# Properties

 Label 99.2.m Level 99 Weight 2 Character orbit m Rep. character $$\chi_{99}(4,\cdot)$$ Character field $$\Q(\zeta_{15})$$ Dimension 80 Newform subspaces 2 Sturm bound 24 Trace bound 1

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$99 = 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 99.m (of order $$15$$ and degree $$8$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$99$$ Character field: $$\Q(\zeta_{15})$$ Newform subspaces: $$2$$ Sturm bound: $$24$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

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

Total New Old
Modular forms 112 112 0
Cusp forms 80 80 0
Eisenstein series 32 32 0

## Trace form

 $$80q - 5q^{2} - 9q^{3} + 5q^{4} - 2q^{5} - 7q^{6} - 3q^{7} - 8q^{8} - 25q^{9} + O(q^{10})$$ $$80q - 5q^{2} - 9q^{3} + 5q^{4} - 2q^{5} - 7q^{6} - 3q^{7} - 8q^{8} - 25q^{9} - 40q^{10} - q^{11} + 2q^{12} - 3q^{13} - 11q^{14} - 8q^{15} + 5q^{16} - 8q^{17} - 24q^{18} + 6q^{19} - 21q^{20} - 16q^{21} - 5q^{22} + 6q^{23} - 3q^{24} + 2q^{25} - 28q^{26} + 27q^{27} - 36q^{28} - 32q^{29} + 12q^{30} + 48q^{32} + 28q^{33} + 2q^{34} + 30q^{35} + 39q^{36} - 18q^{37} - 5q^{38} - 35q^{39} - 13q^{40} + 13q^{41} + 49q^{42} - 2q^{43} - 10q^{44} + 16q^{45} - 11q^{47} - 50q^{48} - 5q^{49} - 24q^{50} + 69q^{51} - 15q^{52} + 32q^{53} + 88q^{54} - 14q^{55} + 102q^{56} - 18q^{57} + 7q^{58} - 13q^{59} - 63q^{60} - 3q^{61} + 190q^{62} - 7q^{63} - 8q^{64} + 52q^{65} + 105q^{66} - 14q^{67} - 32q^{68} + 24q^{69} - 52q^{70} - 6q^{71} + 41q^{72} - 48q^{73} + 81q^{74} + 28q^{75} + 10q^{76} - 37q^{77} - 40q^{78} - 3q^{79} - 4q^{80} + 23q^{81} - 2q^{82} - 10q^{83} + 3q^{84} + 19q^{85} - 56q^{86} - 120q^{87} + 55q^{88} - 128q^{89} - 22q^{90} - 4q^{91} + 29q^{92} - 44q^{93} - 35q^{94} - 81q^{95} - 97q^{96} - 27q^{97} - 256q^{98} - 185q^{99} + O(q^{100})$$

## 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.m.a $$8$$ $$0.791$$ $$\Q(\zeta_{15})$$ None $$-4$$ $$3$$ $$6$$ $$-1$$ $$q+(1-2\zeta_{15}^{2}+\zeta_{15}^{3}-\zeta_{15}^{4}+\zeta_{15}^{5}+\cdots)q^{2}+\cdots$$
99.2.m.b $$72$$ $$0.791$$ None $$-1$$ $$-12$$ $$-8$$ $$-2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$( 1 - T + 4 T^{2} - 2 T^{3} + 9 T^{4} - 4 T^{5} + 16 T^{6} - 8 T^{7} + 16 T^{8} )( 1 + 5 T + 13 T^{2} + 25 T^{3} + 39 T^{4} + 50 T^{5} + 52 T^{6} + 40 T^{7} + 16 T^{8} )$$)
$3$ ($$1 - 3 T + 6 T^{2} - 9 T^{3} + 9 T^{4} - 27 T^{5} + 54 T^{6} - 81 T^{7} + 81 T^{8}$$)
$5$ ($$1 - 6 T + 25 T^{2} - 54 T^{3} + 84 T^{4} - 84 T^{5} + 395 T^{6} - 1746 T^{7} + 5171 T^{8} - 8730 T^{9} + 9875 T^{10} - 10500 T^{11} + 52500 T^{12} - 168750 T^{13} + 390625 T^{14} - 468750 T^{15} + 390625 T^{16}$$)
$7$ ($$( 1 - 4 T + 9 T^{2} - 8 T^{3} - 31 T^{4} - 56 T^{5} + 441 T^{6} - 1372 T^{7} + 2401 T^{8} )( 1 + 5 T + 18 T^{2} + 55 T^{3} + 149 T^{4} + 385 T^{5} + 882 T^{6} + 1715 T^{7} + 2401 T^{8} )$$)
$11$ ($$1 - T - 20 T^{2} + T^{3} + 309 T^{4} + 11 T^{5} - 2420 T^{6} - 1331 T^{7} + 14641 T^{8}$$)
$13$ ($$1 - 8 T + 13 T^{2} + 80 T^{3} - 400 T^{4} + 1168 T^{5} - 2101 T^{6} - 18940 T^{7} + 139519 T^{8} - 246220 T^{9} - 355069 T^{10} + 2566096 T^{11} - 11424400 T^{12} + 29703440 T^{13} + 62748517 T^{14} - 501988136 T^{15} + 815730721 T^{16}$$)
$17$ ($$( 1 - 6 T + 19 T^{2} - 132 T^{3} + 829 T^{4} - 2244 T^{5} + 5491 T^{6} - 29478 T^{7} + 83521 T^{8} )^{2}$$)
$19$ ($$( 1 + T - 18 T^{2} - 37 T^{3} + 305 T^{4} - 703 T^{5} - 6498 T^{6} + 6859 T^{7} + 130321 T^{8} )^{2}$$)
$23$ ($$( 1 + 7 T - 8 T^{2} + 77 T^{3} + 1593 T^{4} + 1771 T^{5} - 4232 T^{6} + 85169 T^{7} + 279841 T^{8} )^{2}$$)
$29$ ($$1 + 9 T + 74 T^{2} + 129 T^{3} - 54 T^{4} - 8388 T^{5} + 18676 T^{6} + 272682 T^{7} + 3330947 T^{8} + 7907778 T^{9} + 15706516 T^{10} - 204574932 T^{11} - 38193174 T^{12} + 2645938221 T^{13} + 44016925754 T^{14} + 155248886781 T^{15} + 500246412961 T^{16}$$)
$31$ ($$1 + 3 T + 36 T^{2} - 23 T^{3} - 84 T^{4} - 5516 T^{5} - 24056 T^{6} - 63114 T^{7} - 303923 T^{8} - 1956534 T^{9} - 23117816 T^{10} - 164327156 T^{11} - 77575764 T^{12} - 658470473 T^{13} + 31950132516 T^{14} + 82537842333 T^{15} + 852891037441 T^{16}$$)
$37$ ($$( 1 - 12 T + 57 T^{2} - 440 T^{3} + 3921 T^{4} - 16280 T^{5} + 78033 T^{6} - 607836 T^{7} + 1874161 T^{8} )^{2}$$)
$41$ ($$1 - 3 T - 59 T^{2} - 42 T^{3} + 1326 T^{4} + 9501 T^{5} + 17474 T^{6} - 345486 T^{7} - 294103 T^{8} - 14164926 T^{9} + 29373794 T^{10} + 654818421 T^{11} + 3746959086 T^{12} - 4865960442 T^{13} - 280256150219 T^{14} - 584262821643 T^{15} + 7984925229121 T^{16}$$)
$43$ ($$( 1 - 3 T - 68 T^{2} + 27 T^{3} + 3693 T^{4} + 1161 T^{5} - 125732 T^{6} - 238521 T^{7} + 3418801 T^{8} )^{2}$$)
$47$ ($$1 - 23 T + 327 T^{2} - 2796 T^{3} + 15428 T^{4} - 41679 T^{5} - 80266 T^{6} + 1119532 T^{7} - 8567157 T^{8} + 52618004 T^{9} - 177307594 T^{10} - 4327238817 T^{11} + 75283718468 T^{12} - 641248639572 T^{13} + 3524803412583 T^{14} - 11652331770649 T^{15} + 23811286661761 T^{16}$$)
$53$ ($$( 1 - 14 T + 43 T^{2} - 160 T^{3} + 2961 T^{4} - 8480 T^{5} + 120787 T^{6} - 2084278 T^{7} + 7890481 T^{8} )^{2}$$)
$59$ ($$1 - 3 T + 64 T^{2} + 51 T^{3} - 168 T^{4} + 20412 T^{5} - 184552 T^{6} + 519906 T^{7} - 7825759 T^{8} + 30674454 T^{9} - 642425512 T^{10} + 4192196148 T^{11} - 2035716648 T^{12} + 36461139249 T^{13} + 2699554153024 T^{14} - 7465954454457 T^{15} + 146830437604321 T^{16}$$)
$61$ ($$1 - 29 T^{2} - 480 T^{3} - 2160 T^{4} + 21600 T^{5} + 186389 T^{6} - 128880 T^{7} - 11482801 T^{8} - 7861680 T^{9} + 693553469 T^{10} + 4902789600 T^{11} - 29907016560 T^{12} - 405406224480 T^{13} - 1494090856469 T^{14} + 191707312997281 T^{16}$$)
$67$ ($$( 1 + 6 T - 31 T^{2} + 402 T^{3} + 4489 T^{4} )^{4}$$)
$71$ ($$( 1 - 21 T + 95 T^{2} + 1371 T^{3} - 21536 T^{4} + 97341 T^{5} + 478895 T^{6} - 7516131 T^{7} + 25411681 T^{8} )^{2}$$)
$73$ ($$( 1 + 4 T - 27 T^{2} + 500 T^{3} + 7301 T^{4} + 36500 T^{5} - 143883 T^{6} + 1556068 T^{7} + 28398241 T^{8} )^{2}$$)
$79$ ($$1 + 22 T + 259 T^{2} + 146 T^{3} - 28378 T^{4} - 420788 T^{5} - 1482397 T^{6} + 19182626 T^{7} + 350855611 T^{8} + 1515427454 T^{9} - 9251639677 T^{10} - 207464894732 T^{11} - 1105325398618 T^{12} + 449250234254 T^{13} + 62959650979939 T^{14} + 422485997695498 T^{15} + 1517108809906561 T^{16}$$)
$83$ ($$1 + 17 T + 213 T^{2} + 3330 T^{3} + 42290 T^{4} + 490473 T^{5} + 4981814 T^{6} + 51032300 T^{7} + 525354219 T^{8} + 4235680900 T^{9} + 34319716646 T^{10} + 280446085251 T^{11} + 2007012395090 T^{12} + 13117005341190 T^{13} + 69638299527597 T^{14} + 461312866823659 T^{15} + 2252292232139041 T^{16}$$)
$89$ ($$( 1 + 18 T + 254 T^{2} + 1602 T^{3} + 7921 T^{4} )^{4}$$)
$97$ ($$1 - 6 T + 97 T^{2} - 1530 T^{3} + 9180 T^{4} + 36876 T^{5} + 316511 T^{6} + 9403380 T^{7} - 144949561 T^{8} + 912127860 T^{9} + 2978051999 T^{10} + 33655729548 T^{11} + 812698799580 T^{12} - 13138630593210 T^{13} + 80798284478113 T^{14} - 484789706868678 T^{15} + 7837433594376961 T^{16}$$)