# Properties

 Label 315.2.p Level 315 Weight 2 Character orbit p Rep. character $$\chi_{315}(118,\cdot)$$ Character field $$\Q(\zeta_{4})$$ Dimension 36 Newform subspaces 5 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.p (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$35$$ Character field: $$\Q(i)$$ Newform subspaces: $$5$$ Sturm bound: $$96$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$2$$, $$17$$

## 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

 $$36q + 4q^{2} - 4q^{7} - 16q^{8} + O(q^{10})$$ $$36q + 4q^{2} - 4q^{7} - 16q^{8} + 20q^{11} - 16q^{16} - 28q^{22} + 32q^{23} - 16q^{25} - 8q^{28} - 48q^{32} - 12q^{35} - 8q^{37} + 4q^{43} - 16q^{46} + 52q^{50} - 28q^{53} - 8q^{56} + 12q^{58} - 40q^{65} + 60q^{67} + 20q^{70} - 40q^{71} + 28q^{77} - 52q^{85} - 88q^{86} + 56q^{88} - 28q^{91} + 40q^{92} + 92q^{95} + 108q^{98} + O(q^{100})$$

## 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.p.a $$4$$ $$2.515$$ $$\Q(i, \sqrt{10})$$ None $$0$$ $$0$$ $$-8$$ $$4$$ $$q+\beta _{1}q^{2}+3\beta _{2}q^{4}+(-2+\beta _{2})q^{5}+\cdots$$
315.2.p.b $$4$$ $$2.515$$ $$\Q(i, \sqrt{10})$$ None $$0$$ $$0$$ $$8$$ $$4$$ $$q+\beta _{1}q^{2}+3\beta _{2}q^{4}+(2-\beta _{2})q^{5}+(1+\cdots)q^{7}+\cdots$$
315.2.p.c $$4$$ $$2.515$$ $$\Q(i, \sqrt{10})$$ None $$4$$ $$0$$ $$0$$ $$4$$ $$q+(1+\beta _{2})q^{2}+\beta _{1}q^{5}+(1+\beta _{2}+\beta _{3})q^{7}+\cdots$$
315.2.p.d $$8$$ $$2.515$$ 8.0.40960000.1 None $$0$$ $$0$$ $$0$$ $$-8$$ $$q+\beta _{1}q^{2}-\beta _{3}q^{4}+\beta _{5}q^{5}+(-1-\beta _{3}+\cdots)q^{7}+\cdots$$
315.2.p.e $$16$$ $$2.515$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$-8$$ $$q-\beta _{2}q^{2}+(-\beta _{6}-\beta _{7}+\beta _{13})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots$$

## 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

$p$ $F_p(T)$
$2$ ($$1 - 7 T^{4} + 16 T^{8}$$)($$1 - 7 T^{4} + 16 T^{8}$$)($$( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} )^{2}$$)($$( 1 + T^{4} + 16 T^{8} )^{2}$$)($$( 1 - 2 T + 4 T^{2} - 6 T^{3} + 9 T^{4} - 12 T^{5} + 16 T^{6} - 16 T^{7} + 16 T^{8} )^{2}( 1 + 2 T + 2 T^{3} + 9 T^{4} + 4 T^{5} + 16 T^{7} + 16 T^{8} )^{2}$$)
$3$ 1
$5$ ($$( 1 + 4 T + 5 T^{2} )^{2}$$)($$( 1 - 4 T + 5 T^{2} )^{2}$$)($$1 + 25 T^{4}$$)($$( 1 + 5 T^{2} )^{4}$$)($$1 + 28 T^{4} - 256 T^{6} - 26 T^{8} - 6400 T^{10} + 17500 T^{12} + 390625 T^{16}$$)
$7$ ($$1 - 4 T + 8 T^{2} - 28 T^{3} + 49 T^{4}$$)($$1 - 4 T + 8 T^{2} - 28 T^{3} + 49 T^{4}$$)($$1 - 4 T + 8 T^{2} - 28 T^{3} + 49 T^{4}$$)($$( 1 + 4 T + 8 T^{2} + 28 T^{3} + 49 T^{4} )^{2}$$)($$1 + 8 T + 32 T^{2} + 88 T^{3} + 196 T^{4} + 248 T^{5} - 416 T^{6} - 2840 T^{7} - 8634 T^{8} - 19880 T^{9} - 20384 T^{10} + 85064 T^{11} + 470596 T^{12} + 1479016 T^{13} + 3764768 T^{14} + 6588344 T^{15} + 5764801 T^{16}$$)
$11$ ($$( 1 + 12 T^{2} + 121 T^{4} )^{2}$$)($$( 1 + 12 T^{2} + 121 T^{4} )^{2}$$)($$( 1 - T + 11 T^{2} )^{4}$$)($$( 1 + 4 T^{2} + 121 T^{4} )^{4}$$)($$( 1 - 4 T + 32 T^{2} - 68 T^{3} + 402 T^{4} - 748 T^{5} + 3872 T^{6} - 5324 T^{7} + 14641 T^{8} )^{4}$$)
$13$ ($$( 1 - 8 T + 32 T^{2} - 104 T^{3} + 169 T^{4} )( 1 + 8 T + 32 T^{2} + 104 T^{3} + 169 T^{4} )$$)($$( 1 - 8 T + 32 T^{2} - 104 T^{3} + 169 T^{4} )( 1 + 8 T + 32 T^{2} + 104 T^{3} + 169 T^{4} )$$)($$1 + 103 T^{4} + 28561 T^{8}$$)($$( 1 - 8 T + 32 T^{2} - 104 T^{3} + 169 T^{4} )^{2}( 1 + 8 T + 32 T^{2} + 104 T^{3} + 169 T^{4} )^{2}$$)($$1 + 424 T^{4} + 47004 T^{8} - 12160488 T^{12} - 4129271418 T^{16} - 347315697768 T^{20} + 38342606809884 T^{24} + 9878388091931944 T^{28} + 665416609183179841 T^{32}$$)
$17$ ($$( 1 - 8 T + 17 T^{2} )^{2}( 1 - 2 T + 17 T^{2} )^{2}$$)($$( 1 + 2 T + 17 T^{2} )^{2}( 1 + 8 T + 17 T^{2} )^{2}$$)($$1 + 263 T^{4} + 83521 T^{8}$$)($$( 1 - 2 T^{4} + 83521 T^{8} )^{2}$$)($$1 + 120 T^{4} + 166556 T^{8} - 9625784 T^{12} + 12237871174 T^{16} - 803955105464 T^{20} + 1161854256343196 T^{24} + 69914668467571320 T^{28} + 48661191875666868481 T^{32}$$)
$19$ ($$( 1 + 28 T^{2} + 361 T^{4} )^{2}$$)($$( 1 + 28 T^{2} + 361 T^{4} )^{2}$$)($$( 1 + 28 T^{2} + 361 T^{4} )^{2}$$)($$( 1 + 28 T^{2} + 361 T^{4} )^{4}$$)($$( 1 + 48 T^{2} + 1524 T^{4} + 40016 T^{6} + 818246 T^{8} + 14445776 T^{10} + 198609204 T^{12} + 2258202288 T^{14} + 16983563041 T^{16} )^{2}$$)
$23$ ($$( 1 - 12 T + 72 T^{2} - 276 T^{3} + 529 T^{4} )( 1 + 12 T + 72 T^{2} + 276 T^{3} + 529 T^{4} )$$)($$( 1 - 12 T + 72 T^{2} - 276 T^{3} + 529 T^{4} )( 1 + 12 T + 72 T^{2} + 276 T^{3} + 529 T^{4} )$$)($$( 1 + 4 T + 8 T^{2} + 92 T^{3} + 529 T^{4} )^{2}$$)($$( 1 + 706 T^{4} + 279841 T^{8} )^{2}$$)($$( 1 - 20 T + 200 T^{2} - 1516 T^{3} + 10388 T^{4} - 65340 T^{5} + 378328 T^{6} - 2076676 T^{7} + 10539814 T^{8} - 47763548 T^{9} + 200135512 T^{10} - 794991780 T^{11} + 2906988308 T^{12} - 9757495988 T^{13} + 29607177800 T^{14} - 68096508940 T^{15} + 78310985281 T^{16} )^{2}$$)
$29$ ($$( 1 - 29 T^{2} )^{4}$$)($$( 1 - 29 T^{2} )^{4}$$)($$( 1 - 49 T^{2} + 841 T^{4} )^{2}$$)($$( 1 - 26 T^{2} + 841 T^{4} )^{4}$$)($$( 1 - 184 T^{2} + 15868 T^{4} - 835400 T^{6} + 29324070 T^{8} - 702571400 T^{10} + 11223134908 T^{12} - 109447491064 T^{14} + 500246412961 T^{16} )^{2}$$)
$31$ ($$( 1 - 52 T^{2} + 961 T^{4} )^{2}$$)($$( 1 - 52 T^{2} + 961 T^{4} )^{2}$$)($$( 1 - 52 T^{2} + 961 T^{4} )^{2}$$)($$( 1 + 28 T^{2} + 961 T^{4} )^{4}$$)($$( 1 - 128 T^{2} + 9396 T^{4} - 463040 T^{6} + 16703398 T^{8} - 444981440 T^{10} + 8677403316 T^{12} - 113600471168 T^{14} + 852891037441 T^{16} )^{2}$$)
$37$ ($$( 1 + 6 T + 18 T^{2} + 222 T^{3} + 1369 T^{4} )^{2}$$)($$( 1 + 6 T + 18 T^{2} + 222 T^{3} + 1369 T^{4} )^{2}$$)($$( 1 + 12 T + 72 T^{2} + 444 T^{3} + 1369 T^{4} )^{2}$$)($$( 1 - 2 T + 2 T^{2} - 74 T^{3} + 1369 T^{4} )^{4}$$)($$( 1 - 16 T + 128 T^{2} - 944 T^{3} + 8860 T^{4} - 73552 T^{5} + 488320 T^{6} - 3175280 T^{7} + 20212134 T^{8} - 117485360 T^{9} + 668510080 T^{10} - 3725629456 T^{11} + 16605066460 T^{12} - 65460695408 T^{13} + 328412980352 T^{14} - 1518910034128 T^{15} + 3512479453921 T^{16} )^{2}$$)
$41$ ($$( 1 - 41 T^{2} )^{4}$$)($$( 1 - 41 T^{2} )^{4}$$)($$( 1 + 8 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 - 2 T^{2} + 1681 T^{4} )^{4}$$)($$( 1 - 144 T^{2} + 12956 T^{4} - 823792 T^{6} + 38320198 T^{8} - 1384794352 T^{10} + 36610559516 T^{12} - 684015010704 T^{14} + 7984925229121 T^{16} )^{2}$$)
$43$ ($$( 1 - 12 T + 72 T^{2} - 516 T^{3} + 1849 T^{4} )^{2}$$)($$( 1 - 12 T + 72 T^{2} - 516 T^{3} + 1849 T^{4} )^{2}$$)($$( 1 + 6 T + 18 T^{2} + 258 T^{3} + 1849 T^{4} )^{2}$$)($$( 1 + 4 T + 8 T^{2} + 172 T^{3} + 1849 T^{4} )^{4}$$)($$( 1 + 8 T + 32 T^{2} + 280 T^{3} + 2788 T^{4} + 14232 T^{5} + 63840 T^{6} + 569416 T^{7} + 5017638 T^{8} + 24484888 T^{9} + 118040160 T^{10} + 1131543624 T^{11} + 9531617188 T^{12} + 41162364040 T^{13} + 202283617568 T^{14} + 2174548888856 T^{15} + 11688200277601 T^{16} )^{2}$$)
$47$ ($$( 1 + 2209 T^{4} )^{2}$$)($$( 1 + 2209 T^{4} )^{2}$$)($$1 - 2017 T^{4} + 4879681 T^{8}$$)($$( 1 - 62 T^{4} + 4879681 T^{8} )^{2}$$)($$1 + 3784 T^{4} + 1124764 T^{8} + 9138019192 T^{12} + 63538455194182 T^{16} + 44590618628837752 T^{20} + 26782078030828949404 T^{24} +$$$$43\!\cdots\!44$$$$T^{28} +$$$$56\!\cdots\!21$$$$T^{32}$$)
$53$ ($$1 + 1778 T^{4} + 7890481 T^{8}$$)($$1 + 1778 T^{4} + 7890481 T^{8}$$)($$( 1 + 2 T + 2 T^{2} + 106 T^{3} + 2809 T^{4} )^{2}$$)($$( 1 - 5582 T^{4} + 7890481 T^{8} )^{2}$$)($$( 1 + 12 T + 72 T^{2} + 572 T^{3} + 1780 T^{4} - 1436 T^{5} + 18200 T^{6} + 441076 T^{7} + 6254598 T^{8} + 23377028 T^{9} + 51123800 T^{10} - 213787372 T^{11} + 14045056180 T^{12} + 239207821996 T^{13} + 1595834001288 T^{14} + 14096533678044 T^{15} + 62259690411361 T^{16} )^{2}$$)
$59$ ($$( 1 + 59 T^{2} )^{4}$$)($$( 1 + 59 T^{2} )^{4}$$)($$( 1 + 28 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 + 59 T^{2} )^{8}$$)($$( 1 + 312 T^{2} + 49660 T^{4} + 5040712 T^{6} + 354176614 T^{8} + 17546718472 T^{10} + 601748147260 T^{12} + 13160326495992 T^{14} + 146830437604321 T^{16} )^{2}$$)
$61$ ($$( 1 + 38 T^{2} + 3721 T^{4} )^{2}$$)($$( 1 + 38 T^{2} + 3721 T^{4} )^{2}$$)($$( 1 - 82 T^{2} + 3721 T^{4} )^{2}$$)($$( 1 - 61 T^{2} )^{8}$$)($$( 1 - 200 T^{2} + 17532 T^{4} - 857912 T^{6} + 37932838 T^{8} - 3192290552 T^{10} + 242745284412 T^{12} - 10304074872200 T^{14} + 191707312997281 T^{16} )^{2}$$)
$67$ ($$( 1 - 16 T + 128 T^{2} - 1072 T^{3} + 4489 T^{4} )^{2}$$)($$( 1 - 16 T + 128 T^{2} - 1072 T^{3} + 4489 T^{4} )^{2}$$)($$( 1 + 2 T + 2 T^{2} + 134 T^{3} + 4489 T^{4} )^{2}$$)($$( 1 - 8 T + 32 T^{2} - 536 T^{3} + 4489 T^{4} )^{4}$$)($$( 1 + 16 T + 128 T^{2} + 1424 T^{3} + 22436 T^{4} + 211280 T^{5} + 1522560 T^{6} + 15870032 T^{7} + 163564774 T^{8} + 1063292144 T^{9} + 6834771840 T^{10} + 63545206640 T^{11} + 452110550756 T^{12} + 1922578152368 T^{13} + 11578672917632 T^{14} + 96971385685168 T^{15} + 406067677556641 T^{16} )^{2}$$)
$71$ ($$( 1 + 52 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 + 52 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 - 6 T + 71 T^{2} )^{4}$$)($$( 1 + 124 T^{2} + 5041 T^{4} )^{4}$$)($$( 1 + 16 T + 232 T^{2} + 2376 T^{3} + 21730 T^{4} + 168696 T^{5} + 1169512 T^{6} + 5726576 T^{7} + 25411681 T^{8} )^{4}$$)
$73$ ($$1 - 9502 T^{4} + 28398241 T^{8}$$)($$1 - 9502 T^{4} + 28398241 T^{8}$$)($$( 1 + 5329 T^{4} )^{2}$$)($$( 1 + 5218 T^{4} + 28398241 T^{8} )^{2}$$)($$1 - 15256 T^{4} + 80862300 T^{8} + 94053698264 T^{12} - 2362018367550906 T^{16} + 2670959590242353624 T^{20} +$$$$65\!\cdots\!00$$$$T^{24} -$$$$34\!\cdots\!76$$$$T^{28} +$$$$65\!\cdots\!61$$$$T^{32}$$)
$79$ ($$( 1 - 142 T^{2} + 6241 T^{4} )^{2}$$)($$( 1 - 142 T^{2} + 6241 T^{4} )^{2}$$)($$( 1 + 11 T^{2} + 6241 T^{4} )^{2}$$)($$( 1 - 14 T^{2} + 6241 T^{4} )^{4}$$)($$( 1 - 312 T^{2} + 58396 T^{4} - 7293320 T^{6} + 672141766 T^{8} - 45517610120 T^{10} + 2274528930076 T^{12} - 75843286122552 T^{14} + 1517108809906561 T^{16} )^{2}$$)
$83$ ($$( 1 + 20 T + 200 T^{2} + 1660 T^{3} + 6889 T^{4} )^{2}$$)($$( 1 - 20 T + 200 T^{2} - 1660 T^{3} + 6889 T^{4} )^{2}$$)($$1 + 7538 T^{4} + 47458321 T^{8}$$)($$( 1 + 2098 T^{4} + 47458321 T^{8} )^{2}$$)($$1 + 5000 T^{4} + 95818588 T^{8} + 596752860728 T^{12} + 4571727903671302 T^{16} + 28320888822097717688 T^{20} +$$$$21\!\cdots\!08$$$$T^{24} +$$$$53\!\cdots\!00$$$$T^{28} +$$$$50\!\cdots\!81$$$$T^{32}$$)
$89$ ($$( 1 - 10 T + 89 T^{2} )^{4}$$)($$( 1 + 10 T + 89 T^{2} )^{4}$$)($$( 1 + 138 T^{2} + 7921 T^{4} )^{2}$$)($$( 1 - 2 T^{2} + 7921 T^{4} )^{4}$$)($$( 1 + 576 T^{2} + 155068 T^{4} + 25338304 T^{6} + 2740378246 T^{8} + 200704705984 T^{10} + 9729313827388 T^{12} + 286261223593536 T^{14} + 3936588805702081 T^{16} )^{2}$$)
$97$ ($$1 + 11458 T^{4} + 88529281 T^{8}$$)($$1 + 11458 T^{4} + 88529281 T^{8}$$)($$1 + 16903 T^{4} + 88529281 T^{8}$$)($$( 1 + 11458 T^{4} + 88529281 T^{8} )^{2}$$)($$1 - 55064 T^{4} + 1465436892 T^{8} - 24336256217256 T^{12} + 274732504520067270 T^{16} -$$$$21\!\cdots\!36$$$$T^{20} +$$$$11\!\cdots\!12$$$$T^{24} -$$$$38\!\cdots\!24$$$$T^{28} +$$$$61\!\cdots\!21$$$$T^{32}$$)