# Properties

 Label 315.2.bf Level 315 Weight 2 Character orbit bf Rep. character $$\chi_{315}(109,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 36 Newform subspaces 3 Sturm bound 96 Trace bound 5

# Related objects

## Defining parameters

 Level: $$N$$ = $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 315.bf (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$35$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$3$$ Sturm bound: $$96$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(315, [\chi])$$.

Total New Old
Modular forms 112 44 68
Cusp forms 80 36 44
Eisenstein series 32 8 24

## Trace form

 $$36q + 14q^{4} + O(q^{10})$$ $$36q + 14q^{4} + 16q^{14} - 14q^{16} + 4q^{19} + 4q^{20} - 14q^{25} + 16q^{26} + 4q^{29} - 28q^{31} - 72q^{34} + 26q^{35} + 28q^{40} - 36q^{41} - 20q^{44} - 18q^{46} - 14q^{49} + 24q^{50} + 8q^{55} - 90q^{56} - 24q^{59} - 22q^{61} + 84q^{64} - 38q^{65} - 50q^{70} + 64q^{71} - 24q^{74} + 56q^{76} - 36q^{79} - 54q^{80} + 62q^{86} - 34q^{89} + 116q^{91} + 10q^{95} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(315, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
315.2.bf.a $$4$$ $$2.515$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$2$$ $$0$$ $$q+\zeta_{12}q^{2}-\zeta_{12}^{2}q^{4}+(1-2\zeta_{12}-\zeta_{12}^{2}+\cdots)q^{5}+\cdots$$
315.2.bf.b $$16$$ $$2.515$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$-2$$ $$0$$ $$q+(-\beta _{5}+\beta _{6}-\beta _{15})q^{2}+(1-\beta _{2}+\beta _{7}+\cdots)q^{4}+\cdots$$
315.2.bf.c $$16$$ $$2.515$$ $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{9}q^{2}+(1+\beta _{5}-\beta _{8})q^{4}+(\beta _{1}+\beta _{9}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(315, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(315, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(35, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(105, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 + 3 T^{2} + 5 T^{4} + 12 T^{6} + 16 T^{8}$$)($$1 + 4 T^{2} + 4 T^{4} - 8 T^{6} - 20 T^{8} + 32 T^{10} + 96 T^{12} - 208 T^{14} - 944 T^{16} - 832 T^{18} + 1536 T^{20} + 2048 T^{22} - 5120 T^{24} - 8192 T^{26} + 16384 T^{28} + 65536 T^{30} + 65536 T^{32}$$)($$( 1 + 2 T^{2} + 2 T^{4} - 12 T^{6} - 28 T^{8} - 48 T^{10} + 32 T^{12} + 128 T^{14} + 256 T^{16} )^{2}$$)
$3$ 1
$5$ ($$1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4}$$)($$1 + 2 T + 4 T^{2} + 20 T^{3} + 62 T^{4} + 118 T^{5} + 200 T^{6} + 670 T^{7} + 1639 T^{8} + 3350 T^{9} + 5000 T^{10} + 14750 T^{11} + 38750 T^{12} + 62500 T^{13} + 62500 T^{14} + 156250 T^{15} + 390625 T^{16}$$)($$1 + 8 T^{2} + 26 T^{4} - 96 T^{6} - 781 T^{8} - 2400 T^{10} + 16250 T^{12} + 125000 T^{14} + 390625 T^{16}$$)
$7$ ($$1 - 13 T^{2} + 49 T^{4}$$)($$1 + 20 T^{2} + 202 T^{4} + 1244 T^{6} + 7267 T^{8} + 60956 T^{10} + 485002 T^{12} + 2352980 T^{14} + 5764801 T^{16}$$)($$( 1 + 91 T^{4} + 2401 T^{8} )^{2}$$)
$11$ ($$( 1 - 11 T^{2} + 121 T^{4} )^{2}$$)($$( 1 - 26 T^{2} + 28 T^{3} + 316 T^{4} - 518 T^{5} - 2872 T^{6} + 2968 T^{7} + 31007 T^{8} + 32648 T^{9} - 347512 T^{10} - 689458 T^{11} + 4626556 T^{12} + 4509428 T^{13} - 46060586 T^{14} + 214358881 T^{16} )^{2}$$)($$( 1 - 2 T^{2} - 176 T^{4} + 124 T^{6} + 17275 T^{8} + 15004 T^{10} - 2576816 T^{12} - 3543122 T^{14} + 214358881 T^{16} )^{2}$$)
$13$ ($$( 1 - 22 T^{2} + 169 T^{4} )^{2}$$)($$( 1 - 44 T^{2} + 1234 T^{4} - 23988 T^{6} + 358939 T^{8} - 4053972 T^{10} + 35244274 T^{12} - 212379596 T^{14} + 815730721 T^{16} )^{2}$$)($$( 1 - 24 T^{2} + 475 T^{4} - 4056 T^{6} + 28561 T^{8} )^{4}$$)
$17$ ($$( 1 - 8 T + 47 T^{2} - 136 T^{3} + 289 T^{4} )( 1 + 8 T + 47 T^{2} + 136 T^{3} + 289 T^{4} )$$)($$1 + 76 T^{2} + 2664 T^{4} + 67392 T^{6} + 1583990 T^{8} + 33141472 T^{10} + 603483136 T^{12} + 11039990476 T^{14} + 198790801859 T^{16} + 3190557247564 T^{18} + 50403515001856 T^{20} + 799954567161568 T^{22} + 11049530028969590 T^{24} + 135861860939059008 T^{26} + 1552105639980083304 T^{28} + 12796714818514470604 T^{30} + 48661191875666868481 T^{32}$$)($$( 1 + 14 T^{2} - 424 T^{4} + 588 T^{6} + 244235 T^{8} + 169932 T^{10} - 35412904 T^{12} + 337925966 T^{14} + 6975757441 T^{16} )^{2}$$)
$19$ ($$( 1 - 6 T + 17 T^{2} - 114 T^{3} + 361 T^{4} )^{2}$$)($$( 1 + 12 T + 30 T^{2} - 80 T^{3} + 689 T^{4} + 9436 T^{5} + 28846 T^{6} + 59944 T^{7} + 291332 T^{8} + 1138936 T^{9} + 10413406 T^{10} + 64721524 T^{11} + 89791169 T^{12} - 198087920 T^{13} + 1411376430 T^{14} + 10726460868 T^{15} + 16983563041 T^{16} )^{2}$$)($$( 1 - 4 T - 19 T^{2} + 12 T^{3} + 560 T^{4} + 228 T^{5} - 6859 T^{6} - 27436 T^{7} + 130321 T^{8} )^{4}$$)
$23$ ($$1 + 37 T^{2} + 840 T^{4} + 19573 T^{6} + 279841 T^{8}$$)($$1 + 20 T^{2} - 1512 T^{4} - 18448 T^{6} + 1612902 T^{8} + 10149224 T^{10} - 1248424544 T^{12} - 1840459676 T^{14} + 770565238915 T^{16} - 973603168604 T^{18} - 349360372817504 T^{20} + 1502449397500136 T^{22} + 126307944781695462 T^{24} - 764236278869396752 T^{26} - 33134912141214725352 T^{28} +$$$$23\!\cdots\!80$$$$T^{30} +$$$$61\!\cdots\!61$$$$T^{32}$$)($$( 1 + 54 T^{2} + 1472 T^{4} + 20844 T^{6} + 262731 T^{8} + 11026476 T^{10} + 411925952 T^{12} + 7993938006 T^{14} + 78310985281 T^{16} )^{2}$$)
$29$ ($$( 1 - 7 T + 29 T^{2} )^{4}$$)($$( 1 + 6 T + 78 T^{2} + 332 T^{3} + 2820 T^{4} + 9628 T^{5} + 65598 T^{6} + 146334 T^{7} + 707281 T^{8} )^{4}$$)($$( 1 + 46 T^{2} + 1364 T^{4} + 38686 T^{6} + 707281 T^{8} )^{4}$$)
$31$ ($$( 1 + 2 T - 27 T^{2} + 62 T^{3} + 961 T^{4} )^{2}$$)($$( 1 - 8 T - 54 T^{2} + 352 T^{3} + 3285 T^{4} - 9272 T^{5} - 157694 T^{6} + 60416 T^{7} + 6248764 T^{8} + 1872896 T^{9} - 151543934 T^{10} - 276222152 T^{11} + 3033766485 T^{12} + 10077461152 T^{13} - 47925198774 T^{14} - 220100912888 T^{15} + 852891037441 T^{16} )^{2}$$)($$( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} )^{8}$$)
$37$ ($$1 + 10 T^{2} - 1269 T^{4} + 13690 T^{6} + 1874161 T^{8}$$)($$1 + 128 T^{2} + 6574 T^{4} + 187656 T^{6} + 5183961 T^{8} + 158877780 T^{10} + 5690568750 T^{12} + 459641012708 T^{14} + 25029766954564 T^{16} + 629248546397252 T^{18} + 10665042019068750 T^{20} + 407636915949292020 T^{22} + 18208556502427761081 T^{24} +$$$$90\!\cdots\!44$$$$T^{26} +$$$$43\!\cdots\!94$$$$T^{28} +$$$$11\!\cdots\!92$$$$T^{30} +$$$$12\!\cdots\!41$$$$T^{32}$$)($$( 1 + 120 T^{2} + 8237 T^{4} + 411000 T^{6} + 16394808 T^{8} + 562659000 T^{10} + 15437464157 T^{12} + 307887169080 T^{14} + 3512479453921 T^{16} )^{2}$$)
$41$ ($$( 1 + 5 T + 41 T^{2} )^{4}$$)($$( 1 + 4 T + 114 T^{2} + 346 T^{3} + 5976 T^{4} + 14186 T^{5} + 191634 T^{6} + 275684 T^{7} + 2825761 T^{8} )^{4}$$)($$( 1 + 10 T^{2} + 860 T^{4} + 16810 T^{6} + 2825761 T^{8} )^{4}$$)
$43$ ($$( 1 - 37 T^{2} + 1849 T^{4} )^{2}$$)($$( 1 - 216 T^{2} + 22138 T^{4} - 1464812 T^{6} + 71619331 T^{8} - 2708437388 T^{10} + 75685416538 T^{12} - 1365414418584 T^{14} + 11688200277601 T^{16} )^{2}$$)($$( 1 + 24 T^{2} + 3499 T^{4} + 44376 T^{6} + 3418801 T^{8} )^{4}$$)
$47$ ($$( 1 + 47 T^{2} + 2209 T^{4} )^{2}$$)($$1 + 308 T^{2} + 50892 T^{4} + 5950136 T^{6} + 547353834 T^{8} + 41768130380 T^{10} + 2726163659344 T^{12} + 155042751818668 T^{14} + 7757667166613347 T^{16} + 342489438767437612 T^{18} + 13302809011391389264 T^{20} +$$$$45\!\cdots\!20$$$$T^{22} +$$$$13\!\cdots\!74$$$$T^{24} +$$$$31\!\cdots\!64$$$$T^{26} +$$$$59\!\cdots\!72$$$$T^{28} +$$$$79\!\cdots\!52$$$$T^{30} +$$$$56\!\cdots\!21$$$$T^{32}$$)($$( 1 + 168 T^{2} + 16778 T^{4} + 1180704 T^{6} + 63784419 T^{8} + 2608175136 T^{10} + 81871287818 T^{12} + 1810908175272 T^{14} + 23811286661761 T^{16} )^{2}$$)
$53$ ($$1 + 70 T^{2} + 2091 T^{4} + 196630 T^{6} + 7890481 T^{8}$$)($$1 + 264 T^{2} + 35652 T^{4} + 3177712 T^{6} + 212585578 T^{8} + 12279652216 T^{10} + 710817442960 T^{12} + 42611108403128 T^{14} + 2398437218467955 T^{16} + 119694603504386552 T^{18} + 5608691528144463760 T^{20} +$$$$27\!\cdots\!64$$$$T^{22} +$$$$13\!\cdots\!58$$$$T^{24} +$$$$55\!\cdots\!88$$$$T^{26} +$$$$17\!\cdots\!32$$$$T^{28} +$$$$36\!\cdots\!16$$$$T^{30} +$$$$38\!\cdots\!21$$$$T^{32}$$)($$( 1 + 108 T^{2} + 3242 T^{4} + 302832 T^{6} + 30275427 T^{8} + 850655088 T^{10} + 25580939402 T^{12} + 2393751001932 T^{14} + 62259690411361 T^{16} )^{2}$$)
$59$ ($$( 1 + 10 T + 41 T^{2} + 590 T^{3} + 3481 T^{4} )^{2}$$)($$( 1 + 2 T - 226 T^{2} - 236 T^{3} + 32016 T^{4} + 19820 T^{5} - 2961940 T^{6} - 407278 T^{7} + 205790979 T^{8} - 24029402 T^{9} - 10310513140 T^{10} + 4070611780 T^{11} + 387949429776 T^{12} - 168722134564 T^{13} - 9532800602866 T^{14} + 4977302969638 T^{15} + 146830437604321 T^{16} )^{2}$$)($$( 1 - 166 T^{2} + 14888 T^{4} - 947196 T^{6} + 52438427 T^{8} - 3297189276 T^{10} + 180403270568 T^{12} - 7001968584406 T^{14} + 146830437604321 T^{16} )^{2}$$)
$61$ ($$( 1 + 7 T - 12 T^{2} + 427 T^{3} + 3721 T^{4} )^{2}$$)($$( 1 - 8 T - 80 T^{2} + 376 T^{3} + 3998 T^{4} + 13396 T^{5} - 247632 T^{6} - 798752 T^{7} + 15355491 T^{8} - 48723872 T^{9} - 921438672 T^{10} + 3040637476 T^{11} + 55355672318 T^{12} + 317568209176 T^{13} - 4121629948880 T^{14} - 25141942688168 T^{15} + 191707312997281 T^{16} )^{2}$$)($$( 1 + 6 T - 88 T^{2} + 12 T^{3} + 9459 T^{4} + 732 T^{5} - 327448 T^{6} + 1361886 T^{7} + 13845841 T^{8} )^{4}$$)
$67$ ($$( 1 - 13 T^{2} + 4489 T^{4} )( 1 + 122 T^{2} + 4489 T^{4} )$$)($$1 + 356 T^{2} + 63502 T^{4} + 8090208 T^{6} + 858425049 T^{8} + 80231005476 T^{10} + 6774300927054 T^{12} + 525083738078264 T^{14} + 37066264561921156 T^{16} + 2357100900233327096 T^{18} +$$$$13\!\cdots\!34$$$$T^{20} +$$$$72\!\cdots\!44$$$$T^{22} +$$$$34\!\cdots\!09$$$$T^{24} +$$$$14\!\cdots\!92$$$$T^{26} +$$$$51\!\cdots\!22$$$$T^{28} +$$$$13\!\cdots\!24$$$$T^{30} +$$$$16\!\cdots\!81$$$$T^{32}$$)($$( 1 + 72 T^{2} + 3485 T^{4} - 524088 T^{6} - 40535496 T^{8} - 2352631032 T^{10} + 70226656685 T^{12} + 6513003516168 T^{14} + 406067677556641 T^{16} )^{2}$$)
$71$ ($$( 1 - 2 T + 71 T^{2} )^{4}$$)($$( 1 - 14 T + 194 T^{2} - 1648 T^{3} + 14264 T^{4} - 117008 T^{5} + 977954 T^{6} - 5010754 T^{7} + 25411681 T^{8} )^{4}$$)($$( 1 + 214 T^{2} + 20684 T^{4} + 1078774 T^{6} + 25411681 T^{8} )^{4}$$)
$73$ ($$( 1 - 16 T + 183 T^{2} - 1168 T^{3} + 5329 T^{4} )( 1 + 16 T + 183 T^{2} + 1168 T^{3} + 5329 T^{4} )$$)($$1 + 312 T^{2} + 46894 T^{4} + 4662008 T^{6} + 358067385 T^{8} + 22613492772 T^{10} + 1186535858958 T^{12} + 57781939528716 T^{14} + 3463839236542756 T^{16} + 307919955748527564 T^{18} + 33695531277831292878 T^{20} +$$$$34\!\cdots\!08$$$$T^{22} +$$$$28\!\cdots\!85$$$$T^{24} +$$$$20\!\cdots\!92$$$$T^{26} +$$$$10\!\cdots\!74$$$$T^{28} +$$$$38\!\cdots\!08$$$$T^{30} +$$$$65\!\cdots\!61$$$$T^{32}$$)($$( 1 + 208 T^{2} + 23365 T^{4} + 1922128 T^{6} + 137018104 T^{8} + 10243020112 T^{10} + 663524900965 T^{12} + 31477519068112 T^{14} + 806460091894081 T^{16} )^{2}$$)
$79$ ($$( 1 + 2 T - 75 T^{2} + 158 T^{3} + 6241 T^{4} )^{2}$$)($$( 1 + 8 T - 98 T^{2} - 1128 T^{3} - 3647 T^{4} - 31028 T^{5} - 546546 T^{6} + 5486636 T^{7} + 152161556 T^{8} + 433444244 T^{9} - 3410993586 T^{10} - 15298014092 T^{11} - 142050945407 T^{12} - 3470919618072 T^{13} - 23822570641058 T^{14} + 153631271889272 T^{15} + 1517108809906561 T^{16} )^{2}$$)($$( 1 + 4 T - 83 T^{2} - 236 T^{3} + 2296 T^{4} - 18644 T^{5} - 518003 T^{6} + 1972156 T^{7} + 38950081 T^{8} )^{4}$$)
$83$ ($$( 1 - 45 T^{2} + 6889 T^{4} )^{2}$$)($$( 1 - 516 T^{2} + 125904 T^{4} - 18859016 T^{6} + 1891189930 T^{8} - 129919761224 T^{10} + 5975192447184 T^{12} - 168701232658404 T^{14} + 2252292232139041 T^{16} )^{2}$$)($$( 1 - 182 T^{2} + 18356 T^{4} - 1253798 T^{6} + 47458321 T^{8} )^{4}$$)
$89$ ($$( 1 + 9 T - 8 T^{2} + 801 T^{3} + 7921 T^{4} )^{2}$$)($$( 1 + 8 T - 98 T^{2} - 1004 T^{3} + 4584 T^{4} + 89290 T^{5} + 1051584 T^{6} - 2927032 T^{7} - 142835833 T^{8} - 260505848 T^{9} + 8329596864 T^{10} + 62946682010 T^{11} + 287610432744 T^{12} - 5606395686796 T^{13} - 48704166514178 T^{14} + 353850679164232 T^{15} + 3936588805702081 T^{16} )^{2}$$)($$( 1 - 314 T^{2} + 58168 T^{4} - 7720004 T^{6} + 790702027 T^{8} - 61150151684 T^{10} + 3649590674488 T^{12} - 156052125361754 T^{14} + 3936588805702081 T^{16} )^{2}$$)
$97$ ($$( 1 + 62 T^{2} + 9409 T^{4} )^{2}$$)($$( 1 - 756 T^{2} + 251932 T^{4} - 48333356 T^{6} + 5838717718 T^{8} - 454768546604 T^{10} + 22303358820892 T^{12} - 629726835726324 T^{14} + 7837433594376961 T^{16} )^{2}$$)($$( 1 - 192 T^{2} + 25234 T^{4} - 1806528 T^{6} + 88529281 T^{8} )^{4}$$)