# Properties

 Label 105.2.q Level 105 Weight 2 Character orbit q Rep. character $$\chi_{105}(4,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 16 Newform subspaces 1 Sturm bound 32 Trace bound 0

# Related objects

## Defining parameters

 Level: $$N$$ = $$105 = 3 \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 105.q (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$35$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$1$$ Sturm bound: $$32$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(105, [\chi])$$.

Total New Old
Modular forms 40 16 24
Cusp forms 24 16 8
Eisenstein series 16 0 16

## Trace form

 $$16q + 8q^{4} + 2q^{5} - 8q^{6} + 8q^{9} + O(q^{10})$$ $$16q + 8q^{4} + 2q^{5} - 8q^{6} + 8q^{9} - 4q^{10} - 24q^{14} - 4q^{15} - 24q^{19} - 8q^{20} - 4q^{21} - 12q^{24} - 4q^{25} - 12q^{26} + 24q^{29} - 12q^{30} + 16q^{31} + 16q^{34} - 10q^{35} + 16q^{36} - 4q^{39} + 32q^{40} + 16q^{41} + 20q^{44} - 2q^{45} - 32q^{46} - 40q^{49} - 40q^{50} + 4q^{51} - 4q^{54} + 8q^{55} + 84q^{56} + 4q^{59} + 16q^{60} + 16q^{61} + 16q^{64} + 30q^{65} + 28q^{66} + 40q^{69} + 16q^{70} - 56q^{71} + 40q^{74} + 8q^{75} - 64q^{76} - 16q^{79} + 52q^{80} - 8q^{81} + 12q^{84} - 64q^{85} - 48q^{86} + 16q^{89} - 8q^{90} + 8q^{91} - 32q^{94} - 22q^{95} + 8q^{96} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(105, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
105.2.q.a $$16$$ $$0.838$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$2$$ $$0$$ $$q+(-\beta _{5}+\beta _{6}-\beta _{15})q^{2}+\beta _{3}q^{3}+(1+\cdots)q^{4}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(105, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(105, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(35, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 4 T^{2} + 4 T^{4} - 8 T^{6} - 20 T^{8} + 32 T^{10} + 96 T^{12} - 208 T^{14} - 944 T^{16} - 832 T^{18} + 1536 T^{20} + 2048 T^{22} - 5120 T^{24} - 8192 T^{26} + 16384 T^{28} + 65536 T^{30} + 65536 T^{32}$$
$3$ $$( 1 - T^{2} + T^{4} )^{4}$$
$5$ $$1 - 2 T + 4 T^{2} - 20 T^{3} + 62 T^{4} - 118 T^{5} + 200 T^{6} - 670 T^{7} + 1639 T^{8} - 3350 T^{9} + 5000 T^{10} - 14750 T^{11} + 38750 T^{12} - 62500 T^{13} + 62500 T^{14} - 156250 T^{15} + 390625 T^{16}$$
$7$ $$1 + 20 T^{2} + 202 T^{4} + 1244 T^{6} + 7267 T^{8} + 60956 T^{10} + 485002 T^{12} + 2352980 T^{14} + 5764801 T^{16}$$
$11$ $$( 1 - 26 T^{2} - 28 T^{3} + 316 T^{4} + 518 T^{5} - 2872 T^{6} - 2968 T^{7} + 31007 T^{8} - 32648 T^{9} - 347512 T^{10} + 689458 T^{11} + 4626556 T^{12} - 4509428 T^{13} - 46060586 T^{14} + 214358881 T^{16} )^{2}$$
$13$ $$( 1 - 44 T^{2} + 1234 T^{4} - 23988 T^{6} + 358939 T^{8} - 4053972 T^{10} + 35244274 T^{12} - 212379596 T^{14} + 815730721 T^{16} )^{2}$$
$17$ $$1 + 76 T^{2} + 2664 T^{4} + 67392 T^{6} + 1583990 T^{8} + 33141472 T^{10} + 603483136 T^{12} + 11039990476 T^{14} + 198790801859 T^{16} + 3190557247564 T^{18} + 50403515001856 T^{20} + 799954567161568 T^{22} + 11049530028969590 T^{24} + 135861860939059008 T^{26} + 1552105639980083304 T^{28} + 12796714818514470604 T^{30} + 48661191875666868481 T^{32}$$
$19$ $$( 1 + 12 T + 30 T^{2} - 80 T^{3} + 689 T^{4} + 9436 T^{5} + 28846 T^{6} + 59944 T^{7} + 291332 T^{8} + 1138936 T^{9} + 10413406 T^{10} + 64721524 T^{11} + 89791169 T^{12} - 198087920 T^{13} + 1411376430 T^{14} + 10726460868 T^{15} + 16983563041 T^{16} )^{2}$$
$23$ $$1 + 20 T^{2} - 1512 T^{4} - 18448 T^{6} + 1612902 T^{8} + 10149224 T^{10} - 1248424544 T^{12} - 1840459676 T^{14} + 770565238915 T^{16} - 973603168604 T^{18} - 349360372817504 T^{20} + 1502449397500136 T^{22} + 126307944781695462 T^{24} - 764236278869396752 T^{26} - 33134912141214725352 T^{28} +$$$$23\!\cdots\!80$$$$T^{30} +$$$$61\!\cdots\!61$$$$T^{32}$$
$29$ $$( 1 - 6 T + 78 T^{2} - 332 T^{3} + 2820 T^{4} - 9628 T^{5} + 65598 T^{6} - 146334 T^{7} + 707281 T^{8} )^{4}$$
$31$ $$( 1 - 8 T - 54 T^{2} + 352 T^{3} + 3285 T^{4} - 9272 T^{5} - 157694 T^{6} + 60416 T^{7} + 6248764 T^{8} + 1872896 T^{9} - 151543934 T^{10} - 276222152 T^{11} + 3033766485 T^{12} + 10077461152 T^{13} - 47925198774 T^{14} - 220100912888 T^{15} + 852891037441 T^{16} )^{2}$$
$37$ $$1 + 128 T^{2} + 6574 T^{4} + 187656 T^{6} + 5183961 T^{8} + 158877780 T^{10} + 5690568750 T^{12} + 459641012708 T^{14} + 25029766954564 T^{16} + 629248546397252 T^{18} + 10665042019068750 T^{20} + 407636915949292020 T^{22} + 18208556502427761081 T^{24} +$$$$90\!\cdots\!44$$$$T^{26} +$$$$43\!\cdots\!94$$$$T^{28} +$$$$11\!\cdots\!92$$$$T^{30} +$$$$12\!\cdots\!41$$$$T^{32}$$
$41$ $$( 1 - 4 T + 114 T^{2} - 346 T^{3} + 5976 T^{4} - 14186 T^{5} + 191634 T^{6} - 275684 T^{7} + 2825761 T^{8} )^{4}$$
$43$ $$( 1 - 216 T^{2} + 22138 T^{4} - 1464812 T^{6} + 71619331 T^{8} - 2708437388 T^{10} + 75685416538 T^{12} - 1365414418584 T^{14} + 11688200277601 T^{16} )^{2}$$
$47$ $$1 + 308 T^{2} + 50892 T^{4} + 5950136 T^{6} + 547353834 T^{8} + 41768130380 T^{10} + 2726163659344 T^{12} + 155042751818668 T^{14} + 7757667166613347 T^{16} + 342489438767437612 T^{18} + 13302809011391389264 T^{20} +$$$$45\!\cdots\!20$$$$T^{22} +$$$$13\!\cdots\!74$$$$T^{24} +$$$$31\!\cdots\!64$$$$T^{26} +$$$$59\!\cdots\!72$$$$T^{28} +$$$$79\!\cdots\!52$$$$T^{30} +$$$$56\!\cdots\!21$$$$T^{32}$$
$53$ $$1 + 264 T^{2} + 35652 T^{4} + 3177712 T^{6} + 212585578 T^{8} + 12279652216 T^{10} + 710817442960 T^{12} + 42611108403128 T^{14} + 2398437218467955 T^{16} + 119694603504386552 T^{18} + 5608691528144463760 T^{20} +$$$$27\!\cdots\!64$$$$T^{22} +$$$$13\!\cdots\!58$$$$T^{24} +$$$$55\!\cdots\!88$$$$T^{26} +$$$$17\!\cdots\!32$$$$T^{28} +$$$$36\!\cdots\!16$$$$T^{30} +$$$$38\!\cdots\!21$$$$T^{32}$$
$59$ $$( 1 - 2 T - 226 T^{2} + 236 T^{3} + 32016 T^{4} - 19820 T^{5} - 2961940 T^{6} + 407278 T^{7} + 205790979 T^{8} + 24029402 T^{9} - 10310513140 T^{10} - 4070611780 T^{11} + 387949429776 T^{12} + 168722134564 T^{13} - 9532800602866 T^{14} - 4977302969638 T^{15} + 146830437604321 T^{16} )^{2}$$
$61$ $$( 1 - 8 T - 80 T^{2} + 376 T^{3} + 3998 T^{4} + 13396 T^{5} - 247632 T^{6} - 798752 T^{7} + 15355491 T^{8} - 48723872 T^{9} - 921438672 T^{10} + 3040637476 T^{11} + 55355672318 T^{12} + 317568209176 T^{13} - 4121629948880 T^{14} - 25141942688168 T^{15} + 191707312997281 T^{16} )^{2}$$
$67$ $$1 + 356 T^{2} + 63502 T^{4} + 8090208 T^{6} + 858425049 T^{8} + 80231005476 T^{10} + 6774300927054 T^{12} + 525083738078264 T^{14} + 37066264561921156 T^{16} + 2357100900233327096 T^{18} +$$$$13\!\cdots\!34$$$$T^{20} +$$$$72\!\cdots\!44$$$$T^{22} +$$$$34\!\cdots\!09$$$$T^{24} +$$$$14\!\cdots\!92$$$$T^{26} +$$$$51\!\cdots\!22$$$$T^{28} +$$$$13\!\cdots\!24$$$$T^{30} +$$$$16\!\cdots\!81$$$$T^{32}$$
$71$ $$( 1 + 14 T + 194 T^{2} + 1648 T^{3} + 14264 T^{4} + 117008 T^{5} + 977954 T^{6} + 5010754 T^{7} + 25411681 T^{8} )^{4}$$
$73$ $$1 + 312 T^{2} + 46894 T^{4} + 4662008 T^{6} + 358067385 T^{8} + 22613492772 T^{10} + 1186535858958 T^{12} + 57781939528716 T^{14} + 3463839236542756 T^{16} + 307919955748527564 T^{18} + 33695531277831292878 T^{20} +$$$$34\!\cdots\!08$$$$T^{22} +$$$$28\!\cdots\!85$$$$T^{24} +$$$$20\!\cdots\!92$$$$T^{26} +$$$$10\!\cdots\!74$$$$T^{28} +$$$$38\!\cdots\!08$$$$T^{30} +$$$$65\!\cdots\!61$$$$T^{32}$$
$79$ $$( 1 + 8 T - 98 T^{2} - 1128 T^{3} - 3647 T^{4} - 31028 T^{5} - 546546 T^{6} + 5486636 T^{7} + 152161556 T^{8} + 433444244 T^{9} - 3410993586 T^{10} - 15298014092 T^{11} - 142050945407 T^{12} - 3470919618072 T^{13} - 23822570641058 T^{14} + 153631271889272 T^{15} + 1517108809906561 T^{16} )^{2}$$
$83$ $$( 1 - 516 T^{2} + 125904 T^{4} - 18859016 T^{6} + 1891189930 T^{8} - 129919761224 T^{10} + 5975192447184 T^{12} - 168701232658404 T^{14} + 2252292232139041 T^{16} )^{2}$$
$89$ $$( 1 - 8 T - 98 T^{2} + 1004 T^{3} + 4584 T^{4} - 89290 T^{5} + 1051584 T^{6} + 2927032 T^{7} - 142835833 T^{8} + 260505848 T^{9} + 8329596864 T^{10} - 62946682010 T^{11} + 287610432744 T^{12} + 5606395686796 T^{13} - 48704166514178 T^{14} - 353850679164232 T^{15} + 3936588805702081 T^{16} )^{2}$$
$97$ $$( 1 - 756 T^{2} + 251932 T^{4} - 48333356 T^{6} + 5838717718 T^{8} - 454768546604 T^{10} + 22303358820892 T^{12} - 629726835726324 T^{14} + 7837433594376961 T^{16} )^{2}$$