1 + 10 T + 60 T^{2} + 277 T^{3} + 999 T^{4} + 2902 T^{5} + 6993 T^{6} + 13573 T^{7} + 20580 T^{8} + 24010 T^{9} + 16807 T^{10} \))
$11$ (\( 1 + 11 T^{2} \))(\( 1 + 6 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( ( 1 + 2 T + 11 T^{2} )^{2} \))(\( 1 + 6 T + 28 T^{2} + 66 T^{3} + 121 T^{4} \))(\( ( 1 + 2 T + 11 T^{2} )^{3} \))(\( ( 1 + 11 T^{2} )^{3} \))(\( ( 1 - 4 T + 11 T^{2} )^{4} \))(\( 1 - 6 T + 37 T^{2} - 140 T^{3} + 672 T^{4} - 2156 T^{5} + 7392 T^{6} - 16940 T^{7} + 49247 T^{8} - 87846 T^{9} + 161051 T^{10} \))
$13$ (\( 1 - 2 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( ( 1 + 2 T + 13 T^{2} )^{2} \))(\( ( 1 - 2 T + 13 T^{2} )^{2} \))(\( 1 - T + 2 T^{2} + 3 T^{3} + 26 T^{4} - 169 T^{5} + 2197 T^{6} \))(\( 1 + T + 6 T^{2} + 17 T^{3} + 78 T^{4} + 169 T^{5} + 2197 T^{6} \))(\( 1 - 3 T + 33 T^{2} - 122 T^{3} + 522 T^{4} - 1586 T^{5} + 5577 T^{6} - 6591 T^{7} + 28561 T^{8} \))(\( 1 + 9 T + 60 T^{2} + 283 T^{3} + 1199 T^{4} + 4280 T^{5} + 15587 T^{6} + 47827 T^{7} + 131820 T^{8} + 257049 T^{9} + 371293 T^{10} \))
$17$ (\( 1 + T + 17 T^{2} \))(\( 1 - 4 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 3 T - 2 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 + 31 T^{2} + 289 T^{4} \))(\( 1 - 2 T + 15 T^{2} - 60 T^{3} + 255 T^{4} - 578 T^{5} + 4913 T^{6} \))(\( 1 - 4 T + 39 T^{2} - 128 T^{3} + 663 T^{4} - 1156 T^{5} + 4913 T^{6} \))(\( 1 + 2 T + 40 T^{2} + 102 T^{3} + 830 T^{4} + 1734 T^{5} + 11560 T^{6} + 9826 T^{7} + 83521 T^{8} \))(\( 1 + 56 T^{2} - 2 T^{3} + 1511 T^{4} + 4 T^{5} + 25687 T^{6} - 578 T^{7} + 275128 T^{8} + 1419857 T^{10} \))
$19$ (\( 1 - T + 19 T^{2} \))(\( 1 + 8 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 7 T + 46 T^{2} + 133 T^{3} + 361 T^{4} \))(\( ( 1 - 5 T + 19 T^{2} )^{2} \))(\( 1 + 5 T + 56 T^{2} + 177 T^{3} + 1064 T^{4} + 1805 T^{5} + 6859 T^{6} \))(\( 1 - 3 T + 40 T^{2} - 119 T^{3} + 760 T^{4} - 1083 T^{5} + 6859 T^{6} \))(\( 1 + T + 23 T^{2} + 48 T^{3} + 220 T^{4} + 912 T^{5} + 8303 T^{6} + 6859 T^{7} + 130321 T^{8} \))(\( 1 - 5 T + 37 T^{2} - 98 T^{3} + 569 T^{4} - 1751 T^{5} + 10811 T^{6} - 35378 T^{7} + 253783 T^{8} - 651605 T^{9} + 2476099 T^{10} \))
$23$ (\( 1 + 9 T + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 - 9 T + 62 T^{2} - 207 T^{3} + 529 T^{4} \))(\( 1 + 19 T^{2} + 529 T^{4} \))(\( 1 + 14 T + 125 T^{2} + 700 T^{3} + 2875 T^{4} + 7406 T^{5} + 12167 T^{6} \))(\( ( 1 - 6 T + 23 T^{2} )^{3} \))(\( ( 1 - 2 T + 23 T^{2} )^{4} \))(\( 1 + 16 T + 194 T^{2} + 1618 T^{3} + 10821 T^{4} + 57492 T^{5} + 248883 T^{6} + 855922 T^{7} + 2360398 T^{8} + 4477456 T^{9} + 6436343 T^{10} \))
$29$ (\( 1 + 9 T + 29 T^{2} \))(\( 1 + 7 T + 29 T^{2} \))(\( 1 - 9 T + 29 T^{2} \))(\( 1 + 12 T + 77 T^{2} + 348 T^{3} + 841 T^{4} \))(\( 1 + 6 T + 55 T^{2} + 174 T^{3} + 841 T^{4} \))(\( 1 - 4 T + 76 T^{2} - 189 T^{3} + 2204 T^{4} - 3364 T^{5} + 24389 T^{6} \))(\( 1 - 8 T + 104 T^{2} - 473 T^{3} + 3016 T^{4} - 6728 T^{5} + 24389 T^{6} \))(\( 1 - T + 21 T^{2} - 9 T^{3} - 455 T^{4} - 261 T^{5} + 17661 T^{6} - 24389 T^{7} + 707281 T^{8} \))(\( 1 + 6 T + 55 T^{2} + 113 T^{3} + 615 T^{4} - 3391 T^{5} + 17835 T^{6} + 95033 T^{7} + 1341395 T^{8} + 4243686 T^{9} + 20511149 T^{10} \))
$31$ (\( 1 - 6 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 + 4 T - 2 T^{2} + 124 T^{3} + 961 T^{4} \))(\( 1 - 10 T + 84 T^{2} - 310 T^{3} + 961 T^{4} \))(\( 1 + 12 T + 113 T^{2} + 752 T^{3} + 3503 T^{4} + 11532 T^{5} + 29791 T^{6} \))(\( 1 - 2 T + 61 T^{2} - 100 T^{3} + 1891 T^{4} - 1922 T^{5} + 29791 T^{6} \))(\( 1 + 2 T + 60 T^{2} + 18 T^{3} + 2086 T^{4} + 558 T^{5} + 57660 T^{6} + 59582 T^{7} + 923521 T^{8} \))(\( 1 - 4 T + 69 T^{2} - 184 T^{3} + 3300 T^{4} - 9112 T^{5} + 102300 T^{6} - 176824 T^{7} + 2055579 T^{8} - 3694084 T^{9} + 28629151 T^{10} \))
$37$ (\( 1 - 7 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 - 4 T + 37 T^{2} \))(\( 1 - 19 T + 160 T^{2} - 703 T^{3} + 1369 T^{4} \))(\( 1 - 4 T + 51 T^{2} - 148 T^{3} + 1369 T^{4} \))(\( 1 + 16 T + 187 T^{2} + 1288 T^{3} + 6919 T^{4} + 21904 T^{5} + 50653 T^{6} \))(\( 1 - 8 T + 63 T^{2} - 528 T^{3} + 2331 T^{4} - 10952 T^{5} + 50653 T^{6} \))(\( ( 1 + 37 T^{2} )^{4} \))(\( 1 + 14 T + 144 T^{2} + 1092 T^{3} + 7647 T^{4} + 47428 T^{5} + 282939 T^{6} + 1494948 T^{7} + 7294032 T^{8} + 26238254 T^{9} + 69343957 T^{10} \))
$41$ (\( 1 + 4 T + 41 T^{2} \))(\( 1 + 4 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 - 6 T + 74 T^{2} - 246 T^{3} + 1681 T^{4} \))(\( 1 + 6 T + 88 T^{2} + 246 T^{3} + 1681 T^{4} \))(\( 1 + 18 T + 203 T^{2} + 1580 T^{3} + 8323 T^{4} + 30258 T^{5} + 68921 T^{6} \))(\( 1 + 10 T + 123 T^{2} + 748 T^{3} + 5043 T^{4} + 16810 T^{5} + 68921 T^{6} \))(\( 1 - 12 T + 144 T^{2} - 1164 T^{3} + 9134 T^{4} - 47724 T^{5} + 242064 T^{6} - 827052 T^{7} + 2825761 T^{8} \))(\( 1 - 10 T + 135 T^{2} - 916 T^{3} + 9144 T^{4} - 50980 T^{5} + 374904 T^{6} - 1539796 T^{7} + 9304335 T^{8} - 28257610 T^{9} + 115856201 T^{10} \))
$43$ (\( 1 - 10 T + 43 T^{2} \))(\( 1 + 11 T + 43 T^{2} \))(\( 1 - T + 43 T^{2} \))(\( 1 + 7 T + 60 T^{2} + 301 T^{3} + 1849 T^{4} \))(\( 1 - 4 T + 42 T^{2} - 172 T^{3} + 1849 T^{4} \))(\( 1 + 12 T + 114 T^{2} + 655 T^{3} + 4902 T^{4} + 22188 T^{5} + 79507 T^{6} \))(\( 1 - 10 T + 138 T^{2} - 779 T^{3} + 5934 T^{4} - 18490 T^{5} + 79507 T^{6} \))(\( 1 - 13 T + 167 T^{2} - 1401 T^{3} + 10967 T^{4} - 60243 T^{5} + 308783 T^{6} - 1033591 T^{7} + 3418801 T^{8} \))(\( 1 + 20 T + 316 T^{2} + 3441 T^{3} + 31079 T^{4} + 221634 T^{5} + 1336397 T^{6} + 6362409 T^{7} + 25124212 T^{8} + 68376020 T^{9} + 147008443 T^{10} \))
$47$ (\( 1 - 9 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 13 T + 98 T^{2} + 611 T^{3} + 2209 T^{4} \))(\( 1 + 67 T^{2} + 2209 T^{4} \))(\( 1 - 14 T + 29 T^{2} + 308 T^{3} + 1363 T^{4} - 30926 T^{5} + 103823 T^{6} \))(\( 1 + 2 T + 101 T^{2} + 260 T^{3} + 4747 T^{4} + 4418 T^{5} + 103823 T^{6} \))(\( 1 + 2 T + 124 T^{2} + 114 T^{3} + 7526 T^{4} + 5358 T^{5} + 273916 T^{6} + 207646 T^{7} + 4879681 T^{8} \))(\( 1 + 24 T + 418 T^{2} + 5110 T^{3} + 48749 T^{4} + 374732 T^{5} + 2291203 T^{6} + 11287990 T^{7} + 43398014 T^{8} + 117112344 T^{9} + 229345007 T^{10} \))
$53$ (\( 1 - 14 T + 53 T^{2} \))(\( 1 - 3 T + 53 T^{2} \))(\( 1 + 13 T + 53 T^{2} \))(\( 1 + 7 T + 114 T^{2} + 371 T^{3} + 2809 T^{4} \))(\( 1 + 6 T + 112 T^{2} + 318 T^{3} + 2809 T^{4} \))(\( 1 - 12 T + 60 T^{2} - 111 T^{3} + 3180 T^{4} - 33708 T^{5} + 148877 T^{6} \))(\( 1 - 16 T + 234 T^{2} - 1805 T^{3} + 12402 T^{4} - 44944 T^{5} + 148877 T^{6} \))(\( 1 - 13 T + 259 T^{2} - 2085 T^{3} + 21755 T^{4} - 110505 T^{5} + 727531 T^{6} - 1935401 T^{7} + 7890481 T^{8} \))(\( 1 + 12 T + 196 T^{2} + 1835 T^{3} + 15809 T^{4} + 126412 T^{5} + 837877 T^{6} + 5154515 T^{7} + 29179892 T^{8} + 94685772 T^{9} + 418195493 T^{10} \))
$59$ (\( 1 - 7 T + 59 T^{2} \))(\( 1 - 9 T + 59 T^{2} \))(\( 1 - T + 59 T^{2} \))(\( 1 + 12 T + 137 T^{2} + 708 T^{3} + 3481 T^{4} \))(\( 1 - 6 T + 19 T^{2} - 354 T^{3} + 3481 T^{4} \))(\( 1 + 16 T + 218 T^{2} + 1691 T^{3} + 12862 T^{4} + 55696 T^{5} + 205379 T^{6} \))(\( 1 + 8 T + 174 T^{2} + 935 T^{3} + 10266 T^{4} + 27848 T^{5} + 205379 T^{6} \))(\( 1 + 7 T + 225 T^{2} + 1209 T^{3} + 19629 T^{4} + 71331 T^{5} + 783225 T^{6} + 1437653 T^{7} + 12117361 T^{8} \))(\( 1 + 6 T + 241 T^{2} + 1177 T^{3} + 25359 T^{4} + 97117 T^{5} + 1496181 T^{6} + 4097137 T^{7} + 49496339 T^{8} + 72704166 T^{9} + 714924299 T^{10} \))
$61$ (\( 1 - 2 T + 61 T^{2} \))(\( 1 + 12 T + 61 T^{2} \))(\( 1 + 4 T + 61 T^{2} \))(\( 1 - 4 T + 58 T^{2} - 244 T^{3} + 3721 T^{4} \))(\( 1 - 10 T + 72 T^{2} - 610 T^{3} + 3721 T^{4} \))(\( 1 + 18 T + 263 T^{2} + 2188 T^{3} + 16043 T^{4} + 66978 T^{5} + 226981 T^{6} \))(\( 1 - 10 T + 119 T^{2} - 1228 T^{3} + 7259 T^{4} - 37210 T^{5} + 226981 T^{6} \))(\( 1 - 2 T + 156 T^{2} - 438 T^{3} + 11782 T^{4} - 26718 T^{5} + 580476 T^{6} - 453962 T^{7} + 13845841 T^{8} \))(\( 1 + 6 T + 127 T^{2} + 492 T^{3} + 11004 T^{4} + 48476 T^{5} + 671244 T^{6} + 1830732 T^{7} + 28826587 T^{8} + 83075046 T^{9} + 844596301 T^{10} \))
$67$ (\( 1 + T \))(\( 1 - T \))(\( 1 + T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{3} \))(\( ( 1 + T )^{3} \))(\( ( 1 - T )^{4} \))(\( ( 1 - T )^{5} \))
$71$ (\( 1 - 8 T + 71 T^{2} \))(\( 1 - 12 T + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( ( 1 + 4 T + 71 T^{2} )^{2} \))(\( 1 + 130 T^{2} + 5041 T^{4} \))(\( 1 + 11 T + 188 T^{2} + 1519 T^{3} + 13348 T^{4} + 55451 T^{5} + 357911 T^{6} \))(\( 1 + 7 T + 148 T^{2} + 563 T^{3} + 10508 T^{4} + 35287 T^{5} + 357911 T^{6} \))(\( 1 - 25 T + 283 T^{2} - 1708 T^{3} + 9480 T^{4} - 121268 T^{5} + 1426603 T^{6} - 8947775 T^{7} + 25411681 T^{8} \))(\( 1 - 9 T + 170 T^{2} - 1223 T^{3} + 16733 T^{4} - 94460 T^{5} + 1188043 T^{6} - 6165143 T^{7} + 60844870 T^{8} - 228705129 T^{9} + 1804229351 T^{10} \))
$73$ (\( 1 + 3 T + 73 T^{2} \))(\( 1 - 14 T + 73 T^{2} \))(\( 1 - 12 T + 73 T^{2} \))(\( 1 - 13 T + 184 T^{2} - 949 T^{3} + 5329 T^{4} \))(\( 1 - 16 T + 207 T^{2} - 1168 T^{3} + 5329 T^{4} \))(\( 1 + 20 T + 287 T^{2} + 2592 T^{3} + 20951 T^{4} + 106580 T^{5} + 389017 T^{6} \))(\( 1 + 16 T + 287 T^{2} + 2408 T^{3} + 20951 T^{4} + 85264 T^{5} + 389017 T^{6} \))(\( 1 + 12 T + 224 T^{2} + 2508 T^{3} + 22014 T^{4} + 183084 T^{5} + 1193696 T^{6} + 4668204 T^{7} + 28398241 T^{8} \))(\( 1 + 10 T + 300 T^{2} + 2516 T^{3} + 41155 T^{4} + 259468 T^{5} + 3004315 T^{6} + 13407764 T^{7} + 116705100 T^{8} + 283982410 T^{9} + 2073071593 T^{10} \))
$79$ (\( 1 - 8 T + 79 T^{2} \))(\( 1 + 6 T + 79 T^{2} \))(\( 1 + 10 T + 79 T^{2} \))(\( 1 + 26 T + 310 T^{2} + 2054 T^{3} + 6241 T^{4} \))(\( 1 + 8 T + 126 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( 1 + 153 T^{2} - 56 T^{3} + 12087 T^{4} + 493039 T^{6} \))(\( ( 1 - 8 T + 79 T^{2} )^{3} \))(\( 1 - 34 T + 548 T^{2} - 5946 T^{3} + 54518 T^{4} - 469734 T^{5} + 3420068 T^{6} - 16763326 T^{7} + 38950081 T^{8} \))(\( 1 + 40 T + 967 T^{2} + 16040 T^{3} + 204630 T^{4} + 2033632 T^{5} + 16165770 T^{6} + 100105640 T^{7} + 476768713 T^{8} + 1558003240 T^{9} + 3077056399 T^{10} \))
$83$ (\( 1 + 4 T + 83 T^{2} \))(\( 1 + 16 T + 83 T^{2} \))(\( 1 - 14 T + 83 T^{2} \))(\( ( 1 - 4 T + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( 1 + 2 T + 17 T^{2} + 436 T^{3} + 1411 T^{4} + 13778 T^{5} + 571787 T^{6} \))(\( 1 + 16 T + 253 T^{2} + 2120 T^{3} + 20999 T^{4} + 110224 T^{5} + 571787 T^{6} \))(\( 1 - 10 T + 72 T^{2} + 630 T^{3} - 5138 T^{4} + 52290 T^{5} + 496008 T^{6} - 5717870 T^{7} + 47458321 T^{8} \))(\( 1 + 18 T + 351 T^{2} + 4128 T^{3} + 50522 T^{4} + 438780 T^{5} + 4193326 T^{6} + 28437792 T^{7} + 200697237 T^{8} + 854249778 T^{9} + 3939040643 T^{10} \))
$89$ (\( 1 + T + 89 T^{2} \))(\( 1 - T + 89 T^{2} \))(\( 1 - 5 T + 89 T^{2} \))(\( 1 - 4 T + 165 T^{2} - 356 T^{3} + 7921 T^{4} \))(\( 1 - 6 T + 175 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 + 4 T + 228 T^{2} + 543 T^{3} + 20292 T^{4} + 31684 T^{5} + 704969 T^{6} \))(\( 1 + 4 T + 264 T^{2} + 703 T^{3} + 23496 T^{4} + 31684 T^{5} + 704969 T^{6} \))(\( 1 + 7 T + 109 T^{2} + 627 T^{3} + 14117 T^{4} + 55803 T^{5} + 863389 T^{6} + 4934783 T^{7} + 62742241 T^{8} \))(\( 1 - 14 T + 103 T^{2} - 575 T^{3} + 13139 T^{4} - 192883 T^{5} + 1169371 T^{6} - 4554575 T^{7} + 72611807 T^{8} - 878391374 T^{9} + 5584059449 T^{10} \))
$97$ (\( 1 - 12 T + 97 T^{2} \))(\( 1 + 11 T + 97 T^{2} \))(\( 1 + 3 T + 97 T^{2} \))(\( 1 + 3 T - 12 T^{2} + 291 T^{3} + 9409 T^{4} \))(\( 1 - 10 T + 216 T^{2} - 970 T^{3} + 9409 T^{4} \))(\( 1 - 24 T + 476 T^{2} - 5105 T^{3} + 46172 T^{4} - 225816 T^{5} + 912673 T^{6} \))(\( 1 - 2 T + 272 T^{2} - 343 T^{3} + 26384 T^{4} - 18818 T^{5} + 912673 T^{6} \))(\( 1 + 23 T + 467 T^{2} + 5805 T^{3} + 68683 T^{4} + 563085 T^{5} + 4394003 T^{6} + 20991479 T^{7} + 88529281 T^{8} \))(\( 1 - 16 T + 356 T^{2} - 3567 T^{3} + 52721 T^{4} - 428704 T^{5} + 5113937 T^{6} - 33561903 T^{7} + 324911588 T^{8} - 1416468496 T^{9} + 8587340257 T^{10} \))
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