Properties

 Label 99.2.j Level 99 Weight 2 Character orbit j Rep. character $$\chi_{99}(8,\cdot)$$ Character field $$\Q(\zeta_{10})$$ Dimension 16 Newform subspaces 1 Sturm bound 24 Trace bound 0

Related objects

Defining parameters

 Level: $$N$$ $$=$$ $$99 = 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 99.j (of order $$10$$ and degree $$4$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$33$$ Character field: $$\Q(\zeta_{10})$$ Newform subspaces: $$1$$ Sturm bound: $$24$$ Trace bound: $$0$$

Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(99, [\chi])$$.

Total New Old
Modular forms 64 16 48
Cusp forms 32 16 16
Eisenstein series 32 0 32

Trace form

 $$16q - 4q^{4} + O(q^{10})$$ $$16q - 4q^{4} - 20q^{16} - 48q^{22} - 32q^{25} + 40q^{28} + 16q^{31} + 40q^{34} - 12q^{37} + 60q^{40} - 40q^{46} - 24q^{49} - 40q^{52} + 16q^{55} + 12q^{58} + 36q^{64} + 96q^{67} + 76q^{70} - 20q^{73} - 12q^{82} - 100q^{85} - 12q^{88} - 72q^{91} - 80q^{94} + 60q^{97} + O(q^{100})$$

Decomposition of $$S_{2}^{\mathrm{new}}(99, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
99.2.j.a $$16$$ $$0.791$$ $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{8}-\beta _{10}-\beta _{11}+\beta _{12}-\beta _{14}+2\beta _{15})q^{2}+\cdots$$

Decomposition of $$S_{2}^{\mathrm{old}}(99, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(99, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(33, [\chi])$$$$^{\oplus 2}$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T^{2} + 5 T^{4} - 20 T^{6} + 30 T^{8} - 106 T^{10} + 212 T^{12} - 350 T^{14} + 1105 T^{16} - 1400 T^{18} + 3392 T^{20} - 6784 T^{22} + 7680 T^{24} - 20480 T^{26} + 20480 T^{28} - 32768 T^{30} + 65536 T^{32}$$
$3$ 1
$5$ $$1 + 26 T^{2} + 267 T^{4} + 1158 T^{6} - 345 T^{8} - 23178 T^{10} - 84123 T^{12} - 137046 T^{14} - 351264 T^{16} - 3426150 T^{18} - 52576875 T^{20} - 362156250 T^{22} - 134765625 T^{24} + 11308593750 T^{26} + 65185546875 T^{28} + 158691406250 T^{30} + 152587890625 T^{32}$$
$7$ $$( 1 + 13 T^{2} + 20 T^{3} + 80 T^{4} + 260 T^{5} + 783 T^{6} + 2170 T^{7} + 6819 T^{8} + 15190 T^{9} + 38367 T^{10} + 89180 T^{11} + 192080 T^{12} + 336140 T^{13} + 1529437 T^{14} + 5764801 T^{16} )^{2}$$
$11$ $$1 - 60 T^{2} + 1779 T^{4} - 33610 T^{6} + 439341 T^{8} - 4066810 T^{10} + 26046339 T^{12} - 106293660 T^{14} + 214358881 T^{16}$$
$13$ $$( 1 + 7 T^{2} + 80 T^{3} + 170 T^{4} + 560 T^{5} + 4587 T^{6} + 13570 T^{7} + 40179 T^{8} + 176410 T^{9} + 775203 T^{10} + 1230320 T^{11} + 4855370 T^{12} + 29703440 T^{13} + 33787663 T^{14} + 815730721 T^{16} )^{2}$$
$17$ $$1 - 58 T^{2} + 1230 T^{4} + 3270 T^{6} - 669405 T^{8} + 13275246 T^{10} - 49036968 T^{12} - 2836388280 T^{14} + 74928517905 T^{16} - 819716212920 T^{18} - 4095616604328 T^{20} + 320432166316974 T^{22} - 4669606909792605 T^{24} + 6592300054468230 T^{26} + 716625351792606030 T^{28} - 9765913940445253882 T^{30} + 48661191875666868481 T^{32}$$
$19$ $$( 1 + 34 T^{2} + 20 T^{3} + 395 T^{4} + 680 T^{5} + 1356 T^{6} + 48280 T^{7} - 47451 T^{8} + 917320 T^{9} + 489516 T^{10} + 4664120 T^{11} + 51476795 T^{12} + 49521980 T^{13} + 1599559954 T^{14} + 16983563041 T^{16} )^{2}$$
$23$ $$( 1 - 102 T^{2} + 5430 T^{4} - 195472 T^{6} + 5177079 T^{8} - 103404688 T^{10} + 1519536630 T^{12} - 15099660678 T^{14} + 78310985281 T^{16} )^{2}$$
$29$ $$1 + 20 T^{2} + 338 T^{4} + 28080 T^{6} - 44437 T^{8} - 6939980 T^{10} + 33962216 T^{12} - 13624904800 T^{14} - 492015847995 T^{16} - 11458544936800 T^{18} + 24020830094696 T^{20} - 4128061951273580 T^{22} - 22229449852747957 T^{24} + 11813459111069644080 T^{26} +$$$$11\!\cdots\!58$$$$T^{28} +$$$$59\!\cdots\!20$$$$T^{30} +$$$$25\!\cdots\!21$$$$T^{32}$$
$31$ $$( 1 - 8 T - 24 T^{2} + 88 T^{3} + 1681 T^{4} - 7064 T^{5} + 54924 T^{6} - 168376 T^{7} - 1019363 T^{8} - 5219656 T^{9} + 52781964 T^{10} - 210443624 T^{11} + 1552438801 T^{12} + 2519365288 T^{13} - 21300088344 T^{14} - 220100912888 T^{15} + 852891037441 T^{16} )^{2}$$
$37$ $$( 1 + 6 T - 7 T^{2} - 118 T^{3} - 254 T^{4} + 3396 T^{5} + 36665 T^{6} - 93294 T^{7} - 1522501 T^{8} - 3451878 T^{9} + 50194385 T^{10} + 172017588 T^{11} - 476036894 T^{12} - 8182586926 T^{13} - 17960084863 T^{14} + 569591262798 T^{15} + 3512479453921 T^{16} )^{2}$$
$41$ $$1 - 138 T^{2} + 4581 T^{4} + 242158 T^{6} - 19370244 T^{8} + 47020276 T^{10} + 27362707279 T^{12} - 302286948126 T^{14} - 26002484153453 T^{16} - 508144359799806 T^{18} + 77320471083414319 T^{20} + 223351212440590516 T^{22} -$$$$15\!\cdots\!24$$$$T^{24} +$$$$32\!\cdots\!58$$$$T^{26} +$$$$10\!\cdots\!61$$$$T^{28} -$$$$52\!\cdots\!18$$$$T^{30} +$$$$63\!\cdots\!41$$$$T^{32}$$
$43$ $$( 1 - 180 T^{2} + 12786 T^{4} - 463360 T^{6} + 14361051 T^{8} - 856752640 T^{10} + 43712789586 T^{12} - 1137845348820 T^{14} + 11688200277601 T^{16} )^{2}$$
$47$ $$1 + 192 T^{2} + 14115 T^{4} + 332470 T^{6} - 12112005 T^{8} - 594955574 T^{10} + 21788151037 T^{12} + 2123156175840 T^{14} + 93108257741080 T^{16} + 4690051992430560 T^{18} + 106319226640379197 T^{20} - 6413154243334793846 T^{22} -$$$$28\!\cdots\!05$$$$T^{24} +$$$$17\!\cdots\!30$$$$T^{26} +$$$$16\!\cdots\!15$$$$T^{28} +$$$$49\!\cdots\!48$$$$T^{30} +$$$$56\!\cdots\!21$$$$T^{32}$$
$53$ $$1 + 166 T^{2} + 9174 T^{4} - 111674 T^{6} - 30841109 T^{8} - 525703922 T^{10} + 67446933816 T^{12} + 3591171944168 T^{14} + 110175435537457 T^{16} + 10087601991167912 T^{18} + 532188749783405496 T^{20} - 11651891574139647938 T^{22} -$$$$19\!\cdots\!49$$$$T^{24} -$$$$19\!\cdots\!26$$$$T^{26} +$$$$45\!\cdots\!34$$$$T^{28} +$$$$22\!\cdots\!54$$$$T^{30} +$$$$38\!\cdots\!21$$$$T^{32}$$
$59$ $$1 + 170 T^{2} + 22518 T^{4} + 2498730 T^{6} + 233218683 T^{8} + 18919071570 T^{10} + 1419551700816 T^{12} + 95367637994400 T^{14} + 5833449282172425 T^{16} + 331974747858506400 T^{18} + 17201220416951466576 T^{20} +$$$$79\!\cdots\!70$$$$T^{22} +$$$$34\!\cdots\!43$$$$T^{24} +$$$$12\!\cdots\!30$$$$T^{26} +$$$$40\!\cdots\!58$$$$T^{28} +$$$$10\!\cdots\!70$$$$T^{30} +$$$$21\!\cdots\!41$$$$T^{32}$$
$61$ $$( 1 + 83 T^{2} + 420 T^{3} + 3413 T^{4} + 34860 T^{5} + 227941 T^{6} + 4235840 T^{7} + 8009300 T^{8} + 258386240 T^{9} + 848168461 T^{10} + 7912557660 T^{11} + 47255855333 T^{12} + 354730446420 T^{13} + 4276191071963 T^{14} + 191707312997281 T^{16} )^{2}$$
$67$ $$( 1 - 24 T + 429 T^{2} - 5008 T^{3} + 47859 T^{4} - 335536 T^{5} + 1925781 T^{6} - 7218312 T^{7} + 20151121 T^{8} )^{4}$$
$71$ $$1 + 338 T^{2} + 50091 T^{4} + 3709542 T^{6} + 63034791 T^{8} - 13931687706 T^{10} - 1397230758891 T^{12} - 60140798780574 T^{14} - 2187888491323968 T^{16} - 303169766652873534 T^{18} - 35505982328326005771 T^{20} -$$$$17\!\cdots\!26$$$$T^{22} +$$$$40\!\cdots\!51$$$$T^{24} +$$$$12\!\cdots\!42$$$$T^{26} +$$$$82\!\cdots\!31$$$$T^{28} +$$$$27\!\cdots\!78$$$$T^{30} +$$$$41\!\cdots\!21$$$$T^{32}$$
$73$ $$( 1 + 10 T + 276 T^{2} + 2170 T^{3} + 24127 T^{4} + 117430 T^{5} - 172242 T^{6} - 4912540 T^{7} - 124201175 T^{8} - 358615420 T^{9} - 917877618 T^{10} + 45682266310 T^{11} + 685164360607 T^{12} + 4498565356810 T^{13} + 41768246455764 T^{14} + 110473985190970 T^{15} + 806460091894081 T^{16} )^{2}$$
$79$ $$( 1 + 203 T^{2} - 180 T^{3} + 12338 T^{4} - 36540 T^{5} - 807689 T^{6} - 4199410 T^{7} - 150806725 T^{8} - 331753390 T^{9} - 5040787049 T^{10} - 18015645060 T^{11} + 480566099378 T^{12} - 553870151820 T^{13} + 49346753470763 T^{14} + 1517108809906561 T^{16} )^{2}$$
$83$ $$1 - 208 T^{2} + 15045 T^{4} + 774660 T^{6} - 236047260 T^{8} + 17546406456 T^{10} - 74692835973 T^{12} - 112052982977490 T^{14} + 12843285776364315 T^{16} - 771932999731928610 T^{18} - 3544796586006981333 T^{20} +$$$$57\!\cdots\!64$$$$T^{22} -$$$$53\!\cdots\!60$$$$T^{24} +$$$$12\!\cdots\!40$$$$T^{26} +$$$$16\!\cdots\!45$$$$T^{28} -$$$$15\!\cdots\!32$$$$T^{30} +$$$$50\!\cdots\!81$$$$T^{32}$$
$89$ $$( 1 - 582 T^{2} + 157158 T^{4} - 25689904 T^{6} + 2778331335 T^{8} - 203489729584 T^{10} + 9860445111078 T^{12} - 289243111339302 T^{14} + 3936588805702081 T^{16} )^{2}$$
$97$ $$( 1 - 30 T + 411 T^{2} - 3150 T^{3} + 11217 T^{4} + 58230 T^{5} - 1794727 T^{6} + 27808950 T^{7} - 324085500 T^{8} + 2697468150 T^{9} - 16886586343 T^{10} + 53144948790 T^{11} + 993032944977 T^{12} - 27050121809550 T^{13} + 342351494025819 T^{14} - 2423948534343390 T^{15} + 7837433594376961 T^{16} )^{2}$$