# Properties

 Label 63.2.o Level 63 Weight 2 Character orbit o Rep. character $$\chi_{63}(20,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 12 Newform subspaces 1 Sturm bound 16 Trace bound 0

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$63 = 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 63.o (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$63$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$1$$ Sturm bound: $$16$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(63, [\chi])$$.

Total New Old
Modular forms 20 20 0
Cusp forms 12 12 0
Eisenstein series 8 8 0

## Trace form

 $$12q - 6q^{2} + 2q^{4} - 2q^{7} - 12q^{9} + O(q^{10})$$ $$12q - 6q^{2} + 2q^{4} - 2q^{7} - 12q^{9} - 12q^{14} + 6q^{15} + 2q^{16} - 6q^{18} - 24q^{21} - 10q^{22} + 24q^{23} - 8q^{28} + 30q^{29} + 48q^{30} - 12q^{32} + 18q^{36} - 4q^{37} + 36q^{42} - 10q^{43} - 40q^{46} + 6q^{49} - 36q^{50} - 42q^{51} + 42q^{56} - 18q^{57} + 2q^{58} - 12q^{60} + 24q^{63} + 16q^{64} - 78q^{65} + 12q^{67} + 18q^{70} - 24q^{72} - 12q^{74} - 24q^{77} - 12q^{78} - 6q^{79} + 24q^{81} - 60q^{84} - 6q^{85} + 96q^{86} + 34q^{88} - 24q^{91} + 30q^{92} + 78q^{93} + 72q^{95} - 24q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(63, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
63.2.o.a $$12$$ $$0.503$$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$-6$$ $$0$$ $$0$$ $$-2$$ $$q-\beta _{1}q^{2}-\beta _{2}q^{3}+(\beta _{1}+\beta _{4}+\beta _{7})q^{4}+\cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 3 T + 7 T^{2} + 12 T^{3} + 17 T^{4} + 24 T^{5} + 31 T^{6} + 48 T^{7} + 68 T^{8} + 96 T^{9} + 112 T^{10} + 96 T^{11} + 64 T^{12} )^{2}$$
$3$ $$1 + 6 T^{2} + 12 T^{4} + 9 T^{6} + 108 T^{8} + 486 T^{10} + 729 T^{12}$$
$5$ $$1 - 15 T^{2} + 84 T^{4} - 457 T^{6} + 3921 T^{8} - 20514 T^{10} + 80841 T^{12} - 512850 T^{14} + 2450625 T^{16} - 7140625 T^{18} + 32812500 T^{20} - 146484375 T^{22} + 244140625 T^{24}$$
$7$ $$1 + 2 T - T^{2} - 30 T^{3} - 50 T^{4} + 58 T^{5} + 715 T^{6} + 406 T^{7} - 2450 T^{8} - 10290 T^{9} - 2401 T^{10} + 33614 T^{11} + 117649 T^{12}$$
$11$ $$( 1 + 22 T^{2} + 242 T^{4} + 66 T^{5} + 2665 T^{6} + 726 T^{7} + 29282 T^{8} + 322102 T^{10} + 1771561 T^{12} )^{2}$$
$13$ $$1 + 33 T^{2} + 351 T^{4} + 2144 T^{6} + 53457 T^{8} + 906351 T^{10} + 9826878 T^{12} + 153173319 T^{14} + 1526785377 T^{16} + 10348678496 T^{18} + 286321483071 T^{20} + 4549330231017 T^{22} + 23298085122481 T^{24}$$
$17$ $$( 1 + 48 T^{2} + 1338 T^{4} + 26845 T^{6} + 386682 T^{8} + 4009008 T^{10} + 24137569 T^{12} )^{2}$$
$19$ $$( 1 - 51 T^{2} + 1599 T^{4} - 35471 T^{6} + 577239 T^{8} - 6646371 T^{10} + 47045881 T^{12} )^{2}$$
$23$ $$( 1 - 12 T + 130 T^{2} - 984 T^{3} + 7082 T^{4} - 39654 T^{5} + 209833 T^{6} - 912042 T^{7} + 3746378 T^{8} - 11972328 T^{9} + 36379330 T^{10} - 77236116 T^{11} + 148035889 T^{12} )^{2}$$
$29$ $$( 1 - 15 T + 184 T^{2} - 1635 T^{3} + 13205 T^{4} - 85158 T^{5} + 500899 T^{6} - 2469582 T^{7} + 11105405 T^{8} - 39876015 T^{9} + 130139704 T^{10} - 307667235 T^{11} + 594823321 T^{12} )^{2}$$
$31$ $$1 + 57 T^{2} + 3216 T^{4} + 97787 T^{6} + 2702697 T^{8} + 47442966 T^{10} + 1072931313 T^{12} + 45592690326 T^{14} + 2495997436137 T^{16} + 86786322453947 T^{18} + 2742897576410256 T^{20} + 46718812357905657 T^{22} + 787662783788549761 T^{24}$$
$37$ $$( 1 + T + 107 T^{2} + 73 T^{3} + 3959 T^{4} + 1369 T^{5} + 50653 T^{6} )^{4}$$
$41$ $$1 - 174 T^{2} + 15930 T^{4} - 1039210 T^{6} + 54933612 T^{8} - 2550462528 T^{10} + 108692576859 T^{12} - 4287327509568 T^{14} + 155229258378732 T^{16} - 4936355828289610 T^{18} + 127199858899897530 T^{20} - 2335542719966517774 T^{22} + 22563490300366186081 T^{24}$$
$43$ $$( 1 + 5 T - 58 T^{2} - 51 T^{3} + 2155 T^{4} - 7106 T^{5} - 129149 T^{6} - 305558 T^{7} + 3984595 T^{8} - 4054857 T^{9} - 198290458 T^{10} + 735042215 T^{11} + 6321363049 T^{12} )^{2}$$
$47$ $$1 - 189 T^{2} + 18726 T^{4} - 1271869 T^{6} + 67940493 T^{8} - 3218131254 T^{10} + 150306249081 T^{12} - 7108851940086 T^{14} + 331527932822733 T^{16} - 13709749821279901 T^{18} + 445890154028136486 T^{20} - 9941235992571879261 T^{22} +$$$$11\!\cdots\!41$$$$T^{24}$$
$53$ $$( 1 - 200 T^{2} + 20924 T^{4} - 1371431 T^{6} + 58775516 T^{8} - 1578096200 T^{10} + 22164361129 T^{12} )^{2}$$
$59$ $$1 - 135 T^{2} + 4578 T^{4} + 47801 T^{6} + 11150547 T^{8} - 1706849076 T^{10} + 108381578457 T^{12} - 5941541633556 T^{14} + 135115203346467 T^{16} + 2016271688573441 T^{18} + 672189743352581538 T^{20} - 69000761695586589135 T^{22} +$$$$17\!\cdots\!81$$$$T^{24}$$
$61$ $$1 + 342 T^{2} + 66840 T^{4} + 9095042 T^{6} + 949804386 T^{8} + 78844563978 T^{10} + 5324903072391 T^{12} + 293380622562138 T^{14} + 13150840509658626 T^{16} + 468579968669018162 T^{18} + 12813716800738262040 T^{20} +$$$$24\!\cdots\!42$$$$T^{22} +$$$$26\!\cdots\!21$$$$T^{24}$$
$67$ $$( 1 - 6 T - 150 T^{2} + 506 T^{3} + 17268 T^{4} - 28236 T^{5} - 1220289 T^{6} - 1891812 T^{7} + 77516052 T^{8} + 152186078 T^{9} - 3022668150 T^{10} - 8100750642 T^{11} + 90458382169 T^{12} )^{2}$$
$71$ $$( 1 - 263 T^{2} + 31517 T^{4} - 2539307 T^{6} + 158877197 T^{8} - 6683272103 T^{10} + 128100283921 T^{12} )^{2}$$
$73$ $$( 1 - 159 T^{2} + 10563 T^{4} - 623423 T^{6} + 56290227 T^{8} - 4515320319 T^{10} + 151334226289 T^{12} )^{2}$$
$79$ $$( 1 + 3 T - 204 T^{2} - 151 T^{3} + 27357 T^{4} + 1896 T^{5} - 2498217 T^{6} + 149784 T^{7} + 170735037 T^{8} - 74448889 T^{9} - 7945816524 T^{10} + 9231169197 T^{11} + 243087455521 T^{12} )^{2}$$
$83$ $$1 - 264 T^{2} + 27306 T^{4} - 2832334 T^{6} + 401548758 T^{8} - 35501027934 T^{10} + 2433955301391 T^{12} - 244566581437326 T^{14} + 19056829854315318 T^{16} - 926004335465713246 T^{18} + 61501091690788653546 T^{20} -$$$$40\!\cdots\!36$$$$T^{22} +$$$$10\!\cdots\!61$$$$T^{24}$$
$89$ $$( 1 + 210 T^{2} + 31902 T^{4} + 3230233 T^{6} + 252695742 T^{8} + 13175870610 T^{10} + 496981290961 T^{12} )^{2}$$
$97$ $$1 + 513 T^{2} + 147894 T^{4} + 30072485 T^{6} + 4748892315 T^{8} + 607745530428 T^{10} + 64510991774769 T^{12} + 5718277695797052 T^{14} + 420416022193375515 T^{16} + 25049538123647278565 T^{18} +$$$$11\!\cdots\!34$$$$T^{20} +$$$$37\!\cdots\!37$$$$T^{22} +$$$$69\!\cdots\!41$$$$T^{24}$$