# Properties

 Label 1350.2.q Level 1350 Weight 2 Character orbit q Rep. character $$\chi_{1350}(143,\cdot)$$ Character field $$\Q(\zeta_{12})$$ Dimension 72 Newform subspaces 8 Sturm bound 540 Trace bound 31

# Related objects

## Defining parameters

 Level: $$N$$ = $$1350 = 2 \cdot 3^{3} \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 1350.q (of order $$12$$ and degree $$4$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$45$$ Character field: $$\Q(\zeta_{12})$$ Newform subspaces: $$8$$ Sturm bound: $$540$$ Trace bound: $$31$$ Distinguishing $$T_p$$: $$7$$, $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(1350, [\chi])$$.

Total New Old
Modular forms 1224 72 1152
Cusp forms 936 72 864
Eisenstein series 288 0 288

## Trace form

 $$72q + O(q^{10})$$ $$72q - 48q^{11} + 36q^{16} - 24q^{23} + 24q^{37} + 36q^{38} + 72q^{41} + 48q^{46} + 48q^{47} + 24q^{56} - 12q^{58} + 24q^{61} + 12q^{67} - 36q^{68} - 48q^{77} + 48q^{82} + 60q^{83} + 36q^{86} - 96q^{91} - 24q^{92} + 36q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(1350, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
1350.2.q.a $$8$$ $$10.780$$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\zeta_{24}^{3}-\zeta_{24}^{7})q^{2}+(\zeta_{24}^{2}-\zeta_{24}^{6}+\cdots)q^{4}+\cdots$$
1350.2.q.b $$8$$ $$10.780$$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{24}^{5}q^{2}+(-\zeta_{24}^{2}+\zeta_{24}^{6})q^{4}+\cdots$$
1350.2.q.c $$8$$ $$10.780$$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{24}q^{2}+\zeta_{24}^{2}q^{4}+\zeta_{24}^{3}q^{8}+(-4+\cdots)q^{11}+\cdots$$
1350.2.q.d $$8$$ $$10.780$$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{24}^{7}q^{2}-\zeta_{24}^{2}q^{4}+(-3\zeta_{24}+2\zeta_{24}^{3}+\cdots)q^{7}+\cdots$$
1350.2.q.e $$8$$ $$10.780$$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{24}^{7}q^{2}-\zeta_{24}^{2}q^{4}+(\zeta_{24}-\zeta_{24}^{5}+\cdots)q^{8}+\cdots$$
1350.2.q.f $$8$$ $$10.780$$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{24}^{5}q^{2}+(-\zeta_{24}^{2}+\zeta_{24}^{6})q^{4}+\cdots$$
1350.2.q.g $$8$$ $$10.780$$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$0$$ $$8$$ $$q+\zeta_{24}^{7}q^{2}-\zeta_{24}^{2}q^{4}+(2\zeta_{24}^{2}+2\zeta_{24}^{3}+\cdots)q^{7}+\cdots$$
1350.2.q.h $$16$$ $$10.780$$ 16.0.$$\cdots$$.9 None $$0$$ $$0$$ $$0$$ $$-8$$ $$q-\beta _{7}q^{2}-\beta _{9}q^{4}+(-1-2\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(1350, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(1350, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(90, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(135, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(225, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(270, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(450, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(675, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 - T^{4} + T^{8}$$)($$1 - T^{4} + T^{8}$$)($$1 - T^{4} + T^{8}$$)($$1 - T^{4} + T^{8}$$)($$1 - T^{4} + T^{8}$$)($$1 - T^{4} + T^{8}$$)($$1 - T^{4} + T^{8}$$)($$( 1 - T^{4} + T^{8} )^{2}$$)
$3$ ()()()()()()()()
$5$ ()()()()()()()()
$7$ ($$1 - 36 T^{2} + 634 T^{4} - 7272 T^{6} + 59571 T^{8} - 356328 T^{10} + 1522234 T^{12} - 4235364 T^{14} + 5764801 T^{16}$$)($$( 1 - 94 T^{4} + 2401 T^{8} )( 1 + 71 T^{4} + 2401 T^{8} )$$)($$( 1 - 49 T^{4} + 2401 T^{8} )^{2}$$)($$1 + 36 T^{2} + 634 T^{4} + 7272 T^{6} + 59571 T^{8} + 356328 T^{10} + 1522234 T^{12} + 4235364 T^{14} + 5764801 T^{16}$$)($$( 1 - 49 T^{4} + 2401 T^{8} )^{2}$$)($$( 1 + 23 T^{4} + 2401 T^{8} )( 1 + 71 T^{4} + 2401 T^{8} )$$)($$1 - 8 T + 32 T^{2} - 64 T^{3} - 7 T^{4} + 464 T^{5} - 1440 T^{6} + 2472 T^{7} - 4016 T^{8} + 17304 T^{9} - 70560 T^{10} + 159152 T^{11} - 16807 T^{12} - 1075648 T^{13} + 3764768 T^{14} - 6588344 T^{15} + 5764801 T^{16}$$)($$1 + 8 T + 32 T^{2} + 32 T^{3} - 168 T^{4} - 752 T^{5} - 128 T^{6} + 4696 T^{7} + 9614 T^{8} - 32960 T^{9} - 159808 T^{10} - 211120 T^{11} + 686208 T^{12} + 2864720 T^{13} + 4398240 T^{14} - 5724096 T^{15} - 29956541 T^{16} - 40068672 T^{17} + 215513760 T^{18} + 982598960 T^{19} + 1647585408 T^{20} - 3548293840 T^{21} - 18801251392 T^{22} - 27143977280 T^{23} + 55422796814 T^{24} + 189500538472 T^{25} - 36156831872 T^{26} - 1486949710736 T^{27} - 2325336249768 T^{28} + 3100448333024 T^{29} + 21703138331168 T^{30} + 37980492079544 T^{31} + 33232930569601 T^{32}$$)
$11$ ($$( 1 + 6 T + 23 T^{2} + 66 T^{3} + 121 T^{4} )^{4}$$)($$( 1 + 6 T + 23 T^{2} + 66 T^{3} + 121 T^{4} )^{4}$$)($$( 1 + 6 T + 23 T^{2} + 66 T^{3} + 121 T^{4} )^{4}$$)($$( 1 + 6 T + 23 T^{2} + 66 T^{3} + 121 T^{4} )^{4}$$)($$( 1 - 3 T + 14 T^{2} - 33 T^{3} + 121 T^{4} )^{4}$$)($$( 1 - 3 T + 14 T^{2} - 33 T^{3} + 121 T^{4} )^{4}$$)($$( 1 - 12 T + 74 T^{2} - 312 T^{3} + 1083 T^{4} - 3432 T^{5} + 8954 T^{6} - 15972 T^{7} + 14641 T^{8} )^{2}$$)($$( 1 + 22 T^{2} + 199 T^{4} + 264 T^{5} + 1138 T^{6} + 6840 T^{7} + 10132 T^{8} + 75240 T^{9} + 137698 T^{10} + 351384 T^{11} + 2913559 T^{12} + 38974342 T^{14} + 214358881 T^{16} )^{2}$$)
$13$ ($$1 - 36 T^{2} + 682 T^{4} - 9000 T^{6} + 106947 T^{8} - 1521000 T^{10} + 19478602 T^{12} - 173765124 T^{14} + 815730721 T^{16}$$)($$1 + 142 T^{4} - 8397 T^{8} + 4055662 T^{12} + 815730721 T^{16}$$)($$1 + 142 T^{4} - 8397 T^{8} + 4055662 T^{12} + 815730721 T^{16}$$)($$1 + 36 T^{2} + 682 T^{4} + 9000 T^{6} + 106947 T^{8} + 1521000 T^{10} + 19478602 T^{12} + 173765124 T^{14} + 815730721 T^{16}$$)($$( 1 - 337 T^{4} + 28561 T^{8} )( 1 + 191 T^{4} + 28561 T^{8} )$$)($$1 + 142 T^{4} - 8397 T^{8} + 4055662 T^{12} + 815730721 T^{16}$$)($$1 + 142 T^{4} - 8397 T^{8} + 4055662 T^{12} + 815730721 T^{16}$$)($$1 - 48 T^{3} - 208 T^{4} - 648 T^{5} + 1152 T^{6} + 768 T^{7} + 58078 T^{8} - 38784 T^{9} + 412704 T^{10} + 164616 T^{11} + 6437888 T^{12} - 1602600 T^{13} + 36461952 T^{14} - 584300928 T^{15} - 665621597 T^{16} - 7595912064 T^{17} + 6162069888 T^{18} - 3520912200 T^{19} + 183872519168 T^{20} + 61120768488 T^{21} + 1992043381536 T^{22} - 2433638483328 T^{23} + 47376008814238 T^{24} + 8144255518464 T^{25} + 158812982610048 T^{26} - 1161319935335976 T^{27} - 4846001705476048 T^{28} - 14538005116428144 T^{29} + 665416609183179841 T^{32}$$)
$17$ ($$( 1 + 289 T^{4} )^{4}$$)($$( 1 - 8 T + 32 T^{2} - 136 T^{3} + 289 T^{4} )^{2}( 1 + 8 T + 32 T^{2} + 136 T^{3} + 289 T^{4} )^{2}$$)($$( 1 - 8 T + 32 T^{2} - 136 T^{3} + 289 T^{4} )^{2}( 1 + 8 T + 32 T^{2} + 136 T^{3} + 289 T^{4} )^{2}$$)($$( 1 + 289 T^{4} )^{4}$$)($$( 1 + 47 T^{4} + 83521 T^{8} )^{2}$$)($$( 1 + 47 T^{4} + 83521 T^{8} )^{2}$$)($$1 + 188 T^{4} - 45306 T^{8} + 15701948 T^{12} + 6975757441 T^{16}$$)($$1 - 356 T^{4} + 24298 T^{8} + 31798576 T^{12} - 15547912973 T^{16} + 2655848866096 T^{20} + 169496954301418 T^{24} - 207413516453794916 T^{28} + 48661191875666868481 T^{32}$$)
$19$ ($$( 1 - 14 T^{2} + 339 T^{4} - 5054 T^{6} + 130321 T^{8} )^{2}$$)($$( 1 - 34 T^{2} + 361 T^{4} )^{4}$$)($$( 1 - 13 T^{2} + 361 T^{4} )^{4}$$)($$( 1 - 14 T^{2} + 339 T^{4} - 5054 T^{6} + 130321 T^{8} )^{2}$$)($$( 1 - 22 T^{2} + 361 T^{4} )^{4}$$)($$( 1 + 11 T^{2} + 361 T^{4} )^{4}$$)($$( 1 - 32 T^{2} + 594 T^{4} - 11552 T^{6} + 130321 T^{8} )^{2}$$)($$( 1 - 100 T^{2} + 5038 T^{4} - 162424 T^{6} + 3656035 T^{8} - 58635064 T^{10} + 656557198 T^{12} - 4704588100 T^{14} + 16983563041 T^{16} )^{2}$$)
$23$ ($$1 - 108 T^{2} + 5818 T^{4} - 208440 T^{6} + 5501811 T^{8} - 110264760 T^{10} + 1628114938 T^{12} - 15987876012 T^{14} + 78310985281 T^{16}$$)($$1 - 311 T^{4} - 183120 T^{8} - 87030551 T^{12} + 78310985281 T^{16}$$)($$1 + 958 T^{4} + 637923 T^{8} + 268087678 T^{12} + 78310985281 T^{16}$$)($$1 + 108 T^{2} + 5818 T^{4} + 208440 T^{6} + 5501811 T^{8} + 110264760 T^{10} + 1628114938 T^{12} + 15987876012 T^{14} + 78310985281 T^{16}$$)($$1 + 958 T^{4} + 637923 T^{8} + 268087678 T^{12} + 78310985281 T^{16}$$)($$1 + 958 T^{4} + 637923 T^{8} + 268087678 T^{12} + 78310985281 T^{16}$$)($$1 - 967 T^{4} + 655248 T^{8} - 270606247 T^{12} + 78310985281 T^{16}$$)($$1 + 24 T + 288 T^{2} + 2304 T^{3} + 14956 T^{4} + 93528 T^{5} + 591552 T^{6} + 3510696 T^{7} + 18949162 T^{8} + 98431632 T^{9} + 520135776 T^{10} + 2742734952 T^{11} + 13919721136 T^{12} + 68549757432 T^{13} + 335617355232 T^{14} + 1641179221488 T^{15} + 7932015899923 T^{16} + 37747122094224 T^{17} + 177541580917728 T^{18} + 834044898675144 T^{19} + 3895308682419376 T^{20} + 17653182909160536 T^{21} + 76998762000864864 T^{22} + 335142525423339504 T^{23} + 1483927546469284522 T^{24} + 6323299443987508248 T^{25} + 24505935561456493248 T^{26} + 89114391038173764456 T^{27} +$$$$32\!\cdots\!76$$$$T^{28} +$$$$11\!\cdots\!32$$$$T^{29} +$$$$33\!\cdots\!92$$$$T^{30} +$$$$63\!\cdots\!68$$$$T^{31} +$$$$61\!\cdots\!61$$$$T^{32}$$)
$29$ ($$( 1 - 18 T + 188 T^{2} - 1404 T^{3} + 8259 T^{4} - 40716 T^{5} + 158108 T^{6} - 439002 T^{7} + 707281 T^{8} )^{2}$$)($$( 1 + 17 T^{2} - 552 T^{4} + 14297 T^{6} + 707281 T^{8} )^{2}$$)($$( 1 - 10 T^{2} - 741 T^{4} - 8410 T^{6} + 707281 T^{8} )^{2}$$)($$( 1 + 18 T + 188 T^{2} + 1404 T^{3} + 8259 T^{4} + 40716 T^{5} + 158108 T^{6} + 439002 T^{7} + 707281 T^{8} )^{2}$$)($$( 1 - 46 T^{2} + 1275 T^{4} - 38686 T^{6} + 707281 T^{8} )^{2}$$)($$( 1 - 46 T^{2} + 1275 T^{4} - 38686 T^{6} + 707281 T^{8} )^{2}$$)($$1 - 106 T^{2} + 6769 T^{4} - 295210 T^{6} + 9797332 T^{8} - 248271610 T^{10} + 4787585089 T^{12} - 63051272026 T^{14} + 500246412961 T^{16}$$)($$1 - 152 T^{2} + 12036 T^{4} - 646864 T^{6} + 26386346 T^{8} - 870522600 T^{10} + 24736791952 T^{12} - 660065237288 T^{14} + 18313327590963 T^{16} - 555114864559208 T^{18} + 17495862948602512 T^{20} - 517807143937554600 T^{22} + 13199674937647830506 T^{24} -$$$$27\!\cdots\!64$$$$T^{26} +$$$$42\!\cdots\!76$$$$T^{28} -$$$$45\!\cdots\!12$$$$T^{30} +$$$$25\!\cdots\!21$$$$T^{32}$$)
$31$ ($$( 1 - 10 T + 40 T^{2} + 20 T^{3} - 461 T^{4} + 620 T^{5} + 38440 T^{6} - 297910 T^{7} + 923521 T^{8} )^{2}$$)($$( 1 + 10 T + 69 T^{2} + 310 T^{3} + 961 T^{4} )^{4}$$)($$( 1 - 7 T + 31 T^{2} )^{4}( 1 + 11 T + 31 T^{2} )^{4}$$)($$( 1 - 10 T + 40 T^{2} + 20 T^{3} - 461 T^{4} + 620 T^{5} + 38440 T^{6} - 297910 T^{7} + 923521 T^{8} )^{2}$$)($$( 1 - 7 T + 31 T^{2} )^{4}( 1 + 11 T + 31 T^{2} )^{4}$$)($$( 1 - 8 T + 33 T^{2} - 248 T^{3} + 961 T^{4} )^{4}$$)($$( 1 - 4 T - 44 T^{2} + 8 T^{3} + 2143 T^{4} + 248 T^{5} - 42284 T^{6} - 119164 T^{7} + 923521 T^{8} )^{2}$$)($$( 1 + 4 T - 36 T^{2} - 280 T^{3} - 244 T^{4} + 4476 T^{5} + 26680 T^{6} + 13564 T^{7} - 525585 T^{8} + 420484 T^{9} + 25639480 T^{10} + 133344516 T^{11} - 225339124 T^{12} - 8016162280 T^{13} - 31950132516 T^{14} + 110050456444 T^{15} + 852891037441 T^{16} )^{2}$$)
$37$ ($$1 - 644 T^{4} - 1762266 T^{8} - 1206959684 T^{12} + 3512479453921 T^{16}$$)($$( 1 + 1106 T^{4} + 1874161 T^{8} )^{2}$$)($$( 1 - 2062 T^{4} + 1874161 T^{8} )^{2}$$)($$1 - 644 T^{4} - 1762266 T^{8} - 1206959684 T^{12} + 3512479453921 T^{16}$$)($$( 1 - 2062 T^{4} + 1874161 T^{8} )^{2}$$)($$( 1 - 2062 T^{4} + 1874161 T^{8} )^{2}$$)($$( 1 - 6 T + 18 T^{2} - 222 T^{3} + 1369 T^{4} )^{4}$$)($$( 1 + 576 T^{3} + 1060 T^{4} - 8640 T^{5} + 165888 T^{6} + 634752 T^{7} - 2979546 T^{8} + 23485824 T^{9} + 227100672 T^{10} - 437641920 T^{11} + 1986610660 T^{12} + 39942119232 T^{13} + 3512479453921 T^{16} )^{2}$$)
$41$ ($$( 1 - 3 T + 44 T^{2} - 123 T^{3} + 1681 T^{4} )^{4}$$)($$( 1 + 15 T + 116 T^{2} + 615 T^{3} + 1681 T^{4} )^{4}$$)($$( 1 - 3 T + 44 T^{2} - 123 T^{3} + 1681 T^{4} )^{4}$$)($$( 1 - 3 T + 44 T^{2} - 123 T^{3} + 1681 T^{4} )^{4}$$)($$( 1 - 12 T + 89 T^{2} - 492 T^{3} + 1681 T^{4} )^{4}$$)($$( 1 - 21 T + 188 T^{2} - 861 T^{3} + 1681 T^{4} )^{4}$$)($$( 1 + 6 T + 65 T^{2} + 318 T^{3} + 1620 T^{4} + 13038 T^{5} + 109265 T^{6} + 413526 T^{7} + 2825761 T^{8} )^{2}$$)($$( 1 + 12 T + 154 T^{2} + 1272 T^{3} + 9001 T^{4} + 62400 T^{5} + 437338 T^{6} + 2866716 T^{7} + 21170932 T^{8} + 117535356 T^{9} + 735165178 T^{10} + 4300670400 T^{11} + 25434674761 T^{12} + 147369087672 T^{13} + 731516053114 T^{14} + 2337051286572 T^{15} + 7984925229121 T^{16} )^{2}$$)
$43$ ($$1 - 72 T^{2} + 1633 T^{4} + 6840 T^{6} - 214704 T^{8} + 12647160 T^{10} + 5582902033 T^{12} - 455138139528 T^{14} + 11688200277601 T^{16}$$)($$( 1 + 23 T^{4} + 3418801 T^{8} )( 1 + 3191 T^{4} + 3418801 T^{8} )$$)($$( 1 - 3214 T^{4} + 3418801 T^{8} )( 1 + 3191 T^{4} + 3418801 T^{8} )$$)($$1 + 72 T^{2} + 1633 T^{4} - 6840 T^{6} - 214704 T^{8} - 12647160 T^{10} + 5582902033 T^{12} + 455138139528 T^{14} + 11688200277601 T^{16}$$)($$1 + 3577 T^{4} + 9376128 T^{8} + 12229051177 T^{12} + 11688200277601 T^{16}$$)($$1 + 217 T^{4} - 3371712 T^{8} + 741879817 T^{12} + 11688200277601 T^{16}$$)($$1 - 1778 T^{4} - 257517 T^{8} - 6078628178 T^{12} + 11688200277601 T^{16}$$)($$1 - 96 T^{3} + 854 T^{4} - 1872 T^{5} + 4608 T^{6} - 53616 T^{7} + 4490401 T^{8} + 2573760 T^{9} + 2964096 T^{10} - 348774432 T^{11} - 9048958570 T^{12} - 6137398128 T^{13} + 6878498688 T^{14} + 451002697968 T^{15} - 494432039036 T^{16} + 19393116012624 T^{17} + 12718344074112 T^{18} - 487966112962896 T^{19} - 30936588608074570 T^{20} - 51272786206529376 T^{21} + 18737126928088704 T^{22} + 699595868522752320 T^{23} + 52484706214739808001 T^{24} - 26947005481605774288 T^{25} + 99585710499613819392 T^{26} -$$$$17\!\cdots\!04$$$$T^{27} +$$$$34\!\cdots\!54$$$$T^{28} -$$$$16\!\cdots\!28$$$$T^{29} +$$$$13\!\cdots\!01$$$$T^{32}$$)
$47$ ($$1 + 1054 T^{4} - 3768765 T^{8} + 5143183774 T^{12} + 23811286661761 T^{16}$$)($$1 - 2807 T^{4} + 2999568 T^{8} - 13697264567 T^{12} + 23811286661761 T^{16}$$)($$1 + 1054 T^{4} - 3768765 T^{8} + 5143183774 T^{12} + 23811286661761 T^{16}$$)($$1 + 1054 T^{4} - 3768765 T^{8} + 5143183774 T^{12} + 23811286661761 T^{16}$$)($$1 + 1918 T^{4} - 1200957 T^{8} + 9359228158 T^{12} + 23811286661761 T^{16}$$)($$1 + 1054 T^{4} - 3768765 T^{8} + 5143183774 T^{12} + 23811286661761 T^{16}$$)($$1 + 4249 T^{4} + 13174320 T^{8} + 20733764569 T^{12} + 23811286661761 T^{16}$$)($$1 - 48 T + 1152 T^{2} - 18432 T^{3} + 221336 T^{4} - 2122632 T^{5} + 16776576 T^{6} - 110138352 T^{7} + 584847502 T^{8} - 2167705440 T^{9} - 101278944 T^{10} + 103905054792 T^{11} - 1313297330560 T^{12} + 11748806482152 T^{13} - 89054821472256 T^{14} + 622050377830560 T^{15} - 4255385068735805 T^{16} + 29236367758036320 T^{17} - 196722100632213504 T^{18} + 1219796335396467096 T^{19} - 6408472031284351360 T^{20} + 23830105518606623544 T^{21} - 1091707545669732576 T^{22} -$$$$10\!\cdots\!20$$$$T^{23} +$$$$13\!\cdots\!22$$$$T^{24} -$$$$12\!\cdots\!84$$$$T^{25} +$$$$88\!\cdots\!24$$$$T^{26} -$$$$52\!\cdots\!96$$$$T^{27} +$$$$25\!\cdots\!76$$$$T^{28} -$$$$10\!\cdots\!64$$$$T^{29} +$$$$29\!\cdots\!88$$$$T^{30} -$$$$57\!\cdots\!64$$$$T^{31} +$$$$56\!\cdots\!21$$$$T^{32}$$)
$53$ ($$1 + 508 T^{4} - 3205722 T^{8} + 4008364348 T^{12} + 62259690411361 T^{16}$$)($$( 1 + 2809 T^{4} )^{4}$$)($$( 1 - 718 T^{4} + 7890481 T^{8} )^{2}$$)($$1 + 508 T^{4} - 3205722 T^{8} + 4008364348 T^{12} + 62259690411361 T^{16}$$)($$( 1 - 4174 T^{4} + 7890481 T^{8} )^{2}$$)($$( 1 + 2809 T^{4} )^{4}$$)($$1 - 4900 T^{4} + 13820838 T^{8} - 38663356900 T^{12} + 62259690411361 T^{16}$$)($$1 + 4192 T^{4} + 3754180 T^{8} - 35115635936 T^{12} - 139198621476794 T^{16} - 277079258155925216 T^{20} +$$$$23\!\cdots\!80$$$$T^{24} +$$$$20\!\cdots\!72$$$$T^{28} +$$$$38\!\cdots\!21$$$$T^{32}$$)
$59$ ($$( 1 + 18 T + 137 T^{2} + 1242 T^{3} + 12372 T^{4} + 73278 T^{5} + 476897 T^{6} + 3696822 T^{7} + 12117361 T^{8} )^{2}$$)($$( 1 - 70 T^{2} + 1419 T^{4} - 243670 T^{6} + 12117361 T^{8} )^{2}$$)($$( 1 - 115 T^{2} + 9744 T^{4} - 400315 T^{6} + 12117361 T^{8} )^{2}$$)($$( 1 - 18 T + 137 T^{2} - 1242 T^{3} + 12372 T^{4} - 73278 T^{5} + 476897 T^{6} - 3696822 T^{7} + 12117361 T^{8} )^{2}$$)($$( 1 - 43 T^{2} - 1632 T^{4} - 149683 T^{6} + 12117361 T^{8} )^{2}$$)($$( 1 + 29 T^{2} - 2640 T^{4} + 100949 T^{6} + 12117361 T^{8} )^{2}$$)($$1 - 16 T^{2} - 5906 T^{4} + 12800 T^{6} + 24961747 T^{8} + 44556800 T^{10} - 71565134066 T^{12} - 674888538256 T^{14} + 146830437604321 T^{16}$$)($$1 - 428 T^{2} + 100818 T^{4} - 16624408 T^{6} + 2120510801 T^{8} - 219902811120 T^{10} + 19093571371522 T^{12} - 1411764424766636 T^{14} + 89719050225623076 T^{16} - 4914351962612659916 T^{18} +$$$$23\!\cdots\!42$$$$T^{20} -$$$$92\!\cdots\!20$$$$T^{22} +$$$$31\!\cdots\!21$$$$T^{24} -$$$$84\!\cdots\!08$$$$T^{26} +$$$$17\!\cdots\!58$$$$T^{28} -$$$$26\!\cdots\!08$$$$T^{30} +$$$$21\!\cdots\!41$$$$T^{32}$$)
$61$ ($$( 1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4} )^{4}$$)($$( 1 - 14 T + 61 T^{2} )^{4}( 1 + T + 61 T^{2} )^{4}$$)($$( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} )^{4}$$)($$( 1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4} )^{4}$$)($$( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} )^{4}$$)($$( 1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4} )^{4}$$)($$( 1 - 6 T - 89 T^{2} - 18 T^{3} + 9708 T^{4} - 1098 T^{5} - 331169 T^{6} - 1361886 T^{7} + 13845841 T^{8} )^{2}$$)($$( 1 + 12 T + 8 T^{2} + 1176 T^{3} + 13384 T^{4} - 12108 T^{5} + 574064 T^{6} + 5555196 T^{7} - 14824577 T^{8} + 338866956 T^{9} + 2136092144 T^{10} - 2748285948 T^{11} + 185312735944 T^{12} + 993245249976 T^{13} + 412162994888 T^{14} + 37712914032252 T^{15} + 191707312997281 T^{16} )^{2}$$)
$67$ ($$1 - 216 T^{2} + 26737 T^{4} - 2415960 T^{6} + 174766032 T^{8} - 10845244440 T^{10} + 538780522177 T^{12} - 19539010548504 T^{14} + 406067677556641 T^{16}$$)($$( 1 + 2903 T^{4} + 20151121 T^{8} )( 1 + 5906 T^{4} + 20151121 T^{8} )$$)($$1 + 5497 T^{4} + 10065888 T^{8} + 110770712137 T^{12} + 406067677556641 T^{16}$$)($$1 + 216 T^{2} + 26737 T^{4} + 2415960 T^{6} + 174766032 T^{8} + 10845244440 T^{10} + 538780522177 T^{12} + 19539010548504 T^{14} + 406067677556641 T^{16}$$)($$( 1 - 8809 T^{4} + 20151121 T^{8} )( 1 + 2903 T^{4} + 20151121 T^{8} )$$)($$1 - 8183 T^{4} + 46810368 T^{8} - 164896623143 T^{12} + 406067677556641 T^{16}$$)($$1 + 4 T + 8 T^{2} - 304 T^{3} - 3511 T^{4} - 10624 T^{5} + 31800 T^{6} + 1654308 T^{7} - 4277600 T^{8} + 110838636 T^{9} + 142750200 T^{10} - 3195306112 T^{11} - 70750585831 T^{12} - 410438032528 T^{13} + 723667057352 T^{14} + 24242846421292 T^{15} + 406067677556641 T^{16}$$)($$1 - 16 T + 128 T^{2} + 128 T^{3} - 5562 T^{4} + 6640 T^{5} + 613888 T^{6} - 5277488 T^{7} + 23820881 T^{8} - 40207664 T^{9} + 355221248 T^{10} - 2724412528 T^{11} + 197658905958 T^{12} - 3038720900128 T^{13} + 23981818128000 T^{14} + 20130978078192 T^{15} - 1000499923621916 T^{16} + 1348775531238864 T^{17} + 107654381576592000 T^{18} - 913934814085197664 T^{19} + 3983048530687278918 T^{20} - 3678297755878140496 T^{21} + 32132739406133126912 T^{22} -$$$$24\!\cdots\!72$$$$T^{23} +$$$$96\!\cdots\!21$$$$T^{24} -$$$$14\!\cdots\!36$$$$T^{25} +$$$$11\!\cdots\!12$$$$T^{26} +$$$$81\!\cdots\!20$$$$T^{27} -$$$$45\!\cdots\!82$$$$T^{28} +$$$$70\!\cdots\!36$$$$T^{29} +$$$$47\!\cdots\!12$$$$T^{30} -$$$$39\!\cdots\!88$$$$T^{31} +$$$$16\!\cdots\!81$$$$T^{32}$$)
$71$ ($$( 1 - 116 T^{2} + 12474 T^{4} - 584756 T^{6} + 25411681 T^{8} )^{2}$$)($$( 1 - 130 T^{2} + 5041 T^{4} )^{4}$$)($$( 1 - 94 T^{2} + 5041 T^{4} )^{4}$$)($$( 1 - 116 T^{2} + 12474 T^{4} - 584756 T^{6} + 25411681 T^{8} )^{2}$$)($$( 1 - 130 T^{2} + 5041 T^{4} )^{4}$$)($$( 1 + 50 T^{2} + 5041 T^{4} )^{4}$$)($$( 1 - 244 T^{2} + 24582 T^{4} - 1230004 T^{6} + 25411681 T^{8} )^{2}$$)($$( 1 - 296 T^{2} + 51256 T^{4} - 5830568 T^{6} + 486270130 T^{8} - 29391893288 T^{10} + 1302501121336 T^{12} - 37917684040616 T^{14} + 645753531245761 T^{16} )^{2}$$)
$73$ ($$( 1 + 3503 T^{4} + 28398241 T^{8} )^{2}$$)($$( 1 - 8542 T^{4} + 28398241 T^{8} )^{2}$$)($$( 1 - 10657 T^{4} + 28398241 T^{8} )^{2}$$)($$( 1 + 3503 T^{4} + 28398241 T^{8} )^{2}$$)($$( 1 - 1054 T^{4} + 28398241 T^{8} )^{2}$$)($$( 1 - 5617 T^{4} + 28398241 T^{8} )^{2}$$)($$( 1 - 8 T + 32 T^{2} - 264 T^{3} + 578 T^{4} - 19272 T^{5} + 170528 T^{6} - 3112136 T^{7} + 28398241 T^{8} )^{2}$$)($$( 1 + 8 T + 32 T^{2} + 736 T^{3} + 2366 T^{4} - 45704 T^{5} - 170496 T^{6} - 3682392 T^{7} - 78261341 T^{8} - 268814616 T^{9} - 908573184 T^{10} - 17779632968 T^{11} + 67190238206 T^{12} + 1525780692448 T^{13} + 4842695241248 T^{14} + 88379188152776 T^{15} + 806460091894081 T^{16} )^{2}$$)
$79$ ($$( 1 + 58 T^{2} - 2877 T^{4} + 361978 T^{6} + 38950081 T^{8} )^{2}$$)($$( 1 + 11 T^{2} + 6241 T^{4} )^{2}( 1 + 131 T^{2} + 6241 T^{4} )^{2}$$)($$( 1 - 38 T^{2} - 4797 T^{4} - 237158 T^{6} + 38950081 T^{8} )^{2}$$)($$( 1 + 58 T^{2} - 2877 T^{4} + 361978 T^{6} + 38950081 T^{8} )^{2}$$)($$( 1 + 11 T^{2} + 6241 T^{4} )^{2}( 1 + 131 T^{2} + 6241 T^{4} )^{2}$$)($$( 1 + 11 T^{2} + 6241 T^{4} )^{2}( 1 + 131 T^{2} + 6241 T^{4} )^{2}$$)($$( 1 + 152 T^{2} + 16863 T^{4} + 948632 T^{6} + 38950081 T^{8} )^{2}$$)($$1 + 200 T^{2} + 7464 T^{4} - 683600 T^{6} + 16634510 T^{8} + 10268111400 T^{10} + 468563441536 T^{12} + 2801113391000 T^{14} + 270299704808259 T^{16} + 17481748673231000 T^{18} + 18250584001465964416 T^{20} +$$$$24\!\cdots\!00$$$$T^{22} +$$$$25\!\cdots\!10$$$$T^{24} -$$$$64\!\cdots\!00$$$$T^{26} +$$$$44\!\cdots\!24$$$$T^{28} +$$$$73\!\cdots\!00$$$$T^{30} +$$$$23\!\cdots\!21$$$$T^{32}$$)
$83$ ($$1 - 432 T^{2} + 91513 T^{4} - 12659760 T^{6} + 1239875616 T^{8} - 87213086640 T^{10} + 4343053329673 T^{12} - 141238241295408 T^{14} + 2252292232139041 T^{16}$$)($$1 - 10871 T^{4} + 70720320 T^{8} - 515919407591 T^{12} + 2252292232139041 T^{16}$$)($$1 + 6553 T^{4} - 4516512 T^{8} + 310994377513 T^{12} + 2252292232139041 T^{16}$$)($$1 + 432 T^{2} + 91513 T^{4} + 12659760 T^{6} + 1239875616 T^{8} + 87213086640 T^{10} + 4343053329673 T^{12} + 141238241295408 T^{14} + 2252292232139041 T^{16}$$)($$1 + 6553 T^{4} - 4516512 T^{8} + 310994377513 T^{12} + 2252292232139041 T^{16}$$)($$1 + 13294 T^{4} + 129272115 T^{8} + 630910919374 T^{12} + 2252292232139041 T^{16}$$)($$1 - 12 T + 72 T^{2} - 288 T^{3} - 8375 T^{4} + 56640 T^{5} - 35208 T^{6} - 3433884 T^{7} + 82789344 T^{8} - 285012372 T^{9} - 242547912 T^{10} + 32386015680 T^{11} - 397463438375 T^{12} - 1134443705184 T^{13} + 23539706882568 T^{14} - 325632611875524 T^{15} + 2252292232139041 T^{16}$$)($$1 - 48 T + 1152 T^{2} - 18432 T^{3} + 215516 T^{4} - 1906416 T^{5} + 13102848 T^{6} - 74216976 T^{7} + 433250986 T^{8} - 3712707360 T^{9} + 40409712000 T^{10} - 415323735120 T^{11} + 3731885962352 T^{12} - 29403431286576 T^{13} + 213762329771904 T^{14} - 1567513821393504 T^{15} + 12987012407281267 T^{16} - 130103647175660832 T^{17} + 1472608689798646656 T^{18} - 16812499765057431312 T^{19} +$$$$17\!\cdots\!92$$$$T^{20} -$$$$16\!\cdots\!60$$$$T^{21} +$$$$13\!\cdots\!00$$$$T^{22} -$$$$10\!\cdots\!20$$$$T^{23} +$$$$97\!\cdots\!26$$$$T^{24} -$$$$13\!\cdots\!28$$$$T^{25} +$$$$20\!\cdots\!52$$$$T^{26} -$$$$24\!\cdots\!72$$$$T^{27} +$$$$23\!\cdots\!76$$$$T^{28} -$$$$16\!\cdots\!16$$$$T^{29} +$$$$84\!\cdots\!08$$$$T^{30} -$$$$29\!\cdots\!36$$$$T^{31} +$$$$50\!\cdots\!81$$$$T^{32}$$)
$89$ ($$( 1 + 103 T^{2} + 7921 T^{4} )^{4}$$)($$( 1 + 175 T^{2} + 7921 T^{4} )^{4}$$)($$( 1 + 31 T^{2} + 7921 T^{4} )^{4}$$)($$( 1 + 103 T^{2} + 7921 T^{4} )^{4}$$)($$( 1 + 175 T^{2} + 7921 T^{4} )^{4}$$)($$( 1 + 130 T^{2} + 7921 T^{4} )^{4}$$)($$( 1 + 286 T^{2} + 35427 T^{4} + 2265406 T^{6} + 62742241 T^{8} )^{2}$$)($$( 1 + 328 T^{2} + 42642 T^{4} + 2598088 T^{6} + 62742241 T^{8} )^{4}$$)
$97$ ($$1 + 360 T^{2} + 65929 T^{4} + 8182440 T^{6} + 834546960 T^{8} + 76988577960 T^{10} + 5836646967049 T^{12} + 299869921774440 T^{14} + 7837433594376961 T^{16}$$)($$1 - 2498 T^{4} - 82289277 T^{8} - 221146143938 T^{12} + 7837433594376961 T^{16}$$)($$( 1 - 18814 T^{4} + 88529281 T^{8} )( 1 + 9743 T^{4} + 88529281 T^{8} )$$)($$1 - 360 T^{2} + 65929 T^{4} - 8182440 T^{6} + 834546960 T^{8} - 76988577960 T^{10} + 5836646967049 T^{12} - 299869921774440 T^{14} + 7837433594376961 T^{16}$$)($$1 + 16417 T^{4} + 180988608 T^{8} + 1453385206177 T^{12} + 7837433594376961 T^{16}$$)($$1 + 4657 T^{4} - 66841632 T^{8} + 412280861617 T^{12} + 7837433594376961 T^{16}$$)($$1 + 12 T + 72 T^{2} - 744 T^{3} - 12542 T^{4} - 57996 T^{5} + 483840 T^{6} + 14345892 T^{7} + 128096019 T^{8} + 1391551524 T^{9} + 4552450560 T^{10} - 52931383308 T^{11} - 1110334242302 T^{12} - 6388981151208 T^{13} + 59973984354888 T^{14} + 969579413737356 T^{15} + 7837433594376961 T^{16}$$)($$1 - 48 T + 1152 T^{2} - 15456 T^{3} + 79442 T^{4} + 1258944 T^{5} - 32502528 T^{6} + 320382528 T^{7} - 602207375 T^{8} - 26997703632 T^{9} + 412336144512 T^{10} - 2856970349424 T^{11} + 3870433274354 T^{12} + 127396070447952 T^{13} - 1412601730375680 T^{14} + 7780535897090304 T^{15} - 41112146089827164 T^{16} + 754711982017759488 T^{17} - 13291169681104773120 T^{18} +$$$$11\!\cdots\!96$$$$T^{19} +$$$$34\!\cdots\!74$$$$T^{20} -$$$$24\!\cdots\!68$$$$T^{21} +$$$$34\!\cdots\!48$$$$T^{22} -$$$$21\!\cdots\!16$$$$T^{23} -$$$$47\!\cdots\!75$$$$T^{24} +$$$$24\!\cdots\!76$$$$T^{25} -$$$$23\!\cdots\!72$$$$T^{26} +$$$$90\!\cdots\!32$$$$T^{27} +$$$$55\!\cdots\!22$$$$T^{28} -$$$$10\!\cdots\!12$$$$T^{29} +$$$$75\!\cdots\!88$$$$T^{30} -$$$$30\!\cdots\!64$$$$T^{31} +$$$$61\!\cdots\!21$$$$T^{32}$$)