Properties

Label 378.2.u
Level 378
Weight 2
Character orbit u
Rep. character \(\chi_{378}(43,\cdot)\)
Character field \(\Q(\zeta_{9})\)
Dimension 108
Newform subspaces 5
Sturm bound 144
Trace bound 1

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Defining parameters

Level: \( N \) = \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 378.u (of order \(9\) and degree \(6\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 27 \)
Character field: \(\Q(\zeta_{9})\)
Newform subspaces: \( 5 \)
Sturm bound: \(144\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(378, [\chi])\).

Total New Old
Modular forms 456 108 348
Cusp forms 408 108 300
Eisenstein series 48 0 48

Trace form

\( 108q + 12q^{5} + 6q^{8} + 12q^{9} + O(q^{10}) \) \( 108q + 12q^{5} + 6q^{8} + 12q^{9} + 18q^{11} + 6q^{12} + 24q^{15} - 12q^{18} - 24q^{20} + 18q^{22} + 12q^{23} + 36q^{25} - 54q^{27} - 72q^{29} + 36q^{31} + 18q^{33} + 18q^{34} + 12q^{35} - 24q^{36} + 24q^{38} + 12q^{39} + 54q^{41} + 18q^{43} + 12q^{44} - 12q^{45} + 72q^{47} + 12q^{48} + 12q^{50} - 36q^{51} - 72q^{53} + 36q^{54} - 42q^{57} - 72q^{59} - 54q^{64} - 84q^{65} - 54q^{67} - 12q^{68} - 96q^{69} - 24q^{71} - 24q^{72} - 96q^{74} - 120q^{75} - 36q^{76} - 36q^{78} - 72q^{79} - 24q^{81} + 12q^{84} - 72q^{85} - 6q^{86} + 60q^{87} - 36q^{88} - 66q^{89} - 36q^{90} - 24q^{92} - 12q^{93} + 12q^{95} - 12q^{96} - 54q^{97} + 6q^{98} + 72q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(378, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
378.2.u.a \(6\) \(3.018\) \(\Q(\zeta_{18})\) None \(0\) \(0\) \(3\) \(0\) \(q-\zeta_{18}q^{2}+(\zeta_{18}-2\zeta_{18}^{4})q^{3}+\zeta_{18}^{2}q^{4}+\cdots\)
378.2.u.b \(12\) \(3.018\) 12.0.\(\cdots\).1 None \(0\) \(3\) \(0\) \(0\) \(q+\beta _{7}q^{2}+(-\beta _{2}+\beta _{4}+\beta _{5}-\beta _{10}+\cdots)q^{3}+\cdots\)
378.2.u.c \(24\) \(3.018\) None \(0\) \(-3\) \(3\) \(0\)
378.2.u.d \(30\) \(3.018\) None \(0\) \(-3\) \(3\) \(0\)
378.2.u.e \(36\) \(3.018\) None \(0\) \(3\) \(3\) \(0\)

Decomposition of \(S_{2}^{\mathrm{old}}(378, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(378, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(189, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T^{3} + T^{6} \))(\( ( 1 + T^{3} + T^{6} )^{2} \))
$3$ (\( 1 + 9 T^{3} + 27 T^{6} \))(\( 1 - 3 T + 3 T^{3} + 9 T^{4} - 45 T^{6} + 81 T^{8} + 81 T^{9} - 729 T^{11} + 729 T^{12} \))
$5$ (\( 1 - 3 T + 18 T^{3} - 36 T^{4} - 75 T^{5} + 379 T^{6} - 375 T^{7} - 900 T^{8} + 2250 T^{9} - 9375 T^{11} + 15625 T^{12} \))(\( 1 + 12 T^{2} - 18 T^{3} + 78 T^{4} - 198 T^{5} + 467 T^{6} - 1233 T^{7} + 2205 T^{8} - 5832 T^{9} + 8847 T^{10} - 26739 T^{11} + 35469 T^{12} - 133695 T^{13} + 221175 T^{14} - 729000 T^{15} + 1378125 T^{16} - 3853125 T^{17} + 7296875 T^{18} - 15468750 T^{19} + 30468750 T^{20} - 35156250 T^{21} + 117187500 T^{22} + 244140625 T^{24} \))
$7$ (\( 1 + T^{3} + T^{6} \))(\( ( 1 + T^{3} + T^{6} )^{2} \))
$11$ (\( 1 - 6 T + 18 T^{2} - 36 T^{3} - 108 T^{4} + 912 T^{5} - 3203 T^{6} + 10032 T^{7} - 13068 T^{8} - 47916 T^{9} + 263538 T^{10} - 966306 T^{11} + 1771561 T^{12} \))(\( 1 - 3 T - 15 T^{2} + 126 T^{3} - 201 T^{4} - 1488 T^{5} + 7145 T^{6} - 1530 T^{7} - 61974 T^{8} + 202716 T^{9} - 19692 T^{10} - 1304451 T^{11} + 4526883 T^{12} - 14348961 T^{13} - 2382732 T^{14} + 269814996 T^{15} - 907361334 T^{16} - 246408030 T^{17} + 12657803345 T^{18} - 28996910448 T^{19} - 43086135081 T^{20} + 297101409066 T^{21} - 389061369015 T^{22} - 855935011833 T^{23} + 3138428376721 T^{24} \))
$13$ (\( 1 + 3 T + 6 T^{2} + 8 T^{3} - 27 T^{4} - 891 T^{5} - 3483 T^{6} - 11583 T^{7} - 4563 T^{8} + 17576 T^{9} + 171366 T^{10} + 1113879 T^{11} + 4826809 T^{12} \))(\( 1 - 12 T + 93 T^{2} - 461 T^{3} + 1830 T^{4} - 6510 T^{5} + 27469 T^{6} - 129519 T^{7} + 566109 T^{8} - 2155844 T^{9} + 7183293 T^{10} - 24098388 T^{11} + 81886711 T^{12} - 313279044 T^{13} + 1213976517 T^{14} - 4736389268 T^{15} + 16168639149 T^{16} - 48089498067 T^{17} + 132587616421 T^{18} - 408492845670 T^{19} + 1492787219430 T^{20} - 4888674210953 T^{21} + 12820839741957 T^{22} - 21505924728444 T^{23} + 23298085122481 T^{24} \))
$17$ (\( 1 - 42 T^{2} + 18 T^{3} + 1050 T^{4} - 378 T^{5} - 20081 T^{6} - 6426 T^{7} + 303450 T^{8} + 88434 T^{9} - 3507882 T^{10} + 24137569 T^{12} \))(\( 1 - 45 T^{2} - 18 T^{3} + 810 T^{4} + 405 T^{5} - 4228 T^{6} + 10935 T^{7} - 8145 T^{8} - 541728 T^{9} - 2753640 T^{10} + 5149980 T^{11} + 87495483 T^{12} + 87549660 T^{13} - 795801960 T^{14} - 2661509664 T^{15} - 680278545 T^{16} + 15526136295 T^{17} - 102053641732 T^{18} + 166187162565 T^{19} + 5650363527210 T^{20} - 2134581776946 T^{21} - 90719725520205 T^{22} + 582622237229761 T^{24} \))
$19$ (\( 1 - 3 T - 24 T^{2} + 23 T^{3} + 279 T^{4} + 666 T^{5} - 6501 T^{6} + 12654 T^{7} + 100719 T^{8} + 157757 T^{9} - 3127704 T^{10} - 7428297 T^{11} + 47045881 T^{12} \))(\( 1 - 12 T + 12 T^{2} + 52 T^{3} + 2649 T^{4} - 10209 T^{5} - 32138 T^{6} - 100341 T^{7} + 1360620 T^{8} + 1473505 T^{9} - 7164300 T^{10} - 30874461 T^{11} + 38067223 T^{12} - 586614759 T^{13} - 2586312300 T^{14} + 10106770795 T^{15} + 177317359020 T^{16} - 248454249759 T^{17} - 1511960523578 T^{18} - 9125536583451 T^{19} + 44989458495609 T^{20} + 16779760284508 T^{21} + 73572795093612 T^{22} - 1397883106778628 T^{23} + 2213314919066161 T^{24} \))
$23$ (\( 1 - 6 T - 36 T^{2} + 396 T^{3} - 810 T^{4} - 5874 T^{5} + 51733 T^{6} - 135102 T^{7} - 428490 T^{8} + 4818132 T^{9} - 10074276 T^{10} - 38618058 T^{11} + 148035889 T^{12} \))(\( 1 - 12 T + 57 T^{2} - 216 T^{3} + 870 T^{4} - 1974 T^{5} + 6398 T^{6} - 34596 T^{7} - 231615 T^{8} + 2798766 T^{9} - 5217057 T^{10} - 7068906 T^{11} - 3870063 T^{12} - 162584838 T^{13} - 2759823153 T^{14} + 34052585922 T^{15} - 64815373215 T^{16} - 222671722428 T^{17} + 947133617822 T^{18} - 6721125432378 T^{19} + 68130557194470 T^{20} - 389048974876008 T^{21} + 2361311139177993 T^{22} - 11433717094967124 T^{23} + 21914624432020321 T^{24} \))
$29$ (\( 1 + 3 T + 108 T^{2} + 90 T^{3} + 4851 T^{4} - 3507 T^{5} + 149293 T^{6} - 101703 T^{7} + 4079691 T^{8} + 2195010 T^{9} + 76386348 T^{10} + 61533447 T^{11} + 594823321 T^{12} \))(\( 1 + 3 T + 30 T^{2} + 207 T^{3} + 978 T^{4} + 4863 T^{5} + 23219 T^{6} + 31419 T^{7} + 849420 T^{8} - 4617 T^{9} - 2257902 T^{10} - 22744773 T^{11} - 94856679 T^{12} - 659598417 T^{13} - 1898895582 T^{14} - 112604013 T^{15} + 600778627020 T^{16} + 644439790431 T^{17} + 13811202690299 T^{18} + 83886148490667 T^{19} + 489240991875858 T^{20} + 3002979217004883 T^{21} + 12621216999006030 T^{22} + 36601529297117487 T^{23} + 353814783205469041 T^{24} \))
$31$ (\( 1 + 3 T + 24 T^{2} + 296 T^{3} + 1530 T^{4} + 7119 T^{5} + 55665 T^{6} + 220689 T^{7} + 1470330 T^{8} + 8818136 T^{9} + 22164504 T^{10} + 85887453 T^{11} + 887503681 T^{12} \))(\( 1 - 21 T + 255 T^{2} - 2414 T^{3} + 18624 T^{4} - 124887 T^{5} + 730225 T^{6} - 3464208 T^{7} + 11411442 T^{8} - 1810082 T^{9} - 324371289 T^{10} + 3091769544 T^{11} - 19862733551 T^{12} + 95844855864 T^{13} - 311720808729 T^{14} - 53924152862 T^{15} + 10538706327282 T^{16} - 99177333927408 T^{17} + 648077375458225 T^{18} - 3435967838480457 T^{19} + 15884242681301184 T^{20} - 63825247895859794 T^{21} + 209005213180104255 T^{22} - 533578014824501451 T^{23} + 787662783788549761 T^{24} \))
$37$ (\( 1 - 12 T + 21 T^{2} + 284 T^{3} + 18 T^{4} - 15444 T^{5} + 118797 T^{6} - 571428 T^{7} + 24642 T^{8} + 14385452 T^{9} + 39357381 T^{10} - 832127484 T^{11} + 2565726409 T^{12} \))(\( 1 - 15 T + 69 T^{2} + 406 T^{3} - 7374 T^{4} + 38802 T^{5} + 15283 T^{6} - 1166850 T^{7} + 3628278 T^{8} + 22593355 T^{9} - 153725946 T^{10} - 917449914 T^{11} + 12989691655 T^{12} - 33945646818 T^{13} - 210450820074 T^{14} + 1144421210815 T^{15} + 6799977124758 T^{16} - 80913996225450 T^{17} + 39211996708747 T^{18} + 3683546696514666 T^{19} - 25901023493213454 T^{20} + 52764466356801262 T^{21} + 331792321696831581 T^{22} - 2668764326691906195 T^{23} + 6582952005840035281 T^{24} \))
$41$ (\( 1 + 12 T + 144 T^{2} + 1044 T^{3} + 9216 T^{4} + 58152 T^{5} + 440263 T^{6} + 2384232 T^{7} + 15492096 T^{8} + 71953524 T^{9} + 406909584 T^{10} + 1390274412 T^{11} + 4750104241 T^{12} \))(\( 1 - 12 T + 39 T^{2} - 252 T^{3} + 6540 T^{4} - 62211 T^{5} + 254375 T^{6} - 1609425 T^{7} + 20028519 T^{8} - 143465580 T^{9} + 690787764 T^{10} - 4925423520 T^{11} + 40342142493 T^{12} - 201942364320 T^{13} + 1161214231284 T^{14} - 9887791239180 T^{15} + 56595807877959 T^{16} - 186461866294425 T^{17} + 1208307766304375 T^{18} - 12115858132410891 T^{19} + 52221410998451340 T^{20} - 82500247467278172 T^{21} + 523483713095943639 T^{22} - 6603948380594981292 T^{23} + 22563490300366186081 T^{24} \))
$43$ (\( 1 + 6 T - 6 T^{2} + 284 T^{3} - 684 T^{4} - 7920 T^{5} + 123837 T^{6} - 340560 T^{7} - 1264716 T^{8} + 22579988 T^{9} - 20512806 T^{10} + 882050658 T^{11} + 6321363049 T^{12} \))(\( 1 - 21 T + 192 T^{2} - 560 T^{3} - 8403 T^{4} + 123036 T^{5} - 639908 T^{6} - 1742472 T^{7} + 55902231 T^{8} - 419028878 T^{9} + 874994820 T^{10} + 12129804183 T^{11} - 134937795011 T^{12} + 521581579869 T^{13} + 1617865422180 T^{14} - 33315729003146 T^{15} + 191118603245031 T^{16} - 256158095691096 T^{17} - 4045090785959492 T^{18} + 33443474636160852 T^{19} - 98215946932681203 T^{20} - 281451862684632080 T^{21} + 4149404604150575808 T^{22} - 19515168528895676847 T^{23} + 39959630797262576401 T^{24} \))
$47$ (\( 1 - 24 T + 306 T^{2} - 2844 T^{3} + 25254 T^{4} - 215106 T^{5} + 1621333 T^{6} - 10109982 T^{7} + 55786086 T^{8} - 295272612 T^{9} + 1493182386 T^{10} - 5504280168 T^{11} + 10779215329 T^{12} \))(\( 1 - 12 T - 72 T^{2} + 1215 T^{3} + 6273 T^{4} - 85908 T^{5} - 561724 T^{6} + 5249610 T^{7} + 35240229 T^{8} - 201170169 T^{9} - 2024333352 T^{10} + 3145093866 T^{11} + 108872443875 T^{12} + 147819411702 T^{13} - 4471752374568 T^{14} - 20886090456087 T^{15} + 171961075886949 T^{16} + 1203971842197270 T^{17} - 6054943951467196 T^{18} - 43522979032735404 T^{19} + 149368201229226753 T^{20} + 1359743524819861905 T^{21} - 3787137520979763528 T^{22} - 29665910581008147636 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \))
$53$ (\( ( 1 + 42 T^{2} + 153 T^{3} + 2226 T^{4} + 148877 T^{6} )^{2} \))(\( ( 1 + 21 T + 450 T^{2} + 5868 T^{3} + 70182 T^{4} + 637455 T^{5} + 5208409 T^{6} + 33785115 T^{7} + 197141238 T^{8} + 873610236 T^{9} + 3550716450 T^{10} + 8782105353 T^{11} + 22164361129 T^{12} )^{2} \))
$59$ (\( 1 + 24 T + 252 T^{2} + 1395 T^{3} - 2475 T^{4} - 150699 T^{5} - 1633823 T^{6} - 8891241 T^{7} - 8615475 T^{8} + 286503705 T^{9} + 3053574972 T^{10} + 17158183176 T^{11} + 42180533641 T^{12} \))(\( 1 - 18 T + 72 T^{2} + 3006 T^{4} - 28638 T^{5} + 390494 T^{6} - 4658796 T^{7} + 22409190 T^{8} - 176439384 T^{9} + 1983739302 T^{10} - 9795803364 T^{11} + 37789076343 T^{12} - 577952398476 T^{13} + 6905396510262 T^{14} - 36236944246536 T^{15} + 271540244947590 T^{16} - 3330686464484004 T^{17} + 16471245303608654 T^{18} - 71270001222246522 T^{19} + 441372295438588926 T^{20} + 36800406237646180872 T^{22} - \)\(54\!\cdots\!62\)\( T^{23} + \)\(17\!\cdots\!81\)\( T^{24} \))
$61$ (\( 1 + 3 T + 114 T^{2} + 926 T^{3} + 12897 T^{4} + 70029 T^{5} + 1006281 T^{6} + 4271769 T^{7} + 47989737 T^{8} + 210184406 T^{9} + 1578425874 T^{10} + 2533788903 T^{11} + 51520374361 T^{12} \))(\( 1 + 189 T^{2} + 130 T^{3} + 25974 T^{4} - 17334 T^{5} + 2536246 T^{6} - 3252339 T^{7} + 204923790 T^{8} - 455484935 T^{9} + 14770394064 T^{10} - 36254574540 T^{11} + 930935536957 T^{12} - 2211529046940 T^{13} + 54960636312144 T^{14} - 103386426031235 T^{15} + 2837342213457390 T^{16} - 2746913488998039 T^{17} + 130668343391588806 T^{18} - 54476304319588014 T^{19} + 4979405747791376694 T^{20} + 1520238992068438330 T^{21} + \)\(13\!\cdots\!89\)\( T^{22} + \)\(26\!\cdots\!21\)\( T^{24} \))
$67$ (\( 1 - 3 T - 96 T^{2} + 1094 T^{3} - 3735 T^{4} - 48879 T^{5} + 940191 T^{6} - 3274893 T^{7} - 16766415 T^{8} + 329034722 T^{9} - 1934507616 T^{10} - 4050375321 T^{11} + 90458382169 T^{12} \))(\( 1 + 33 T + 642 T^{2} + 8368 T^{3} + 69393 T^{4} + 186288 T^{5} - 4701338 T^{6} - 91813068 T^{7} - 902003391 T^{8} - 4727097308 T^{9} + 9492444366 T^{10} + 478832565675 T^{11} + 5293151069269 T^{12} + 32081781900225 T^{13} + 42611582758974 T^{14} - 1421735967646004 T^{15} - 18176379474451311 T^{16} - 123959128257498276 T^{17} - 425275429509642122 T^{18} + 1129037843532411024 T^{19} + 28178254348687988913 T^{20} + \)\(22\!\cdots\!96\)\( T^{21} + \)\(11\!\cdots\!58\)\( T^{22} + \)\(40\!\cdots\!39\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \))
$71$ (\( 1 + 12 T - 24 T^{2} - 738 T^{3} - 228 T^{4} + 5556 T^{5} - 117857 T^{6} + 394476 T^{7} - 1149348 T^{8} - 264138318 T^{9} - 609880344 T^{10} + 21650752212 T^{11} + 128100283921 T^{12} \))(\( 1 - 24 T + 105 T^{2} + 2718 T^{3} - 33666 T^{4} - 35205 T^{5} + 2568722 T^{6} - 8972217 T^{7} - 55594071 T^{8} - 248099436 T^{9} + 7228158138 T^{10} + 28994592564 T^{11} - 876571769937 T^{12} + 2058616072044 T^{13} + 36437145173658 T^{14} - 88797517238196 T^{15} - 1412738797743351 T^{16} - 16187937254941167 T^{17} + 329054017514118962 T^{18} - 320193705176155155 T^{19} - 21739938382919789826 T^{20} + \)\(12\!\cdots\!58\)\( T^{21} + \)\(34\!\cdots\!05\)\( T^{22} - \)\(55\!\cdots\!04\)\( T^{23} + \)\(16\!\cdots\!41\)\( T^{24} \))
$73$ (\( 1 - 3 T - 96 T^{2} + 635 T^{3} + 1935 T^{4} - 19872 T^{5} + 150873 T^{6} - 1450656 T^{7} + 10311615 T^{8} + 247025795 T^{9} - 2726231136 T^{10} - 6219214779 T^{11} + 151334226289 T^{12} \))(\( 1 - 9 T - 162 T^{2} + 1543 T^{3} + 6453 T^{4} - 30168 T^{5} - 1029257 T^{6} - 3127977 T^{7} + 217574586 T^{8} - 589643561 T^{9} - 12568109643 T^{10} + 46186633458 T^{11} + 321235971409 T^{12} + 3371624242434 T^{13} - 66975456287547 T^{14} - 229381369169537 T^{15} + 6178735528703226 T^{16} - 6484520262257361 T^{17} - 155761811747537273 T^{18} - 333277918524118296 T^{19} + 5204086972992504693 T^{20} + 90838858290857389759 T^{21} - \)\(69\!\cdots\!38\)\( T^{22} - \)\(28\!\cdots\!93\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \))
$79$ (\( 1 - 33 T + 492 T^{2} - 4402 T^{3} + 17532 T^{4} + 134901 T^{5} - 2452347 T^{6} + 10657179 T^{7} + 109417212 T^{8} - 2170357678 T^{9} + 19163439852 T^{10} - 101542861167 T^{11} + 243087455521 T^{12} \))(\( 1 - 135 T^{2} + 1993 T^{3} - 2052 T^{4} - 232362 T^{5} + 3259333 T^{6} - 10002285 T^{7} - 260701254 T^{8} + 3944140684 T^{9} - 13669666056 T^{10} - 186122056770 T^{11} + 3171346188679 T^{12} - 14703642484830 T^{13} - 85312385855496 T^{14} + 1944615178698676 T^{15} - 10154334960101574 T^{16} - 30777595063871715 T^{17} + 792302965665627493 T^{18} - 4462258699841877558 T^{19} - 3113107277928263172 T^{20} + \)\(23\!\cdots\!67\)\( T^{21} - \)\(12\!\cdots\!35\)\( T^{22} + \)\(59\!\cdots\!41\)\( T^{24} \))
$83$ (\( 1 + 144 T^{2} + 774 T^{3} + 20916 T^{4} + 72810 T^{5} + 2086741 T^{6} + 6043230 T^{7} + 144090324 T^{8} + 442563138 T^{9} + 6833998224 T^{10} + 326940373369 T^{12} \))(\( 1 - 15 T + 147 T^{2} - 1854 T^{3} + 25071 T^{4} - 149289 T^{5} + 1163042 T^{6} - 9519381 T^{7} + 3954789 T^{8} + 847137582 T^{9} - 4335015195 T^{10} + 98252949789 T^{11} - 1318582794483 T^{12} + 8154994832487 T^{13} - 29863919678355 T^{14} + 484382256599034 T^{15} + 187687645849269 T^{16} - 37497228655201983 T^{17} + 380245385723828498 T^{18} - 4051113916190425203 T^{19} + 56467218551957896911 T^{20} - \)\(34\!\cdots\!62\)\( T^{21} + \)\(22\!\cdots\!03\)\( T^{22} - \)\(19\!\cdots\!05\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \))
$89$ (\( 1 - 21 T + 138 T^{2} + 891 T^{3} - 11325 T^{4} - 132708 T^{5} + 2900905 T^{6} - 11811012 T^{7} - 89705325 T^{8} + 628127379 T^{9} + 8658429258 T^{10} - 117265248429 T^{11} + 496981290961 T^{12} \))(\( 1 + 51 T + 1197 T^{2} + 18234 T^{3} + 216342 T^{4} + 2052636 T^{5} + 13370033 T^{6} + 34362828 T^{7} - 391121892 T^{8} - 7793030061 T^{9} - 89642793852 T^{10} - 757259381556 T^{11} - 6184239975177 T^{12} - 67396084958484 T^{13} - 710060570101692 T^{14} - 5493844609073109 T^{15} - 24539864008239972 T^{16} + 191884074387761772 T^{17} + 6644656260531171713 T^{18} + 90790830334619064444 T^{19} + \)\(85\!\cdots\!02\)\( T^{20} + \)\(63\!\cdots\!06\)\( T^{21} + \)\(37\!\cdots\!97\)\( T^{22} + \)\(14\!\cdots\!39\)\( T^{23} + \)\(24\!\cdots\!21\)\( T^{24} \))
$97$ (\( 1 + 15 T + 120 T^{2} + 2138 T^{3} + 14679 T^{4} + 127917 T^{5} + 2485869 T^{6} + 12407949 T^{7} + 138114711 T^{8} + 1951294874 T^{9} + 10623513720 T^{10} + 128810103855 T^{11} + 832972004929 T^{12} \))(\( 1 - 48 T + 1326 T^{2} - 27506 T^{3} + 457320 T^{4} - 6561246 T^{5} + 84857617 T^{6} - 1022695011 T^{7} + 11895196383 T^{8} - 134109734294 T^{9} + 1470887073393 T^{10} - 15554728611663 T^{11} + 156712969708897 T^{12} - 1508808675331311 T^{13} + 13839576473554737 T^{14} - 122398333527307862 T^{15} + 1053073183140790623 T^{16} - 8782230038593357827 T^{17} + 70684019365987194193 T^{18} - \)\(53\!\cdots\!98\)\( T^{19} + \)\(35\!\cdots\!20\)\( T^{20} - \)\(20\!\cdots\!02\)\( T^{21} + \)\(97\!\cdots\!74\)\( T^{22} - \)\(34\!\cdots\!44\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \))
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