Properties

Label 6036.2.a
Level 6036
Weight 2
Character orbit a
Rep. character \(\chi_{6036}(1,\cdot)\)
Character field \(\Q\)
Dimension 84
Newform subspaces 9
Sturm bound 2016
Trace bound 5

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Defining parameters

Level: \( N \) \(=\) \( 6036 = 2^{2} \cdot 3 \cdot 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6036.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 9 \)
Sturm bound: \(2016\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(6036))\).

Total New Old
Modular forms 1014 84 930
Cusp forms 1003 84 919
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.

\(2\)\(3\)\(503\)FrickeDim.
\(-\)\(+\)\(+\)\(-\)\(26\)
\(-\)\(+\)\(-\)\(+\)\(16\)
\(-\)\(-\)\(+\)\(+\)\(16\)
\(-\)\(-\)\(-\)\(-\)\(26\)
Plus space\(+\)\(32\)
Minus space\(-\)\(52\)

Trace form

\( 84q + 4q^{5} + 4q^{7} + 84q^{9} + O(q^{10}) \) \( 84q + 4q^{5} + 4q^{7} + 84q^{9} - 4q^{11} + 4q^{13} + 4q^{15} + 4q^{17} - 8q^{19} - 8q^{23} + 92q^{25} + 8q^{29} + 4q^{31} + 8q^{33} - 8q^{35} + 12q^{37} - 8q^{39} + 8q^{41} + 4q^{45} + 28q^{47} + 104q^{49} + 8q^{51} + 16q^{53} + 16q^{55} - 4q^{57} + 4q^{59} + 20q^{61} + 4q^{63} + 40q^{65} + 12q^{67} + 16q^{69} + 20q^{71} + 8q^{73} + 44q^{77} + 8q^{79} + 84q^{81} - 8q^{83} + 36q^{85} + 32q^{89} - 12q^{91} - 12q^{93} + 28q^{95} + 12q^{97} - 4q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(6036))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 503
6036.2.a.a \(1\) \(48.198\) \(\Q\) None \(0\) \(-1\) \(-3\) \(3\) \(-\) \(+\) \(-\) \(q-q^{3}-3q^{5}+3q^{7}+q^{9}+6q^{11}+\cdots\)
6036.2.a.b \(1\) \(48.198\) \(\Q\) None \(0\) \(-1\) \(3\) \(1\) \(-\) \(+\) \(-\) \(q-q^{3}+3q^{5}+q^{7}+q^{9}-2q^{11}-5q^{13}+\cdots\)
6036.2.a.c \(1\) \(48.198\) \(\Q\) None \(0\) \(1\) \(-3\) \(-5\) \(-\) \(-\) \(-\) \(q+q^{3}-3q^{5}-5q^{7}+q^{9}-6q^{11}+\cdots\)
6036.2.a.d \(1\) \(48.198\) \(\Q\) None \(0\) \(1\) \(-1\) \(1\) \(-\) \(-\) \(-\) \(q+q^{3}-q^{5}+q^{7}+q^{9}+6q^{11}+3q^{13}+\cdots\)
6036.2.a.e \(1\) \(48.198\) \(\Q\) None \(0\) \(1\) \(1\) \(1\) \(-\) \(-\) \(+\) \(q+q^{3}+q^{5}+q^{7}+q^{9}-2q^{11}+3q^{13}+\cdots\)
6036.2.a.f \(14\) \(48.198\) \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None \(0\) \(-14\) \(-6\) \(-7\) \(-\) \(+\) \(-\) \(q-q^{3}-\beta _{1}q^{5}+\beta _{2}q^{7}+q^{9}+\beta _{3}q^{11}+\cdots\)
6036.2.a.g \(15\) \(48.198\) \(\mathbb{Q}[x]/(x^{15} - \cdots)\) None \(0\) \(15\) \(-11\) \(-4\) \(-\) \(-\) \(+\) \(q+q^{3}+(-1+\beta _{1})q^{5}-\beta _{8}q^{7}+q^{9}+\cdots\)
6036.2.a.h \(24\) \(48.198\) None \(0\) \(24\) \(18\) \(9\) \(-\) \(-\) \(-\)
6036.2.a.i \(26\) \(48.198\) None \(0\) \(-26\) \(6\) \(5\) \(-\) \(+\) \(+\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(6036))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(6036)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(503))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1006))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1509))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2012))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3018))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( ( 1 + T )^{14} \))(\( ( 1 - T )^{15} \))
$5$ (\( 1 + 3 T + 5 T^{2} \))(\( 1 - 3 T + 5 T^{2} \))(\( 1 + 3 T + 5 T^{2} \))(\( 1 + T + 5 T^{2} \))(\( 1 - T + 5 T^{2} \))(\( 1 + 6 T + 58 T^{2} + 278 T^{3} + 1562 T^{4} + 6250 T^{5} + 26331 T^{6} + 90600 T^{7} + 313910 T^{8} + 947418 T^{9} + 2821173 T^{10} + 7562470 T^{11} + 19808806 T^{12} + 47466067 T^{13} + 110699411 T^{14} + 237330335 T^{15} + 495220150 T^{16} + 945308750 T^{17} + 1763233125 T^{18} + 2960681250 T^{19} + 4904843750 T^{20} + 7078125000 T^{21} + 10285546875 T^{22} + 12207031250 T^{23} + 15253906250 T^{24} + 13574218750 T^{25} + 14160156250 T^{26} + 7324218750 T^{27} + 6103515625 T^{28} \))(\( 1 + 11 T + 97 T^{2} + 616 T^{3} + 3396 T^{4} + 15980 T^{5} + 68027 T^{6} + 261135 T^{7} + 927821 T^{8} + 3047022 T^{9} + 9381405 T^{10} + 27034937 T^{11} + 73561421 T^{12} + 188579157 T^{13} + 458143188 T^{14} + 1051725229 T^{15} + 2290715940 T^{16} + 4714478925 T^{17} + 9195177625 T^{18} + 16896835625 T^{19} + 29316890625 T^{20} + 47609718750 T^{21} + 72486015625 T^{22} + 102005859375 T^{23} + 132865234375 T^{24} + 156054687500 T^{25} + 165820312500 T^{26} + 150390625000 T^{27} + 118408203125 T^{28} + 67138671875 T^{29} + 30517578125 T^{30} \))
$7$ (\( 1 - 3 T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 + 5 T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 + 7 T + 83 T^{2} + 460 T^{3} + 3137 T^{4} + 14373 T^{5} + 72788 T^{6} + 284365 T^{7} + 1177120 T^{8} + 4014686 T^{9} + 14276610 T^{10} + 43294167 T^{11} + 136252341 T^{12} + 372092100 T^{13} + 1053965492 T^{14} + 2604644700 T^{15} + 6676364709 T^{16} + 14849899281 T^{17} + 34278140610 T^{18} + 67474827602 T^{19} + 138486990880 T^{20} + 234186805195 T^{21} + 419608335188 T^{22} + 580002393411 T^{23} + 886124856113 T^{24} + 909570301780 T^{25} + 1148826837683 T^{26} + 678223072849 T^{27} + 678223072849 T^{28} \))(\( 1 + 4 T + 65 T^{2} + 228 T^{3} + 2114 T^{4} + 6590 T^{5} + 45416 T^{6} + 127540 T^{7} + 721881 T^{8} + 1842909 T^{9} + 9006868 T^{10} + 21021823 T^{11} + 91252090 T^{12} + 195193938 T^{13} + 765153143 T^{14} + 1498840444 T^{15} + 5356072001 T^{16} + 9564502962 T^{17} + 31299466870 T^{18} + 50473397023 T^{19} + 151378430476 T^{20} + 216816400941 T^{21} + 594500044383 T^{22} + 735242719540 T^{23} + 1832699415512 T^{24} + 1861511890910 T^{25} + 4180068734702 T^{26} + 3155813481828 T^{27} + 6297785676455 T^{28} + 2712892291396 T^{29} + 4747561509943 T^{30} \))
$11$ (\( 1 - 6 T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 + 6 T + 11 T^{2} \))(\( 1 - 6 T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 - T + 111 T^{2} - 135 T^{3} + 6023 T^{4} - 8048 T^{5} + 212027 T^{6} - 292342 T^{7} + 5403338 T^{8} - 7361881 T^{9} + 105288081 T^{10} - 136977781 T^{11} + 1615950645 T^{12} - 1945732808 T^{13} + 19832017913 T^{14} - 21403060888 T^{15} + 195530028045 T^{16} - 182317426511 T^{17} + 1541522793921 T^{18} - 1185638296931 T^{19} + 9572342870618 T^{20} - 5696918544482 T^{21} + 45449870461787 T^{22} - 18976763017168 T^{23} + 156221108371823 T^{24} - 38517075532485 T^{25} + 348365549816031 T^{26} - 34522712143931 T^{27} + 379749833583241 T^{28} \))(\( 1 + 5 T + 96 T^{2} + 414 T^{3} + 4510 T^{4} + 17115 T^{5} + 138898 T^{6} + 470898 T^{7} + 3168001 T^{8} + 9715381 T^{9} + 57310873 T^{10} + 160705352 T^{11} + 858815354 T^{12} + 2222037371 T^{13} + 10952894932 T^{14} + 26308166854 T^{15} + 120481844252 T^{16} + 268866521891 T^{17} + 1143083236174 T^{18} + 2352887058632 T^{19} + 9229973407523 T^{20} + 17211390079741 T^{21} + 61735377215171 T^{22} + 100941168345138 T^{23} + 327514218384518 T^{24} + 443919022046115 T^{25} + 1286755634455610 T^{26} + 1299309347962494 T^{27} + 3314180365817376 T^{28} + 1898749167916205 T^{29} + 4177248169415651 T^{30} \))
$13$ (\( 1 + T + 13 T^{2} \))(\( 1 + 5 T + 13 T^{2} \))(\( 1 + T + 13 T^{2} \))(\( 1 - 3 T + 13 T^{2} \))(\( 1 - 3 T + 13 T^{2} \))(\( 1 + T + 122 T^{2} + 112 T^{3} + 7274 T^{4} + 6282 T^{5} + 282377 T^{6} + 233640 T^{7} + 8007151 T^{8} + 6381741 T^{9} + 176012700 T^{10} + 133882094 T^{11} + 3100320477 T^{12} + 2201881653 T^{13} + 44518848131 T^{14} + 28624461489 T^{15} + 523954160613 T^{16} + 294138960518 T^{17} + 5027098724700 T^{18} + 2369495761113 T^{19} + 38648988511159 T^{20} + 14660563511880 T^{21} + 230343593803817 T^{22} + 66617465061186 T^{23} + 1002782669709626 T^{24} + 200721964132144 T^{25} + 2842366384942682 T^{26} + 302875106592253 T^{27} + 3937376385699289 T^{28} \))(\( 1 + 14 T + 204 T^{2} + 1859 T^{3} + 16640 T^{4} + 117460 T^{5} + 807785 T^{6} + 4750311 T^{7} + 27210272 T^{8} + 138491856 T^{9} + 687684056 T^{10} + 3097624923 T^{11} + 13636038487 T^{12} + 55072726104 T^{13} + 217638242505 T^{14} + 793114521453 T^{15} + 2829297152565 T^{16} + 9307290711576 T^{17} + 29958376555939 T^{18} + 88471265425803 T^{19} + 255332276204408 T^{20} + 668473736967504 T^{21} + 1707404215166624 T^{22} + 3874974617004231 T^{23} + 8566155526018805 T^{24} + 16192858452583540 T^{25} + 29821548956775680 T^{26} + 43311140242692179 T^{27} + 61786521744819612 T^{28} + 55123269399790046 T^{29} + 51185893014090757 T^{30} \))
$17$ (\( 1 + 5 T + 17 T^{2} \))(\( 1 + 3 T + 17 T^{2} \))(\( 1 + T + 17 T^{2} \))(\( 1 - 5 T + 17 T^{2} \))(\( 1 + 5 T + 17 T^{2} \))(\( 1 + 6 T + 196 T^{2} + 1095 T^{3} + 18171 T^{4} + 94871 T^{5} + 1061309 T^{6} + 5176901 T^{7} + 43827969 T^{8} + 198828909 T^{9} + 1359235081 T^{10} + 5682927218 T^{11} + 32753076644 T^{12} + 124466281164 T^{13} + 624296368626 T^{14} + 2115926779788 T^{15} + 9465639150116 T^{16} + 27920221422034 T^{17} + 113524673200201 T^{18} + 282308618246013 T^{19} + 1057900625867361 T^{20} + 2124282686592373 T^{21} + 7403434153950269 T^{22} + 11250550431146887 T^{23} + 36632625165058779 T^{24} + 37527726456858135 T^{25} + 114193958497033156 T^{26} + 59427468197435622 T^{27} + 168377826559400929 T^{28} \))(\( 1 + 5 T + 131 T^{2} + 447 T^{3} + 7616 T^{4} + 16419 T^{5} + 277311 T^{6} + 298729 T^{7} + 7593465 T^{8} + 1108333 T^{9} + 173452021 T^{10} - 111422650 T^{11} + 3443718565 T^{12} - 4326811700 T^{13} + 61687686178 T^{14} - 92569555454 T^{15} + 1048690665026 T^{16} - 1250448581300 T^{17} + 16918989309845 T^{18} - 9306131150650 T^{19} + 246277066180997 T^{20} + 26752464262477 T^{21} + 3115892351571945 T^{22} + 2083861044592489 T^{23} + 32885722619259567 T^{24} + 33100603851472131 T^{25} + 261014762278932928 T^{26} + 260432140041703167 T^{27} + 1297499722310677747 T^{28} + 841889132797004645 T^{29} + 2862423051509815793 T^{30} \))
$19$ (\( 1 + 2 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 + 8 T + 19 T^{2} \))(\( 1 - 6 T + 19 T^{2} \))(\( 1 + 6 T + 19 T^{2} \))(\( 1 - T + 166 T^{2} - 21 T^{3} + 13038 T^{4} + 10350 T^{5} + 647630 T^{6} + 1159858 T^{7} + 23021057 T^{8} + 65256595 T^{9} + 634111607 T^{10} + 2393747074 T^{11} + 14482366836 T^{12} + 62244693079 T^{13} + 290239897357 T^{14} + 1182649168501 T^{15} + 5228134427796 T^{16} + 16418711180566 T^{17} + 82638058735847 T^{18} + 161581789622905 T^{19} + 1083045908116217 T^{20} + 1036764287453062 T^{21} + 10999064932242830 T^{22} + 3339817672012650 T^{23} + 79936841869209438 T^{24} - 2446295436862599 T^{25} + 367410276564982726 T^{26} - 42052983462257059 T^{27} + 799006685782884121 T^{28} \))(\( 1 - 3 T + 179 T^{2} - 522 T^{3} + 15816 T^{4} - 43527 T^{5} + 914280 T^{6} - 2325442 T^{7} + 38758469 T^{8} - 89897111 T^{9} + 1283413374 T^{10} - 2699693157 T^{11} + 34558912227 T^{12} - 66104402547 T^{13} + 776572610391 T^{14} - 1361691218254 T^{15} + 14754879597429 T^{16} - 23863689319467 T^{17} + 237039578964993 T^{18} - 351826711913397 T^{19} + 3177858571948026 T^{20} - 4229288786349791 T^{21} + 34645100086007591 T^{22} - 39494290805189122 T^{23} + 295026908325384120 T^{24} - 266866921003304127 T^{25} + 1842409934734231704 T^{26} - 1155350387752536042 T^{27} + 7527484039744013561 T^{28} - 2397020057348652363 T^{29} + 15181127029874798299 T^{30} \))
$23$ (\( 1 - 2 T + 23 T^{2} \))(\( 1 + 2 T + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 - 2 T + 23 T^{2} \))(\( 1 - 10 T + 216 T^{2} - 1939 T^{3} + 23449 T^{4} - 186556 T^{5} + 1671187 T^{6} - 11795776 T^{7} + 86901151 T^{8} - 546862368 T^{9} + 3479750684 T^{10} - 19624403658 T^{11} + 110515924766 T^{12} - 560445205284 T^{13} + 2826170678691 T^{14} - 12890239721532 T^{15} + 58462924201214 T^{16} - 238770119306886 T^{17} + 973776911161244 T^{18} - 3519793774240224 T^{19} + 12864489143408239 T^{20} - 40162558291911872 T^{21} + 130872300558798547 T^{22} - 336015835911891428 T^{23} + 971410261448855401 T^{24} - 1847498120595104453 T^{25} + 4733558877316389336 T^{26} - 5040363619364673830 T^{27} + 11592836324538749809 T^{28} \))(\( 1 + 32 T + 675 T^{2} + 10221 T^{3} + 125843 T^{4} + 1293129 T^{5} + 11560958 T^{6} + 91264720 T^{7} + 651244412 T^{8} + 4251771928 T^{9} + 25854263491 T^{10} + 147969413794 T^{11} + 807530442302 T^{12} + 4226179381988 T^{13} + 21345250240231 T^{14} + 104073730215610 T^{15} + 490940755525313 T^{16} + 2235648893071652 T^{17} + 9825222891488434 T^{18} + 41407908725526754 T^{19} + 166406907840453413 T^{20} + 629414837186723992 T^{21} + 2217373546194152164 T^{22} + 7147030144594586320 T^{23} + 20823050270761961554 T^{24} + 53569823019194717721 T^{25} + \)\(11\!\cdots\!61\)\( T^{26} + \)\(22\!\cdots\!41\)\( T^{27} + \)\(34\!\cdots\!25\)\( T^{28} + \)\(37\!\cdots\!88\)\( T^{29} + \)\(26\!\cdots\!07\)\( T^{30} \))
$29$ (\( 1 + 5 T + 29 T^{2} \))(\( 1 - 9 T + 29 T^{2} \))(\( 1 + 9 T + 29 T^{2} \))(\( 1 + 7 T + 29 T^{2} \))(\( 1 + T + 29 T^{2} \))(\( 1 + 6 T + 227 T^{2} + 1229 T^{3} + 25746 T^{4} + 124551 T^{5} + 1937152 T^{6} + 8437387 T^{7} + 109285471 T^{8} + 433876095 T^{9} + 4943002742 T^{10} + 18056046552 T^{11} + 185509142815 T^{12} + 624907920941 T^{13} + 5856466452307 T^{14} + 18122329707289 T^{15} + 156013189107415 T^{16} + 440368919356728 T^{17} + 3496091922364502 T^{18} + 8899297232083155 T^{19} + 65005546797269191 T^{20} + 145543882121164583 T^{21} + 969053339360227072 T^{22} + 1806879538440459819 T^{23} + 10831528428546974946 T^{24} + 14994426502052463841 T^{25} + 80315955787641472307 T^{26} + 61563772277751613134 T^{27} + \)\(29\!\cdots\!81\)\( T^{28} \))(\( 1 + 23 T + 514 T^{2} + 7334 T^{3} + 98542 T^{4} + 1046586 T^{5} + 10531319 T^{6} + 90084320 T^{7} + 738210906 T^{8} + 5328365805 T^{9} + 37357344720 T^{10} + 236020665719 T^{11} + 1473381984059 T^{12} + 8440342260900 T^{13} + 48606194334639 T^{14} + 260032255592519 T^{15} + 1409579635704531 T^{16} + 7098327841416900 T^{17} + 35934313209214951 T^{18} + 166932932470400039 T^{19} + 766242063796283280 T^{20} + 3169436243632938405 T^{21} + 12734046818454825954 T^{22} + 45064357944030871520 T^{23} + \)\(15\!\cdots\!11\)\( T^{24} + \)\(44\!\cdots\!86\)\( T^{25} + \)\(12\!\cdots\!18\)\( T^{26} + \)\(25\!\cdots\!94\)\( T^{27} + \)\(52\!\cdots\!46\)\( T^{28} + \)\(68\!\cdots\!63\)\( T^{29} + \)\(86\!\cdots\!49\)\( T^{30} \))
$31$ (\( 1 + 8 T + 31 T^{2} \))(\( 1 - 2 T + 31 T^{2} \))(\( 1 + 6 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 + 5 T + 261 T^{2} + 1099 T^{3} + 33795 T^{4} + 121395 T^{5} + 2893326 T^{6} + 8978864 T^{7} + 184068577 T^{8} + 500190054 T^{9} + 9241536839 T^{10} + 22312475042 T^{11} + 378425193101 T^{12} + 823683012922 T^{13} + 12849822369873 T^{14} + 25534173400582 T^{15} + 363666610570061 T^{16} + 664710943976222 T^{17} + 8534753343090119 T^{18} + 14320016584664154 T^{19} + 163361539643931937 T^{20} + 247032020387149904 T^{21} + 2467691813795018766 T^{22} + 3209637932194656045 T^{23} + 27699337958516169795 T^{24} + 27923916109148909269 T^{25} + \)\(20\!\cdots\!21\)\( T^{26} + \)\(12\!\cdots\!55\)\( T^{27} + \)\(75\!\cdots\!21\)\( T^{28} \))(\( 1 + 13 T + 350 T^{2} + 3724 T^{3} + 57830 T^{4} + 525958 T^{5} + 6095013 T^{6} + 48672201 T^{7} + 463198093 T^{8} + 3302098968 T^{9} + 27052271652 T^{10} + 173874277858 T^{11} + 1258629282260 T^{12} + 7330062865708 T^{13} + 47586918523056 T^{14} + 251344206014494 T^{15} + 1475194474214736 T^{16} + 7044190413945388 T^{17} + 37495824947807660 T^{18} + 160576546961698018 T^{19} + 774483570018127452 T^{20} + 2930624989126301208 T^{21} + 12743790389660090323 T^{22} + 41512084005426877641 T^{23} + \)\(16\!\cdots\!23\)\( T^{24} + \)\(43\!\cdots\!58\)\( T^{25} + \)\(14\!\cdots\!30\)\( T^{26} + \)\(29\!\cdots\!64\)\( T^{27} + \)\(85\!\cdots\!50\)\( T^{28} + \)\(98\!\cdots\!73\)\( T^{29} + \)\(23\!\cdots\!51\)\( T^{30} \))
$37$ (\( 1 - 4 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 + 4 T + 37 T^{2} \))(\( 1 + 4 T + 37 T^{2} \))(\( 1 + 12 T + 324 T^{2} + 2770 T^{3} + 42753 T^{4} + 276775 T^{5} + 3235986 T^{6} + 16488388 T^{7} + 169349259 T^{8} + 721009651 T^{9} + 7294547160 T^{10} + 28871221876 T^{11} + 296641242514 T^{12} + 1160563633924 T^{13} + 11462334151770 T^{14} + 42940854455188 T^{15} + 406101861001666 T^{16} + 1462414001685028 T^{17} + 13671155799932760 T^{18} + 49997662235529007 T^{19} + 434503866160880931 T^{20} + 1565273623737231604 T^{21} + 11366334338176001106 T^{22} + 35970160531782436675 T^{23} + \)\(20\!\cdots\!97\)\( T^{24} + \)\(49\!\cdots\!10\)\( T^{25} + \)\(21\!\cdots\!44\)\( T^{26} + \)\(29\!\cdots\!64\)\( T^{27} + \)\(90\!\cdots\!89\)\( T^{28} \))(\( 1 + 10 T + 329 T^{2} + 3282 T^{3} + 53337 T^{4} + 520011 T^{5} + 5751981 T^{6} + 53146601 T^{7} + 467221691 T^{8} + 3955514081 T^{9} + 30368986355 T^{10} + 229905059087 T^{11} + 1623277397842 T^{12} + 10934109413996 T^{13} + 72084819753152 T^{14} + 438327255088672 T^{15} + 2667138330866624 T^{16} + 14968795787760524 T^{17} + 82223870032890826 T^{18} + 430879095443551007 T^{19} + 2105905683934706735 T^{20} + 10148766938793065129 T^{21} + 44354232163884491903 T^{22} + \)\(18\!\cdots\!21\)\( T^{23} + \)\(74\!\cdots\!37\)\( T^{24} + \)\(25\!\cdots\!39\)\( T^{25} + \)\(94\!\cdots\!81\)\( T^{26} + \)\(21\!\cdots\!42\)\( T^{27} + \)\(80\!\cdots\!13\)\( T^{28} + \)\(90\!\cdots\!90\)\( T^{29} + \)\(33\!\cdots\!93\)\( T^{30} \))
$41$ (\( 1 - 9 T + 41 T^{2} \))(\( 1 + 9 T + 41 T^{2} \))(\( 1 + 11 T + 41 T^{2} \))(\( 1 + T + 41 T^{2} \))(\( 1 + 7 T + 41 T^{2} \))(\( 1 + 21 T + 481 T^{2} + 6357 T^{3} + 86387 T^{4} + 838884 T^{5} + 8353230 T^{6} + 61726005 T^{7} + 474602859 T^{8} + 2503687769 T^{9} + 14273222280 T^{10} + 30075360938 T^{11} + 41972232506 T^{12} - 2096259939142 T^{13} - 10495300292395 T^{14} - 85946657504822 T^{15} + 70555322842586 T^{16} + 2072823951207898 T^{17} + 40332714863155080 T^{18} + 290067753406505569 T^{19} + 2254413053326625019 T^{20} + 12021403283349975405 T^{21} + 66699916971650410830 T^{22} + \)\(27\!\cdots\!24\)\( T^{23} + \)\(11\!\cdots\!87\)\( T^{24} + \)\(34\!\cdots\!37\)\( T^{25} + \)\(10\!\cdots\!61\)\( T^{26} + \)\(19\!\cdots\!41\)\( T^{27} + \)\(37\!\cdots\!61\)\( T^{28} \))(\( 1 + 14 T + 479 T^{2} + 5551 T^{3} + 106705 T^{4} + 1052098 T^{5} + 14823137 T^{6} + 126912011 T^{7} + 1452705140 T^{8} + 10982758517 T^{9} + 107907749962 T^{10} + 731233210951 T^{11} + 6383015747014 T^{12} + 39284435845650 T^{13} + 311349214849579 T^{14} + 1756057900273881 T^{15} + 12765317808832739 T^{16} + 66037136656537650 T^{17} + 439923828299951894 T^{18} + 2066290289410108711 T^{19} + 12501781969055214362 T^{20} + 52169247809480570597 T^{21} + \)\(28\!\cdots\!40\)\( T^{22} + \)\(10\!\cdots\!31\)\( T^{23} + \)\(48\!\cdots\!57\)\( T^{24} + \)\(14\!\cdots\!98\)\( T^{25} + \)\(58\!\cdots\!05\)\( T^{26} + \)\(12\!\cdots\!31\)\( T^{27} + \)\(44\!\cdots\!59\)\( T^{28} + \)\(53\!\cdots\!54\)\( T^{29} + \)\(15\!\cdots\!01\)\( T^{30} \))
$43$ (\( 1 + 8 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))(\( 1 + 8 T + 43 T^{2} \))(\( 1 - 8 T + 354 T^{2} - 1938 T^{3} + 56760 T^{4} - 219355 T^{5} + 5976154 T^{6} - 17262883 T^{7} + 486530298 T^{8} - 1127367194 T^{9} + 32370895769 T^{10} - 62776797506 T^{11} + 1788115244316 T^{12} - 3011573704262 T^{13} + 83329654210912 T^{14} - 129497669283266 T^{15} + 3306225086740284 T^{16} - 4991194839309542 T^{17} + 110669650825952969 T^{18} - 165732495879218942 T^{19} + 3075534647996158602 T^{20} - 4692372880762641481 T^{21} + 69850484841786326554 T^{22} - \)\(11\!\cdots\!65\)\( T^{23} + \)\(12\!\cdots\!40\)\( T^{24} - \)\(18\!\cdots\!66\)\( T^{25} + \)\(14\!\cdots\!54\)\( T^{26} - \)\(13\!\cdots\!44\)\( T^{27} + \)\(73\!\cdots\!49\)\( T^{28} \))(\( 1 - 4 T + 421 T^{2} - 1830 T^{3} + 87666 T^{4} - 399757 T^{5} + 12027826 T^{6} - 55696692 T^{7} + 1217627048 T^{8} - 5559377275 T^{9} + 96277781559 T^{10} - 422260041628 T^{11} + 6132011183759 T^{12} - 25246133662968 T^{13} + 319768346240448 T^{14} - 1209410534970396 T^{15} + 13750038888339264 T^{16} - 46680101142827832 T^{17} + 487537813187126813 T^{18} - 1443623052577848028 T^{19} + 14153646762482702637 T^{20} - 35142842081635311475 T^{21} + \)\(33\!\cdots\!36\)\( T^{22} - \)\(65\!\cdots\!92\)\( T^{23} + \)\(60\!\cdots\!18\)\( T^{24} - \)\(86\!\cdots\!93\)\( T^{25} + \)\(81\!\cdots\!62\)\( T^{26} - \)\(73\!\cdots\!30\)\( T^{27} + \)\(72\!\cdots\!03\)\( T^{28} - \)\(29\!\cdots\!96\)\( T^{29} + \)\(31\!\cdots\!07\)\( T^{30} \))
$47$ (\( 1 + 6 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 - 12 T + 47 T^{2} \))(\( 1 - 2 T + 47 T^{2} \))(\( 1 + 10 T + 47 T^{2} \))(\( 1 - 18 T + 473 T^{2} - 5721 T^{3} + 88561 T^{4} - 838235 T^{5} + 9953843 T^{6} - 80091839 T^{7} + 801886449 T^{8} - 5698841202 T^{9} + 50405804023 T^{10} - 326828281946 T^{11} + 2678298763406 T^{12} - 16489842494004 T^{13} + 129893488380185 T^{14} - 775022597218188 T^{15} + 5916361968363854 T^{16} - 33932292716479558 T^{17} + 245964244180756663 T^{18} - 1307000775364578414 T^{19} + 8643706703178176721 T^{20} - 40576377397800201457 T^{21} + \)\(23\!\cdots\!23\)\( T^{22} - \)\(93\!\cdots\!45\)\( T^{23} + \)\(46\!\cdots\!89\)\( T^{24} - \)\(14\!\cdots\!63\)\( T^{25} + \)\(54\!\cdots\!93\)\( T^{26} - \)\(98\!\cdots\!86\)\( T^{27} + \)\(25\!\cdots\!69\)\( T^{28} \))(\( 1 + 20 T + 642 T^{2} + 9279 T^{3} + 172060 T^{4} + 1952844 T^{5} + 26758946 T^{6} + 248999182 T^{7} + 2776360166 T^{8} + 21763431995 T^{9} + 209481578906 T^{10} + 1419201259152 T^{11} + 12424584353155 T^{12} + 75576607562502 T^{13} + 632424307836997 T^{14} + 3639160633167956 T^{15} + 29723942468338859 T^{16} + 166948726105566918 T^{17} + 1289957621297611565 T^{18} + 6925249419460090512 T^{19} + 48043554180567622342 T^{20} + \)\(23\!\cdots\!55\)\( T^{21} + \)\(14\!\cdots\!58\)\( T^{22} + \)\(59\!\cdots\!02\)\( T^{23} + \)\(29\!\cdots\!82\)\( T^{24} + \)\(10\!\cdots\!56\)\( T^{25} + \)\(42\!\cdots\!80\)\( T^{26} + \)\(10\!\cdots\!39\)\( T^{27} + \)\(35\!\cdots\!34\)\( T^{28} + \)\(51\!\cdots\!80\)\( T^{29} + \)\(12\!\cdots\!43\)\( T^{30} \))
$53$ (\( 1 - 9 T + 53 T^{2} \))(\( 1 + 5 T + 53 T^{2} \))(\( 1 + 3 T + 53 T^{2} \))(\( 1 + 5 T + 53 T^{2} \))(\( 1 + 3 T + 53 T^{2} \))(\( 1 + T + 390 T^{2} - 262 T^{3} + 75166 T^{4} - 163394 T^{5} + 9790723 T^{6} - 32589030 T^{7} + 983567676 T^{8} - 3933196150 T^{9} + 80880002764 T^{10} - 339939296676 T^{11} + 5561314668583 T^{12} - 22658964842730 T^{13} + 321398332607647 T^{14} - 1200925136664690 T^{15} + 15621732904049647 T^{16} - 50609142671232852 T^{17} + 638182125089289484 T^{18} - 1644844903014951950 T^{19} + 21800149165675266204 T^{20} - 38282696577482188110 T^{21} + \)\(60\!\cdots\!03\)\( T^{22} - \)\(53\!\cdots\!02\)\( T^{23} + \)\(13\!\cdots\!34\)\( T^{24} - \)\(24\!\cdots\!14\)\( T^{25} + \)\(19\!\cdots\!90\)\( T^{26} + \)\(26\!\cdots\!73\)\( T^{27} + \)\(13\!\cdots\!69\)\( T^{28} \))(\( 1 + 30 T + 966 T^{2} + 18783 T^{3} + 359830 T^{4} + 5293042 T^{5} + 75851543 T^{6} + 908859649 T^{7} + 10641202903 T^{8} + 108623061234 T^{9} + 1090764414128 T^{10} + 9780044092726 T^{11} + 86854118392617 T^{12} + 698283563027325 T^{13} + 5587637615344376 T^{14} + 40738950287754171 T^{15} + 296144793613251928 T^{16} + 1961478528543755925 T^{17} + 12930580583937641109 T^{18} + 77169252092816741206 T^{19} + \)\(45\!\cdots\!04\)\( T^{20} + \)\(24\!\cdots\!86\)\( T^{21} + \)\(12\!\cdots\!11\)\( T^{22} + \)\(56\!\cdots\!89\)\( T^{23} + \)\(25\!\cdots\!19\)\( T^{24} + \)\(92\!\cdots\!58\)\( T^{25} + \)\(33\!\cdots\!10\)\( T^{26} + \)\(92\!\cdots\!03\)\( T^{27} + \)\(25\!\cdots\!18\)\( T^{28} + \)\(41\!\cdots\!70\)\( T^{29} + \)\(73\!\cdots\!57\)\( T^{30} \))
$59$ (\( 1 - 6 T + 59 T^{2} \))(\( 1 + 6 T + 59 T^{2} \))(\( 1 + 14 T + 59 T^{2} \))(\( 1 + 10 T + 59 T^{2} \))(\( 1 - 2 T + 59 T^{2} \))(\( 1 - 14 T + 452 T^{2} - 4322 T^{3} + 78638 T^{4} - 521049 T^{5} + 6923590 T^{6} - 28320427 T^{7} + 310018991 T^{8} - 344454210 T^{9} + 4093714768 T^{10} + 29171121678 T^{11} - 250440926296 T^{12} + 967433674354 T^{13} - 19218817079819 T^{14} + 57078586786886 T^{15} - 871784864436376 T^{16} + 5991135799105962 T^{17} + 49605019674887248 T^{18} - 246258684621848790 T^{19} + 13076766479224376231 T^{20} - 70479672704258097713 T^{21} + \)\(10\!\cdots\!90\)\( T^{22} - \)\(45\!\cdots\!11\)\( T^{23} + \)\(40\!\cdots\!38\)\( T^{24} - \)\(13\!\cdots\!98\)\( T^{25} + \)\(80\!\cdots\!12\)\( T^{26} - \)\(14\!\cdots\!06\)\( T^{27} + \)\(61\!\cdots\!61\)\( T^{28} \))(\( 1 + 14 T + 707 T^{2} + 8920 T^{3} + 240924 T^{4} + 2735653 T^{5} + 52405270 T^{6} + 536415800 T^{7} + 8137407681 T^{8} + 75244007539 T^{9} + 956253552971 T^{10} + 7995971749086 T^{11} + 87940876062190 T^{12} + 664344623419966 T^{13} + 6449903247407469 T^{14} + 43860651689011190 T^{15} + 380544291597040671 T^{16} + 2312583634124901646 T^{17} + 18061209184776520010 T^{18} + 96890076229476482046 T^{19} + \)\(68\!\cdots\!29\)\( T^{20} + \)\(31\!\cdots\!99\)\( T^{21} + \)\(20\!\cdots\!39\)\( T^{22} + \)\(78\!\cdots\!00\)\( T^{23} + \)\(45\!\cdots\!30\)\( T^{24} + \)\(13\!\cdots\!53\)\( T^{25} + \)\(72\!\cdots\!16\)\( T^{26} + \)\(15\!\cdots\!20\)\( T^{27} + \)\(74\!\cdots\!53\)\( T^{28} + \)\(86\!\cdots\!54\)\( T^{29} + \)\(36\!\cdots\!99\)\( T^{30} \))
$61$ (\( 1 - T + 61 T^{2} \))(\( 1 + 11 T + 61 T^{2} \))(\( 1 - T + 61 T^{2} \))(\( 1 - 5 T + 61 T^{2} \))(\( 1 - 5 T + 61 T^{2} \))(\( 1 + 19 T + 583 T^{2} + 8507 T^{3} + 156404 T^{4} + 1889437 T^{5} + 26505968 T^{6} + 276232681 T^{7} + 3238426200 T^{8} + 29919430179 T^{9} + 307821241368 T^{10} + 2574193615139 T^{11} + 23964991029635 T^{12} + 184198427484908 T^{13} + 1579458247956783 T^{14} + 11236104076579388 T^{15} + 89173731621271835 T^{16} + 584293040957865359 T^{17} + 4262043964403950488 T^{18} + 25269840057211167879 T^{19} + \)\(16\!\cdots\!00\)\( T^{20} + \)\(86\!\cdots\!01\)\( T^{21} + \)\(50\!\cdots\!08\)\( T^{22} + \)\(22\!\cdots\!17\)\( T^{23} + \)\(11\!\cdots\!04\)\( T^{24} + \)\(37\!\cdots\!27\)\( T^{25} + \)\(15\!\cdots\!43\)\( T^{26} + \)\(30\!\cdots\!39\)\( T^{27} + \)\(98\!\cdots\!41\)\( T^{28} \))(\( 1 + 38 T + 1249 T^{2} + 28683 T^{3} + 583564 T^{4} + 9978576 T^{5} + 155131571 T^{6} + 2150978410 T^{7} + 27589205231 T^{8} + 324297245352 T^{9} + 3565921240973 T^{10} + 36457211714766 T^{11} + 351121289554624 T^{12} + 3169009981931552 T^{13} + 27047697578657986 T^{14} + 217098127688769025 T^{15} + 1649909552298137146 T^{16} + 11791886142767304992 T^{17} + 79697861424398110144 T^{18} + \)\(50\!\cdots\!06\)\( T^{19} + \)\(30\!\cdots\!73\)\( T^{20} + \)\(16\!\cdots\!72\)\( T^{21} + \)\(86\!\cdots\!51\)\( T^{22} + \)\(41\!\cdots\!10\)\( T^{23} + \)\(18\!\cdots\!11\)\( T^{24} + \)\(71\!\cdots\!76\)\( T^{25} + \)\(25\!\cdots\!04\)\( T^{26} + \)\(76\!\cdots\!43\)\( T^{27} + \)\(20\!\cdots\!69\)\( T^{28} + \)\(37\!\cdots\!58\)\( T^{29} + \)\(60\!\cdots\!01\)\( T^{30} \))
$67$ (\( 1 + 9 T + 67 T^{2} \))(\( 1 + 3 T + 67 T^{2} \))(\( 1 - 3 T + 67 T^{2} \))(\( 1 + 15 T + 67 T^{2} \))(\( 1 - 9 T + 67 T^{2} \))(\( 1 - 17 T + 743 T^{2} - 10871 T^{3} + 265080 T^{4} - 3383136 T^{5} + 60227118 T^{6} - 678275198 T^{7} + 9745973268 T^{8} - 97652114071 T^{9} + 1189453063149 T^{10} - 10651817747195 T^{11} + 112990683237438 T^{12} - 905106476523372 T^{13} + 8490493017296618 T^{14} - 60642133927065924 T^{15} + 507215177052859182 T^{16} - 3203672661099609785 T^{17} + 23968812599336140029 T^{18} - \)\(13\!\cdots\!97\)\( T^{19} + \)\(88\!\cdots\!92\)\( T^{20} - \)\(41\!\cdots\!54\)\( T^{21} + \)\(24\!\cdots\!38\)\( T^{22} - \)\(92\!\cdots\!92\)\( T^{23} + \)\(48\!\cdots\!20\)\( T^{24} - \)\(13\!\cdots\!93\)\( T^{25} + \)\(60\!\cdots\!23\)\( T^{26} - \)\(93\!\cdots\!79\)\( T^{27} + \)\(36\!\cdots\!29\)\( T^{28} \))(\( 1 + 8 T + 645 T^{2} + 5241 T^{3} + 201142 T^{4} + 1664395 T^{5} + 40358982 T^{6} + 341133804 T^{7} + 5867089076 T^{8} + 50650011539 T^{9} + 662220847794 T^{10} + 5790514870397 T^{11} + 60933255794926 T^{12} + 527945746988497 T^{13} + 4736397212919384 T^{14} + 39119402570507002 T^{15} + 317338613265598728 T^{16} + 2369948458231363033 T^{17} + 18326468812649328538 T^{18} + \)\(11\!\cdots\!37\)\( T^{19} + \)\(89\!\cdots\!58\)\( T^{20} + \)\(45\!\cdots\!91\)\( T^{21} + \)\(35\!\cdots\!48\)\( T^{22} + \)\(13\!\cdots\!64\)\( T^{23} + \)\(10\!\cdots\!54\)\( T^{24} + \)\(30\!\cdots\!55\)\( T^{25} + \)\(24\!\cdots\!86\)\( T^{26} + \)\(42\!\cdots\!01\)\( T^{27} + \)\(35\!\cdots\!15\)\( T^{28} + \)\(29\!\cdots\!32\)\( T^{29} + \)\(24\!\cdots\!43\)\( T^{30} \))
$71$ (\( 1 - 4 T + 71 T^{2} \))(\( 1 - 16 T + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 - 12 T + 71 T^{2} \))(\( 1 - 4 T + 71 T^{2} \))(\( 1 + 13 T + 414 T^{2} + 3380 T^{3} + 68701 T^{4} + 375534 T^{5} + 7416577 T^{6} + 31234087 T^{7} + 725206675 T^{8} + 2896681910 T^{9} + 67031322512 T^{10} + 237112194174 T^{11} + 5222205503552 T^{12} + 14827269124340 T^{13} + 368025813733024 T^{14} + 1052736107828140 T^{15} + 26325137943405632 T^{16} + 84865062529010514 T^{17} + 1703378584683062672 T^{18} + 5226278522532740410 T^{19} + 92899180968904372675 T^{20} + \)\(28\!\cdots\!17\)\( T^{21} + \)\(47\!\cdots\!97\)\( T^{22} + \)\(17\!\cdots\!54\)\( T^{23} + \)\(22\!\cdots\!01\)\( T^{24} + \)\(78\!\cdots\!80\)\( T^{25} + \)\(67\!\cdots\!74\)\( T^{26} + \)\(15\!\cdots\!43\)\( T^{27} + \)\(82\!\cdots\!81\)\( T^{28} \))(\( 1 + 41 T + 1359 T^{2} + 32319 T^{3} + 673145 T^{4} + 11908316 T^{5} + 191259886 T^{6} + 2756273537 T^{7} + 36801863178 T^{8} + 452479565833 T^{9} + 5220162399675 T^{10} + 56265067088856 T^{11} + 573713331700676 T^{12} + 5511186382209690 T^{13} + 50321251921007876 T^{14} + 434571199063636192 T^{15} + 3572808886391559196 T^{16} + 27781890552719047290 T^{17} + \)\(20\!\cdots\!36\)\( T^{18} + \)\(14\!\cdots\!36\)\( T^{19} + \)\(94\!\cdots\!25\)\( T^{20} + \)\(57\!\cdots\!93\)\( T^{21} + \)\(33\!\cdots\!98\)\( T^{22} + \)\(17\!\cdots\!57\)\( T^{23} + \)\(87\!\cdots\!66\)\( T^{24} + \)\(38\!\cdots\!16\)\( T^{25} + \)\(15\!\cdots\!95\)\( T^{26} + \)\(53\!\cdots\!79\)\( T^{27} + \)\(15\!\cdots\!49\)\( T^{28} + \)\(33\!\cdots\!21\)\( T^{29} + \)\(58\!\cdots\!51\)\( T^{30} \))
$73$ (\( 1 - T + 73 T^{2} \))(\( 1 + 7 T + 73 T^{2} \))(\( 1 - T + 73 T^{2} \))(\( 1 + 7 T + 73 T^{2} \))(\( 1 - T + 73 T^{2} \))(\( 1 + 12 T + 541 T^{2} + 5015 T^{3} + 139284 T^{4} + 1030216 T^{5} + 23298209 T^{6} + 138310964 T^{7} + 2899077033 T^{8} + 13784429115 T^{9} + 291536859944 T^{10} + 1125711264561 T^{11} + 25142453455456 T^{12} + 83468872411010 T^{13} + 1931710095428119 T^{14} + 6093227686003730 T^{15} + 133984134464125024 T^{16} + 437920819005726537 T^{17} + 8279134009072958504 T^{18} + 28576108424028630195 T^{19} + \)\(43\!\cdots\!37\)\( T^{20} + \)\(15\!\cdots\!08\)\( T^{21} + \)\(18\!\cdots\!29\)\( T^{22} + \)\(60\!\cdots\!08\)\( T^{23} + \)\(59\!\cdots\!16\)\( T^{24} + \)\(15\!\cdots\!55\)\( T^{25} + \)\(12\!\cdots\!61\)\( T^{26} + \)\(20\!\cdots\!96\)\( T^{27} + \)\(12\!\cdots\!09\)\( T^{28} \))(\( 1 + 19 T + 650 T^{2} + 9798 T^{3} + 201934 T^{4} + 2585383 T^{5} + 41250209 T^{6} + 466546467 T^{7} + 6310478170 T^{8} + 64540557982 T^{9} + 771996028786 T^{10} + 7231790197260 T^{11} + 78233997131423 T^{12} + 675655659989171 T^{13} + 6697432416039189 T^{14} + 53458301377682301 T^{15} + 488912566370860797 T^{16} + 3600569012082292259 T^{17} + 30434354862074781191 T^{18} + \)\(20\!\cdots\!60\)\( T^{19} + \)\(16\!\cdots\!98\)\( T^{20} + \)\(97\!\cdots\!98\)\( T^{21} + \)\(69\!\cdots\!90\)\( T^{22} + \)\(37\!\cdots\!27\)\( T^{23} + \)\(24\!\cdots\!17\)\( T^{24} + \)\(11\!\cdots\!67\)\( T^{25} + \)\(63\!\cdots\!18\)\( T^{26} + \)\(22\!\cdots\!58\)\( T^{27} + \)\(10\!\cdots\!50\)\( T^{28} + \)\(23\!\cdots\!71\)\( T^{29} + \)\(89\!\cdots\!57\)\( T^{30} \))
$79$ (\( 1 + 11 T + 79 T^{2} \))(\( 1 - 3 T + 79 T^{2} \))(\( 1 + 7 T + 79 T^{2} \))(\( 1 - 15 T + 79 T^{2} \))(\( 1 - 15 T + 79 T^{2} \))(\( 1 + 8 T + 615 T^{2} + 3845 T^{3} + 180030 T^{4} + 873585 T^{5} + 34014811 T^{6} + 124470739 T^{7} + 4745819278 T^{8} + 12494836529 T^{9} + 530294511516 T^{10} + 964343335912 T^{11} + 50231604288570 T^{12} + 66951805620120 T^{13} + 4194826354383810 T^{14} + 5289192643989480 T^{15} + 313495442364965370 T^{16} + 475458873994716568 T^{17} + 20655014177403632796 T^{18} + 38447316696018399071 T^{19} + \)\(11\!\cdots\!38\)\( T^{20} + \)\(23\!\cdots\!01\)\( T^{21} + \)\(51\!\cdots\!71\)\( T^{22} + \)\(10\!\cdots\!15\)\( T^{23} + \)\(17\!\cdots\!30\)\( T^{24} + \)\(28\!\cdots\!55\)\( T^{25} + \)\(36\!\cdots\!15\)\( T^{26} + \)\(37\!\cdots\!12\)\( T^{27} + \)\(36\!\cdots\!81\)\( T^{28} \))(\( 1 + 27 T + 1050 T^{2} + 22566 T^{3} + 518502 T^{4} + 9196810 T^{5} + 160555048 T^{6} + 2416244311 T^{7} + 34979676440 T^{8} + 456116023448 T^{9} + 5694059388913 T^{10} + 65303238285547 T^{11} + 716843029794602 T^{12} + 7299877657233390 T^{13} + 71170413195654204 T^{14} + 646469808318244446 T^{15} + 5622462642456682116 T^{16} + 45558536458793586990 T^{17} + \)\(35\!\cdots\!78\)\( T^{18} + \)\(25\!\cdots\!07\)\( T^{19} + \)\(17\!\cdots\!87\)\( T^{20} + \)\(11\!\cdots\!08\)\( T^{21} + \)\(67\!\cdots\!60\)\( T^{22} + \)\(36\!\cdots\!71\)\( T^{23} + \)\(19\!\cdots\!12\)\( T^{24} + \)\(87\!\cdots\!10\)\( T^{25} + \)\(38\!\cdots\!58\)\( T^{26} + \)\(13\!\cdots\!06\)\( T^{27} + \)\(49\!\cdots\!50\)\( T^{28} + \)\(99\!\cdots\!87\)\( T^{29} + \)\(29\!\cdots\!99\)\( T^{30} \))
$83$ (\( 1 - 8 T + 83 T^{2} \))(\( 1 - 2 T + 83 T^{2} \))(\( 1 + 2 T + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 + 8 T + 83 T^{2} \))(\( 1 - 11 T + 656 T^{2} - 5056 T^{3} + 192317 T^{4} - 959632 T^{5} + 34657195 T^{6} - 87214970 T^{7} + 4558328492 T^{8} - 1765125372 T^{9} + 501708447367 T^{10} + 494206905927 T^{11} + 49776214226460 T^{12} + 72247657065423 T^{13} + 4414188170499223 T^{14} + 5996555536430109 T^{15} + 342908339806082940 T^{16} + 282581084119281549 T^{17} + 23810240543554690807 T^{18} - 6952900580298494196 T^{19} + \)\(14\!\cdots\!48\)\( T^{20} - \)\(23\!\cdots\!90\)\( T^{21} + \)\(78\!\cdots\!95\)\( T^{22} - \)\(17\!\cdots\!96\)\( T^{23} + \)\(29\!\cdots\!33\)\( T^{24} - \)\(65\!\cdots\!52\)\( T^{25} + \)\(70\!\cdots\!16\)\( T^{26} - \)\(97\!\cdots\!93\)\( T^{27} + \)\(73\!\cdots\!29\)\( T^{28} \))(\( 1 + 17 T + 895 T^{2} + 12903 T^{3} + 380827 T^{4} + 4778944 T^{5} + 103534520 T^{6} + 1153374804 T^{7} + 20353992719 T^{8} + 204192205836 T^{9} + 3094174002223 T^{10} + 28210128415843 T^{11} + 378014911572701 T^{12} + 3144855770268248 T^{13} + 37895831441235575 T^{14} + 287535856869273600 T^{15} + 3145354009622552725 T^{16} + 21664911401377960472 T^{17} + \)\(21\!\cdots\!87\)\( T^{18} + \)\(13\!\cdots\!03\)\( T^{19} + \)\(12\!\cdots\!89\)\( T^{20} + \)\(66\!\cdots\!84\)\( T^{21} + \)\(55\!\cdots\!13\)\( T^{22} + \)\(25\!\cdots\!64\)\( T^{23} + \)\(19\!\cdots\!60\)\( T^{24} + \)\(74\!\cdots\!56\)\( T^{25} + \)\(49\!\cdots\!09\)\( T^{26} + \)\(13\!\cdots\!83\)\( T^{27} + \)\(79\!\cdots\!85\)\( T^{28} + \)\(12\!\cdots\!93\)\( T^{29} + \)\(61\!\cdots\!07\)\( T^{30} \))
$89$ (\( 1 + 2 T + 89 T^{2} \))(\( 1 + 14 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 + 10 T + 89 T^{2} \))(\( 1 + 9 T + 719 T^{2} + 5654 T^{3} + 250799 T^{4} + 1744954 T^{5} + 57326462 T^{6} + 355900040 T^{7} + 9754216470 T^{8} + 54483622147 T^{9} + 1325899407645 T^{10} + 6727184441231 T^{11} + 150016815772611 T^{12} + 698396014652661 T^{13} + 14432101932699162 T^{14} + 62157245304086829 T^{15} + 1188283197734851731 T^{16} + 4742456488350176839 T^{17} + 83189900176219832445 T^{18} + \)\(30\!\cdots\!03\)\( T^{19} + \)\(48\!\cdots\!70\)\( T^{20} + \)\(15\!\cdots\!60\)\( T^{21} + \)\(22\!\cdots\!22\)\( T^{22} + \)\(61\!\cdots\!86\)\( T^{23} + \)\(78\!\cdots\!99\)\( T^{24} + \)\(15\!\cdots\!06\)\( T^{25} + \)\(17\!\cdots\!99\)\( T^{26} + \)\(19\!\cdots\!21\)\( T^{27} + \)\(19\!\cdots\!41\)\( T^{28} \))(\( 1 + 23 T + 912 T^{2} + 15483 T^{3} + 363334 T^{4} + 5035500 T^{5} + 89766795 T^{6} + 1076202962 T^{7} + 16100580714 T^{8} + 174169158601 T^{9} + 2306508428369 T^{10} + 23131651321022 T^{11} + 278472354864998 T^{12} + 2610951093355554 T^{13} + 28861293727805253 T^{14} + 251706236092035502 T^{15} + 2568655141774667517 T^{16} + 20681343610469343234 T^{17} + \)\(19\!\cdots\!62\)\( T^{18} + \)\(14\!\cdots\!02\)\( T^{19} + \)\(12\!\cdots\!81\)\( T^{20} + \)\(86\!\cdots\!61\)\( T^{21} + \)\(71\!\cdots\!06\)\( T^{22} + \)\(42\!\cdots\!22\)\( T^{23} + \)\(31\!\cdots\!55\)\( T^{24} + \)\(15\!\cdots\!00\)\( T^{25} + \)\(10\!\cdots\!26\)\( T^{26} + \)\(38\!\cdots\!43\)\( T^{27} + \)\(20\!\cdots\!28\)\( T^{28} + \)\(44\!\cdots\!43\)\( T^{29} + \)\(17\!\cdots\!49\)\( T^{30} \))
$97$ (\( 1 + T + 97 T^{2} \))(\( 1 + 5 T + 97 T^{2} \))(\( 1 - 3 T + 97 T^{2} \))(\( 1 + 17 T + 97 T^{2} \))(\( 1 + T + 97 T^{2} \))(\( 1 + 25 T + 794 T^{2} + 14844 T^{3} + 289128 T^{4} + 4261325 T^{5} + 63287826 T^{6} + 758679247 T^{7} + 9154798945 T^{8} + 90406995183 T^{9} + 916189532616 T^{10} + 7540054718201 T^{11} + 69062971714944 T^{12} + 524234361513402 T^{13} + 5466935127344123 T^{14} + 50850733066799994 T^{15} + 649813500865908096 T^{16} + 6881604359824661273 T^{17} + 81109600582220529096 T^{18} + \)\(77\!\cdots\!31\)\( T^{19} + \)\(76\!\cdots\!05\)\( T^{20} + \)\(61\!\cdots\!11\)\( T^{21} + \)\(49\!\cdots\!86\)\( T^{22} + \)\(32\!\cdots\!25\)\( T^{23} + \)\(21\!\cdots\!72\)\( T^{24} + \)\(10\!\cdots\!32\)\( T^{25} + \)\(55\!\cdots\!54\)\( T^{26} + \)\(16\!\cdots\!25\)\( T^{27} + \)\(65\!\cdots\!69\)\( T^{28} \))(\( 1 + 18 T + 944 T^{2} + 14099 T^{3} + 423722 T^{4} + 5568897 T^{5} + 123896511 T^{6} + 1479342346 T^{7} + 26781985186 T^{8} + 295205318823 T^{9} + 4561806513400 T^{10} + 46677685715020 T^{11} + 633987153680113 T^{12} + 6013643233271767 T^{13} + 73285993532563989 T^{14} + 640258982828617327 T^{15} + 7108741372658706933 T^{16} + 56582369181854055703 T^{17} + \)\(57\!\cdots\!49\)\( T^{18} + \)\(41\!\cdots\!20\)\( T^{19} + \)\(39\!\cdots\!00\)\( T^{20} + \)\(24\!\cdots\!67\)\( T^{21} + \)\(21\!\cdots\!18\)\( T^{22} + \)\(11\!\cdots\!06\)\( T^{23} + \)\(94\!\cdots\!87\)\( T^{24} + \)\(41\!\cdots\!53\)\( T^{25} + \)\(30\!\cdots\!66\)\( T^{26} + \)\(97\!\cdots\!59\)\( T^{27} + \)\(63\!\cdots\!88\)\( T^{28} + \)\(11\!\cdots\!42\)\( T^{29} + \)\(63\!\cdots\!93\)\( T^{30} \))
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