# Properties

 Label 252.2.x Level 252 Weight 2 Character orbit x Rep. character $$\chi_{252}(41,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 16 Newform subspaces 1 Sturm bound 96 Trace bound 0

# Related objects

## Defining parameters

 Level: $$N$$ = $$252 = 2^{2} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 252.x (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$63$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$1$$ Sturm bound: $$96$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(252, [\chi])$$.

Total New Old
Modular forms 108 16 92
Cusp forms 84 16 68
Eisenstein series 24 0 24

## Trace form

 $$16q - q^{7} + O(q^{10})$$ $$16q - q^{7} + 6q^{11} - 12q^{15} + 9q^{21} + 6q^{23} - 8q^{25} - 12q^{29} + 4q^{37} + 18q^{39} + 4q^{43} - 5q^{49} - 18q^{51} - 42q^{57} - 27q^{63} - 24q^{65} + 14q^{67} - 21q^{77} + 20q^{79} - 36q^{81} + 6q^{85} - 18q^{91} - 24q^{93} - 60q^{95} + 90q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(252, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
252.2.x.a $$16$$ $$2.012$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$-1$$ $$q+(-\beta _{1}-\beta _{8})q^{3}+\beta _{15}q^{5}+(-\beta _{1}+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(252, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(252, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(63, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(126, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 9 T^{4} - 45 T^{6} - 18 T^{8} - 405 T^{10} + 729 T^{12} + 6561 T^{16}$$
$5$ $$1 - 16 T^{2} + 159 T^{4} - 998 T^{6} + 4298 T^{8} - 12066 T^{10} + 21316 T^{12} - 64633 T^{14} + 271566 T^{16} - 1615825 T^{18} + 13322500 T^{20} - 188531250 T^{22} + 1678906250 T^{24} - 9746093750 T^{26} + 38818359375 T^{28} - 97656250000 T^{30} + 152587890625 T^{32}$$
$7$ $$1 + T + 3 T^{2} + 20 T^{3} + 53 T^{4} + 21 T^{5} - 74 T^{6} + 355 T^{7} - 1314 T^{8} + 2485 T^{9} - 3626 T^{10} + 7203 T^{11} + 127253 T^{12} + 336140 T^{13} + 352947 T^{14} + 823543 T^{15} + 5764801 T^{16}$$
$11$ $$( 1 - 3 T + 26 T^{2} - 69 T^{3} + 265 T^{4} - 756 T^{5} + 2585 T^{6} - 8259 T^{7} + 34846 T^{8} - 90849 T^{9} + 312785 T^{10} - 1006236 T^{11} + 3879865 T^{12} - 11112519 T^{13} + 46060586 T^{14} - 58461513 T^{15} + 214358881 T^{16} )^{2}$$
$13$ $$1 + 56 T^{2} + 1590 T^{4} + 26848 T^{6} + 274865 T^{8} + 1615656 T^{10} + 12863830 T^{12} + 363405056 T^{14} + 6601006116 T^{16} + 61415454464 T^{18} + 367403848630 T^{20} + 7798462921704 T^{22} + 224215824627665 T^{24} + 3701224789161952 T^{26} + 37043955344744790 T^{28} + 220493077599160184 T^{30} + 665416609183179841 T^{32}$$
$17$ $$( 1 + 58 T^{2} + 1603 T^{4} + 30013 T^{6} + 497674 T^{8} + 8673757 T^{10} + 133884163 T^{12} + 1399979002 T^{14} + 6975757441 T^{16} )^{2}$$
$19$ $$( 1 - 77 T^{2} + 3034 T^{4} - 84368 T^{6} + 1811980 T^{8} - 30456848 T^{10} + 395393914 T^{12} - 3622532837 T^{14} + 16983563041 T^{16} )^{2}$$
$23$ $$( 1 - 3 T + 56 T^{2} - 159 T^{3} + 1321 T^{4} - 3510 T^{5} + 30863 T^{6} - 67173 T^{7} + 842170 T^{8} - 1544979 T^{9} + 16326527 T^{10} - 42706170 T^{11} + 369669961 T^{12} - 1023378537 T^{13} + 8290009784 T^{14} - 10214476341 T^{15} + 78310985281 T^{16} )^{2}$$
$29$ $$( 1 + 6 T + 53 T^{2} + 246 T^{3} + 580 T^{4} + 540 T^{5} - 22066 T^{6} - 299625 T^{7} - 1268282 T^{8} - 8689125 T^{9} - 18557506 T^{10} + 13170060 T^{11} + 410222980 T^{12} + 5045742654 T^{13} + 31525636013 T^{14} + 103499257854 T^{15} + 500246412961 T^{16} )^{2}$$
$31$ $$1 + 176 T^{2} + 16407 T^{4} + 1045570 T^{6} + 50906954 T^{8} + 2036295270 T^{10} + 71233973380 T^{12} + 2306080860179 T^{14} + 72156636517806 T^{16} + 2216143706632019 T^{18} + 65786070329870980 T^{20} + 1807219547727888870 T^{22} + 43418084810021264714 T^{24} +$$$$85\!\cdots\!70$$$$T^{26} +$$$$12\!\cdots\!27$$$$T^{28} +$$$$13\!\cdots\!96$$$$T^{30} +$$$$72\!\cdots\!81$$$$T^{32}$$
$37$ $$( 1 - T + 82 T^{2} - 88 T^{3} + 3940 T^{4} - 3256 T^{5} + 112258 T^{6} - 50653 T^{7} + 1874161 T^{8} )^{4}$$
$41$ $$1 - 151 T^{2} + 9564 T^{4} - 353897 T^{6} + 12352577 T^{8} - 588962616 T^{10} + 30503470855 T^{12} - 1573469011633 T^{14} + 72313421958936 T^{16} - 2645001408555073 T^{18} + 86195518306695655 T^{20} - 2797633820052054456 T^{22} + 98634403731959794817 T^{24} -$$$$47\!\cdots\!97$$$$T^{26} +$$$$21\!\cdots\!84$$$$T^{28} -$$$$57\!\cdots\!11$$$$T^{30} +$$$$63\!\cdots\!41$$$$T^{32}$$
$43$ $$( 1 - 2 T - 93 T^{2} - 64 T^{3} + 3956 T^{4} + 12192 T^{5} - 158990 T^{6} - 321485 T^{7} + 7972668 T^{8} - 13823855 T^{9} - 293972510 T^{10} + 969349344 T^{11} + 13524776756 T^{12} - 9408540352 T^{13} - 587886763557 T^{14} - 543637222214 T^{15} + 11688200277601 T^{16} )^{2}$$
$47$ $$1 - 154 T^{2} + 7683 T^{4} - 245888 T^{6} + 25028264 T^{8} - 1538143278 T^{10} + 38475997042 T^{12} - 2602469435029 T^{14} + 205422570296046 T^{16} - 5748854981979061 T^{18} + 187750591721903602 T^{20} - 16579977600415908462 T^{22} +$$$$59\!\cdots\!04$$$$T^{24} -$$$$12\!\cdots\!12$$$$T^{26} +$$$$89\!\cdots\!03$$$$T^{28} -$$$$39\!\cdots\!26$$$$T^{30} +$$$$56\!\cdots\!21$$$$T^{32}$$
$53$ $$( 1 - 10 T^{2} + 3133 T^{4} + 54785 T^{6} + 5778574 T^{8} + 153891065 T^{10} + 24720876973 T^{12} - 221643611290 T^{14} + 62259690411361 T^{16} )^{2}$$
$59$ $$1 - 376 T^{2} + 75453 T^{4} - 10695122 T^{6} + 1193881406 T^{8} - 111003832932 T^{10} + 8882998756942 T^{12} - 624280670070511 T^{14} + 38988958528434078 T^{16} - 2173121012515448791 T^{18} +$$$$10\!\cdots\!62$$$$T^{20} -$$$$46\!\cdots\!12$$$$T^{22} +$$$$17\!\cdots\!26$$$$T^{24} -$$$$54\!\cdots\!22$$$$T^{26} +$$$$13\!\cdots\!93$$$$T^{28} -$$$$23\!\cdots\!36$$$$T^{30} +$$$$21\!\cdots\!41$$$$T^{32}$$
$61$ $$1 + 137 T^{2} + 306 T^{4} - 368483 T^{6} + 33183791 T^{8} + 2225199582 T^{10} - 143899132121 T^{12} + 1205057298539 T^{14} + 1133086682080440 T^{16} + 4484018207863619 T^{18} - 1992404503385358761 T^{20} +$$$$11\!\cdots\!02$$$$T^{22} +$$$$63\!\cdots\!71$$$$T^{24} -$$$$26\!\cdots\!83$$$$T^{26} +$$$$81\!\cdots\!26$$$$T^{28} +$$$$13\!\cdots\!17$$$$T^{30} +$$$$36\!\cdots\!61$$$$T^{32}$$
$67$ $$( 1 - 7 T - 108 T^{2} - 293 T^{3} + 11753 T^{4} + 50778 T^{5} - 198827 T^{6} - 2620663 T^{7} - 10701486 T^{8} - 175584421 T^{9} - 892534403 T^{10} + 15272143614 T^{11} + 236836125113 T^{12} - 395586656351 T^{13} - 9769505274252 T^{14} - 42424981237261 T^{15} + 406067677556641 T^{16} )^{2}$$
$71$ $$( 1 - 361 T^{2} + 64216 T^{4} - 7488268 T^{6} + 623512600 T^{8} - 37748358988 T^{10} + 1631836507096 T^{12} - 46244202495481 T^{14} + 645753531245761 T^{16} )^{2}$$
$73$ $$( 1 - 341 T^{2} + 62398 T^{4} - 7562288 T^{6} + 647645452 T^{8} - 40299432752 T^{10} + 1771993441918 T^{12} - 51604971164549 T^{14} + 806460091894081 T^{16} )^{2}$$
$79$ $$( 1 - 10 T - 123 T^{2} + 844 T^{3} + 11042 T^{4} + 2166 T^{5} - 1268066 T^{6} - 531391 T^{7} + 106354368 T^{8} - 41979889 T^{9} - 7913999906 T^{10} + 1067922474 T^{11} + 430086794402 T^{12} + 2597035600756 T^{13} - 29899757029083 T^{14} - 192039089861590 T^{15} + 1517108809906561 T^{16} )^{2}$$
$83$ $$1 - 397 T^{2} + 87990 T^{4} - 12501089 T^{6} + 1185510017 T^{8} - 57837981066 T^{10} - 2507933243939 T^{12} + 849901181197043 T^{14} - 92913215010718536 T^{16} + 5854969237266429227 T^{18} -$$$$11\!\cdots\!19$$$$T^{20} -$$$$18\!\cdots\!54$$$$T^{22} +$$$$26\!\cdots\!97$$$$T^{24} -$$$$19\!\cdots\!61$$$$T^{26} +$$$$94\!\cdots\!90$$$$T^{28} -$$$$29\!\cdots\!13$$$$T^{30} +$$$$50\!\cdots\!81$$$$T^{32}$$
$89$ $$( 1 + 64 T^{2} + 9487 T^{4} + 662335 T^{6} + 143358802 T^{8} + 5246355535 T^{10} + 595235640367 T^{12} + 31806802621504 T^{14} + 3936588805702081 T^{16} )^{2}$$
$97$ $$1 + 404 T^{2} + 81609 T^{4} + 10963666 T^{6} + 1046175986 T^{8} + 57359540784 T^{10} - 1663618077194 T^{12} - 785812116013063 T^{14} - 99406691908304130 T^{16} - 7393706199566909767 T^{18} -$$$$14\!\cdots\!14$$$$T^{20} +$$$$47\!\cdots\!36$$$$T^{22} +$$$$81\!\cdots\!46$$$$T^{24} +$$$$80\!\cdots\!34$$$$T^{26} +$$$$56\!\cdots\!69$$$$T^{28} +$$$$26\!\cdots\!76$$$$T^{30} +$$$$61\!\cdots\!21$$$$T^{32}$$