Properties

Label 365.2.a
Level 365
Weight 2
Character orbit a
Rep. character \(\chi_{365}(1,\cdot)\)
Character field \(\Q\)
Dimension 25
Newform subspaces 5
Sturm bound 74
Trace bound 1

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Defining parameters

Level: \( N \) = \( 365 = 5 \cdot 73 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 365.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(74\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(365))\).

Total New Old
Modular forms 38 25 13
Cusp forms 35 25 10
Eisenstein series 3 0 3

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

\(5\)\(73\)FrickeDim.
\(+\)\(+\)\(+\)\(5\)
\(+\)\(-\)\(-\)\(8\)
\(-\)\(+\)\(-\)\(9\)
\(-\)\(-\)\(+\)\(3\)
Plus space\(+\)\(8\)
Minus space\(-\)\(17\)

Trace form

\( 25q - q^{2} + 4q^{3} + 23q^{4} - q^{5} + 4q^{6} + 4q^{7} + 3q^{8} + 33q^{9} + O(q^{10}) \) \( 25q - q^{2} + 4q^{3} + 23q^{4} - q^{5} + 4q^{6} + 4q^{7} + 3q^{8} + 33q^{9} - 3q^{10} - 4q^{11} + 8q^{12} + 6q^{13} - 16q^{14} + 15q^{16} - 2q^{17} - 17q^{18} + 8q^{19} + q^{20} - 12q^{21} - 24q^{22} - 8q^{23} - 24q^{24} + 25q^{25} - 22q^{26} + 16q^{27} - 16q^{28} - 22q^{29} - 12q^{30} - 5q^{32} - 14q^{34} + 51q^{36} + 6q^{37} + 24q^{38} + 36q^{39} - 15q^{40} + 2q^{41} - 32q^{42} + 8q^{43} - 28q^{44} - 5q^{45} - 32q^{46} - 12q^{47} + 40q^{48} + 29q^{49} - q^{50} - 4q^{51} + 46q^{52} - 22q^{53} - 16q^{54} + 4q^{55} + 44q^{57} + 22q^{58} + 28q^{59} + 16q^{60} + 6q^{61} - 28q^{62} + 3q^{64} + 2q^{65} + 20q^{66} - 54q^{68} - 36q^{69} + 12q^{70} + 16q^{71} - 105q^{72} - 3q^{73} + 6q^{74} + 4q^{75} + 52q^{76} - 32q^{77} - 84q^{78} - 20q^{79} + q^{80} + 73q^{81} + 2q^{82} - 36q^{83} - 68q^{84} - 2q^{85} - 4q^{86} - 16q^{87} - 44q^{88} - 26q^{89} - 23q^{90} + 16q^{91} - 52q^{92} + 44q^{93} + 24q^{94} - 4q^{95} - 68q^{96} + 6q^{97} + 19q^{98} - 32q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(365))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 5 73
365.2.a.a \(2\) \(2.915\) \(\Q(\sqrt{3}) \) None \(0\) \(4\) \(2\) \(6\) \(-\) \(+\) \(q+\beta q^{2}+2q^{3}+q^{4}+q^{5}+2\beta q^{6}+\cdots\)
365.2.a.b \(3\) \(2.915\) \(\Q(\zeta_{14})^+\) None \(-1\) \(-4\) \(3\) \(-1\) \(-\) \(-\) \(q-\beta _{1}q^{2}+(-1+\beta _{2})q^{3}+\beta _{2}q^{4}+q^{5}+\cdots\)
365.2.a.c \(5\) \(2.915\) 5.5.117688.1 None \(-1\) \(-6\) \(-5\) \(-5\) \(+\) \(+\) \(q-\beta _{1}q^{2}+(-1-\beta _{2})q^{3}+\beta _{2}q^{4}-q^{5}+\cdots\)
365.2.a.d \(7\) \(2.915\) \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-1\) \(2\) \(7\) \(-3\) \(-\) \(+\) \(q-\beta _{1}q^{2}+\beta _{2}q^{3}+(1+\beta _{1}-\beta _{3}+\beta _{5}+\cdots)q^{4}+\cdots\)
365.2.a.e \(8\) \(2.915\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(2\) \(8\) \(-8\) \(7\) \(+\) \(-\) \(q+\beta _{1}q^{2}+(1-\beta _{4})q^{3}+(1+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(365))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(365)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(73))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T^{2} + 4 T^{4} \))(\( 1 + T + 4 T^{2} + 3 T^{3} + 8 T^{4} + 4 T^{5} + 8 T^{6} \))(\( 1 + T + 5 T^{2} + 4 T^{3} + 14 T^{4} + 9 T^{5} + 28 T^{6} + 16 T^{7} + 40 T^{8} + 16 T^{9} + 32 T^{10} \))(\( 1 + T + 2 T^{2} + 3 T^{3} + 3 T^{4} + 7 T^{5} + 18 T^{6} + 17 T^{7} + 36 T^{8} + 28 T^{9} + 24 T^{10} + 48 T^{11} + 64 T^{12} + 64 T^{13} + 128 T^{14} \))(\( 1 - 2 T + 5 T^{2} - 9 T^{3} + 16 T^{4} - 24 T^{5} + 35 T^{6} - 51 T^{7} + 63 T^{8} - 102 T^{9} + 140 T^{10} - 192 T^{11} + 256 T^{12} - 288 T^{13} + 320 T^{14} - 256 T^{15} + 256 T^{16} \))
$3$ (\( ( 1 - 2 T + 3 T^{2} )^{2} \))(\( 1 + 4 T + 12 T^{2} + 23 T^{3} + 36 T^{4} + 36 T^{5} + 27 T^{6} \))(\( 1 + 6 T + 22 T^{2} + 63 T^{3} + 145 T^{4} + 274 T^{5} + 435 T^{6} + 567 T^{7} + 594 T^{8} + 486 T^{9} + 243 T^{10} \))(\( 1 - 2 T + 7 T^{2} - 19 T^{3} + 43 T^{4} - 97 T^{5} + 184 T^{6} - 331 T^{7} + 552 T^{8} - 873 T^{9} + 1161 T^{10} - 1539 T^{11} + 1701 T^{12} - 1458 T^{13} + 2187 T^{14} \))(\( 1 - 8 T + 38 T^{2} - 133 T^{3} + 380 T^{4} - 940 T^{5} + 2077 T^{6} - 4150 T^{7} + 7544 T^{8} - 12450 T^{9} + 18693 T^{10} - 25380 T^{11} + 30780 T^{12} - 32319 T^{13} + 27702 T^{14} - 17496 T^{15} + 6561 T^{16} \))
$5$ (\( ( 1 - T )^{2} \))(\( ( 1 - T )^{3} \))(\( ( 1 + T )^{5} \))(\( ( 1 - T )^{7} \))(\( ( 1 + T )^{8} \))
$7$ (\( 1 - 6 T + 20 T^{2} - 42 T^{3} + 49 T^{4} \))(\( 1 + T + 5 T^{2} - 15 T^{3} + 35 T^{4} + 49 T^{5} + 343 T^{6} \))(\( 1 + 5 T + 37 T^{2} + 125 T^{3} + 516 T^{4} + 1258 T^{5} + 3612 T^{6} + 6125 T^{7} + 12691 T^{8} + 12005 T^{9} + 16807 T^{10} \))(\( 1 + 3 T + 27 T^{2} + 73 T^{3} + 407 T^{4} + 917 T^{5} + 3981 T^{6} + 7790 T^{7} + 27867 T^{8} + 44933 T^{9} + 139601 T^{10} + 175273 T^{11} + 453789 T^{12} + 352947 T^{13} + 823543 T^{14} \))(\( 1 - 7 T + 44 T^{2} - 172 T^{3} + 702 T^{4} - 2198 T^{5} + 7476 T^{6} - 20329 T^{7} + 59938 T^{8} - 142303 T^{9} + 366324 T^{10} - 753914 T^{11} + 1685502 T^{12} - 2890804 T^{13} + 5176556 T^{14} - 5764801 T^{15} + 5764801 T^{16} \))
$11$ (\( 1 + 6 T + 28 T^{2} + 66 T^{3} + 121 T^{4} \))(\( ( 1 + 3 T + 11 T^{2} )^{3} \))(\( 1 - 3 T + 24 T^{2} - 83 T^{3} + 361 T^{4} - 1378 T^{5} + 3971 T^{6} - 10043 T^{7} + 31944 T^{8} - 43923 T^{9} + 161051 T^{10} \))(\( 1 - 15 T + 155 T^{2} - 1140 T^{3} + 6859 T^{4} - 33713 T^{5} + 141878 T^{6} - 505739 T^{7} + 1560658 T^{8} - 4079273 T^{9} + 9129329 T^{10} - 16690740 T^{11} + 24962905 T^{12} - 26573415 T^{13} + 19487171 T^{14} \))(\( 1 + 7 T + 56 T^{2} + 309 T^{3} + 1672 T^{4} + 7481 T^{5} + 31013 T^{6} + 117514 T^{7} + 407154 T^{8} + 1292654 T^{9} + 3752573 T^{10} + 9957211 T^{11} + 24479752 T^{12} + 49764759 T^{13} + 99207416 T^{14} + 136410197 T^{15} + 214358881 T^{16} \))
$13$ (\( 1 + 14 T^{2} + 169 T^{4} \))(\( 1 + T + 23 T^{2} + 39 T^{3} + 299 T^{4} + 169 T^{5} + 2197 T^{6} \))(\( 1 + 9 T + 89 T^{2} + 477 T^{3} + 2598 T^{4} + 9356 T^{5} + 33774 T^{6} + 80613 T^{7} + 195533 T^{8} + 257049 T^{9} + 371293 T^{10} \))(\( 1 - 5 T + 54 T^{2} - 220 T^{3} + 1371 T^{4} - 4919 T^{5} + 23954 T^{6} - 75531 T^{7} + 311402 T^{8} - 831311 T^{9} + 3012087 T^{10} - 6283420 T^{11} + 20049822 T^{12} - 24134045 T^{13} + 62748517 T^{14} \))(\( 1 - 11 T + 121 T^{2} - 801 T^{3} + 5169 T^{4} - 24993 T^{5} + 120143 T^{6} - 467694 T^{7} + 1851074 T^{8} - 6080022 T^{9} + 20304167 T^{10} - 54909621 T^{11} + 147631809 T^{12} - 297405693 T^{13} + 584043889 T^{14} - 690233687 T^{15} + 815730721 T^{16} \))
$17$ (\( 1 + 22 T^{2} + 289 T^{4} \))(\( 1 - 3 T + 47 T^{2} - 103 T^{3} + 799 T^{4} - 867 T^{5} + 4913 T^{6} \))(\( 1 + 5 T + 29 T^{2} + 79 T^{3} + 782 T^{4} + 2888 T^{5} + 13294 T^{6} + 22831 T^{7} + 142477 T^{8} + 417605 T^{9} + 1419857 T^{10} \))(\( 1 + 5 T + 90 T^{2} + 376 T^{3} + 3771 T^{4} + 13391 T^{5} + 97266 T^{6} + 287171 T^{7} + 1653522 T^{8} + 3869999 T^{9} + 18526923 T^{10} + 31403896 T^{11} + 127787130 T^{12} + 120687845 T^{13} + 410338673 T^{14} \))(\( 1 - 5 T + 69 T^{2} - 207 T^{3} + 1685 T^{4} - 923 T^{5} + 14127 T^{6} + 83586 T^{7} + 15846 T^{8} + 1420962 T^{9} + 4082703 T^{10} - 4534699 T^{11} + 140732885 T^{12} - 293910399 T^{13} + 1665492261 T^{14} - 2051693365 T^{15} + 6975757441 T^{16} \))
$19$ (\( 1 - 4 T + 30 T^{2} - 76 T^{3} + 361 T^{4} \))(\( 1 + 15 T + 125 T^{2} + 667 T^{3} + 2375 T^{4} + 5415 T^{5} + 6859 T^{6} \))(\( 1 + 15 T + 151 T^{2} + 1129 T^{3} + 6730 T^{4} + 32064 T^{5} + 127870 T^{6} + 407569 T^{7} + 1035709 T^{8} + 1954815 T^{9} + 2476099 T^{10} \))(\( 1 - 13 T + 177 T^{2} - 1417 T^{3} + 11157 T^{4} - 64511 T^{5} + 364189 T^{6} - 1606742 T^{7} + 6919591 T^{8} - 23288471 T^{9} + 76525863 T^{10} - 184664857 T^{11} + 438269523 T^{12} - 611596453 T^{13} + 893871739 T^{14} \))(\( 1 - 21 T + 280 T^{2} - 2732 T^{3} + 21872 T^{4} - 147662 T^{5} + 869880 T^{6} - 4507635 T^{7} + 20832990 T^{8} - 85645065 T^{9} + 314026680 T^{10} - 1012813658 T^{11} + 2850380912 T^{12} - 6764702468 T^{13} + 13172846680 T^{14} - 18771306519 T^{15} + 16983563041 T^{16} \))
$23$ (\( 1 + 4 T + 2 T^{2} + 92 T^{3} + 529 T^{4} \))(\( 1 + T + 53 T^{2} + 59 T^{3} + 1219 T^{4} + 529 T^{5} + 12167 T^{6} \))(\( 1 + 3 T + 47 T^{2} + 169 T^{3} + 1698 T^{4} + 5124 T^{5} + 39054 T^{6} + 89401 T^{7} + 571849 T^{8} + 839523 T^{9} + 6436343 T^{10} \))(\( 1 + 5 T + 126 T^{2} + 538 T^{3} + 7425 T^{4} + 26705 T^{5} + 263322 T^{6} + 779991 T^{7} + 6056406 T^{8} + 14126945 T^{9} + 90339975 T^{10} + 150554458 T^{11} + 810979218 T^{12} + 740179445 T^{13} + 3404825447 T^{14} \))(\( 1 - 5 T + 113 T^{2} - 491 T^{3} + 6547 T^{4} - 24555 T^{5} + 248665 T^{6} - 810322 T^{7} + 6718136 T^{8} - 18637406 T^{9} + 131543785 T^{10} - 298760685 T^{11} + 1832119027 T^{12} - 3160244413 T^{13} + 16728055457 T^{14} - 17024127235 T^{15} + 78310985281 T^{16} \))
$29$ (\( 1 + 8 T + 62 T^{2} + 232 T^{3} + 841 T^{4} \))(\( 1 + 11 T + 125 T^{2} + 679 T^{3} + 3625 T^{4} + 9251 T^{5} + 24389 T^{6} \))(\( 1 - 3 T + 93 T^{2} - 275 T^{3} + 4066 T^{4} - 10972 T^{5} + 117914 T^{6} - 231275 T^{7} + 2268177 T^{8} - 2121843 T^{9} + 20511149 T^{10} \))(\( 1 + 3 T + 123 T^{2} + 255 T^{3} + 7269 T^{4} + 13433 T^{5} + 295511 T^{6} + 500890 T^{7} + 8569819 T^{8} + 11297153 T^{9} + 177283641 T^{10} + 180356655 T^{11} + 2522871327 T^{12} + 1784469963 T^{13} + 17249876309 T^{14} \))(\( 1 + 3 T + 148 T^{2} + 502 T^{3} + 11058 T^{4} + 36172 T^{5} + 539876 T^{6} + 1567679 T^{7} + 18511690 T^{8} + 45462691 T^{9} + 454035716 T^{10} + 882198908 T^{11} + 7821113298 T^{12} + 10296596798 T^{13} + 88033851508 T^{14} + 51749628927 T^{15} + 500246412961 T^{16} \))
$31$ (\( 1 - 14 T + 108 T^{2} - 434 T^{3} + 961 T^{4} \))(\( 1 + 21 T + 233 T^{2} + 1589 T^{3} + 7223 T^{4} + 20181 T^{5} + 29791 T^{6} \))(\( 1 + 7 T + 143 T^{2} + 673 T^{3} + 8014 T^{4} + 27938 T^{5} + 248434 T^{6} + 646753 T^{7} + 4260113 T^{8} + 6464647 T^{9} + 28629151 T^{10} \))(\( 1 - 11 T + 156 T^{2} - 1138 T^{3} + 10783 T^{4} - 65287 T^{5} + 488162 T^{6} - 2460011 T^{7} + 15133022 T^{8} - 62740807 T^{9} + 321236353 T^{10} - 1050966898 T^{11} + 4466147556 T^{12} - 9762540491 T^{13} + 27512614111 T^{14} \))(\( 1 - 3 T + 179 T^{2} - 415 T^{3} + 14521 T^{4} - 25401 T^{5} + 729049 T^{6} - 992532 T^{7} + 26109490 T^{8} - 30768492 T^{9} + 700616089 T^{10} - 756721191 T^{11} + 13410448441 T^{12} - 11881097665 T^{13} + 158863158899 T^{14} - 82537842333 T^{15} + 852891037441 T^{16} \))
$37$ (\( ( 1 - 8 T + 37 T^{2} )^{2} \))(\( 1 - 4 T + 72 T^{2} - 127 T^{3} + 2664 T^{4} - 5476 T^{5} + 50653 T^{6} \))(\( 1 + 6 T + 122 T^{2} + 677 T^{3} + 7649 T^{4} + 35054 T^{5} + 283013 T^{6} + 926813 T^{7} + 6179666 T^{8} + 11244966 T^{9} + 69343957 T^{10} \))(\( 1 + 26 T + 374 T^{2} + 3933 T^{3} + 34028 T^{4} + 255610 T^{5} + 1748385 T^{6} + 11054126 T^{7} + 64690245 T^{8} + 349930090 T^{9} + 1723620284 T^{10} + 7371075213 T^{11} + 25934639918 T^{12} + 66708886634 T^{13} + 94931877133 T^{14} \))(\( 1 - 18 T + 359 T^{2} - 4151 T^{3} + 48808 T^{4} - 421051 T^{5} + 3629545 T^{6} - 24715664 T^{7} + 167543454 T^{8} - 914479568 T^{9} + 4968847105 T^{10} - 21327496303 T^{11} + 91474050088 T^{12} - 287846765507 T^{13} + 921095780831 T^{14} - 1708773788394 T^{15} + 3512479453921 T^{16} \))
$41$ (\( 1 + 4 T + 74 T^{2} + 164 T^{3} + 1681 T^{4} \))(\( 1 + 7 T + 123 T^{2} + 525 T^{3} + 5043 T^{4} + 11767 T^{5} + 68921 T^{6} \))(\( 1 - 13 T + 193 T^{2} - 1697 T^{3} + 14254 T^{4} - 94760 T^{5} + 584414 T^{6} - 2852657 T^{7} + 13301753 T^{8} - 36734893 T^{9} + 115856201 T^{10} \))(\( 1 - 7 T + 204 T^{2} - 1112 T^{3} + 19679 T^{4} - 89389 T^{5} + 1194186 T^{6} - 4527293 T^{7} + 48961626 T^{8} - 150262909 T^{9} + 1356296359 T^{10} - 3142246232 T^{11} + 23634665004 T^{12} - 33250729687 T^{13} + 194754273881 T^{14} \))(\( 1 + 7 T + 203 T^{2} + 1437 T^{3} + 20679 T^{4} + 143301 T^{5} + 1385191 T^{6} + 8866890 T^{7} + 66374554 T^{8} + 363542490 T^{9} + 2328506071 T^{10} + 9876448221 T^{11} + 58433911719 T^{12} + 166485360837 T^{13} + 964271160923 T^{14} + 1363279917167 T^{15} + 7984925229121 T^{16} \))
$43$ (\( 1 - 6 T + 68 T^{2} - 258 T^{3} + 1849 T^{4} \))(\( 1 + 18 T + 216 T^{2} + 1631 T^{3} + 9288 T^{4} + 33282 T^{5} + 79507 T^{6} \))(\( 1 + 10 T + 144 T^{2} + 1167 T^{3} + 9929 T^{4} + 67740 T^{5} + 426947 T^{6} + 2157783 T^{7} + 11449008 T^{8} + 34188010 T^{9} + 147008443 T^{10} \))(\( 1 - 16 T + 254 T^{2} - 2805 T^{3} + 29348 T^{4} - 241848 T^{5} + 1935619 T^{6} - 13001894 T^{7} + 83231617 T^{8} - 447176952 T^{9} + 2333371436 T^{10} - 9589736805 T^{11} + 37340144522 T^{12} - 101141808784 T^{13} + 271818611107 T^{14} \))(\( 1 - 14 T + 305 T^{2} - 3133 T^{3} + 40528 T^{4} - 331299 T^{5} + 3200311 T^{6} - 21509968 T^{7} + 166962526 T^{8} - 924928624 T^{9} + 5917375039 T^{10} - 26340589593 T^{11} + 138557166928 T^{12} - 460577451919 T^{13} + 1928015729945 T^{14} - 3805460555498 T^{15} + 11688200277601 T^{16} \))
$47$ (\( 1 + 10 T + 116 T^{2} + 470 T^{3} + 2209 T^{4} \))(\( 1 - T + 76 T^{2} - 261 T^{3} + 3572 T^{4} - 2209 T^{5} + 103823 T^{6} \))(\( 1 + 11 T + 204 T^{2} + 1535 T^{3} + 17437 T^{4} + 99734 T^{5} + 819539 T^{6} + 3390815 T^{7} + 21179892 T^{8} + 53676491 T^{9} + 229345007 T^{10} \))(\( 1 + 3 T + 68 T^{2} - 249 T^{3} + 5318 T^{4} - 4559 T^{5} + 333059 T^{6} - 749110 T^{7} + 15653773 T^{8} - 10070831 T^{9} + 552130714 T^{10} - 1215040569 T^{11} + 15595460476 T^{12} + 32337645987 T^{13} + 506623120463 T^{14} \))(\( 1 - 11 T + 357 T^{2} - 3212 T^{3} + 56404 T^{4} - 418098 T^{5} + 5169443 T^{6} - 31465297 T^{7} + 301103878 T^{8} - 1478868959 T^{9} + 11419299587 T^{10} - 43408188654 T^{11} + 275233527124 T^{12} - 736656162484 T^{13} + 3848179872453 T^{14} - 5572854325093 T^{15} + 23811286661761 T^{16} \))
$53$ (\( 1 - 12 T + 94 T^{2} - 636 T^{3} + 2809 T^{4} \))(\( 1 - 6 T + 80 T^{2} - 175 T^{3} + 4240 T^{4} - 16854 T^{5} + 148877 T^{6} \))(\( 1 + 6 T + 210 T^{2} + 913 T^{3} + 19305 T^{4} + 64626 T^{5} + 1023165 T^{6} + 2564617 T^{7} + 31264170 T^{8} + 47342886 T^{9} + 418195493 T^{10} \))(\( 1 + 34 T + 729 T^{2} + 11293 T^{3} + 142615 T^{4} + 1497547 T^{5} + 13543332 T^{6} + 105510405 T^{7} + 717796596 T^{8} + 4206609523 T^{9} + 21232093355 T^{10} + 89107201933 T^{11} + 304864514397 T^{12} + 753588278386 T^{13} + 1174711139837 T^{14} \))(\( 1 + 302 T^{2} - 175 T^{3} + 43298 T^{4} - 37854 T^{5} + 3916629 T^{6} - 3585572 T^{7} + 246047150 T^{8} - 190035316 T^{9} + 11001810861 T^{10} - 5635589958 T^{11} + 341642046338 T^{12} - 73184211275 T^{13} + 6693637060958 T^{14} + 62259690411361 T^{16} \))
$59$ (\( 1 + 6 T + 124 T^{2} + 354 T^{3} + 3481 T^{4} \))(\( 1 + 5 T + 36 T^{2} + 213 T^{3} + 2124 T^{4} + 17405 T^{5} + 205379 T^{6} \))(\( 1 + T + 208 T^{2} + 273 T^{3} + 21145 T^{4} + 21734 T^{5} + 1247555 T^{6} + 950313 T^{7} + 42718832 T^{8} + 12117361 T^{9} + 714924299 T^{10} \))(\( 1 - 27 T + 472 T^{2} - 5903 T^{3} + 66350 T^{4} - 641693 T^{5} + 5724283 T^{6} - 45137410 T^{7} + 337732697 T^{8} - 2233733333 T^{9} + 13626896650 T^{10} - 71528781983 T^{11} + 337444269128 T^{12} - 1138874408307 T^{13} + 2488651484819 T^{14} \))(\( 1 - 13 T + 245 T^{2} - 1216 T^{3} + 13336 T^{4} + 29402 T^{5} + 311075 T^{6} + 4429465 T^{7} + 21889518 T^{8} + 261338435 T^{9} + 1082852075 T^{10} + 6038553358 T^{11} + 161597126296 T^{12} - 869347947584 T^{13} + 10334230742045 T^{14} - 32352469302647 T^{15} + 146830437604321 T^{16} \))
$61$ (\( 1 + 12 T + 146 T^{2} + 732 T^{3} + 3721 T^{4} \))(\( 1 + 3 T + 95 T^{2} - 11 T^{3} + 5795 T^{4} + 11163 T^{5} + 226981 T^{6} \))(\( 1 - T + 137 T^{2} - 289 T^{3} + 13490 T^{4} - 17384 T^{5} + 822890 T^{6} - 1075369 T^{7} + 31096397 T^{8} - 13845841 T^{9} + 844596301 T^{10} \))(\( 1 - 15 T + 388 T^{2} - 4040 T^{3} + 61267 T^{4} - 494597 T^{5} + 5643906 T^{6} - 37142013 T^{7} + 344278266 T^{8} - 1840395437 T^{9} + 13906444927 T^{10} - 55937197640 T^{11} + 327703364788 T^{12} - 772805615415 T^{13} + 3142742836021 T^{14} \))(\( 1 - 5 T + 223 T^{2} - 1783 T^{3} + 27703 T^{4} - 241639 T^{5} + 2707195 T^{6} - 19711810 T^{7} + 197164694 T^{8} - 1202420410 T^{9} + 10073472595 T^{10} - 54847461859 T^{11} + 383571333223 T^{12} - 1505915204683 T^{13} + 11489043482503 T^{14} - 15713714180105 T^{15} + 191707312997281 T^{16} \))
$67$ (\( ( 1 + 2 T + 67 T^{2} )^{2} \))(\( 1 + 7 T + 173 T^{2} + 945 T^{3} + 11591 T^{4} + 31423 T^{5} + 300763 T^{6} \))(\( 1 + 29 T + 481 T^{2} + 5119 T^{3} + 43600 T^{4} + 340388 T^{5} + 2921200 T^{6} + 22979191 T^{7} + 144667003 T^{8} + 584382509 T^{9} + 1350125107 T^{10} \))(\( 1 - 11 T + 232 T^{2} - 2368 T^{3} + 29249 T^{4} - 256117 T^{5} + 2421762 T^{6} - 19790711 T^{7} + 162258054 T^{8} - 1149709213 T^{9} + 8797016987 T^{10} - 47717854528 T^{11} + 313229024824 T^{12} - 995042203859 T^{13} + 6060711605323 T^{14} \))(\( 1 - 29 T + 657 T^{2} - 10941 T^{3} + 155823 T^{4} - 1892583 T^{5} + 20485229 T^{6} - 195853182 T^{7} + 1698608332 T^{8} - 13122163194 T^{9} + 91958192981 T^{10} - 569218940829 T^{11} + 3140008127583 T^{12} - 14771718795687 T^{13} + 59431157085033 T^{14} - 175760636554367 T^{15} + 406067677556641 T^{16} \))
$71$ (\( ( 1 - 8 T + 71 T^{2} )^{2} \))(\( 1 + 6 T + 218 T^{2} + 839 T^{3} + 15478 T^{4} + 30246 T^{5} + 357911 T^{6} \))(\( 1 - 26 T + 564 T^{2} - 7665 T^{3} + 91823 T^{4} - 816826 T^{5} + 6519433 T^{6} - 38639265 T^{7} + 201861804 T^{8} - 660703706 T^{9} + 1804229351 T^{10} \))(\( 1 - 26 T + 618 T^{2} - 8977 T^{3} + 123668 T^{4} - 1283482 T^{5} + 13223443 T^{6} - 110479830 T^{7} + 938864453 T^{8} - 6470032762 T^{9} + 44262137548 T^{10} - 228120660337 T^{11} + 1115013738918 T^{12} - 3330607381946 T^{13} + 9095120158391 T^{14} \))(\( 1 + 46 T + 1389 T^{2} + 29813 T^{3} + 514602 T^{4} + 7298579 T^{5} + 88291971 T^{6} + 916776324 T^{7} + 8291095402 T^{8} + 65091119004 T^{9} + 445079825811 T^{10} + 2612241708469 T^{11} + 13076901865962 T^{12} + 53789489641363 T^{13} + 177931294366269 T^{14} + 418375527285986 T^{15} + 645753531245761 T^{16} \))
$73$ (\( ( 1 + T )^{2} \))(\( ( 1 - T )^{3} \))(\( ( 1 + T )^{5} \))(\( ( 1 + T )^{7} \))(\( ( 1 - T )^{8} \))
$79$ (\( 1 + 20 T + 246 T^{2} + 1580 T^{3} + 6241 T^{4} \))(\( 1 + 5 T + 180 T^{2} + 497 T^{3} + 14220 T^{4} + 31205 T^{5} + 493039 T^{6} \))(\( 1 + 5 T + 146 T^{2} + 1391 T^{3} + 15045 T^{4} + 133184 T^{5} + 1188555 T^{6} + 8681231 T^{7} + 71983694 T^{8} + 194750405 T^{9} + 3077056399 T^{10} \))(\( 1 - 19 T + 504 T^{2} - 6615 T^{3} + 104554 T^{4} - 1053989 T^{5} + 12528803 T^{6} - 102660626 T^{7} + 989775437 T^{8} - 6577945349 T^{9} + 51549199606 T^{10} - 257654785815 T^{11} + 1550836425096 T^{12} - 4618661654899 T^{13} + 19203908986159 T^{14} \))(\( 1 + 9 T + 331 T^{2} + 2028 T^{3} + 46550 T^{4} + 162618 T^{5} + 3958965 T^{6} + 5680055 T^{7} + 292792530 T^{8} + 448724345 T^{9} + 24707900565 T^{10} + 80177016102 T^{11} + 1813126270550 T^{12} + 6240270377172 T^{13} + 80461947777451 T^{14} + 172835180875431 T^{15} + 1517108809906561 T^{16} \))
$83$ (\( 1 - 2 T + 92 T^{2} - 166 T^{3} + 6889 T^{4} \))(\( 1 - 13 T + 303 T^{2} - 2229 T^{3} + 25149 T^{4} - 89557 T^{5} + 571787 T^{6} \))(\( 1 + 23 T + 551 T^{2} + 7403 T^{3} + 99422 T^{4} + 908006 T^{5} + 8252026 T^{6} + 50999267 T^{7} + 315054637 T^{8} + 1091541383 T^{9} + 3939040643 T^{10} \))(\( 1 + 17 T + 635 T^{2} + 8171 T^{3} + 166131 T^{4} + 1661103 T^{5} + 23498377 T^{6} + 182600442 T^{7} + 1950365291 T^{8} + 11443338567 T^{9} + 94991546097 T^{10} + 387781940891 T^{11} + 2501290808305 T^{12} + 5557986347273 T^{13} + 27136050989627 T^{14} \))(\( 1 + 11 T + 140 T^{2} + 1964 T^{3} + 27954 T^{4} + 278110 T^{5} + 2907028 T^{6} + 28238993 T^{7} + 293241466 T^{8} + 2343836419 T^{9} + 20026515892 T^{10} + 159019682570 T^{11} + 1326649905234 T^{12} + 7736275822852 T^{13} + 45771652271660 T^{14} + 298496560885897 T^{15} + 2252292232139041 T^{16} \))
$89$ (\( 1 + 4 T - 10 T^{2} + 356 T^{3} + 7921 T^{4} \))(\( 1 - 6 T^{2} + 889 T^{3} - 534 T^{4} + 704969 T^{6} \))(\( 1 - 4 T + 72 T^{2} + 73 T^{3} + 12499 T^{4} - 53506 T^{5} + 1112411 T^{6} + 578233 T^{7} + 50757768 T^{8} - 250968964 T^{9} + 5584059449 T^{10} \))(\( 1 - 2 T + 405 T^{2} - 927 T^{3} + 84763 T^{4} - 175333 T^{5} + 11174168 T^{6} - 20184215 T^{7} + 994500952 T^{8} - 1388812693 T^{9} + 59755287347 T^{10} - 58162057407 T^{11} + 2261544076845 T^{12} - 993962581922 T^{13} + 44231334895529 T^{14} \))(\( 1 + 28 T + 706 T^{2} + 12257 T^{3} + 198590 T^{4} + 2624734 T^{5} + 32716237 T^{6} + 349369884 T^{7} + 3533388446 T^{8} + 31093919676 T^{9} + 259145313277 T^{10} + 1850356103246 T^{11} + 12459981640190 T^{12} + 68443816666393 T^{13} + 350868791418466 T^{14} + 1238477377074812 T^{15} + 3936588805702081 T^{16} \))
$97$ (\( 1 - 8 T + 162 T^{2} - 776 T^{3} + 9409 T^{4} \))(\( 1 + 179 T^{2} - 448 T^{3} + 17363 T^{4} + 912673 T^{6} \))(\( 1 + 8 T + 317 T^{2} + 2464 T^{3} + 50450 T^{4} + 346416 T^{5} + 4893650 T^{6} + 23183776 T^{7} + 289317341 T^{8} + 708234248 T^{9} + 8587340257 T^{10} \))(\( 1 + 4 T + 287 T^{2} + 712 T^{3} + 54013 T^{4} + 136060 T^{5} + 7030435 T^{6} + 13600624 T^{7} + 681952195 T^{8} + 1280188540 T^{9} + 49296206749 T^{10} + 63032848072 T^{11} + 2464566653759 T^{12} + 3331888019716 T^{13} + 80798284478113 T^{14} \))(\( 1 - 10 T + 248 T^{2} - 1126 T^{3} + 16684 T^{4} + 8430 T^{5} + 353800 T^{6} + 2014210 T^{7} + 27830950 T^{8} + 195378370 T^{9} + 3328904200 T^{10} + 7693833390 T^{11} + 1477022524204 T^{12} - 9669345129382 T^{13} + 206577057222392 T^{14} - 807982844781130 T^{15} + 7837433594376961 T^{16} \))
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