Properties

Label 1002.2.a
Level 1002
Weight 2
Character orbit a
Rep. character \(\chi_{1002}(1,\cdot)\)
Character field \(\Q\)
Dimension 29
Newform subspaces 11
Sturm bound 336
Trace bound 5

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Defining parameters

Level: \( N \) = \( 1002 = 2 \cdot 3 \cdot 167 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1002.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 11 \)
Sturm bound: \(336\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1002))\).

Total New Old
Modular forms 172 29 143
Cusp forms 165 29 136
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.

\(2\)\(3\)\(167\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(5\)
\(+\)\(+\)\(-\)\(-\)\(2\)
\(+\)\(-\)\(+\)\(-\)\(5\)
\(+\)\(-\)\(-\)\(+\)\(2\)
\(-\)\(+\)\(+\)\(-\)\(4\)
\(-\)\(+\)\(-\)\(+\)\(3\)
\(-\)\(-\)\(+\)\(+\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(7\)
Plus space\(+\)\(11\)
Minus space\(-\)\(18\)

Trace form

\( 29q + q^{2} + q^{3} + 29q^{4} + 6q^{5} + q^{6} + 8q^{7} + q^{8} + 29q^{9} + O(q^{10}) \) \( 29q + q^{2} + q^{3} + 29q^{4} + 6q^{5} + q^{6} + 8q^{7} + q^{8} + 29q^{9} + 2q^{10} - 4q^{11} + q^{12} - 6q^{13} - 8q^{14} - 2q^{15} + 29q^{16} - 6q^{17} + q^{18} - 4q^{19} + 6q^{20} + 8q^{21} + 4q^{22} - 8q^{23} + q^{24} + 35q^{25} + 6q^{26} + q^{27} + 8q^{28} - 2q^{29} + 2q^{30} + 8q^{31} + q^{32} - 4q^{33} + 2q^{34} - 16q^{35} + 29q^{36} - 6q^{37} - 12q^{38} + 6q^{39} + 2q^{40} - 14q^{41} + 8q^{42} - 16q^{43} - 4q^{44} + 6q^{45} - 8q^{46} - 32q^{47} + q^{48} + 53q^{49} - q^{50} + 6q^{51} - 6q^{52} - 10q^{53} + q^{54} - 8q^{55} - 8q^{56} + 4q^{57} + 30q^{58} - 12q^{59} - 2q^{60} + 6q^{61} + 8q^{63} + 29q^{64} - 12q^{65} + 4q^{66} - 16q^{67} - 6q^{68} + 8q^{69} + 8q^{70} - 24q^{71} + q^{72} + 26q^{73} - 2q^{74} + 15q^{75} - 4q^{76} - 32q^{77} - 6q^{78} - 16q^{79} + 6q^{80} + 29q^{81} - 6q^{82} - 20q^{83} + 8q^{84} - 12q^{85} + 4q^{86} - 18q^{87} + 4q^{88} - 22q^{89} + 2q^{90} + 24q^{91} - 8q^{92} + 32q^{93} - 24q^{95} + q^{96} - 6q^{97} + 9q^{98} - 4q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1002))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 167
1002.2.a.a \(1\) \(8.001\) \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
1002.2.a.b \(1\) \(8.001\) \(\Q\) None \(-1\) \(-1\) \(3\) \(3\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}+3q^{5}+q^{6}+3q^{7}+\cdots\)
1002.2.a.c \(1\) \(8.001\) \(\Q\) None \(-1\) \(1\) \(-1\) \(-1\) \(+\) \(-\) \(-\) \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}-q^{7}+\cdots\)
1002.2.a.d \(1\) \(8.001\) \(\Q\) None \(-1\) \(1\) \(2\) \(-4\) \(+\) \(-\) \(-\) \(q-q^{2}+q^{3}+q^{4}+2q^{5}-q^{6}-4q^{7}+\cdots\)
1002.2.a.e \(1\) \(8.001\) \(\Q\) None \(1\) \(1\) \(-3\) \(-3\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{3}+q^{4}-3q^{5}+q^{6}-3q^{7}+\cdots\)
1002.2.a.f \(2\) \(8.001\) \(\Q(\sqrt{2}) \) None \(-2\) \(-2\) \(-4\) \(0\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}+q^{4}+(-2+\beta )q^{5}+q^{6}+\cdots\)
1002.2.a.g \(3\) \(8.001\) 3.3.1300.1 None \(-3\) \(-3\) \(3\) \(1\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}+q^{4}+(1-\beta _{1})q^{5}+q^{6}+\cdots\)
1002.2.a.h \(3\) \(8.001\) 3.3.148.1 None \(3\) \(-3\) \(-3\) \(-5\) \(-\) \(+\) \(-\) \(q+q^{2}-q^{3}+q^{4}+(-1+\beta _{2})q^{5}-q^{6}+\cdots\)
1002.2.a.i \(4\) \(8.001\) 4.4.2777.1 None \(4\) \(-4\) \(5\) \(1\) \(-\) \(+\) \(+\) \(q+q^{2}-q^{3}+q^{4}+(1-\beta _{1})q^{5}-q^{6}+\cdots\)
1002.2.a.j \(5\) \(8.001\) 5.5.11256624.1 None \(-5\) \(5\) \(-1\) \(9\) \(+\) \(-\) \(+\) \(q-q^{2}+q^{3}+q^{4}+\beta _{2}q^{5}-q^{6}+(2+\cdots)q^{7}+\cdots\)
1002.2.a.k \(7\) \(8.001\) \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(7\) \(7\) \(5\) \(7\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{3}+q^{4}+(1-\beta _{1})q^{5}+q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1002))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1002)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(167))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(334))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(501))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{3} \))(\( ( 1 - T )^{3} \))(\( ( 1 - T )^{4} \))(\( ( 1 + T )^{5} \))(\( ( 1 - T )^{7} \))
$3$ (\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{3} \))(\( ( 1 + T )^{3} \))(\( ( 1 + T )^{4} \))(\( ( 1 - T )^{5} \))(\( ( 1 - T )^{7} \))
$5$ (\( 1 + 5 T^{2} \))(\( 1 - 3 T + 5 T^{2} \))(\( 1 + T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 + 3 T + 5 T^{2} \))(\( 1 + 4 T + 12 T^{2} + 20 T^{3} + 25 T^{4} \))(\( 1 - 3 T + 8 T^{2} - 11 T^{3} + 40 T^{4} - 75 T^{5} + 125 T^{6} \))(\( 1 + 3 T + 14 T^{2} + 29 T^{3} + 70 T^{4} + 75 T^{5} + 125 T^{6} \))(\( 1 - 5 T + 20 T^{2} - 55 T^{3} + 142 T^{4} - 275 T^{5} + 500 T^{6} - 625 T^{7} + 625 T^{8} \))(\( 1 + T + 6 T^{2} - T^{3} + 25 T^{4} - 36 T^{5} + 125 T^{6} - 25 T^{7} + 750 T^{8} + 625 T^{9} + 3125 T^{10} \))(\( 1 - 5 T + 24 T^{2} - 57 T^{3} + 140 T^{4} - 125 T^{5} + 183 T^{6} + 286 T^{7} + 915 T^{8} - 3125 T^{9} + 17500 T^{10} - 35625 T^{11} + 75000 T^{12} - 78125 T^{13} + 78125 T^{14} \))
$7$ (\( 1 + 7 T^{2} \))(\( 1 - 3 T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 + 3 T + 7 T^{2} \))(\( 1 + 6 T^{2} + 49 T^{4} \))(\( 1 - T + 8 T^{2} + 3 T^{3} + 56 T^{4} - 49 T^{5} + 343 T^{6} \))(\( 1 + 5 T + 20 T^{2} + 69 T^{3} + 140 T^{4} + 245 T^{5} + 343 T^{6} \))(\( 1 - T + 8 T^{2} - 21 T^{3} + 78 T^{4} - 147 T^{5} + 392 T^{6} - 343 T^{7} + 2401 T^{8} \))(\( 1 - 9 T + 50 T^{2} - 203 T^{3} + 673 T^{4} - 1888 T^{5} + 4711 T^{6} - 9947 T^{7} + 17150 T^{8} - 21609 T^{9} + 16807 T^{10} \))(\( 1 - 7 T + 44 T^{2} - 195 T^{3} + 826 T^{4} - 2821 T^{5} + 9063 T^{6} - 24610 T^{7} + 63441 T^{8} - 138229 T^{9} + 283318 T^{10} - 468195 T^{11} + 739508 T^{12} - 823543 T^{13} + 823543 T^{14} \))
$11$ (\( 1 + 11 T^{2} \))(\( 1 - 6 T + 11 T^{2} \))(\( 1 - 2 T + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( ( 1 - 2 T + 11 T^{2} )^{2} \))(\( 1 + 10 T + 53 T^{2} + 200 T^{3} + 583 T^{4} + 1210 T^{5} + 1331 T^{6} \))(\( 1 + 5 T^{2} + 52 T^{3} + 55 T^{4} + 1331 T^{6} \))(\( ( 1 + 11 T^{2} )^{4} \))(\( 1 + 2 T + 27 T^{2} + 28 T^{3} + 430 T^{4} + 420 T^{5} + 4730 T^{6} + 3388 T^{7} + 35937 T^{8} + 29282 T^{9} + 161051 T^{10} \))(\( 1 + 29 T^{2} + 12 T^{3} + 525 T^{4} + 160 T^{5} + 7753 T^{6} + 1768 T^{7} + 85283 T^{8} + 19360 T^{9} + 698775 T^{10} + 175692 T^{11} + 4670479 T^{12} + 19487171 T^{14} \))
$13$ (\( 1 + 4 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + 6 T + 13 T^{2} \))(\( 1 + 13 T^{2} \))(\( 1 + 6 T + 13 T^{2} \))(\( 1 + 18 T^{2} + 169 T^{4} \))(\( 1 + 4 T + 11 T^{2} + 36 T^{3} + 143 T^{4} + 676 T^{5} + 2197 T^{6} \))(\( 1 + 8 T + 51 T^{2} + 212 T^{3} + 663 T^{4} + 1352 T^{5} + 2197 T^{6} \))(\( 1 - 8 T + 56 T^{2} - 256 T^{3} + 1102 T^{4} - 3328 T^{5} + 9464 T^{6} - 17576 T^{7} + 28561 T^{8} \))(\( 1 - 6 T + 63 T^{2} - 260 T^{3} + 1564 T^{4} - 4740 T^{5} + 20332 T^{6} - 43940 T^{7} + 138411 T^{8} - 171366 T^{9} + 371293 T^{10} \))(\( 1 - 6 T + 61 T^{2} - 224 T^{3} + 1311 T^{4} - 3074 T^{5} + 16051 T^{6} - 30960 T^{7} + 208663 T^{8} - 519506 T^{9} + 2880267 T^{10} - 6397664 T^{11} + 22648873 T^{12} - 28960854 T^{13} + 62748517 T^{14} \))
$17$ (\( 1 - 6 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 4 T + 17 T^{2} \))(\( 1 + 4 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 + 8 T + 48 T^{2} + 136 T^{3} + 289 T^{4} \))(\( 1 - 2 T + 19 T^{2} - 72 T^{3} + 323 T^{4} - 578 T^{5} + 4913 T^{6} \))(\( 1 + 16 T + 131 T^{2} + 668 T^{3} + 2227 T^{4} + 4624 T^{5} + 4913 T^{6} \))(\( 1 - 10 T + 88 T^{2} - 494 T^{3} + 2398 T^{4} - 8398 T^{5} + 25432 T^{6} - 49130 T^{7} + 83521 T^{8} \))(\( 1 - 4 T + 75 T^{2} - 248 T^{3} + 2404 T^{4} - 6144 T^{5} + 40868 T^{6} - 71672 T^{7} + 368475 T^{8} - 334084 T^{9} + 1419857 T^{10} \))(\( 1 - 6 T + 63 T^{2} - 428 T^{3} + 2417 T^{4} - 13658 T^{5} + 63663 T^{6} - 277752 T^{7} + 1082271 T^{8} - 3947162 T^{9} + 11874721 T^{10} - 35746988 T^{11} + 89450991 T^{12} - 144825414 T^{13} + 410338673 T^{14} \))
$19$ (\( 1 + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))(\( 1 - 8 T + 46 T^{2} - 152 T^{3} + 361 T^{4} \))(\( 1 + 8 T + 65 T^{2} + 300 T^{3} + 1235 T^{4} + 2888 T^{5} + 6859 T^{6} \))(\( 1 + 2 T + 25 T^{2} + 128 T^{3} + 475 T^{4} + 722 T^{5} + 6859 T^{6} \))(\( 1 + 2 T + 44 T^{2} + 82 T^{3} + 1078 T^{4} + 1558 T^{5} + 15884 T^{6} + 13718 T^{7} + 130321 T^{8} \))(\( 1 - 12 T + 123 T^{2} - 812 T^{3} + 4918 T^{4} - 22368 T^{5} + 93442 T^{6} - 293132 T^{7} + 843657 T^{8} - 1563852 T^{9} + 2476099 T^{10} \))(\( 1 + 2 T + 17 T^{2} + 184 T^{3} + 497 T^{4} + 4174 T^{5} + 22745 T^{6} + 45392 T^{7} + 432155 T^{8} + 1506814 T^{9} + 3408923 T^{10} + 23979064 T^{11} + 42093683 T^{12} + 94091762 T^{13} + 893871739 T^{14} \))
$23$ (\( 1 - 4 T + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 - 8 T + 23 T^{2} \))(\( 1 + 4 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 + 14 T^{2} + 529 T^{4} \))(\( 1 + 14 T + 121 T^{2} + 696 T^{3} + 2783 T^{4} + 7406 T^{5} + 12167 T^{6} \))(\( 1 + 4 T + 65 T^{2} + 164 T^{3} + 1495 T^{4} + 2116 T^{5} + 12167 T^{6} \))(\( 1 - 2 T + 60 T^{2} - 106 T^{3} + 1830 T^{4} - 2438 T^{5} + 31740 T^{6} - 24334 T^{7} + 279841 T^{8} \))(\( 1 - 2 T + 87 T^{2} - 124 T^{3} + 3502 T^{4} - 3876 T^{5} + 80546 T^{6} - 65596 T^{7} + 1058529 T^{8} - 559682 T^{9} + 6436343 T^{10} \))(\( 1 + 69 T^{2} - 140 T^{3} + 2337 T^{4} - 9584 T^{5} + 59309 T^{6} - 291720 T^{7} + 1364107 T^{8} - 5069936 T^{9} + 28434279 T^{10} - 39177740 T^{11} + 444107667 T^{12} + 3404825447 T^{14} \))
$29$ (\( 1 - 6 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 - 4 T + 29 T^{2} \))(\( ( 1 - 2 T + 29 T^{2} )^{2} \))(\( 1 + 10 T + 107 T^{2} + 560 T^{3} + 3103 T^{4} + 8410 T^{5} + 24389 T^{6} \))(\( 1 + 59 T^{2} - 52 T^{3} + 1711 T^{4} + 24389 T^{6} \))(\( 1 - 14 T + 152 T^{2} - 1162 T^{3} + 7102 T^{4} - 33698 T^{5} + 127832 T^{6} - 341446 T^{7} + 707281 T^{8} \))(\( 1 + 6 T + 61 T^{2} + 396 T^{3} + 2830 T^{4} + 11148 T^{5} + 82070 T^{6} + 333036 T^{7} + 1487729 T^{8} + 4243686 T^{9} + 20511149 T^{10} \))(\( 1 + 4 T + 43 T^{2} + 300 T^{3} + 2397 T^{4} + 13692 T^{5} + 90543 T^{6} + 417320 T^{7} + 2625747 T^{8} + 11514972 T^{9} + 58460433 T^{10} + 212184300 T^{11} + 881979407 T^{12} + 2379293284 T^{13} + 17249876309 T^{14} \))
$31$ (\( 1 + 31 T^{2} \))(\( 1 + 9 T + 31 T^{2} \))(\( 1 + 5 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( 1 - 5 T + 31 T^{2} \))(\( 1 - 8 T + 70 T^{2} - 248 T^{3} + 961 T^{4} \))(\( ( 1 + T + 31 T^{2} )^{3} \))(\( 1 + 9 T + 8 T^{2} - 167 T^{3} + 248 T^{4} + 8649 T^{5} + 29791 T^{6} \))(\( 1 - T + 88 T^{2} + 3 T^{3} + 3470 T^{4} + 93 T^{5} + 84568 T^{6} - 29791 T^{7} + 923521 T^{8} \))(\( 1 - 9 T + 150 T^{2} - 887 T^{3} + 8569 T^{4} - 37440 T^{5} + 265639 T^{6} - 852407 T^{7} + 4468650 T^{8} - 8311689 T^{9} + 28629151 T^{10} \))(\( 1 - 7 T + 148 T^{2} - 931 T^{3} + 11198 T^{4} - 58509 T^{5} + 519739 T^{6} - 2272786 T^{7} + 16111909 T^{8} - 56227149 T^{9} + 333599618 T^{10} - 859798051 T^{11} + 4237114348 T^{12} - 6212525767 T^{13} + 27512614111 T^{14} \))
$37$ (\( 1 + 4 T + 37 T^{2} \))(\( 1 + T + 37 T^{2} \))(\( 1 - 3 T + 37 T^{2} \))(\( 1 + 12 T + 37 T^{2} \))(\( 1 + T + 37 T^{2} \))(\( ( 1 + 4 T + 37 T^{2} )^{2} \))(\( 1 - 15 T + 176 T^{2} - 1175 T^{3} + 6512 T^{4} - 20535 T^{5} + 50653 T^{6} \))(\( 1 + 5 T + 82 T^{2} + 263 T^{3} + 3034 T^{4} + 6845 T^{5} + 50653 T^{6} \))(\( 1 - 5 T + 82 T^{2} - 371 T^{3} + 3898 T^{4} - 13727 T^{5} + 112258 T^{6} - 253265 T^{7} + 1874161 T^{8} \))(\( 1 - 5 T + 118 T^{2} - 325 T^{3} + 5909 T^{4} - 10586 T^{5} + 218633 T^{6} - 444925 T^{7} + 5977054 T^{8} - 9370805 T^{9} + 69343957 T^{10} \))(\( 1 + 3 T + 114 T^{2} + 211 T^{3} + 8476 T^{4} + 16119 T^{5} + 421173 T^{6} + 585550 T^{7} + 15583401 T^{8} + 22066911 T^{9} + 429334828 T^{10} + 395447971 T^{11} + 7905211098 T^{12} + 7697179227 T^{13} + 94931877133 T^{14} \))
$41$ (\( 1 + 10 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 - 12 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 + 8 T + 96 T^{2} + 328 T^{3} + 1681 T^{4} \))(\( 1 - 2 T + 71 T^{2} - 28 T^{3} + 2911 T^{4} - 3362 T^{5} + 68921 T^{6} \))(\( 1 + 14 T + 151 T^{2} + 996 T^{3} + 6191 T^{4} + 23534 T^{5} + 68921 T^{6} \))(\( 1 - 14 T + 152 T^{2} - 1242 T^{3} + 8030 T^{4} - 50922 T^{5} + 255512 T^{6} - 964894 T^{7} + 2825761 T^{8} \))(\( 1 - 4 T + 95 T^{2} - 788 T^{3} + 5080 T^{4} - 46992 T^{5} + 208280 T^{6} - 1324628 T^{7} + 6547495 T^{8} - 11303044 T^{9} + 115856201 T^{10} \))(\( 1 + 12 T + 183 T^{2} + 1700 T^{3} + 17105 T^{4} + 131452 T^{5} + 1040039 T^{6} + 6545832 T^{7} + 42641599 T^{8} + 220970812 T^{9} + 1178893705 T^{10} + 4803793700 T^{11} + 21201684783 T^{12} + 57001250892 T^{13} + 194754273881 T^{14} \))
$43$ (\( 1 - 10 T + 43 T^{2} \))(\( 1 + 8 T + 43 T^{2} \))(\( 1 + 8 T + 43 T^{2} \))(\( 1 + 8 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 + 8 T + 84 T^{2} + 344 T^{3} + 1849 T^{4} \))(\( 1 + 6 T + 21 T^{2} - 56 T^{3} + 903 T^{4} + 11094 T^{5} + 79507 T^{6} \))(\( 1 + 2 T + 85 T^{2} + 152 T^{3} + 3655 T^{4} + 3698 T^{5} + 79507 T^{6} \))(\( 1 - 2 T + 140 T^{2} - 298 T^{3} + 8374 T^{4} - 12814 T^{5} + 258860 T^{6} - 159014 T^{7} + 3418801 T^{8} \))(\( 1 - 18 T + 191 T^{2} - 1052 T^{3} + 3586 T^{4} - 6484 T^{5} + 154198 T^{6} - 1945148 T^{7} + 15185837 T^{8} - 61538418 T^{9} + 147008443 T^{10} \))(\( 1 + 2 T + 167 T^{2} + 660 T^{3} + 14931 T^{4} + 67206 T^{5} + 899789 T^{6} + 3782856 T^{7} + 38690927 T^{8} + 124263894 T^{9} + 1187119017 T^{10} + 2256408660 T^{11} + 24550409981 T^{12} + 12642726098 T^{13} + 271818611107 T^{14} \))
$47$ (\( 1 - 8 T + 47 T^{2} \))(\( 1 + T + 47 T^{2} \))(\( 1 - 3 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 9 T + 47 T^{2} \))(\( 1 + 20 T + 186 T^{2} + 940 T^{3} + 2209 T^{4} \))(\( 1 - T + 128 T^{2} - 77 T^{3} + 6016 T^{4} - 2209 T^{5} + 103823 T^{6} \))(\( 1 + 3 T + 60 T^{2} - 69 T^{3} + 2820 T^{4} + 6627 T^{5} + 103823 T^{6} \))(\( 1 - 7 T + 156 T^{2} - 779 T^{3} + 10182 T^{4} - 36613 T^{5} + 344604 T^{6} - 726761 T^{7} + 4879681 T^{8} \))(\( 1 + 7 T + 174 T^{2} + 1073 T^{3} + 14593 T^{4} + 69780 T^{5} + 685871 T^{6} + 2370257 T^{7} + 18065202 T^{8} + 34157767 T^{9} + 229345007 T^{10} \))(\( 1 + 11 T + 192 T^{2} + 1847 T^{3} + 17634 T^{4} + 150193 T^{5} + 1120915 T^{6} + 8253386 T^{7} + 52683005 T^{8} + 331776337 T^{9} + 1830814782 T^{10} + 9012770807 T^{11} + 44034241344 T^{12} + 118571368619 T^{13} + 506623120463 T^{14} \))
$53$ (\( 1 - 12 T + 53 T^{2} \))(\( 1 + 3 T + 53 T^{2} \))(\( 1 + 7 T + 53 T^{2} \))(\( 1 - 14 T + 53 T^{2} \))(\( 1 - 3 T + 53 T^{2} \))(\( 1 + 4 T + 92 T^{2} + 212 T^{3} + 2809 T^{4} \))(\( 1 + 17 T + 242 T^{2} + 1891 T^{3} + 12826 T^{4} + 47753 T^{5} + 148877 T^{6} \))(\( 1 + 15 T + 164 T^{2} + 1451 T^{3} + 8692 T^{4} + 42135 T^{5} + 148877 T^{6} \))(\( 1 - 3 T + 172 T^{2} - 441 T^{3} + 12622 T^{4} - 23373 T^{5} + 483148 T^{6} - 446631 T^{7} + 7890481 T^{8} \))(\( 1 - 3 T + 56 T^{2} + 81 T^{3} + 5467 T^{4} - 17952 T^{5} + 289751 T^{6} + 227529 T^{7} + 8337112 T^{8} - 23671443 T^{9} + 418195493 T^{10} \))(\( 1 - T + 270 T^{2} - 231 T^{3} + 34242 T^{4} - 26021 T^{5} + 2690283 T^{6} - 1753150 T^{7} + 142584999 T^{8} - 73092989 T^{9} + 5097846234 T^{10} - 1822701111 T^{11} + 112912783110 T^{12} - 22164361129 T^{13} + 1174711139837 T^{14} \))
$59$ (\( 1 - 6 T + 59 T^{2} \))(\( 1 - 3 T + 59 T^{2} \))(\( 1 - 11 T + 59 T^{2} \))(\( 1 - 2 T + 59 T^{2} \))(\( 1 - 7 T + 59 T^{2} \))(\( 1 + 20 T + 210 T^{2} + 1180 T^{3} + 3481 T^{4} \))(\( 1 + 5 T + 102 T^{2} + 525 T^{3} + 6018 T^{4} + 17405 T^{5} + 205379 T^{6} \))(\( 1 + 5 T + 120 T^{2} + 615 T^{3} + 7080 T^{4} + 17405 T^{5} + 205379 T^{6} \))(\( 1 + 5 T + 120 T^{2} + 921 T^{3} + 8006 T^{4} + 54339 T^{5} + 417720 T^{6} + 1026895 T^{7} + 12117361 T^{8} \))(\( 1 - 13 T + 168 T^{2} - 773 T^{3} + 5239 T^{4} - 4014 T^{5} + 309101 T^{6} - 2690813 T^{7} + 34503672 T^{8} - 157525693 T^{9} + 714924299 T^{10} \))(\( 1 + 19 T + 364 T^{2} + 4667 T^{3} + 57934 T^{4} + 561883 T^{5} + 5308991 T^{6} + 41425142 T^{7} + 313230469 T^{8} + 1955914723 T^{9} + 11898426986 T^{10} + 56551723787 T^{11} + 260232444836 T^{12} + 801430139179 T^{13} + 2488651484819 T^{14} \))
$61$ (\( 1 - 6 T + 61 T^{2} \))(\( 1 + 12 T + 61 T^{2} \))(\( 1 - 4 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( 1 + 8 T + 61 T^{2} \))(\( 1 - 78 T^{2} + 3721 T^{4} \))(\( 1 + 8 T - 9 T^{2} - 688 T^{3} - 549 T^{4} + 29768 T^{5} + 226981 T^{6} \))(\( 1 + 4 T + 103 T^{2} + 424 T^{3} + 6283 T^{4} + 14884 T^{5} + 226981 T^{6} \))(\( 1 - 2 T + 208 T^{2} - 294 T^{3} + 18062 T^{4} - 17934 T^{5} + 773968 T^{6} - 453962 T^{7} + 13845841 T^{8} \))(\( 1 - 16 T + 277 T^{2} - 2880 T^{3} + 32214 T^{4} - 247136 T^{5} + 1965054 T^{6} - 10716480 T^{7} + 62873737 T^{8} - 221533456 T^{9} + 844596301 T^{10} \))(\( 1 - 12 T + 295 T^{2} - 3288 T^{3} + 43865 T^{4} - 410260 T^{5} + 4116935 T^{6} - 31032016 T^{7} + 251133035 T^{8} - 1526577460 T^{9} + 9956521565 T^{10} - 45525125208 T^{11} + 249155908795 T^{12} - 618244492332 T^{13} + 3142742836021 T^{14} \))
$67$ (\( 1 - 10 T + 67 T^{2} \))(\( 1 - T + 67 T^{2} \))(\( 1 + 7 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 - 3 T + 67 T^{2} \))(\( 1 + 132 T^{2} + 4489 T^{4} \))(\( 1 + 7 T + 134 T^{2} + 687 T^{3} + 8978 T^{4} + 31423 T^{5} + 300763 T^{6} \))(\( 1 - 3 T + 84 T^{2} + 219 T^{3} + 5628 T^{4} - 13467 T^{5} + 300763 T^{6} \))(\( 1 + 7 T + 82 T^{2} - 441 T^{3} - 1198 T^{4} - 29547 T^{5} + 368098 T^{6} + 2105341 T^{7} + 20151121 T^{8} \))(\( 1 - 9 T + 180 T^{2} - 827 T^{3} + 16051 T^{4} - 66492 T^{5} + 1075417 T^{6} - 3712403 T^{7} + 54137340 T^{8} - 181360089 T^{9} + 1350125107 T^{10} \))(\( 1 + 17 T + 98 T^{2} + 137 T^{3} + 4010 T^{4} + 81897 T^{5} + 561533 T^{6} + 2400242 T^{7} + 37622711 T^{8} + 367635633 T^{9} + 1206059630 T^{10} + 2760703577 T^{11} + 132312260486 T^{12} + 1537792496873 T^{13} + 6060711605323 T^{14} \))
$71$ (\( 1 + 4 T + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 + 4 T + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 - 14 T + 71 T^{2} \))(\( 1 + 8 T + 86 T^{2} + 568 T^{3} + 5041 T^{4} \))(\( 1 + 6 T + 125 T^{2} + 1000 T^{3} + 8875 T^{4} + 30246 T^{5} + 357911 T^{6} \))(\( 1 + 4 T + 173 T^{2} + 500 T^{3} + 12283 T^{4} + 20164 T^{5} + 357911 T^{6} \))(\( 1 - 2 T + 116 T^{2} - 298 T^{3} + 13174 T^{4} - 21158 T^{5} + 584756 T^{6} - 715822 T^{7} + 25411681 T^{8} \))(\( 1 + 10 T + 171 T^{2} + 1364 T^{3} + 15538 T^{4} + 96324 T^{5} + 1103198 T^{6} + 6875924 T^{7} + 61202781 T^{8} + 254116810 T^{9} + 1804229351 T^{10} \))(\( 1 + 20 T + 529 T^{2} + 7100 T^{3} + 107685 T^{4} + 1088764 T^{5} + 12096037 T^{6} + 97336680 T^{7} + 858818627 T^{8} + 5488459324 T^{9} + 38541646035 T^{10} + 180422935100 T^{11} + 954437326679 T^{12} + 2562005678420 T^{13} + 9095120158391 T^{14} \))
$73$ (\( 1 - 10 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 - 6 T + 73 T^{2} \))(\( 1 - 6 T + 73 T^{2} \))(\( 1 + 8 T + 73 T^{2} \))(\( 1 + 4 T - 50 T^{2} + 292 T^{3} + 5329 T^{4} \))(\( 1 + 20 T + 279 T^{2} + 2780 T^{3} + 20367 T^{4} + 106580 T^{5} + 389017 T^{6} \))(\( 1 - 6 T + 71 T^{2} - 136 T^{3} + 5183 T^{4} - 31974 T^{5} + 389017 T^{6} \))(\( 1 - 2 T + 160 T^{2} - 366 T^{3} + 13214 T^{4} - 26718 T^{5} + 852640 T^{6} - 778034 T^{7} + 28398241 T^{8} \))(\( 1 - 8 T + 261 T^{2} - 1684 T^{3} + 32642 T^{4} - 163368 T^{5} + 2382866 T^{6} - 8974036 T^{7} + 101533437 T^{8} - 227185928 T^{9} + 2073071593 T^{10} \))(\( 1 - 10 T + 331 T^{2} - 3088 T^{3} + 57889 T^{4} - 453094 T^{5} + 6297923 T^{6} - 41415872 T^{7} + 459748379 T^{8} - 2414537926 T^{9} + 22519805113 T^{10} - 87693768208 T^{11} + 686186697283 T^{12} - 1513342262890 T^{13} + 11047398519097 T^{14} \))
$79$ (\( 1 + 79 T^{2} \))(\( 1 + 12 T + 79 T^{2} \))(\( 1 - 4 T + 79 T^{2} \))(\( 1 + 14 T + 79 T^{2} \))(\( 1 + 16 T + 79 T^{2} \))(\( 1 + 12 T + 176 T^{2} + 948 T^{3} + 6241 T^{4} \))(\( 1 - 4 T + 189 T^{2} - 664 T^{3} + 14931 T^{4} - 24964 T^{5} + 493039 T^{6} \))(\( 1 - 32 T + 557 T^{2} - 6048 T^{3} + 44003 T^{4} - 199712 T^{5} + 493039 T^{6} \))(\( 1 + 10 T + 260 T^{2} + 1994 T^{3} + 30070 T^{4} + 157526 T^{5} + 1622660 T^{6} + 4930390 T^{7} + 38950081 T^{8} \))(\( 1 - 6 T + 293 T^{2} - 1088 T^{3} + 38140 T^{4} - 103348 T^{5} + 3013060 T^{6} - 6790208 T^{7} + 144460427 T^{8} - 233700486 T^{9} + 3077056399 T^{10} \))(\( 1 - 2 T + 293 T^{2} - 672 T^{3} + 48493 T^{4} - 101038 T^{5} + 5408089 T^{6} - 10050656 T^{7} + 427239031 T^{8} - 630578158 T^{9} + 23908940227 T^{10} - 26174454432 T^{11} + 901577524907 T^{12} - 486174911042 T^{13} + 19203908986159 T^{14} \))
$83$ (\( 1 - 14 T + 83 T^{2} \))(\( 1 - 17 T + 83 T^{2} \))(\( 1 + 15 T + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))(\( 1 + 3 T + 83 T^{2} \))(\( 1 + 12 T + 170 T^{2} + 996 T^{3} + 6889 T^{4} \))(\( 1 + 37 T + 672 T^{2} + 7551 T^{3} + 55776 T^{4} + 254893 T^{5} + 571787 T^{6} \))(\( 1 - 3 T + 222 T^{2} - 467 T^{3} + 18426 T^{4} - 20667 T^{5} + 571787 T^{6} \))(\( 1 - 13 T + 236 T^{2} - 2465 T^{3} + 29470 T^{4} - 204595 T^{5} + 1625804 T^{6} - 7433231 T^{7} + 47458321 T^{8} \))(\( 1 - T + 134 T^{2} - 167 T^{3} + 15877 T^{4} - 3678 T^{5} + 1317791 T^{6} - 1150463 T^{7} + 76619458 T^{8} - 47458321 T^{9} + 3939040643 T^{10} \))(\( 1 + 7 T + 406 T^{2} + 2165 T^{3} + 78228 T^{4} + 332467 T^{5} + 9448267 T^{6} + 33037258 T^{7} + 784206161 T^{8} + 2290365163 T^{9} + 44729753436 T^{10} + 102747264965 T^{11} + 1599250501058 T^{12} + 2288582613583 T^{13} + 27136050989627 T^{14} \))
$89$ (\( 1 + 6 T + 89 T^{2} \))(\( 1 + 15 T + 89 T^{2} \))(\( 1 - 9 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 - 15 T + 89 T^{2} \))(\( 1 + 4 T + 54 T^{2} + 356 T^{3} + 7921 T^{4} \))(\( 1 - 7 T + 150 T^{2} - 375 T^{3} + 13350 T^{4} - 55447 T^{5} + 704969 T^{6} \))(\( 1 + 27 T + 410 T^{2} + 4367 T^{3} + 36490 T^{4} + 213867 T^{5} + 704969 T^{6} \))(\( 1 - 13 T + 354 T^{2} - 2955 T^{3} + 45722 T^{4} - 262995 T^{5} + 2804034 T^{6} - 9164597 T^{7} + 62742241 T^{8} \))(\( 1 + 5 T + 240 T^{2} + 643 T^{3} + 33427 T^{4} + 85512 T^{5} + 2975003 T^{6} + 5093203 T^{7} + 169192560 T^{8} + 313711205 T^{9} + 5584059449 T^{10} \))(\( 1 + 3 T + 142 T^{2} + 1563 T^{3} + 15208 T^{4} + 131465 T^{5} + 2000929 T^{6} + 8498354 T^{7} + 178082681 T^{8} + 1041334265 T^{9} + 10721168552 T^{10} + 98066122683 T^{11} + 792936441758 T^{12} + 1490943872883 T^{13} + 44231334895529 T^{14} \))
$97$ (\( 1 - 14 T + 97 T^{2} \))(\( 1 + 13 T + 97 T^{2} \))(\( 1 + 13 T + 97 T^{2} \))(\( 1 - 14 T + 97 T^{2} \))(\( 1 + T + 97 T^{2} \))(\( 1 + 12 T + 102 T^{2} + 1164 T^{3} + 9409 T^{4} \))(\( 1 - 5 T + 26 T^{2} - 1065 T^{3} + 2522 T^{4} - 47045 T^{5} + 912673 T^{6} \))(\( 1 - 5 T + 254 T^{2} - 909 T^{3} + 24638 T^{4} - 47045 T^{5} + 912673 T^{6} \))(\( 1 - 5 T + 242 T^{2} - 827 T^{3} + 31386 T^{4} - 80219 T^{5} + 2276978 T^{6} - 4563365 T^{7} + 88529281 T^{8} \))(\( 1 + 7 T + 232 T^{2} + 881 T^{3} + 26207 T^{4} + 67036 T^{5} + 2542079 T^{6} + 8289329 T^{7} + 211740136 T^{8} + 619704967 T^{9} + 8587340257 T^{10} \))(\( 1 + 3 T + 542 T^{2} + 1923 T^{3} + 132516 T^{4} + 492061 T^{5} + 19451309 T^{6} + 64748442 T^{7} + 1886776973 T^{8} + 4629801949 T^{9} + 120943775268 T^{10} + 170241807363 T^{11} + 4654338419294 T^{12} + 2498916014787 T^{13} + 80798284478113 T^{14} \))
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