# Properties

 Label 525.2.m Level 525 Weight 2 Character orbit m Rep. character $$\chi_{525}(118,\cdot)$$ Character field $$\Q(\zeta_{4})$$ Dimension 48 Newform subspaces 3 Sturm bound 160 Trace bound 1

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 525.m (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$35$$ Character field: $$\Q(i)$$ Newform subspaces: $$3$$ Sturm bound: $$160$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(525, [\chi])$$.

Total New Old
Modular forms 184 48 136
Cusp forms 136 48 88
Eisenstein series 48 0 48

## Trace form

 $$48q + 8q^{7} - 24q^{8} + O(q^{10})$$ $$48q + 8q^{7} - 24q^{8} + 32q^{11} + 16q^{16} + 4q^{21} + 16q^{22} + 40q^{23} - 24q^{28} - 48q^{32} - 48q^{36} - 32q^{37} + 16q^{42} + 16q^{43} - 8q^{46} + 32q^{51} - 24q^{53} - 48q^{56} - 8q^{57} - 32q^{58} - 8q^{63} + 32q^{67} - 128q^{71} - 24q^{72} + 24q^{77} + 8q^{78} - 48q^{81} - 8q^{86} + 64q^{88} - 4q^{91} + 40q^{92} - 24q^{93} + 96q^{98} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(525, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
525.2.m.a $$8$$ $$4.192$$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{24}^{2}q^{2}-\zeta_{24}q^{3}+\zeta_{24}^{3}q^{4}-\zeta_{24}^{4}q^{6}+\cdots$$
525.2.m.b $$16$$ $$4.192$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$8$$ $$q-\beta _{5}q^{2}+\beta _{9}q^{3}+(\beta _{2}+\beta _{7}-\beta _{11}+\cdots)q^{4}+\cdots$$
525.2.m.c $$24$$ $$4.192$$ None $$0$$ $$0$$ $$0$$ $$0$$

## Decomposition of $$S_{2}^{\mathrm{old}}(525, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(525, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(35, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(105, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(175, [\chi])$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$( 1 - 7 T^{4} + 16 T^{8} )^{2}$$)($$( 1 - 2 T + 4 T^{2} - 6 T^{3} + 9 T^{4} - 12 T^{5} + 16 T^{6} - 16 T^{7} + 16 T^{8} )^{2}( 1 + 2 T + 2 T^{3} + 9 T^{4} + 4 T^{5} + 16 T^{7} + 16 T^{8} )^{2}$$)
$3$ ($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{4}$$)
$5$ 1
$7$ ($$1 - 94 T^{4} + 2401 T^{8}$$)($$1 - 8 T + 32 T^{2} - 88 T^{3} + 196 T^{4} - 248 T^{5} - 416 T^{6} + 2840 T^{7} - 8634 T^{8} + 19880 T^{9} - 20384 T^{10} - 85064 T^{11} + 470596 T^{12} - 1479016 T^{13} + 3764768 T^{14} - 6588344 T^{15} + 5764801 T^{16}$$)
$11$ ($$( 1 + 11 T^{2} )^{8}$$)($$( 1 + 4 T + 32 T^{2} + 68 T^{3} + 402 T^{4} + 748 T^{5} + 3872 T^{6} + 5324 T^{7} + 14641 T^{8} )^{4}$$)
$13$ ($$( 1 - 24 T^{2} + 169 T^{4} )^{2}( 1 + 24 T^{2} + 169 T^{4} )^{2}$$)($$1 + 424 T^{4} + 47004 T^{8} - 12160488 T^{12} - 4129271418 T^{16} - 347315697768 T^{20} + 38342606809884 T^{24} + 9878388091931944 T^{28} + 665416609183179841 T^{32}$$)
$17$ ($$( 1 + 289 T^{4} )^{4}$$)($$1 + 120 T^{4} + 166556 T^{8} - 9625784 T^{12} + 12237871174 T^{16} - 803955105464 T^{20} + 1161854256343196 T^{24} + 69914668467571320 T^{28} + 48661191875666868481 T^{32}$$)
$19$ ($$( 1 - 10 T^{2} + 361 T^{4} )^{4}$$)($$( 1 + 48 T^{2} + 1524 T^{4} + 40016 T^{6} + 818246 T^{8} + 14445776 T^{10} + 198609204 T^{12} + 2258202288 T^{14} + 16983563041 T^{16} )^{2}$$)
$23$ ($$( 1 + 98 T^{4} + 279841 T^{8} )^{2}$$)($$( 1 - 20 T + 200 T^{2} - 1516 T^{3} + 10388 T^{4} - 65340 T^{5} + 378328 T^{6} - 2076676 T^{7} + 10539814 T^{8} - 47763548 T^{9} + 200135512 T^{10} - 794991780 T^{11} + 2906988308 T^{12} - 9757495988 T^{13} + 29607177800 T^{14} - 68096508940 T^{15} + 78310985281 T^{16} )^{2}$$)
$29$ ($$( 1 - 22 T^{2} + 841 T^{4} )^{4}$$)($$( 1 - 184 T^{2} + 15868 T^{4} - 835400 T^{6} + 29324070 T^{8} - 702571400 T^{10} + 11223134908 T^{12} - 109447491064 T^{14} + 500246412961 T^{16} )^{2}$$)
$31$ ($$( 1 - 14 T^{2} + 961 T^{4} )^{4}$$)($$( 1 - 128 T^{2} + 9396 T^{4} - 463040 T^{6} + 16703398 T^{8} - 444981440 T^{10} + 8677403316 T^{12} - 113600471168 T^{14} + 852891037441 T^{16} )^{2}$$)
$37$ ($$( 1 - 2062 T^{4} + 1874161 T^{8} )^{2}$$)($$( 1 + 16 T + 128 T^{2} + 944 T^{3} + 8860 T^{4} + 73552 T^{5} + 488320 T^{6} + 3175280 T^{7} + 20212134 T^{8} + 117485360 T^{9} + 668510080 T^{10} + 3725629456 T^{11} + 16605066460 T^{12} + 65460695408 T^{13} + 328412980352 T^{14} + 1518910034128 T^{15} + 3512479453921 T^{16} )^{2}$$)
$41$ ($$( 1 - 41 T^{2} )^{8}$$)($$( 1 - 144 T^{2} + 12956 T^{4} - 823792 T^{6} + 38320198 T^{8} - 1384794352 T^{10} + 36610559516 T^{12} - 684015010704 T^{14} + 7984925229121 T^{16} )^{2}$$)
$43$ ($$( 1 + 1778 T^{4} + 3418801 T^{8} )^{2}$$)($$( 1 - 8 T + 32 T^{2} - 280 T^{3} + 2788 T^{4} - 14232 T^{5} + 63840 T^{6} - 569416 T^{7} + 5017638 T^{8} - 24484888 T^{9} + 118040160 T^{10} - 1131543624 T^{11} + 9531617188 T^{12} - 41162364040 T^{13} + 202283617568 T^{14} - 2174548888856 T^{15} + 11688200277601 T^{16} )^{2}$$)
$47$ ($$( 1 - 1918 T^{4} + 4879681 T^{8} )^{2}$$)($$1 + 3784 T^{4} + 1124764 T^{8} + 9138019192 T^{12} + 63538455194182 T^{16} + 44590618628837752 T^{20} + 26782078030828949404 T^{24} +$$$$43\!\cdots\!44$$$$T^{28} +$$$$56\!\cdots\!21$$$$T^{32}$$)
$53$ ($$( 1 + 2809 T^{4} )^{4}$$)($$( 1 + 12 T + 72 T^{2} + 572 T^{3} + 1780 T^{4} - 1436 T^{5} + 18200 T^{6} + 441076 T^{7} + 6254598 T^{8} + 23377028 T^{9} + 51123800 T^{10} - 213787372 T^{11} + 14045056180 T^{12} + 239207821996 T^{13} + 1595834001288 T^{14} + 14096533678044 T^{15} + 62259690411361 T^{16} )^{2}$$)
$59$ ($$( 1 - 74 T^{2} + 3481 T^{4} )^{4}$$)($$( 1 + 312 T^{2} + 49660 T^{4} + 5040712 T^{6} + 354176614 T^{8} + 17546718472 T^{10} + 601748147260 T^{12} + 13160326495992 T^{14} + 146830437604321 T^{16} )^{2}$$)
$61$ ($$( 1 - 61 T^{2} )^{8}$$)($$( 1 - 200 T^{2} + 17532 T^{4} - 857912 T^{6} + 37932838 T^{8} - 3192290552 T^{10} + 242745284412 T^{12} - 10304074872200 T^{14} + 191707312997281 T^{16} )^{2}$$)
$67$ ($$( 1 - 8302 T^{4} + 20151121 T^{8} )^{2}$$)($$( 1 - 16 T + 128 T^{2} - 1424 T^{3} + 22436 T^{4} - 211280 T^{5} + 1522560 T^{6} - 15870032 T^{7} + 163564774 T^{8} - 1063292144 T^{9} + 6834771840 T^{10} - 63545206640 T^{11} + 452110550756 T^{12} - 1922578152368 T^{13} + 11578672917632 T^{14} - 96971385685168 T^{15} + 406067677556641 T^{16} )^{2}$$)
$71$ ($$( 1 + 12 T + 71 T^{2} )^{8}$$)($$( 1 - 16 T + 232 T^{2} - 2376 T^{3} + 21730 T^{4} - 168696 T^{5} + 1169512 T^{6} - 5726576 T^{7} + 25411681 T^{8} )^{4}$$)
$73$ ($$( 1 + 6242 T^{4} + 28398241 T^{8} )^{2}$$)($$1 - 15256 T^{4} + 80862300 T^{8} + 94053698264 T^{12} - 2362018367550906 T^{16} + 2670959590242353624 T^{20} +$$$$65\!\cdots\!00$$$$T^{24} -$$$$34\!\cdots\!76$$$$T^{28} +$$$$65\!\cdots\!61$$$$T^{32}$$)
$79$ ($$( 1 - 94 T^{2} + 6241 T^{4} )^{4}$$)($$( 1 - 312 T^{2} + 58396 T^{4} - 7293320 T^{6} + 672141766 T^{8} - 45517610120 T^{10} + 2274528930076 T^{12} - 75843286122552 T^{14} + 1517108809906561 T^{16} )^{2}$$)
$83$ ($$( 1 - 13294 T^{4} + 47458321 T^{8} )^{2}$$)($$1 + 5000 T^{4} + 95818588 T^{8} + 596752860728 T^{12} + 4571727903671302 T^{16} + 28320888822097717688 T^{20} +$$$$21\!\cdots\!08$$$$T^{24} +$$$$53\!\cdots\!00$$$$T^{28} +$$$$50\!\cdots\!81$$$$T^{32}$$)
$89$ ($$( 1 - 14 T^{2} + 7921 T^{4} )^{4}$$)($$( 1 + 576 T^{2} + 155068 T^{4} + 25338304 T^{6} + 2740378246 T^{8} + 200704705984 T^{10} + 9729313827388 T^{12} + 286261223593536 T^{14} + 3936588805702081 T^{16} )^{2}$$)
$97$ ($$( 1 + 12866 T^{4} + 88529281 T^{8} )^{2}$$)($$1 - 55064 T^{4} + 1465436892 T^{8} - 24336256217256 T^{12} + 274732504520067270 T^{16} -$$$$21\!\cdots\!36$$$$T^{20} +$$$$11\!\cdots\!12$$$$T^{24} -$$$$38\!\cdots\!24$$$$T^{28} +$$$$61\!\cdots\!21$$$$T^{32}$$)