# Properties

 Label 126.4.g Level $126$ Weight $4$ Character orbit 126.g Rep. character $\chi_{126}(37,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $20$ Newform subspaces $7$ Sturm bound $96$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$126 = 2 \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 126.g (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$7$$ Sturm bound: $$96$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(126, [\chi])$$.

Total New Old
Modular forms 160 20 140
Cusp forms 128 20 108
Eisenstein series 32 0 32

## Trace form

 $$20q - 40q^{4} - 18q^{5} - 8q^{7} + O(q^{10})$$ $$20q - 40q^{4} - 18q^{5} - 8q^{7} + 8q^{10} + 6q^{11} - 192q^{13} - 72q^{14} - 160q^{16} - 150q^{17} + 46q^{19} + 144q^{20} + 256q^{22} + 210q^{23} - 68q^{25} - 120q^{26} + 40q^{28} - 336q^{29} - 138q^{31} - 512q^{34} - 426q^{35} - 242q^{37} - 192q^{38} + 32q^{40} + 2016q^{41} + 1288q^{43} + 24q^{44} + 40q^{46} + 102q^{47} + 1412q^{49} - 1824q^{50} + 384q^{52} - 1182q^{53} - 1916q^{55} + 288q^{56} - 136q^{58} + 90q^{59} - 294q^{61} + 3744q^{62} + 1280q^{64} + 1524q^{65} + 1226q^{67} - 600q^{68} - 1520q^{70} - 672q^{71} + 1142q^{73} - 312q^{74} - 368q^{76} - 2226q^{77} + 1266q^{79} - 288q^{80} + 1200q^{82} + 2568q^{83} - 4892q^{85} - 984q^{86} - 512q^{88} - 2754q^{89} + 2100q^{91} - 1680q^{92} - 216q^{94} - 1974q^{95} - 160q^{97} + 2880q^{98} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(126, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
126.4.g.a $$2$$ $$7.434$$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$0$$ $$-15$$ $$35$$ $$q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+(-15+\cdots)q^{5}+\cdots$$
126.4.g.b $$2$$ $$7.434$$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$0$$ $$-6$$ $$-7$$ $$q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+(-6+6\zeta_{6})q^{5}+\cdots$$
126.4.g.c $$2$$ $$7.434$$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$0$$ $$7$$ $$-20$$ $$q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+(7-7\zeta_{6})q^{5}+\cdots$$
126.4.g.d $$2$$ $$7.434$$ $$\Q(\sqrt{-3})$$ None $$2$$ $$0$$ $$-9$$ $$-28$$ $$q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+(-9+9\zeta_{6})q^{5}+\cdots$$
126.4.g.e $$4$$ $$7.434$$ $$\Q(\sqrt{-3}, \sqrt{193})$$ None $$-4$$ $$0$$ $$7$$ $$6$$ $$q-2\beta _{2}q^{2}+(-4+4\beta _{2})q^{4}+(\beta _{1}+3\beta _{2}+\cdots)q^{5}+\cdots$$
126.4.g.f $$4$$ $$7.434$$ $$\Q(\sqrt{-3}, \sqrt{193})$$ None $$4$$ $$0$$ $$-7$$ $$6$$ $$q+2\beta _{2}q^{2}+(-4+4\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots$$
126.4.g.g $$4$$ $$7.434$$ $$\Q(\sqrt{-3}, \sqrt{1345})$$ None $$4$$ $$0$$ $$5$$ $$0$$ $$q+2\beta _{2}q^{2}+(-4+4\beta _{2})q^{4}+(\beta _{1}+2\beta _{2}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(126, [\chi])$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(126, [\chi]) \cong$$ $$S_{4}^{\mathrm{new}}(7, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(14, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(21, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(42, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(63, [\chi])$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 + 2 T + 4 T^{2}$$)($$1 + 2 T + 4 T^{2}$$)($$1 + 2 T + 4 T^{2}$$)($$1 - 2 T + 4 T^{2}$$)($$( 1 + 2 T + 4 T^{2} )^{2}$$)($$( 1 - 2 T + 4 T^{2} )^{2}$$)($$( 1 - 2 T + 4 T^{2} )^{2}$$)
$3$ 1
$5$ ($$1 + 15 T + 100 T^{2} + 1875 T^{3} + 15625 T^{4}$$)($$1 + 6 T - 89 T^{2} + 750 T^{3} + 15625 T^{4}$$)($$1 - 7 T - 76 T^{2} - 875 T^{3} + 15625 T^{4}$$)($$1 + 9 T - 44 T^{2} + 1125 T^{3} + 15625 T^{4}$$)($$1 - 7 T - 165 T^{2} + 252 T^{3} + 24046 T^{4} + 31500 T^{5} - 2578125 T^{6} - 13671875 T^{7} + 244140625 T^{8}$$)($$1 + 7 T - 165 T^{2} - 252 T^{3} + 24046 T^{4} - 31500 T^{5} - 2578125 T^{6} + 13671875 T^{7} + 244140625 T^{8}$$)($$1 - 5 T + 105 T^{2} + 1650 T^{3} - 12350 T^{4} + 206250 T^{5} + 1640625 T^{6} - 9765625 T^{7} + 244140625 T^{8}$$)
$7$ ($$1 - 35 T + 343 T^{2}$$)($$1 + 7 T + 343 T^{2}$$)($$1 + 20 T + 343 T^{2}$$)($$1 + 28 T + 343 T^{2}$$)($$1 - 6 T - 77 T^{2} - 2058 T^{3} + 117649 T^{4}$$)($$1 - 6 T - 77 T^{2} - 2058 T^{3} + 117649 T^{4}$$)($$1 - 659 T^{2} + 117649 T^{4}$$)
$11$ ($$1 + 9 T - 1250 T^{2} + 11979 T^{3} + 1771561 T^{4}$$)($$1 + 30 T - 431 T^{2} + 39930 T^{3} + 1771561 T^{4}$$)($$1 - 35 T - 106 T^{2} - 46585 T^{3} + 1771561 T^{4}$$)($$1 + 57 T + 1918 T^{2} + 75867 T^{3} + 1771561 T^{4}$$)($$1 + 25 T + 171 T^{2} - 55200 T^{3} - 2397320 T^{4} - 73471200 T^{5} + 302936931 T^{6} + 58948692275 T^{7} + 3138428376721 T^{8}$$)($$1 - 25 T + 171 T^{2} + 55200 T^{3} - 2397320 T^{4} + 73471200 T^{5} + 302936931 T^{6} - 58948692275 T^{7} + 3138428376721 T^{8}$$)($$1 - 67 T + 1041 T^{2} - 52662 T^{3} + 4142284 T^{4} - 70093122 T^{5} + 1844195001 T^{6} - 157982495297 T^{7} + 3138428376721 T^{8}$$)
$13$ ($$( 1 + 88 T + 2197 T^{2} )^{2}$$)($$( 1 - 53 T + 2197 T^{2} )^{2}$$)($$( 1 - 66 T + 2197 T^{2} )^{2}$$)($$( 1 + 70 T + 2197 T^{2} )^{2}$$)($$( 1 + 49 T + 3788 T^{2} + 107653 T^{3} + 4826809 T^{4} )^{2}$$)($$( 1 + 49 T + 3788 T^{2} + 107653 T^{3} + 4826809 T^{4} )^{2}$$)($$( 1 - 41 T + 4478 T^{2} - 90077 T^{3} + 4826809 T^{4} )^{2}$$)
$17$ ($$1 + 84 T + 2143 T^{2} + 412692 T^{3} + 24137569 T^{4}$$)($$1 + 84 T + 2143 T^{2} + 412692 T^{3} + 24137569 T^{4}$$)($$1 - 59 T - 1432 T^{2} - 289867 T^{3} + 24137569 T^{4}$$)($$1 - 51 T - 2312 T^{2} - 250563 T^{3} + 24137569 T^{4}$$)($$1 - 98 T + 2202 T^{2} + 237552 T^{3} - 16532417 T^{4} + 1167092976 T^{5} + 53150926938 T^{6} - 11621611896706 T^{7} + 582622237229761 T^{8}$$)($$1 + 98 T + 2202 T^{2} - 237552 T^{3} - 16532417 T^{4} - 1167092976 T^{5} + 53150926938 T^{6} + 11621611896706 T^{7} + 582622237229761 T^{8}$$)($$1 + 92 T + 1902 T^{2} - 300288 T^{3} - 22661357 T^{4} - 1475314944 T^{5} + 45909656238 T^{6} + 10910084637724 T^{7} + 582622237229761 T^{8}$$)
$19$ ($$1 + 104 T + 3957 T^{2} + 713336 T^{3} + 47045881 T^{4}$$)($$1 - 97 T + 2550 T^{2} - 665323 T^{3} + 47045881 T^{4}$$)($$1 + 137 T + 11910 T^{2} + 939683 T^{3} + 47045881 T^{4}$$)($$1 + 5 T - 6834 T^{2} + 34295 T^{3} + 47045881 T^{4}$$)($$1 - 119 T - 2663 T^{2} - 369614 T^{3} + 138870796 T^{4} - 2535182426 T^{5} - 125283181103 T^{6} - 38399836035701 T^{7} + 2213314919066161 T^{8}$$)($$1 - 119 T - 2663 T^{2} - 369614 T^{3} + 138870796 T^{4} - 2535182426 T^{5} - 125283181103 T^{6} - 38399836035701 T^{7} + 2213314919066161 T^{8}$$)($$1 + 43 T - 9305 T^{2} - 110252 T^{3} + 64683544 T^{4} - 756218468 T^{5} - 437761922705 T^{6} + 13875571004497 T^{7} + 2213314919066161 T^{8}$$)
$23$ ($$1 + 84 T - 5111 T^{2} + 1022028 T^{3} + 148035889 T^{4}$$)($$1 - 84 T - 5111 T^{2} - 1022028 T^{3} + 148035889 T^{4}$$)($$1 + 7 T - 12118 T^{2} + 85169 T^{3} + 148035889 T^{4}$$)($$1 - 69 T - 7406 T^{2} - 839523 T^{3} + 148035889 T^{4}$$)($$1 - 122 T - 3714 T^{2} + 699792 T^{3} + 16756087 T^{4} + 8514369264 T^{5} - 549805291746 T^{6} - 219740624698486 T^{7} + 21914624432020321 T^{8}$$)($$1 + 122 T - 3714 T^{2} - 699792 T^{3} + 16756087 T^{4} - 8514369264 T^{5} - 549805291746 T^{6} + 219740624698486 T^{7} + 21914624432020321 T^{8}$$)($$1 - 148 T - 2526 T^{2} - 14208 T^{3} + 182283043 T^{4} - 172868736 T^{5} - 373938655614 T^{6} - 266570593896524 T^{7} + 21914624432020321 T^{8}$$)
$29$ ($$( 1 + 51 T + 24389 T^{2} )^{2}$$)($$( 1 - 180 T + 24389 T^{2} )^{2}$$)($$( 1 + 106 T + 24389 T^{2} )^{2}$$)($$( 1 + 114 T + 24389 T^{2} )^{2}$$)($$( 1 + 73 T + 47746 T^{2} + 1780397 T^{3} + 594823321 T^{4} )^{2}$$)($$( 1 - 73 T + 47746 T^{2} - 1780397 T^{3} + 594823321 T^{4} )^{2}$$)($$( 1 + 77 T + 9574 T^{2} + 1877953 T^{3} + 594823321 T^{4} )^{2}$$)
$31$ ($$1 + 185 T + 4434 T^{2} + 5511335 T^{3} + 887503681 T^{4}$$)($$1 + 179 T + 2250 T^{2} + 5332589 T^{3} + 887503681 T^{4}$$)($$1 + 75 T - 24166 T^{2} + 2234325 T^{3} + 887503681 T^{4}$$)($$1 + 23 T - 29262 T^{2} + 685193 T^{3} + 887503681 T^{4}$$)($$1 + 98 T - 14551 T^{2} - 3471846 T^{3} - 590152420 T^{4} - 103429764186 T^{5} - 12914066062231 T^{6} + 2591082971745758 T^{7} + 787662783788549761 T^{8}$$)($$1 + 98 T - 14551 T^{2} - 3471846 T^{3} - 590152420 T^{4} - 103429764186 T^{5} - 12914066062231 T^{6} + 2591082971745758 T^{7} + 787662783788549761 T^{8}$$)($$1 - 520 T + 144563 T^{2} - 34452600 T^{3} + 6891960488 T^{4} - 1026377406600 T^{5} + 128300194636403 T^{6} - 13748603523548920 T^{7} + 787662783788549761 T^{8}$$)
$37$ ($$1 + 44 T - 48717 T^{2} + 2228732 T^{3} + 2565726409 T^{4}$$)($$1 - 145 T - 29628 T^{2} - 7344685 T^{3} + 2565726409 T^{4}$$)($$1 + 11 T - 50532 T^{2} + 557183 T^{3} + 2565726409 T^{4}$$)($$1 - 253 T + 13356 T^{2} - 12815209 T^{3} + 2565726409 T^{4}$$)($$1 + 289 T - 17387 T^{2} - 115022 T^{3} + 3386108842 T^{4} - 5826209366 T^{5} - 44610285073283 T^{6} + 37558942800777253 T^{7} + 6582952005840035281 T^{8}$$)($$1 + 289 T - 17387 T^{2} - 115022 T^{3} + 3386108842 T^{4} - 5826209366 T^{5} - 44610285073283 T^{6} + 37558942800777253 T^{7} + 6582952005840035281 T^{8}$$)($$1 + 7 T - 74033 T^{2} - 190568 T^{3} + 2919934318 T^{4} - 9652840904 T^{5} - 189948423237497 T^{6} + 909732178565539 T^{7} + 6582952005840035281 T^{8}$$)
$41$ ($$( 1 - 168 T + 68921 T^{2} )^{2}$$)($$( 1 + 126 T + 68921 T^{2} )^{2}$$)($$( 1 - 498 T + 68921 T^{2} )^{2}$$)($$( 1 - 42 T + 68921 T^{2} )^{2}$$)($$( 1 + 168 T + 68921 T^{2} )^{4}$$)($$( 1 - 168 T + 68921 T^{2} )^{4}$$)($$( 1 - 426 T + 171106 T^{2} - 29360346 T^{3} + 4750104241 T^{4} )^{2}$$)
$43$ ($$( 1 - 326 T + 79507 T^{2} )^{2}$$)($$( 1 + 325 T + 79507 T^{2} )^{2}$$)($$( 1 - 260 T + 79507 T^{2} )^{2}$$)($$( 1 + 124 T + 79507 T^{2} )^{2}$$)($$( 1 - 307 T + 161298 T^{2} - 24408649 T^{3} + 6321363049 T^{4} )^{2}$$)($$( 1 - 307 T + 161298 T^{2} - 24408649 T^{3} + 6321363049 T^{4} )^{2}$$)($$( 1 + 107 T + 86220 T^{2} + 8507249 T^{3} + 6321363049 T^{4} )^{2}$$)
$47$ ($$1 + 138 T - 84779 T^{2} + 14327574 T^{3} + 10779215329 T^{4}$$)($$1 + 366 T + 30133 T^{2} + 37999218 T^{3} + 10779215329 T^{4}$$)($$1 + 171 T - 74582 T^{2} + 17753733 T^{3} + 10779215329 T^{4}$$)($$1 - 201 T - 63422 T^{2} - 20868423 T^{3} + 10779215329 T^{4}$$)($$1 - 672 T + 158834 T^{2} - 57189888 T^{3} + 28038541539 T^{4} - 5937625741824 T^{5} + 1712105887566386 T^{6} - 752055677925059424 T^{7} +$$$$11\!\cdots\!41$$$$T^{8}$$)($$1 + 672 T + 158834 T^{2} + 57189888 T^{3} + 28038541539 T^{4} + 5937625741824 T^{5} + 1712105887566386 T^{6} + 752055677925059424 T^{7} +$$$$11\!\cdots\!41$$$$T^{8}$$)($$1 - 576 T + 89606 T^{2} - 19885824 T^{3} + 13421113923 T^{4} - 2064605905152 T^{5} + 965882368770374 T^{6} - 644619152507193792 T^{7} +$$$$11\!\cdots\!41$$$$T^{8}$$)
$53$ ($$1 - 639 T + 259444 T^{2} - 95132403 T^{3} + 22164361129 T^{4}$$)($$1 + 768 T + 440947 T^{2} + 114337536 T^{3} + 22164361129 T^{4}$$)($$1 + 417 T + 25012 T^{2} + 62081709 T^{3} + 22164361129 T^{4}$$)($$1 + 393 T + 5572 T^{2} + 58508661 T^{3} + 22164361129 T^{4}$$)($$1 + 375 T - 781 T^{2} - 58630500 T^{3} - 23104532418 T^{4} - 8728732948500 T^{5} - 17310366041749 T^{6} + 1237411346925799875 T^{7} +$$$$49\!\cdots\!41$$$$T^{8}$$)($$1 - 375 T - 781 T^{2} + 58630500 T^{3} - 23104532418 T^{4} + 8728732948500 T^{5} - 17310366041749 T^{6} - 1237411346925799875 T^{7} +$$$$49\!\cdots\!41$$$$T^{8}$$)($$1 + 243 T - 250441 T^{2} + 2851848 T^{3} + 64828660998 T^{4} + 424574574696 T^{5} - 5550864765507889 T^{6} + 801842552807918319 T^{7} +$$$$49\!\cdots\!41$$$$T^{8}$$)
$59$ ($$1 - 159 T - 180098 T^{2} - 32655261 T^{3} + 42180533641 T^{4}$$)($$1 + 264 T - 135683 T^{2} + 54220056 T^{3} + 42180533641 T^{4}$$)($$1 + 17 T - 205090 T^{2} + 3491443 T^{3} + 42180533641 T^{4}$$)($$1 - 219 T - 157418 T^{2} - 44978001 T^{3} + 42180533641 T^{4}$$)($$1 - 763 T + 205407 T^{2} + 25938948 T^{3} - 19796216048 T^{4} + 5327315201292 T^{5} + 8664176873596887 T^{6} - 6609865809633718457 T^{7} +$$$$17\!\cdots\!81$$$$T^{8}$$)($$1 + 763 T + 205407 T^{2} - 25938948 T^{3} - 19796216048 T^{4} - 5327315201292 T^{5} + 8664176873596887 T^{6} + 6609865809633718457 T^{7} +$$$$17\!\cdots\!81$$$$T^{8}$$)($$1 + 7 T - 200565 T^{2} - 1471008 T^{3} - 1944620216 T^{4} - 302114152032 T^{5} - 8459938729707165 T^{6} + 60640970730584573 T^{7} +$$$$17\!\cdots\!81$$$$T^{8}$$)
$61$ ($$1 + 722 T + 294303 T^{2} + 163880282 T^{3} + 51520374361 T^{4}$$)($$1 + 818 T + 442143 T^{2} + 185670458 T^{3} + 51520374361 T^{4}$$)($$1 + 51 T - 224380 T^{2} + 11576031 T^{3} + 51520374361 T^{4}$$)($$1 - 709 T + 275700 T^{2} - 160929529 T^{3} + 51520374361 T^{4}$$)($$1 - 406 T - 274558 T^{2} + 5914608 T^{3} + 104132072759 T^{4} + 1342503638448 T^{5} - 14145330943807438 T^{6} - 4747823313690661246 T^{7} +$$$$26\!\cdots\!21$$$$T^{8}$$)($$1 - 406 T - 274558 T^{2} + 5914608 T^{3} + 104132072759 T^{4} + 1342503638448 T^{5} - 14145330943807438 T^{6} - 4747823313690661246 T^{7} +$$$$26\!\cdots\!21$$$$T^{8}$$)($$1 + 224 T - 410950 T^{2} + 1604736 T^{3} + 149727814859 T^{4} + 364244582016 T^{5} - 21172297843652950 T^{6} + 2619488724794847584 T^{7} +$$$$26\!\cdots\!21$$$$T^{8}$$)
$67$ ($$1 - 166 T - 273207 T^{2} - 49926658 T^{3} + 90458382169 T^{4}$$)($$1 - 523 T - 27234 T^{2} - 157299049 T^{3} + 90458382169 T^{4}$$)($$1 + 439 T - 108042 T^{2} + 132034957 T^{3} + 90458382169 T^{4}$$)($$1 + 419 T - 125202 T^{2} + 126019697 T^{3} + 90458382169 T^{4}$$)($$1 - 1041 T + 270341 T^{2} - 220498374 T^{3} + 245132424828 T^{4} - 66317752459362 T^{5} + 24454609493949629 T^{6} - 28322002306543039827 T^{7} +$$$$81\!\cdots\!61$$$$T^{8}$$)($$1 - 1041 T + 270341 T^{2} - 220498374 T^{3} + 245132424828 T^{4} - 66317752459362 T^{5} + 24454609493949629 T^{6} - 28322002306543039827 T^{7} +$$$$81\!\cdots\!61$$$$T^{8}$$)($$1 + 687 T - 206863 T^{2} + 53109222 T^{3} + 228403689708 T^{4} + 15973288936386 T^{5} - 18712492310625847 T^{6} + 18690889130254628589 T^{7} +$$$$81\!\cdots\!61$$$$T^{8}$$)
$71$ ($$( 1 + 1086 T + 357911 T^{2} )^{2}$$)($$( 1 - 342 T + 357911 T^{2} )^{2}$$)($$( 1 - 784 T + 357911 T^{2} )^{2}$$)($$( 1 - 96 T + 357911 T^{2} )^{2}$$)($$( 1 + 1652 T + 1360270 T^{2} + 591268972 T^{3} + 128100283921 T^{4} )^{2}$$)($$( 1 - 1652 T + 1360270 T^{2} - 591268972 T^{3} + 128100283921 T^{4} )^{2}$$)($$( 1 + 472 T + 637018 T^{2} + 168933992 T^{3} + 128100283921 T^{4} )^{2}$$)
$73$ ($$1 + 218 T - 341493 T^{2} + 84805706 T^{3} + 151334226289 T^{4}$$)($$1 - 43 T - 387168 T^{2} - 16727731 T^{3} + 151334226289 T^{4}$$)($$1 + 295 T - 301992 T^{2} + 114760015 T^{3} + 151334226289 T^{4}$$)($$1 - 313 T - 291048 T^{2} - 121762321 T^{3} + 151334226289 T^{4}$$)($$1 - 189 T - 297259 T^{2} + 84115206 T^{3} - 54354622146 T^{4} + 32722245092502 T^{5} - 44985460772441851 T^{6} - 11126729887862635557 T^{7} +$$$$22\!\cdots\!21$$$$T^{8}$$)($$1 - 189 T - 297259 T^{2} + 84115206 T^{3} - 54354622146 T^{4} + 32722245092502 T^{5} - 44985460772441851 T^{6} - 11126729887862635557 T^{7} +$$$$22\!\cdots\!21$$$$T^{8}$$)($$1 - 921 T + 270053 T^{2} + 184058166 T^{3} - 147013032042 T^{4} + 71601755562822 T^{5} + 40868261812023317 T^{6} - 54220731358314747873 T^{7} +$$$$22\!\cdots\!21$$$$T^{8}$$)
$79$ ($$1 - 583 T - 153150 T^{2} - 287441737 T^{3} + 243087455521 T^{4}$$)($$1 - 1171 T + 878202 T^{2} - 577348669 T^{3} + 243087455521 T^{4}$$)($$1 - 495 T - 248014 T^{2} - 244054305 T^{3} + 243087455521 T^{4}$$)($$1 + 461 T - 280518 T^{2} + 227290979 T^{3} + 243087455521 T^{4}$$)($$1 + 524 T - 316753 T^{2} - 206848476 T^{3} - 28794145744 T^{4} - 101984365758564 T^{5} - 76998680798643313 T^{6} + 62802236294891999156 T^{7} +$$$$59\!\cdots\!41$$$$T^{8}$$)($$1 + 524 T - 316753 T^{2} - 206848476 T^{3} - 28794145744 T^{4} - 101984365758564 T^{5} - 76998680798643313 T^{6} + 62802236294891999156 T^{7} +$$$$59\!\cdots\!41$$$$T^{8}$$)($$1 - 526 T - 757051 T^{2} - 25063374 T^{3} + 689091996644 T^{4} - 12357220853586 T^{5} - 184029601289628571 T^{6} - 63041939486857235794 T^{7} +$$$$59\!\cdots\!41$$$$T^{8}$$)
$83$ ($$( 1 - 597 T + 571787 T^{2} )^{2}$$)($$( 1 - 810 T + 571787 T^{2} )^{2}$$)($$( 1 + 932 T + 571787 T^{2} )^{2}$$)($$( 1 - 588 T + 571787 T^{2} )^{2}$$)($$( 1 + 287 T + 781978 T^{2} + 164102869 T^{3} + 326940373369 T^{4} )^{2}$$)($$( 1 - 287 T + 781978 T^{2} - 164102869 T^{3} + 326940373369 T^{4} )^{2}$$)($$( 1 - 221 T + 945628 T^{2} - 126364927 T^{3} + 326940373369 T^{4} )^{2}$$)
$89$ ($$1 + 1038 T + 372475 T^{2} + 731757822 T^{3} + 496981290961 T^{4}$$)($$1 + 600 T - 344969 T^{2} + 422981400 T^{3} + 496981290961 T^{4}$$)($$1 + 873 T + 57160 T^{2} + 615437937 T^{3} + 496981290961 T^{4}$$)($$1 + 1017 T + 329320 T^{2} + 716953473 T^{3} + 496981290961 T^{4}$$)($$1 - 2394 T + 3029186 T^{2} - 3093316128 T^{3} + 2763749199855 T^{4} - 2180691977440032 T^{5} + 1505448768840987746 T^{6} -$$$$83\!\cdots\!46$$$$T^{7} +$$$$24\!\cdots\!21$$$$T^{8}$$)($$1 + 2394 T + 3029186 T^{2} + 3093316128 T^{3} + 2763749199855 T^{4} + 2180691977440032 T^{5} + 1505448768840987746 T^{6} +$$$$83\!\cdots\!46$$$$T^{7} +$$$$24\!\cdots\!21$$$$T^{8}$$)($$1 - 774 T - 367486 T^{2} + 343173024 T^{3} + 14930800239 T^{4} + 241926343556256 T^{5} - 182633666690094046 T^{6} -$$$$27\!\cdots\!66$$$$T^{7} +$$$$24\!\cdots\!21$$$$T^{8}$$)
$97$ ($$( 1 + 169 T + 912673 T^{2} )^{2}$$)($$( 1 - 386 T + 912673 T^{2} )^{2}$$)($$( 1 + 290 T + 912673 T^{2} )^{2}$$)($$( 1 + 1834 T + 912673 T^{2} )^{2}$$)($$( 1 + 63 T + 667132 T^{2} + 57498399 T^{3} + 832972004929 T^{4} )^{2}$$)($$( 1 + 63 T + 667132 T^{2} + 57498399 T^{3} + 832972004929 T^{4} )^{2}$$)($$( 1 - 1953 T + 2366992 T^{2} - 1782450369 T^{3} + 832972004929 T^{4} )^{2}$$)