# Properties

 Label 30.3 Level 30 Weight 3 Dimension 12 Nonzero newspaces 3 Newform subspaces 3 Sturm bound 144 Trace bound 3

## Defining parameters

 Level: $$N$$ = $$30 = 2 \cdot 3 \cdot 5$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$3$$ Newform subspaces: $$3$$ Sturm bound: $$144$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(30))$$.

Total New Old
Modular forms 64 12 52
Cusp forms 32 12 20
Eisenstein series 32 0 32

## Trace form

 $$12q + 4q^{2} + 4q^{3} - 8q^{6} - 8q^{7} - 8q^{8} - 40q^{9} + O(q^{10})$$ $$12q + 4q^{2} + 4q^{3} - 8q^{6} - 8q^{7} - 8q^{8} - 40q^{9} - 12q^{10} - 16q^{11} - 8q^{12} - 28q^{13} + 4q^{15} + 16q^{16} + 44q^{17} + 44q^{18} + 80q^{19} + 8q^{20} + 64q^{21} + 32q^{22} - 16q^{23} + 36q^{25} + 24q^{26} + 28q^{27} + 16q^{28} - 176q^{31} - 16q^{32} - 24q^{33} - 160q^{34} - 112q^{35} - 24q^{36} - 196q^{37} - 64q^{38} - 40q^{39} - 24q^{40} + 32q^{41} - 80q^{42} + 104q^{43} - 60q^{45} + 80q^{46} + 96q^{47} + 16q^{48} + 240q^{49} + 92q^{50} + 152q^{51} + 104q^{52} + 100q^{53} + 240q^{54} + 464q^{55} + 64q^{56} + 200q^{57} + 64q^{58} - 72q^{60} - 224q^{61} - 80q^{62} - 304q^{63} - 204q^{65} - 320q^{66} - 344q^{67} - 88q^{68} - 200q^{69} - 48q^{70} + 64q^{71} - 40q^{72} + 12q^{73} - 244q^{75} - 96q^{76} - 128q^{77} + 136q^{78} - 400q^{79} + 180q^{81} + 224q^{82} + 192q^{83} + 240q^{84} - 44q^{85} + 96q^{86} + 192q^{87} - 64q^{88} + 36q^{90} + 208q^{91} - 32q^{92} - 64q^{93} - 320q^{94} + 64q^{95} - 32q^{96} + 428q^{97} - 124q^{98} + 80q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(30))$$

We only show spaces with odd parity, since no modular forms exist when this condition is not satisfied. Within each space $$S_k^{\mathrm{new}}(N, \chi)$$ we list the newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
30.3.b $$\chi_{30}(29, \cdot)$$ 30.3.b.a 4 1
30.3.d $$\chi_{30}(11, \cdot)$$ 30.3.d.a 4 1
30.3.f $$\chi_{30}(7, \cdot)$$ 30.3.f.a 4 2

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(30))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(30)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$( 1 - 2 T^{2} )^{2}$$)($$( 1 + 2 T^{2} )^{2}$$)($$( 1 - 2 T + 2 T^{2} )^{2}$$)
$3$ ($$1 + 16 T^{2} + 81 T^{4}$$)($$1 - 4 T + 12 T^{2} - 36 T^{3} + 81 T^{4}$$)($$1 + 9 T^{4}$$)
$5$ ($$1 + 18 T^{2} + 625 T^{4}$$)($$( 1 + 5 T^{2} )^{2}$$)($$1 - 46 T^{2} + 625 T^{4}$$)
$7$ ($$( 1 - 64 T^{2} + 2401 T^{4} )^{2}$$)($$( 1 - 4 T + 12 T^{2} - 196 T^{3} + 2401 T^{4} )^{2}$$)($$1 + 16 T + 128 T^{2} + 528 T^{3} + 1922 T^{4} + 25872 T^{5} + 307328 T^{6} + 1882384 T^{7} + 5764801 T^{8}$$)
$11$ ($$( 1 + 30 T^{2} + 14641 T^{4} )^{2}$$)($$( 1 - 170 T^{2} + 14641 T^{4} )^{2}$$)($$( 1 + 8 T + 162 T^{2} + 968 T^{3} + 14641 T^{4} )^{2}$$)
$13$ ($$( 1 - 13 T )^{4}( 1 + 13 T )^{4}$$)($$( 1 + 10 T + 169 T^{2} )^{4}$$)($$1 - 12 T + 72 T^{2} + 60 T^{3} - 30226 T^{4} + 10140 T^{5} + 2056392 T^{6} - 57921708 T^{7} + 815730721 T^{8}$$)
$17$ ($$( 1 + 450 T^{2} + 83521 T^{4} )^{2}$$)($$1 - 220 T^{2} - 28218 T^{4} - 18374620 T^{6} + 6975757441 T^{8}$$)($$1 - 44 T + 968 T^{2} - 21252 T^{3} + 428942 T^{4} - 6141828 T^{5} + 80848328 T^{6} - 1062053036 T^{7} + 6975757441 T^{8}$$)
$19$ ($$( 1 - 12 T + 361 T^{2} )^{4}$$)($$( 1 - 16 T + 426 T^{2} - 5776 T^{3} + 130321 T^{4} )^{2}$$)($$1 - 740 T^{2} + 299238 T^{4} - 96437540 T^{6} + 16983563041 T^{8}$$)
$23$ ($$( 1 + 480 T^{2} + 279841 T^{4} )^{2}$$)($$1 - 1720 T^{2} + 1286322 T^{4} - 481326520 T^{6} + 78310985281 T^{8}$$)($$1 + 16 T + 128 T^{2} + 2064 T^{3} - 126718 T^{4} + 1091856 T^{5} + 35819648 T^{6} + 2368574224 T^{7} + 78310985281 T^{8}$$)
$29$ ($$( 1 - 29 T )^{4}( 1 + 29 T )^{4}$$)($$( 1 - 962 T^{2} + 707281 T^{4} )^{2}$$)($$1 - 2660 T^{2} + 3085158 T^{4} - 1881367460 T^{6} + 500246412961 T^{8}$$)
$31$ ($$( 1 + 32 T + 961 T^{2} )^{4}$$)($$( 1 - 8 T + 961 T^{2} )^{4}$$)($$( 1 + 40 T + 1938 T^{2} + 38440 T^{3} + 923521 T^{4} )^{2}$$)
$37$ ($$( 1 - 2194 T^{2} + 1874161 T^{4} )^{2}$$)($$( 1 + 44 T + 1782 T^{2} + 60236 T^{3} + 1874161 T^{4} )^{2}$$)($$( 1 + 54 T + 1458 T^{2} + 73926 T^{3} + 1874161 T^{4} )^{2}$$)
$41$ ($$( 1 - 30 T^{2} + 2825761 T^{4} )^{2}$$)($$1 - 4060 T^{2} + 8942982 T^{4} - 11472589660 T^{6} + 7984925229121 T^{8}$$)($$( 1 - 16 T + 3330 T^{2} - 26896 T^{3} + 2825761 T^{4} )^{2}$$)
$43$ ($$( 1 - 2032 T^{2} + 3418801 T^{4} )^{2}$$)($$( 1 - 28 T + 3084 T^{2} - 51772 T^{3} + 3418801 T^{4} )^{2}$$)($$1 - 48 T + 1152 T^{2} - 44976 T^{3} + 924194 T^{4} - 83160624 T^{5} + 3938458752 T^{6} - 303425426352 T^{7} + 11688200277601 T^{8}$$)
$47$ ($$( 1 + 3168 T^{2} + 4879681 T^{4} )^{2}$$)($$1 + 1064 T^{2} + 1942386 T^{4} + 5191980584 T^{6} + 23811286661761 T^{8}$$)($$1 - 96 T + 4608 T^{2} - 281184 T^{3} + 16639682 T^{4} - 621135456 T^{5} + 22485570048 T^{6} - 1034804671584 T^{7} + 23811286661761 T^{8}$$)
$53$ ($$( 1 + 1010 T^{2} + 7890481 T^{4} )^{2}$$)($$1 - 10300 T^{2} + 42096102 T^{4} - 81271954300 T^{6} + 62259690411361 T^{8}$$)($$1 - 100 T + 5000 T^{2} - 17100 T^{3} - 6900562 T^{4} - 48033900 T^{5} + 39452405000 T^{6} - 2216436112900 T^{7} + 62259690411361 T^{8}$$)
$59$ ($$( 1 - 6690 T^{2} + 12117361 T^{4} )^{2}$$)($$1 - 7300 T^{2} + 30092262 T^{4} - 88456735300 T^{6} + 146830437604321 T^{8}$$)($$( 1 - 6562 T^{2} + 12117361 T^{4} )^{2}$$)
$61$ ($$( 1 + 16 T + 3721 T^{2} )^{4}$$)($$( 1 + 32 T + 6258 T^{2} + 119072 T^{3} + 13845841 T^{4} )^{2}$$)($$( 1 + 48 T + 6482 T^{2} + 178608 T^{3} + 13845841 T^{4} )^{2}$$)
$67$ ($$( 1 - 8944 T^{2} + 20151121 T^{4} )^{2}$$)($$( 1 + 164 T + 14892 T^{2} + 736196 T^{3} + 20151121 T^{4} )^{2}$$)($$1 + 16 T + 128 T^{2} + 10128 T^{3} - 14067358 T^{4} + 45464592 T^{5} + 2579343488 T^{6} + 1447334114704 T^{7} + 406067677556641 T^{8}$$)
$71$ ($$( 1 - 71 T )^{4}( 1 + 71 T )^{4}$$)($$1 - 15124 T^{2} + 102823206 T^{4} - 384326263444 T^{6} + 645753531245761 T^{8}$$)($$( 1 - 32 T + 10242 T^{2} - 161312 T^{3} + 25411681 T^{4} )^{2}$$)
$73$ ($$( 1 + 2942 T^{2} + 28398241 T^{4} )^{2}$$)($$( 1 - 100 T + 11718 T^{2} - 532900 T^{3} + 28398241 T^{4} )^{2}$$)($$1 + 188 T + 17672 T^{2} + 1796340 T^{3} + 164736974 T^{4} + 9572695860 T^{5} + 501853714952 T^{6} + 28450834542332 T^{7} + 806460091894081 T^{8}$$)
$79$ ($$( 1 + 72 T + 6241 T^{2} )^{4}$$)($$( 1 + 56 T + 12906 T^{2} + 349496 T^{3} + 38950081 T^{4} )^{2}$$)($$1 + 7772 T^{2} + 83656134 T^{4} + 302720029532 T^{6} + 1517108809906561 T^{8}$$)
$83$ ($$( 1 + 11856 T^{2} + 47458321 T^{4} )^{2}$$)($$1 - 26872 T^{2} + 275326098 T^{4} - 1275300001912 T^{6} + 2252292232139041 T^{8}$$)($$1 - 192 T + 18432 T^{2} - 1755840 T^{3} + 162172514 T^{4} - 12095981760 T^{5} + 874751772672 T^{6} - 62772551686848 T^{7} + 2252292232139041 T^{8}$$)
$89$ ($$( 1 - 11490 T^{2} + 62742241 T^{4} )^{2}$$)($$1 - 27940 T^{2} + 317327622 T^{4} - 1753018213540 T^{6} + 3936588805702081 T^{8}$$)($$1 - 14468 T^{2} + 151051974 T^{4} - 907754742788 T^{6} + 3936588805702081 T^{8}$$)
$97$ ($$( 1 + 7838 T^{2} + 88529281 T^{4} )^{2}$$)($$( 1 - 148 T + 22854 T^{2} - 1392532 T^{3} + 88529281 T^{4} )^{2}$$)($$1 - 132 T + 8712 T^{2} - 895884 T^{3} + 85251854 T^{4} - 8429372556 T^{5} + 771267096072 T^{6} - 109952304650628 T^{7} + 7837433594376961 T^{8}$$)