Properties

Label 350.2.o
Level 350
Weight 2
Character orbit o
Rep. character \(\chi_{350}(143,\cdot)\)
Character field \(\Q(\zeta_{12})\)
Dimension 48
Newform subspaces 4
Sturm bound 120
Trace bound 9

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Defining parameters

Level: \( N \) = \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 350.o (of order \(12\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 35 \)
Character field: \(\Q(\zeta_{12})\)
Newform subspaces: \( 4 \)
Sturm bound: \(120\)
Trace bound: \(9\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(350, [\chi])\).

Total New Old
Modular forms 288 48 240
Cusp forms 192 48 144
Eisenstein series 96 0 96

Trace form

\( 48q - 8q^{7} + O(q^{10}) \) \( 48q - 8q^{7} + 24q^{11} + 24q^{16} + 36q^{17} + 8q^{18} + 16q^{21} + 8q^{22} + 4q^{23} - 24q^{26} - 4q^{28} - 48q^{31} - 48q^{33} - 64q^{36} - 4q^{37} - 24q^{38} - 36q^{42} + 8q^{43} + 16q^{46} - 12q^{47} + 32q^{51} + 28q^{53} + 8q^{56} - 8q^{57} + 32q^{58} - 96q^{61} + 36q^{63} - 32q^{67} + 36q^{68} - 32q^{71} + 8q^{72} + 12q^{73} - 16q^{77} - 16q^{78} + 40q^{81} + 48q^{82} - 24q^{86} + 24q^{87} + 4q^{88} - 8q^{91} - 8q^{92} - 28q^{93} - 24q^{96} - 40q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(350, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
350.2.o.a \(8\) \(2.795\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{24}^{7}q^{2}+(2\zeta_{24}-\zeta_{24}^{3}-\zeta_{24}^{5}+\cdots)q^{3}+\cdots\)
350.2.o.b \(8\) \(2.795\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{24}^{7}q^{2}+(2\zeta_{24}+\zeta_{24}^{3}-\zeta_{24}^{5}+\cdots)q^{3}+\cdots\)
350.2.o.c \(16\) \(2.795\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(-8\) \(q+\beta _{13}q^{2}+(\beta _{3}-\beta _{7}-\beta _{15})q^{3}+(\beta _{11}+\cdots)q^{4}+\cdots\)
350.2.o.d \(16\) \(2.795\) 16.0.\(\cdots\).1 None \(0\) \(0\) \(0\) \(0\) \(q+(-\beta _{5}+\beta _{7})q^{2}+(-\beta _{4}-\beta _{13}+\beta _{14}+\cdots)q^{3}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(350, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(350, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(175, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - T^{4} + T^{8} \))(\( 1 - T^{4} + T^{8} \))(\( ( 1 - T^{4} + T^{8} )^{2} \))(\( ( 1 - T^{4} + T^{8} )^{2} \))
$3$ (\( 1 + 12 T^{2} + 74 T^{4} + 312 T^{6} + 1027 T^{8} + 2808 T^{10} + 5994 T^{12} + 8748 T^{14} + 6561 T^{16} \))(\( 1 - 12 T^{2} + 74 T^{4} - 312 T^{6} + 1027 T^{8} - 2808 T^{10} + 5994 T^{12} - 8748 T^{14} + 6561 T^{16} \))(\( 1 + 3 T^{4} + 24 T^{5} - 48 T^{7} + 5 T^{8} + 72 T^{9} + 288 T^{10} - 120 T^{11} - 570 T^{12} + 720 T^{13} + 2016 T^{14} + 2136 T^{15} - 2570 T^{16} + 6408 T^{17} + 18144 T^{18} + 19440 T^{19} - 46170 T^{20} - 29160 T^{21} + 209952 T^{22} + 157464 T^{23} + 32805 T^{24} - 944784 T^{25} + 4251528 T^{27} + 1594323 T^{28} + 43046721 T^{32} \))(\( 1 + 7 T^{4} - 111 T^{8} - 14 T^{12} + 15070 T^{16} - 1134 T^{20} - 728271 T^{24} + 3720087 T^{28} + 43046721 T^{32} \))
$5$ 1
$7$ (\( 1 + 71 T^{4} + 2401 T^{8} \))(\( 1 + 71 T^{4} + 2401 T^{8} \))(\( 1 + 8 T + 32 T^{2} + 80 T^{3} + 73 T^{4} - 160 T^{5} - 416 T^{6} + 696 T^{7} + 4944 T^{8} + 4872 T^{9} - 20384 T^{10} - 54880 T^{11} + 175273 T^{12} + 1344560 T^{13} + 3764768 T^{14} + 6588344 T^{15} + 5764801 T^{16} \))(\( 1 + 73 T^{4} + 2928 T^{8} + 175273 T^{12} + 5764801 T^{16} \))
$11$ (\( ( 1 - 6 T + 8 T^{2} - 36 T^{3} + 267 T^{4} - 396 T^{5} + 968 T^{6} - 7986 T^{7} + 14641 T^{8} )^{2} \))(\( ( 1 - 6 T + 8 T^{2} - 36 T^{3} + 267 T^{4} - 396 T^{5} + 968 T^{6} - 7986 T^{7} + 14641 T^{8} )^{2} \))(\( ( 1 + 6 T + 5 T^{2} - 18 T^{3} - 29 T^{4} - 252 T^{5} - 544 T^{6} - 516 T^{7} - 6650 T^{8} - 5676 T^{9} - 65824 T^{10} - 335412 T^{11} - 424589 T^{12} - 2898918 T^{13} + 8857805 T^{14} + 116923026 T^{15} + 214358881 T^{16} )^{2} \))(\( ( 1 - 3 T - T^{2} + 36 T^{3} - 120 T^{4} + 396 T^{5} - 121 T^{6} - 3993 T^{7} + 14641 T^{8} )^{4} \))
$13$ (\( 1 - 452 T^{4} + 106470 T^{8} - 12909572 T^{12} + 815730721 T^{16} \))(\( 1 - 452 T^{4} + 106470 T^{8} - 12909572 T^{12} + 815730721 T^{16} \))(\( 1 + 714 T^{4} + 266977 T^{8} + 68653746 T^{12} + 13237862628 T^{16} + 1960819639506 T^{20} + 217781340700417 T^{24} + 16634832777451434 T^{28} + 665416609183179841 T^{32} \))(\( ( 1 + T^{4} - 50736 T^{8} + 28561 T^{12} + 815730721 T^{16} )^{2} \))
$17$ (\( 1 - 72 T^{2} + 2569 T^{4} - 60552 T^{6} + 1123152 T^{8} - 17499528 T^{10} + 214565449 T^{12} - 1737904968 T^{14} + 6975757441 T^{16} \))(\( 1 + 72 T^{2} + 2569 T^{4} + 60552 T^{6} + 1123152 T^{8} + 17499528 T^{10} + 214565449 T^{12} + 1737904968 T^{14} + 6975757441 T^{16} \))(\( 1 - 36 T + 648 T^{2} - 7776 T^{3} + 70080 T^{4} - 507420 T^{5} + 3088368 T^{6} - 16357284 T^{7} + 77768530 T^{8} - 343436904 T^{9} + 1468660248 T^{10} - 6351048828 T^{11} + 28493424864 T^{12} - 131692528476 T^{13} + 608521516920 T^{14} - 2723900163624 T^{15} + 11569567164915 T^{16} - 46306302781608 T^{17} + 175862718389880 T^{18} - 647005392402588 T^{19} + 2379799338066144 T^{20} - 9017581135777596 T^{21} + 35449888073657112 T^{22} - 140925443446588392 T^{23} + 542494401823131730 T^{24} - 1939775574818354148 T^{25} + 6226131050341877232 T^{26} - 17390245624419136860 T^{27} + 40830166385061650880 T^{28} - 77017998783876566112 T^{29} + \)\(10\!\cdots\!92\)\( T^{30} - \)\(10\!\cdots\!48\)\( T^{31} + 48661191875666868481 T^{32} \))(\( ( 1 + 94 T^{4} - 74685 T^{8} + 7850974 T^{12} + 6975757441 T^{16} )^{2} \))
$19$ (\( ( 1 - 19 T^{2} + 361 T^{4} )^{4} \))(\( ( 1 - 19 T^{2} + 361 T^{4} )^{4} \))(\( 1 - 90 T^{2} + 4011 T^{4} - 123414 T^{6} + 3034057 T^{8} - 61998180 T^{10} + 1094493006 T^{12} - 18780034728 T^{14} + 344074548210 T^{16} - 6779592536808 T^{18} + 142635423034926 T^{20} - 2916758998496580 T^{22} + 51529098329487337 T^{24} - 756659411140252614 T^{26} + 8877606140374371771 T^{28} - 71910601720459570890 T^{30} + \)\(28\!\cdots\!81\)\( T^{32} \))(\( ( 1 + 11 T^{2} - 503 T^{4} - 1078 T^{6} + 215374 T^{8} - 389158 T^{10} - 65551463 T^{12} + 517504691 T^{14} + 16983563041 T^{16} )^{2} \))
$23$ (\( 1 + 697 T^{4} + 205968 T^{8} + 195049177 T^{12} + 78310985281 T^{16} \))(\( 1 + 697 T^{4} + 205968 T^{8} + 195049177 T^{12} + 78310985281 T^{16} \))(\( 1 - 4 T + 8 T^{2} + 128 T^{3} - 198 T^{4} + 292 T^{5} + 8608 T^{6} - 3564 T^{7} - 17335 T^{8} - 1017836 T^{9} + 5500848 T^{10} + 425132 T^{11} + 20115594 T^{12} - 929862264 T^{13} + 4560471720 T^{14} + 14572270428 T^{15} - 102219015884 T^{16} + 335162219844 T^{17} + 2412489539880 T^{18} - 11313634166088 T^{19} + 5629167940554 T^{20} + 2736295372276 T^{21} + 814322923933872 T^{22} - 3465553913672692 T^{23} - 1357520929846135 T^{24} - 6419308085454132 T^{25} + 356599408527090592 T^{26} + 278220449310866684 T^{27} - 4339095637540023558 T^{28} + 64516654327867825024 T^{29} + 92742690596309998472 T^{30} - \)\(10\!\cdots\!28\)\( T^{31} + \)\(61\!\cdots\!61\)\( T^{32} \))(\( ( 1 - 311 T^{4} - 183120 T^{8} - 87030551 T^{12} + 78310985281 T^{16} )^{2} \))
$29$ (\( ( 1 - 46 T^{2} + 841 T^{4} )^{4} \))(\( ( 1 - 46 T^{2} + 841 T^{4} )^{4} \))(\( ( 1 - 70 T^{2} + 2685 T^{4} - 80906 T^{6} + 2505752 T^{8} - 68041946 T^{10} + 1899049485 T^{12} - 41637632470 T^{14} + 500246412961 T^{16} )^{2} \))(\( ( 1 - 83 T^{2} + 3276 T^{4} - 69803 T^{6} + 707281 T^{8} )^{4} \))
$31$ (\( ( 1 - 4 T + 31 T^{2} )^{4}( 1 + 7 T + 31 T^{2} )^{4} \))(\( ( 1 - 4 T + 31 T^{2} )^{4}( 1 + 7 T + 31 T^{2} )^{4} \))(\( ( 1 - 12 T + 156 T^{2} - 1296 T^{3} + 10686 T^{4} - 72996 T^{5} + 489600 T^{6} - 2919012 T^{7} + 17046947 T^{8} - 90489372 T^{9} + 470505600 T^{10} - 2174623836 T^{11} + 9868745406 T^{12} - 37103379696 T^{13} + 138450574236 T^{14} - 330151369332 T^{15} + 852891037441 T^{16} )^{2} \))(\( ( 1 + 6 T + 43 T^{2} + 186 T^{3} + 961 T^{4} )^{8} \))
$37$ (\( ( 1 - 529 T^{4} + 1874161 T^{8} )( 1 + 2591 T^{4} + 1874161 T^{8} ) \))(\( ( 1 - 529 T^{4} + 1874161 T^{8} )( 1 + 2591 T^{4} + 1874161 T^{8} ) \))(\( 1 + 4 T + 8 T^{2} - 240 T^{3} - 3049 T^{4} - 13544 T^{5} - 984 T^{6} + 514972 T^{7} + 3800209 T^{8} + 12256568 T^{9} - 16248928 T^{10} - 295990824 T^{11} + 119383314 T^{12} + 6433305488 T^{13} + 16607500784 T^{14} - 478842888760 T^{15} - 3828375485890 T^{16} - 17717186884120 T^{17} + 22735668573296 T^{18} + 325866222883664 T^{19} + 223743551149554 T^{20} - 20525174971850568 T^{21} - 41690303687539552 T^{22} + 1163539007448259544 T^{23} + 13348156033105669489 T^{24} + 66926657065750392844 T^{25} - 4731647022459163416 T^{26} - \)\(24\!\cdots\!72\)\( T^{27} - \)\(20\!\cdots\!69\)\( T^{28} - \)\(58\!\cdots\!80\)\( T^{29} + \)\(72\!\cdots\!12\)\( T^{30} + \)\(13\!\cdots\!72\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \))(\( 1 - 1393 T^{4} - 596879 T^{8} + 1686914642 T^{12} - 711159912626 T^{16} + 3161549632365362 T^{20} - 2096525223976912559 T^{24} - \)\(91\!\cdots\!33\)\( T^{28} + \)\(12\!\cdots\!41\)\( T^{32} \))
$41$ (\( ( 1 - 38 T^{2} - 165 T^{4} - 63878 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 - 38 T^{2} - 165 T^{4} - 63878 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 - 188 T^{2} + 17802 T^{4} - 1146544 T^{6} + 54447827 T^{8} - 1927340464 T^{10} + 50304197322 T^{12} - 893019597308 T^{14} + 7984925229121 T^{16} )^{2} \))(\( ( 1 - 55 T^{2} + 1681 T^{4} )^{8} \))
$43$ (\( 1 - 5444 T^{4} + 14231334 T^{8} - 18611952644 T^{12} + 11688200277601 T^{16} \))(\( 1 - 5444 T^{4} + 14231334 T^{8} - 18611952644 T^{12} + 11688200277601 T^{16} \))(\( ( 1 - 4 T + 8 T^{2} - 32 T^{3} + 3425 T^{4} - 20696 T^{5} + 55896 T^{6} - 651228 T^{7} + 6527152 T^{8} - 28002804 T^{9} + 103351704 T^{10} - 1645476872 T^{11} + 11709393425 T^{12} - 4704270176 T^{13} + 50570904392 T^{14} - 1087274444428 T^{15} + 11688200277601 T^{16} )^{2} \))(\( ( 1 + 217 T^{4} - 5396064 T^{8} + 741879817 T^{12} + 11688200277601 T^{16} )^{2} \))
$47$ (\( 1 + 144 T^{2} + 11689 T^{4} + 687888 T^{6} + 33208656 T^{8} + 1519544592 T^{10} + 57038591209 T^{12} + 1552207007376 T^{14} + 23811286661761 T^{16} \))(\( 1 - 144 T^{2} + 11689 T^{4} - 687888 T^{6} + 33208656 T^{8} - 1519544592 T^{10} + 57038591209 T^{12} - 1552207007376 T^{14} + 23811286661761 T^{16} \))(\( 1 + 12 T + 72 T^{2} + 288 T^{3} - 449 T^{4} - 2880 T^{5} + 39240 T^{6} + 500820 T^{7} + 44673 T^{8} - 37644384 T^{9} - 288946944 T^{10} - 1723474464 T^{11} + 9984376994 T^{12} + 179746348152 T^{13} + 1044056404368 T^{14} + 4096130163936 T^{15} + 8350203627806 T^{16} + 192518117704992 T^{17} + 2306320597248912 T^{18} + 18661805104185096 T^{19} + 48720574714458914 T^{20} - 395270263010401248 T^{21} - 3114621328032504576 T^{22} - 19071515289987429792 T^{23} + 1063721609040849153 T^{24} + \)\(56\!\cdots\!40\)\( T^{25} + \)\(20\!\cdots\!60\)\( T^{26} - \)\(71\!\cdots\!40\)\( T^{27} - \)\(52\!\cdots\!09\)\( T^{28} + \)\(15\!\cdots\!76\)\( T^{29} + \)\(18\!\cdots\!68\)\( T^{30} + \)\(14\!\cdots\!16\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \))(\( 1 - 4153 T^{4} + 8461249 T^{8} + 4041707906 T^{12} - 30776533112690 T^{16} + 19722245276457986 T^{20} + \)\(20\!\cdots\!89\)\( T^{24} - \)\(48\!\cdots\!73\)\( T^{28} + \)\(56\!\cdots\!21\)\( T^{32} \))
$53$ (\( 1 + 108 T^{2} + 5578 T^{4} + 182520 T^{6} + 5887011 T^{8} + 512698680 T^{10} + 44013103018 T^{12} + 2393751001932 T^{14} + 62259690411361 T^{16} \))(\( 1 - 108 T^{2} + 5578 T^{4} - 182520 T^{6} + 5887011 T^{8} - 512698680 T^{10} + 44013103018 T^{12} - 2393751001932 T^{14} + 62259690411361 T^{16} \))(\( 1 - 28 T + 392 T^{2} - 1808 T^{3} - 25321 T^{4} + 567704 T^{5} - 4335448 T^{6} - 1435172 T^{7} + 392677969 T^{8} - 4105212392 T^{9} + 14977011104 T^{10} + 124625519288 T^{11} - 2189877391278 T^{12} + 14281594759536 T^{13} - 14623477702416 T^{14} - 648224117593688 T^{15} + 7103959219709822 T^{16} - 34355878232465464 T^{17} - 41077348866086544 T^{18} + 2126200983015441072 T^{19} - 17279185948208624718 T^{20} + 52117830479026168984 T^{21} + \)\(33\!\cdots\!16\)\( T^{22} - \)\(48\!\cdots\!04\)\( T^{23} + \)\(24\!\cdots\!09\)\( T^{24} - \)\(47\!\cdots\!76\)\( T^{25} - \)\(75\!\cdots\!52\)\( T^{26} + \)\(52\!\cdots\!88\)\( T^{27} - \)\(12\!\cdots\!61\)\( T^{28} - \)\(47\!\cdots\!84\)\( T^{29} + \)\(54\!\cdots\!48\)\( T^{30} - \)\(20\!\cdots\!96\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \))(\( 1 + 7727 T^{4} + 31770289 T^{8} + 93923833106 T^{12} + 247060582275790 T^{16} + 741104220570063986 T^{20} + \)\(19\!\cdots\!29\)\( T^{24} + \)\(37\!\cdots\!07\)\( T^{28} + \)\(38\!\cdots\!21\)\( T^{32} \))
$59$ (\( ( 1 - 18 T + 152 T^{2} - 972 T^{3} + 6987 T^{4} - 57348 T^{5} + 529112 T^{6} - 3696822 T^{7} + 12117361 T^{8} )^{2} \))(\( ( 1 + 18 T + 152 T^{2} + 972 T^{3} + 6987 T^{4} + 57348 T^{5} + 529112 T^{6} + 3696822 T^{7} + 12117361 T^{8} )^{2} \))(\( 1 - 320 T^{2} + 54244 T^{4} - 6209920 T^{6} + 532238186 T^{8} - 36082424000 T^{10} + 2045329177232 T^{12} - 105925693186240 T^{14} + 5808829694261683 T^{16} - 368727337981301440 T^{18} + 24783992004353124752 T^{20} - \)\(15\!\cdots\!00\)\( T^{22} + \)\(78\!\cdots\!06\)\( T^{24} - \)\(31\!\cdots\!20\)\( T^{26} + \)\(96\!\cdots\!64\)\( T^{28} - \)\(19\!\cdots\!20\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \))(\( ( 1 - 10 T^{2} - 3381 T^{4} - 34810 T^{6} + 12117361 T^{8} )^{4} \))
$61$ (\( ( 1 - 6 T + 56 T^{2} - 264 T^{3} - 1053 T^{4} - 16104 T^{5} + 208376 T^{6} - 1361886 T^{7} + 13845841 T^{8} )^{2} \))(\( ( 1 - 6 T + 56 T^{2} - 264 T^{3} - 1053 T^{4} - 16104 T^{5} + 208376 T^{6} - 1361886 T^{7} + 13845841 T^{8} )^{2} \))(\( ( 1 + 6 T + 207 T^{2} + 1170 T^{3} + 23079 T^{4} + 119796 T^{5} + 1969284 T^{6} + 9244740 T^{7} + 135493478 T^{8} + 563929140 T^{9} + 7327705764 T^{10} + 27191415876 T^{11} + 319548164439 T^{12} + 988177672170 T^{13} + 10664717492727 T^{14} + 18856457016126 T^{15} + 191707312997281 T^{16} )^{2} \))(\( ( 1 + 27 T + 383 T^{2} + 3780 T^{3} + 30702 T^{4} + 230580 T^{5} + 1425143 T^{6} + 6128487 T^{7} + 13845841 T^{8} )^{4} \))
$67$ (\( 1 + 324 T^{2} + 52042 T^{4} + 5524200 T^{6} + 427630467 T^{8} + 24798133800 T^{10} + 1048704639082 T^{12} + 29308515822756 T^{14} + 406067677556641 T^{16} \))(\( 1 - 324 T^{2} + 52042 T^{4} - 5524200 T^{6} + 427630467 T^{8} - 24798133800 T^{10} + 1048704639082 T^{12} - 29308515822756 T^{14} + 406067677556641 T^{16} \))(\( 1 + 32 T + 512 T^{2} + 3896 T^{3} - 10509 T^{4} - 663104 T^{5} - 8249312 T^{6} - 44834240 T^{7} + 165963557 T^{8} + 5505231928 T^{9} + 49681319072 T^{10} + 159136747544 T^{11} - 1412717145882 T^{12} - 21627811976512 T^{13} - 103082665870656 T^{14} + 335467607852040 T^{15} + 7552610469365206 T^{16} + 22476329726086680 T^{17} - 462738087093374784 T^{18} - 6504845613491678656 T^{19} - 28467834145442833722 T^{20} + \)\(21\!\cdots\!08\)\( T^{21} + \)\(44\!\cdots\!68\)\( T^{22} + \)\(33\!\cdots\!44\)\( T^{23} + \)\(67\!\cdots\!37\)\( T^{24} - \)\(12\!\cdots\!80\)\( T^{25} - \)\(15\!\cdots\!88\)\( T^{26} - \)\(80\!\cdots\!32\)\( T^{27} - \)\(85\!\cdots\!49\)\( T^{28} + \)\(21\!\cdots\!52\)\( T^{29} + \)\(18\!\cdots\!48\)\( T^{30} + \)\(78\!\cdots\!76\)\( T^{31} + \)\(16\!\cdots\!81\)\( T^{32} \))(\( 1 + 3479 T^{4} + 20488945 T^{8} - 169384668334 T^{12} - 579648904329986 T^{16} - 3413290947143302414 T^{20} + \)\(83\!\cdots\!45\)\( T^{24} + \)\(28\!\cdots\!19\)\( T^{28} + \)\(16\!\cdots\!81\)\( T^{32} \))
$71$ (\( ( 1 - 6 T + 43 T^{2} - 426 T^{3} + 5041 T^{4} )^{4} \))(\( ( 1 - 6 T + 43 T^{2} - 426 T^{3} + 5041 T^{4} )^{4} \))(\( ( 1 - 4 T + 94 T^{2} + 964 T^{3} - 1158 T^{4} + 68444 T^{5} + 473854 T^{6} - 1431644 T^{7} + 25411681 T^{8} )^{4} \))(\( ( 1 + 6 T + 71 T^{2} )^{16} \))
$73$ (\( 1 + 3934 T^{4} - 12921885 T^{8} + 111718680094 T^{12} + 806460091894081 T^{16} \))(\( 1 + 3934 T^{4} - 12921885 T^{8} + 111718680094 T^{12} + 806460091894081 T^{16} \))(\( 1 - 12 T + 72 T^{2} - 288 T^{3} + 12064 T^{4} - 83700 T^{5} + 177264 T^{6} + 3613524 T^{7} + 19255890 T^{8} + 96865032 T^{9} - 2225167272 T^{10} + 39441388620 T^{11} + 6056513120 T^{12} + 1050679100940 T^{13} - 2551121523144 T^{14} + 71385024146568 T^{15} + 865631164724531 T^{16} + 5211106762699464 T^{17} - 13594926596834376 T^{18} + 408732031810375980 T^{19} + 171994319201421920 T^{20} + 81764822336595471660 T^{21} - \)\(33\!\cdots\!08\)\( T^{22} + \)\(10\!\cdots\!04\)\( T^{23} + \)\(15\!\cdots\!90\)\( T^{24} + \)\(21\!\cdots\!12\)\( T^{25} + \)\(76\!\cdots\!36\)\( T^{26} - \)\(26\!\cdots\!00\)\( T^{27} + \)\(27\!\cdots\!44\)\( T^{28} - \)\(48\!\cdots\!04\)\( T^{29} + \)\(87\!\cdots\!48\)\( T^{30} - \)\(10\!\cdots\!84\)\( T^{31} + \)\(65\!\cdots\!61\)\( T^{32} \))(\( 1 - 2788 T^{4} + 31113226 T^{8} + 223421298032 T^{12} - 482429265046061 T^{16} + 6344771866045561712 T^{20} + \)\(25\!\cdots\!06\)\( T^{24} - \)\(63\!\cdots\!48\)\( T^{28} + \)\(65\!\cdots\!61\)\( T^{32} \))
$79$ (\( ( 1 - 36 T + 697 T^{2} - 9540 T^{3} + 98112 T^{4} - 753660 T^{5} + 4349977 T^{6} - 17749404 T^{7} + 38950081 T^{8} )^{2} \))(\( ( 1 + 36 T + 697 T^{2} + 9540 T^{3} + 98112 T^{4} + 753660 T^{5} + 4349977 T^{6} + 17749404 T^{7} + 38950081 T^{8} )^{2} \))(\( 1 + 344 T^{2} + 56836 T^{4} + 6614224 T^{6} + 630756554 T^{8} + 45125337896 T^{10} + 2052833339792 T^{12} + 54520165812136 T^{14} + 1660868031271123 T^{16} + 340260354833540776 T^{18} + 79958024864398923152 T^{20} + \)\(10\!\cdots\!16\)\( T^{22} + \)\(95\!\cdots\!94\)\( T^{24} + \)\(62\!\cdots\!24\)\( T^{26} + \)\(33\!\cdots\!76\)\( T^{28} + \)\(12\!\cdots\!64\)\( T^{30} + \)\(23\!\cdots\!21\)\( T^{32} \))(\( ( 1 + 58 T^{2} - 2877 T^{4} + 361978 T^{6} + 38950081 T^{8} )^{4} \))
$83$ (\( ( 1 + 3122 T^{4} + 47458321 T^{8} )^{2} \))(\( ( 1 + 3122 T^{4} + 47458321 T^{8} )^{2} \))(\( 1 - 25638 T^{4} + 318852241 T^{8} - 2957648820318 T^{12} + 22747905510477444 T^{16} - \)\(14\!\cdots\!78\)\( T^{20} + \)\(71\!\cdots\!81\)\( T^{24} - \)\(27\!\cdots\!18\)\( T^{28} + \)\(50\!\cdots\!81\)\( T^{32} \))(\( ( 1 + 1033 T^{4} - 67132992 T^{8} + 49024445593 T^{12} + 2252292232139041 T^{16} )^{2} \))
$89$ (\( ( 1 + 12 T - 43 T^{2} + 108 T^{3} + 14232 T^{4} + 9612 T^{5} - 340603 T^{6} + 8459628 T^{7} + 62742241 T^{8} )^{2} \))(\( ( 1 - 12 T - 43 T^{2} - 108 T^{3} + 14232 T^{4} - 9612 T^{5} - 340603 T^{6} - 8459628 T^{7} + 62742241 T^{8} )^{2} \))(\( 1 - 522 T^{2} + 144915 T^{4} - 27754278 T^{6} + 4091335849 T^{8} - 494840689332 T^{10} + 51768593923950 T^{12} - 4926383904774360 T^{14} + 445675116984224850 T^{16} - 39021886909717705560 T^{18} + \)\(32\!\cdots\!50\)\( T^{20} - \)\(24\!\cdots\!52\)\( T^{22} + \)\(16\!\cdots\!69\)\( T^{24} - \)\(86\!\cdots\!78\)\( T^{26} + \)\(35\!\cdots\!15\)\( T^{28} - \)\(10\!\cdots\!02\)\( T^{30} + \)\(15\!\cdots\!61\)\( T^{32} \))(\( ( 1 - 257 T^{2} + 34849 T^{4} - 3947006 T^{6} + 387523630 T^{8} - 31264234526 T^{10} + 2186504356609 T^{12} - 127724191776977 T^{14} + 3936588805702081 T^{16} )^{2} \))
$97$ (\( 1 - 22322 T^{4} + 289141683 T^{8} - 1976150610482 T^{12} + 7837433594376961 T^{16} \))(\( 1 - 22322 T^{4} + 289141683 T^{8} - 1976150610482 T^{12} + 7837433594376961 T^{16} \))(\( ( 1 + 3868 T^{4} + 90482118 T^{8} + 342431258908 T^{12} + 7837433594376961 T^{16} )^{2} \))(\( ( 1 + 28228 T^{4} + 363130758 T^{8} + 2499004544068 T^{12} + 7837433594376961 T^{16} )^{2} \))
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