# Properties

 Label 224.3.o Level 224 Weight 3 Character orbit o Rep. character $$\chi_{224}(79,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 28 Newform subspaces 4 Sturm bound 96 Trace bound 5

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$224 = 2^{5} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 224.o (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$56$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$4$$ Sturm bound: $$96$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$3$$, $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(224, [\chi])$$.

Total New Old
Modular forms 144 36 108
Cusp forms 112 28 84
Eisenstein series 32 8 24

## Trace form

 $$28q + 2q^{3} - 32q^{9} + O(q^{10})$$ $$28q + 2q^{3} - 32q^{9} + 18q^{11} - 2q^{17} + 2q^{19} + 28q^{25} - 28q^{27} + 34q^{33} + 6q^{35} - 8q^{41} + 104q^{43} - 20q^{49} + 46q^{51} - 68q^{57} - 62q^{59} + 48q^{65} - 222q^{67} - 82q^{73} - 84q^{75} + 14q^{81} + 488q^{83} + 46q^{89} - 288q^{91} - 424q^{97} - 832q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(224, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
224.3.o.a $$2$$ $$6.104$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$1$$ $$-9$$ $$2$$ $$q+\zeta_{6}q^{3}+(-3-3\zeta_{6})q^{5}+(-3+8\zeta_{6})q^{7}+\cdots$$
224.3.o.b $$2$$ $$6.104$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$1$$ $$9$$ $$-2$$ $$q+\zeta_{6}q^{3}+(3+3\zeta_{6})q^{5}+(3-8\zeta_{6})q^{7}+\cdots$$
224.3.o.c $$12$$ $$6.104$$ $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ None $$0$$ $$-6$$ $$0$$ $$0$$ $$q+(-1+\beta _{2}-\beta _{8})q^{3}+(\beta _{1}-\beta _{4}+\beta _{5}+\cdots)q^{5}+\cdots$$
224.3.o.d $$12$$ $$6.104$$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$0$$ $$6$$ $$0$$ $$0$$ $$q+(1-\beta _{3}+\beta _{6}+\beta _{7}-\beta _{10})q^{3}+(-\beta _{1}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{3}^{\mathrm{old}}(224, [\chi])$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(224, [\chi]) \cong$$ $$S_{3}^{\mathrm{new}}(56, [\chi])$$$$^{\oplus 3}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ ($$1 - T - 8 T^{2} - 9 T^{3} + 81 T^{4}$$)($$1 - T - 8 T^{2} - 9 T^{3} + 81 T^{4}$$)($$( 1 + 3 T - 7 T^{2} - 22 T^{3} + 7 T^{4} - 61 T^{5} - 350 T^{6} - 549 T^{7} + 567 T^{8} - 16038 T^{9} - 45927 T^{10} + 177147 T^{11} + 531441 T^{12} )^{2}$$)($$( 1 - 3 T + T^{2} + 14 T^{3} - 65 T^{4} + 37 T^{5} + 514 T^{6} + 333 T^{7} - 5265 T^{8} + 10206 T^{9} + 6561 T^{10} - 177147 T^{11} + 531441 T^{12} )^{2}$$)
$5$ ($$1 + 9 T + 52 T^{2} + 225 T^{3} + 625 T^{4}$$)($$1 - 9 T + 52 T^{2} - 225 T^{3} + 625 T^{4}$$)($$1 + 17 T^{2} - 437 T^{4} - 28164 T^{6} - 270975 T^{8} + 5916803 T^{10} + 456152134 T^{12} + 3698001875 T^{14} - 105849609375 T^{16} - 6875976562500 T^{18} - 66680908203125 T^{20} + 1621246337890625 T^{22} + 59604644775390625 T^{24}$$)($$1 + 121 T^{2} + 7979 T^{4} + 380588 T^{6} + 14474897 T^{8} + 457025259 T^{10} + 12295779174 T^{12} + 285640786875 T^{14} + 5654256640625 T^{16} + 92916992187500 T^{18} + 1217498779296875 T^{20} + 11539459228515625 T^{22} + 59604644775390625 T^{24}$$)
$7$ ($$1 - 2 T + 49 T^{2}$$)($$1 + 2 T + 49 T^{2}$$)($$1 + 102 T^{2} + 8463 T^{4} + 426692 T^{6} + 20319663 T^{8} + 588009702 T^{10} + 13841287201 T^{12}$$)($$1 - 186 T^{2} + 16527 T^{4} - 956284 T^{6} + 39681327 T^{8} - 1072252986 T^{10} + 13841287201 T^{12}$$)
$11$ ($$1 - 17 T + 168 T^{2} - 2057 T^{3} + 14641 T^{4}$$)($$1 - 17 T + 168 T^{2} - 2057 T^{3} + 14641 T^{4}$$)($$( 1 - 7 T - 193 T^{2} + 2268 T^{3} + 15415 T^{4} - 171941 T^{5} - 336170 T^{6} - 20804861 T^{7} + 225691015 T^{8} + 4017900348 T^{9} - 41371264033 T^{10} - 181561972207 T^{11} + 3138428376721 T^{12} )^{2}$$)($$( 1 + 15 T - 129 T^{2} - 924 T^{3} + 35727 T^{4} + 38013 T^{5} - 5198650 T^{6} + 4599573 T^{7} + 523079007 T^{8} - 1636922364 T^{9} - 27652295649 T^{10} + 389061369015 T^{11} + 3138428376721 T^{12} )^{2}$$)
$13$ ($$( 1 - 22 T + 169 T^{2} )( 1 + 22 T + 169 T^{2} )$$)($$( 1 - 22 T + 169 T^{2} )( 1 + 22 T + 169 T^{2} )$$)($$( 1 - 950 T^{2} + 386271 T^{4} - 85941828 T^{6} + 11032286031 T^{8} - 774944184950 T^{10} + 23298085122481 T^{12} )^{2}$$)($$( 1 - 86 T^{2} + 60895 T^{4} - 4902788 T^{6} + 1739222095 T^{8} - 70152842006 T^{10} + 23298085122481 T^{12} )^{2}$$)
$17$ ($$1 - 25 T + 336 T^{2} - 7225 T^{3} + 83521 T^{4}$$)($$1 - 25 T + 336 T^{2} - 7225 T^{3} + 83521 T^{4}$$)($$( 1 + 41 T + 471 T^{2} + 5456 T^{3} + 209253 T^{4} + 1462951 T^{5} - 27854634 T^{6} + 422792839 T^{7} + 17477019813 T^{8} + 131694576464 T^{9} + 3285581754711 T^{10} + 82655749918409 T^{11} + 582622237229761 T^{12} )^{2}$$)($$( 1 - 15 T - 665 T^{2} + 3920 T^{3} + 405365 T^{4} - 1221665 T^{5} - 124870762 T^{6} - 353061185 T^{7} + 33856490165 T^{8} + 94619270480 T^{9} - 4638878698265 T^{10} - 30239908506735 T^{11} + 582622237229761 T^{12} )^{2}$$)
$19$ ($$1 + 7 T - 312 T^{2} + 2527 T^{3} + 130321 T^{4}$$)($$1 + 7 T - 312 T^{2} + 2527 T^{3} + 130321 T^{4}$$)($$( 1 - 47 T + 487 T^{2} - 8532 T^{3} + 642195 T^{4} - 10028861 T^{5} + 46257550 T^{6} - 3620418821 T^{7} + 83691494595 T^{8} - 401395456692 T^{9} + 8270995200967 T^{10} - 288160114116647 T^{11} + 2213314919066161 T^{12} )^{2}$$)($$( 1 + 39 T + 151 T^{2} - 3964 T^{3} + 263147 T^{4} + 5820949 T^{5} + 41830430 T^{6} + 2101362589 T^{7} + 34293580187 T^{8} - 186489872284 T^{9} + 2564518019191 T^{10} + 239111584054239 T^{11} + 2213314919066161 T^{12} )^{2}$$)
$23$ ($$1 + 9 T + 556 T^{2} + 4761 T^{3} + 279841 T^{4}$$)($$1 - 9 T + 556 T^{2} - 4761 T^{3} + 279841 T^{4}$$)($$1 + 2381 T^{2} + 2947123 T^{4} + 2827993812 T^{6} + 2298643835613 T^{8} + 1531108455942719 T^{10} + 862820921151363886 T^{12} +$$$$42\!\cdots\!79$$$$T^{14} +$$$$18\!\cdots\!53$$$$T^{16} +$$$$61\!\cdots\!52$$$$T^{18} +$$$$18\!\cdots\!03$$$$T^{20} +$$$$40\!\cdots\!81$$$$T^{22} +$$$$48\!\cdots\!41$$$$T^{24}$$)($$1 + 693 T^{2} - 15581 T^{4} - 315659500 T^{6} - 121343121715 T^{8} + 25399919636503 T^{10} + 46081428374834798 T^{12} + 7107938910998636023 T^{14} -$$$$95\!\cdots\!15$$$$T^{16} -$$$$69\!\cdots\!00$$$$T^{18} -$$$$95\!\cdots\!41$$$$T^{20} +$$$$11\!\cdots\!93$$$$T^{22} +$$$$48\!\cdots\!41$$$$T^{24}$$)
$29$ ($$1 - 1490 T^{2} + 707281 T^{4}$$)($$1 - 1490 T^{2} + 707281 T^{4}$$)($$( 1 - 1838 T^{2} + 2723567 T^{4} - 2423469812 T^{6} + 1926327191327 T^{8} - 919452907022318 T^{10} + 353814783205469041 T^{12} )^{2}$$)($$( 1 - 3662 T^{2} + 6471151 T^{4} - 6858243380 T^{6} + 4576922150431 T^{8} - 1831902364263182 T^{10} + 353814783205469041 T^{12} )^{2}$$)
$31$ ($$1 + 57 T + 2044 T^{2} + 54777 T^{3} + 923521 T^{4}$$)($$1 - 57 T + 2044 T^{2} - 54777 T^{3} + 923521 T^{4}$$)($$1 + 3329 T^{2} + 4631887 T^{4} + 6475839496 T^{6} + 10686875408317 T^{8} + 11546553377441207 T^{10} + 9928800654054771830 T^{12} +$$$$10\!\cdots\!47$$$$T^{14} +$$$$91\!\cdots\!97$$$$T^{16} +$$$$51\!\cdots\!56$$$$T^{18} +$$$$33\!\cdots\!47$$$$T^{20} +$$$$22\!\cdots\!29$$$$T^{22} +$$$$62\!\cdots\!21$$$$T^{24}$$)($$1 + 3561 T^{2} + 5838879 T^{4} + 8280992264 T^{6} + 11856779149293 T^{8} + 13054196823637455 T^{10} + 12193464685853753718 T^{12} +$$$$12\!\cdots\!55$$$$T^{14} +$$$$10\!\cdots\!13$$$$T^{16} +$$$$65\!\cdots\!04$$$$T^{18} +$$$$42\!\cdots\!99$$$$T^{20} +$$$$23\!\cdots\!61$$$$T^{22} +$$$$62\!\cdots\!21$$$$T^{24}$$)
$37$ ($$1 - 15 T + 1444 T^{2} - 20535 T^{3} + 1874161 T^{4}$$)($$1 + 15 T + 1444 T^{2} + 20535 T^{3} + 1874161 T^{4}$$)($$1 + 6401 T^{2} + 22091395 T^{4} + 55331146644 T^{6} + 111395987301489 T^{8} + 188098942339055675 T^{10} +$$$$27\!\cdots\!06$$$$T^{12} +$$$$35\!\cdots\!75$$$$T^{14} +$$$$39\!\cdots\!69$$$$T^{16} +$$$$36\!\cdots\!64$$$$T^{18} +$$$$27\!\cdots\!95$$$$T^{20} +$$$$14\!\cdots\!01$$$$T^{22} +$$$$43\!\cdots\!61$$$$T^{24}$$)($$1 + 5785 T^{2} + 17889827 T^{4} + 38047414052 T^{6} + 62842022068961 T^{8} + 88030006321881747 T^{10} +$$$$11\!\cdots\!14$$$$T^{12} +$$$$16\!\cdots\!67$$$$T^{14} +$$$$22\!\cdots\!81$$$$T^{16} +$$$$25\!\cdots\!12$$$$T^{18} +$$$$22\!\cdots\!07$$$$T^{20} +$$$$13\!\cdots\!85$$$$T^{22} +$$$$43\!\cdots\!61$$$$T^{24}$$)
$41$ ($$( 1 - 26 T + 1681 T^{2} )^{2}$$)($$( 1 - 26 T + 1681 T^{2} )^{2}$$)($$( 1 - 30 T + 4755 T^{2} - 100496 T^{3} + 7993155 T^{4} - 84772830 T^{5} + 4750104241 T^{6} )^{4}$$)($$( 1 + 58 T + 3139 T^{2} + 88960 T^{3} + 5276659 T^{4} + 163894138 T^{5} + 4750104241 T^{6} )^{4}$$)
$43$ ($$( 1 + 14 T + 1849 T^{2} )^{2}$$)($$( 1 + 14 T + 1849 T^{2} )^{2}$$)($$( 1 + 10 T + 3855 T^{2} + 53388 T^{3} + 7127895 T^{4} + 34188010 T^{5} + 6321363049 T^{6} )^{4}$$)($$( 1 - 50 T + 4047 T^{2} - 107900 T^{3} + 7482903 T^{4} - 170940050 T^{5} + 6321363049 T^{6} )^{4}$$)
$47$ ($$1 - 87 T + 4732 T^{2} - 192183 T^{3} + 4879681 T^{4}$$)($$1 + 87 T + 4732 T^{2} + 192183 T^{3} + 4879681 T^{4}$$)($$1 + 8313 T^{2} + 34702703 T^{4} + 101154992824 T^{6} + 247422958854189 T^{8} + 586359622432093519 T^{10} +$$$$13\!\cdots\!26$$$$T^{12} +$$$$28\!\cdots\!39$$$$T^{14} +$$$$58\!\cdots\!29$$$$T^{16} +$$$$11\!\cdots\!84$$$$T^{18} +$$$$19\!\cdots\!63$$$$T^{20} +$$$$22\!\cdots\!13$$$$T^{22} +$$$$13\!\cdots\!81$$$$T^{24}$$)($$1 + 3905 T^{2} + 6537087 T^{4} + 2079696952 T^{6} - 28977286497379 T^{8} - 99900087904130553 T^{10} -$$$$24\!\cdots\!46$$$$T^{12} -$$$$48\!\cdots\!93$$$$T^{14} -$$$$68\!\cdots\!19$$$$T^{16} +$$$$24\!\cdots\!32$$$$T^{18} +$$$$37\!\cdots\!27$$$$T^{20} +$$$$10\!\cdots\!05$$$$T^{22} +$$$$13\!\cdots\!81$$$$T^{24}$$)
$53$ ($$( 1 - 53 T )^{2}( 1 - 53 T + 2809 T^{2} )$$)($$( 1 + 53 T )^{2}( 1 + 53 T + 2809 T^{2} )$$)($$1 + 9273 T^{2} + 34839939 T^{4} + 130830291652 T^{6} + 620089921398753 T^{8} + 1876428047292173523 T^{10} +$$$$44\!\cdots\!94$$$$T^{12} +$$$$14\!\cdots\!63$$$$T^{14} +$$$$38\!\cdots\!33$$$$T^{16} +$$$$64\!\cdots\!32$$$$T^{18} +$$$$13\!\cdots\!19$$$$T^{20} +$$$$28\!\cdots\!73$$$$T^{22} +$$$$24\!\cdots\!81$$$$T^{24}$$)($$1 + 10561 T^{2} + 59733731 T^{4} + 209884512884 T^{6} + 494098524300977 T^{8} + 757507316642438811 T^{10} +$$$$12\!\cdots\!10$$$$T^{12} +$$$$59\!\cdots\!91$$$$T^{14} +$$$$30\!\cdots\!97$$$$T^{16} +$$$$10\!\cdots\!44$$$$T^{18} +$$$$23\!\cdots\!51$$$$T^{20} +$$$$32\!\cdots\!61$$$$T^{22} +$$$$24\!\cdots\!81$$$$T^{24}$$)
$59$ ($$1 + 55 T - 456 T^{2} + 191455 T^{3} + 12117361 T^{4}$$)($$1 + 55 T - 456 T^{2} + 191455 T^{3} + 12117361 T^{4}$$)($$( 1 + 31 T - 7567 T^{2} - 176030 T^{3} + 36214647 T^{4} + 412965247 T^{5} - 131958052142 T^{6} + 1437532024807 T^{7} + 438825951186567 T^{8} - 7425039336825230 T^{9} - 1111065921351897007 T^{10} + 15844619352319883431 T^{11} +$$$$17\!\cdots\!81$$$$T^{12} )^{2}$$)($$( 1 - 55 T - 7367 T^{2} + 181142 T^{3} + 51119807 T^{4} - 598293727 T^{5} - 186579818926 T^{6} - 2082660463687 T^{7} + 619437155669327 T^{8} + 7640666224798022 T^{9} - 1081699833831032807 T^{10} - 28111421431535277055 T^{11} +$$$$17\!\cdots\!81$$$$T^{12} )^{2}$$)
$61$ ($$1 - 39 T + 4228 T^{2} - 145119 T^{3} + 13845841 T^{4}$$)($$1 + 39 T + 4228 T^{2} + 145119 T^{3} + 13845841 T^{4}$$)($$1 - 503 T^{2} - 13520125 T^{4} - 91919186268 T^{6} + 20373908662561 T^{8} + 680714329682975843 T^{10} +$$$$44\!\cdots\!06$$$$T^{12} +$$$$94\!\cdots\!63$$$$T^{14} +$$$$39\!\cdots\!41$$$$T^{16} -$$$$24\!\cdots\!28$$$$T^{18} -$$$$49\!\cdots\!25$$$$T^{20} -$$$$25\!\cdots\!03$$$$T^{22} +$$$$70\!\cdots\!41$$$$T^{24}$$)($$1 + 9201 T^{2} + 56941411 T^{4} + 154999649300 T^{6} - 75695642240335 T^{8} - 3911957905117164149 T^{10} -$$$$19\!\cdots\!14$$$$T^{12} -$$$$54\!\cdots\!09$$$$T^{14} -$$$$14\!\cdots\!35$$$$T^{16} +$$$$41\!\cdots\!00$$$$T^{18} +$$$$20\!\cdots\!71$$$$T^{20} +$$$$46\!\cdots\!01$$$$T^{22} +$$$$70\!\cdots\!41$$$$T^{24}$$)
$67$ ($$1 - 17 T - 4200 T^{2} - 76313 T^{3} + 20151121 T^{4}$$)($$1 - 17 T - 4200 T^{2} - 76313 T^{3} + 20151121 T^{4}$$)($$( 1 - 89 T - 19 T^{2} + 251662 T^{3} - 15244389 T^{4} + 746405371 T^{5} - 32107721582 T^{6} + 3350613710419 T^{7} - 307191527310069 T^{8} + 22764937373414878 T^{9} - 7715285873576179 T^{10} -$$$$16\!\cdots\!61$$$$T^{11} +$$$$81\!\cdots\!61$$$$T^{12} )^{2}$$)($$( 1 + 217 T + 18053 T^{2} + 1666970 T^{3} + 209974835 T^{4} + 15078221277 T^{5} + 815515698066 T^{6} + 67686135312453 T^{7} + 4231228307040035 T^{8} + 150791409324257930 T^{9} + 7330739782930039973 T^{10} +$$$$39\!\cdots\!33$$$$T^{11} +$$$$81\!\cdots\!61$$$$T^{12} )^{2}$$)
$71$ ($$( 1 - 71 T )^{2}( 1 + 71 T )^{2}$$)($$( 1 - 71 T )^{2}( 1 + 71 T )^{2}$$)($$( 1 - 7830 T^{2} + 72414895 T^{4} - 385201987444 T^{6} + 1840184211388495 T^{8} - 5056250149654308630 T^{10} +$$$$16\!\cdots\!41$$$$T^{12} )^{2}$$)($$( 1 - 23062 T^{2} + 245626031 T^{4} - 1559141837940 T^{6} + 6241770345068111 T^{8} - 14892367937589740182 T^{10} +$$$$16\!\cdots\!41$$$$T^{12} )^{2}$$)
$73$ ($$1 + 119 T + 8832 T^{2} + 634151 T^{3} + 28398241 T^{4}$$)($$1 + 119 T + 8832 T^{2} + 634151 T^{3} + 28398241 T^{4}$$)($$( 1 - 27 T - 12877 T^{2} + 216188 T^{3} + 104738937 T^{4} - 821298961 T^{5} - 619422939130 T^{6} - 4376702163169 T^{7} + 2974401575009817 T^{8} + 32716643712966332 T^{9} - 10384786603320081037 T^{10} -$$$$11\!\cdots\!23$$$$T^{11} +$$$$22\!\cdots\!21$$$$T^{12} )^{2}$$)($$( 1 - 51 T - 9165 T^{2} + 579660 T^{3} + 45654729 T^{4} - 1993134921 T^{5} - 160997144218 T^{6} - 10621415994009 T^{7} + 1296513996931689 T^{8} + 87722397610681740 T^{9} - 7391206742209252365 T^{10} -$$$$21\!\cdots\!99$$$$T^{11} +$$$$22\!\cdots\!21$$$$T^{12} )^{2}$$)
$79$ ($$1 + 129 T + 11788 T^{2} + 805089 T^{3} + 38950081 T^{4}$$)($$1 - 129 T + 11788 T^{2} - 805089 T^{3} + 38950081 T^{4}$$)($$1 + 27485 T^{2} + 389982499 T^{4} + 4285373244996 T^{6} + 40173777270526125 T^{8} +$$$$31\!\cdots\!47$$$$T^{10} +$$$$20\!\cdots\!50$$$$T^{12} +$$$$12\!\cdots\!07$$$$T^{14} +$$$$60\!\cdots\!25$$$$T^{16} +$$$$25\!\cdots\!36$$$$T^{18} +$$$$89\!\cdots\!79$$$$T^{20} +$$$$24\!\cdots\!85$$$$T^{22} +$$$$34\!\cdots\!81$$$$T^{24}$$)($$1 + 16693 T^{2} + 149251283 T^{4} + 640487711012 T^{6} - 978609215845699 T^{8} - 44625107831251066425 T^{10} -$$$$37\!\cdots\!74$$$$T^{12} -$$$$17\!\cdots\!25$$$$T^{14} -$$$$14\!\cdots\!39$$$$T^{16} +$$$$37\!\cdots\!92$$$$T^{18} +$$$$34\!\cdots\!43$$$$T^{20} +$$$$14\!\cdots\!93$$$$T^{22} +$$$$34\!\cdots\!81$$$$T^{24}$$)
$83$ ($$( 1 + 110 T + 6889 T^{2} )^{2}$$)($$( 1 + 110 T + 6889 T^{2} )^{2}$$)($$( 1 - 98 T + 12183 T^{2} - 526716 T^{3} + 83928687 T^{4} - 4650915458 T^{5} + 326940373369 T^{6} )^{4}$$)($$( 1 - 134 T + 22583 T^{2} - 1661172 T^{3} + 155574287 T^{4} - 6359415014 T^{5} + 326940373369 T^{6} )^{4}$$)
$89$ ($$1 + 71 T - 2880 T^{2} + 562391 T^{3} + 62742241 T^{4}$$)($$1 + 71 T - 2880 T^{2} + 562391 T^{3} + 62742241 T^{4}$$)($$( 1 + 13 T - 19181 T^{2} - 141748 T^{3} + 218594713 T^{4} + 635752183 T^{5} - 1943770650106 T^{6} + 5035793041543 T^{7} + 13715122164371833 T^{8} - 70446104031139828 T^{9} - 75507709882171615661 T^{10} +$$$$40\!\cdots\!13$$$$T^{11} +$$$$24\!\cdots\!21$$$$T^{12} )^{2}$$)($$( 1 - 107 T + 1395 T^{2} + 497084 T^{3} - 62702935 T^{4} + 2422278015 T^{5} + 93422604838 T^{6} + 19186864156815 T^{7} - 3934122659177335 T^{8} + 247041448036057724 T^{9} + 5491541383954402995 T^{10} -$$$$33\!\cdots\!07$$$$T^{11} +$$$$24\!\cdots\!21$$$$T^{12} )^{2}$$)
$97$ ($$( 1 + 22 T + 9409 T^{2} )^{2}$$)($$( 1 + 22 T + 9409 T^{2} )^{2}$$)($$( 1 + 46 T + 18423 T^{2} + 458352 T^{3} + 173342007 T^{4} + 4072346926 T^{5} + 832972004929 T^{6} )^{4}$$)($$( 1 + 38 T + 25191 T^{2} + 597344 T^{3} + 237022119 T^{4} + 3364112678 T^{5} + 832972004929 T^{6} )^{4}$$)