# Properties

 Label 20.4.e Level 20 Weight 4 Character orbit e Rep. character $$\chi_{20}(3,\cdot)$$ Character field $$\Q(\zeta_{4})$$ Dimension 14 Newform subspaces 2 Sturm bound 12 Trace bound 1

# Related objects

## Defining parameters

 Level: $$N$$ = $$20 = 2^{2} \cdot 5$$ Weight: $$k$$ = $$4$$ Character orbit: $$[\chi]$$ = 20.e (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$20$$ Character field: $$\Q(i)$$ Newform subspaces: $$2$$ Sturm bound: $$12$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(20, [\chi])$$.

Total New Old
Modular forms 22 22 0
Cusp forms 14 14 0
Eisenstein series 8 8 0

## Trace form

 $$14q - 2q^{2} - 4q^{5} + 8q^{6} - 44q^{8} + O(q^{10})$$ $$14q - 2q^{2} - 4q^{5} + 8q^{6} - 44q^{8} - 74q^{10} - 80q^{12} + 42q^{13} + 184q^{16} - 134q^{17} + 306q^{18} + 316q^{20} - 144q^{21} + 360q^{22} + 106q^{25} - 460q^{26} - 880q^{28} - 1240q^{30} - 632q^{32} + 80q^{33} + 892q^{36} + 326q^{37} + 1600q^{38} + 1836q^{40} + 288q^{41} + 1160q^{42} + 586q^{45} - 1432q^{46} - 2720q^{48} - 2214q^{50} - 1524q^{52} - 698q^{53} + 2048q^{56} - 960q^{57} + 2712q^{58} + 3280q^{60} - 1832q^{61} + 2440q^{62} - 1778q^{65} - 1680q^{66} - 2428q^{68} - 3040q^{70} - 2172q^{72} + 1942q^{73} + 800q^{76} + 3120q^{77} + 3720q^{78} + 2096q^{80} + 4530q^{81} + 536q^{82} + 2282q^{85} - 2552q^{86} - 2400q^{88} - 2154q^{90} - 1840q^{92} - 3280q^{93} + 1088q^{96} - 5994q^{97} + 326q^{98} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(20, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
20.4.e.a $$2$$ $$1.180$$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-1})$$ $$4$$ $$0$$ $$-4$$ $$0$$ $$q+(2+2i)q^{2}+8iq^{4}+(-2-11i)q^{5}+\cdots$$
20.4.e.b $$12$$ $$1.180$$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$-6$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}-\beta _{5})q^{2}-\beta _{9}q^{3}+(2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 - 4 T + 8 T^{2}$$)($$1 + 6 T + 18 T^{2} + 40 T^{3} - 256 T^{5} - 736 T^{6} - 2048 T^{7} + 20480 T^{9} + 73728 T^{10} + 196608 T^{11} + 262144 T^{12}$$)
$3$ ($$1 + 729 T^{4}$$)($$1 - 2226 T^{4} + 2588175 T^{8} - 2136402140 T^{12} + 1375462310175 T^{16} - 628688148206706 T^{20} + 150094635296999121 T^{24}$$)
$5$ ($$1 + 4 T + 125 T^{2}$$)($$( 1 - 85 T^{2} - 400 T^{3} - 10625 T^{4} + 1953125 T^{6} )^{2}$$)
$7$ ($$1 + 117649 T^{4}$$)($$1 - 163746 T^{4} - 12219131425 T^{8} + 4249089576371140 T^{12} -$$$$16\!\cdots\!25$$$$T^{16} -$$$$31\!\cdots\!46$$$$T^{20} +$$$$26\!\cdots\!01$$$$T^{24}$$)
$11$ ($$( 1 - 1331 T^{2} )^{2}$$)($$( 1 - 4986 T^{2} + 12380375 T^{4} - 19918383340 T^{6} + 21932589515375 T^{8} - 15648203886330906 T^{10} + 5559917313492231481 T^{12} )^{2}$$)
$13$ ($$( 1 - 18 T + 2197 T^{2} )( 1 + 92 T + 2197 T^{2} )$$)($$( 1 - 58 T + 1682 T^{2} - 72450 T^{3} + 7507335 T^{4} - 469328492 T^{5} + 17218216316 T^{6} - 1031114696924 T^{7} + 36236472144015 T^{8} - 768295979573850 T^{9} + 39187379176013042 T^{10} - 2968781794817263906 T^{11} +$$$$11\!\cdots\!29$$$$T^{12} )^{2}$$)
$17$ ($$( 1 - 104 T + 4913 T^{2} )( 1 - 94 T + 4913 T^{2} )$$)($$( 1 + 166 T + 13778 T^{2} + 1478470 T^{3} + 180009375 T^{4} + 13146146484 T^{5} + 795027918044 T^{6} + 64587017675892 T^{7} + 4344988709709375 T^{8} + 175328617764519590 T^{9} + 8027369184551647058 T^{10} +$$$$47\!\cdots\!38$$$$T^{11} +$$$$14\!\cdots\!09$$$$T^{12} )^{2}$$)
$19$ ($$( 1 + 6859 T^{2} )^{2}$$)($$( 1 + 16994 T^{2} + 156289815 T^{4} + 1042328090300 T^{6} + 7352792038002015 T^{8} + 37613073734610340034 T^{10} +$$$$10\!\cdots\!41$$$$T^{12} )^{2}$$)
$23$ ($$1 + 148035889 T^{4}$$)($$1 + 86914334 T^{4} + 54372416869500575 T^{8} +$$$$33\!\cdots\!60$$$$T^{12} +$$$$11\!\cdots\!75$$$$T^{16} +$$$$41\!\cdots\!94$$$$T^{20} +$$$$10\!\cdots\!61$$$$T^{24}$$)
$29$ ($$( 1 - 130 T + 24389 T^{2} )( 1 + 130 T + 24389 T^{2} )$$)($$( 1 - 80686 T^{2} + 2793486135 T^{4} - 69052346800420 T^{6} + 1661630699988154335 T^{8} -$$$$28\!\cdots\!26$$$$T^{10} +$$$$21\!\cdots\!61$$$$T^{12} )^{2}$$)
$31$ ($$( 1 - 29791 T^{2} )^{2}$$)($$( 1 - 103106 T^{2} + 5864634815 T^{4} - 214293224466300 T^{6} + 5204884986033254015 T^{8} -$$$$81\!\cdots\!66$$$$T^{10} +$$$$69\!\cdots\!41$$$$T^{12} )^{2}$$)
$37$ ($$( 1 - 214 T + 50653 T^{2} )( 1 + 396 T + 50653 T^{2} )$$)($$( 1 - 254 T + 32258 T^{2} + 4711770 T^{3} - 385957065 T^{4} - 961296674116 T^{5} + 267719946494684 T^{6} - 48692560433997748 T^{7} - 990260234410629585 T^{8} +$$$$61\!\cdots\!90$$$$T^{9} +$$$$21\!\cdots\!98$$$$T^{10} -$$$$84\!\cdots\!22$$$$T^{11} +$$$$16\!\cdots\!29$$$$T^{12} )^{2}$$)
$41$ ($$( 1 - 472 T + 68921 T^{2} )^{2}$$)($$( 1 + 164 T + 188335 T^{2} + 20815080 T^{3} + 12980236535 T^{4} + 779017095524 T^{5} + 327381934393961 T^{6} )^{4}$$)
$43$ ($$1 + 6321363049 T^{4}$$)($$1 + 9995711534 T^{4} - 4601157531678236945 T^{8} -$$$$44\!\cdots\!80$$$$T^{12} -$$$$18\!\cdots\!45$$$$T^{16} +$$$$15\!\cdots\!34$$$$T^{20} +$$$$63\!\cdots\!01$$$$T^{24}$$)
$47$ ($$1 + 10779215329 T^{4}$$)($$1 - 14480422466 T^{4} +$$$$18\!\cdots\!35$$$$T^{8} -$$$$27\!\cdots\!80$$$$T^{12} +$$$$21\!\cdots\!35$$$$T^{16} -$$$$19\!\cdots\!46$$$$T^{20} +$$$$15\!\cdots\!21$$$$T^{24}$$)
$53$ ($$( 1 - 518 T + 148877 T^{2} )( 1 + 572 T + 148877 T^{2} )$$)($$( 1 + 322 T + 51842 T^{2} + 24976890 T^{3} + 1487618135 T^{4} + 291824210108 T^{5} + 328768813336156 T^{6} + 43445912928248716 T^{7} + 32972105566189474415 T^{8} +$$$$82\!\cdots\!70$$$$T^{9} +$$$$25\!\cdots\!22$$$$T^{10} +$$$$23\!\cdots\!54$$$$T^{11} +$$$$10\!\cdots\!89$$$$T^{12} )^{2}$$)
$59$ ($$( 1 + 205379 T^{2} )^{2}$$)($$( 1 + 543794 T^{2} + 146689289095 T^{4} + 30962597079669980 T^{6} +$$$$61\!\cdots\!95$$$$T^{8} +$$$$96\!\cdots\!14$$$$T^{10} +$$$$75\!\cdots\!21$$$$T^{12} )^{2}$$)
$61$ ($$( 1 + 468 T + 226981 T^{2} )^{2}$$)($$( 1 + 224 T + 625475 T^{2} + 89988720 T^{3} + 141970940975 T^{4} + 11540563856864 T^{5} + 11694146092834141 T^{6} )^{4}$$)
$67$ ($$1 + 90458382169 T^{4}$$)($$1 + 152118288974 T^{4} -$$$$67\!\cdots\!65$$$$T^{8} -$$$$22\!\cdots\!80$$$$T^{12} -$$$$55\!\cdots\!65$$$$T^{16} +$$$$10\!\cdots\!54$$$$T^{20} +$$$$54\!\cdots\!81$$$$T^{24}$$)
$71$ ($$( 1 - 357911 T^{2} )^{2}$$)($$( 1 - 1431506 T^{2} + 1032027741935 T^{4} - 460759845466007580 T^{6} +$$$$13\!\cdots\!35$$$$T^{8} -$$$$23\!\cdots\!46$$$$T^{10} +$$$$21\!\cdots\!61$$$$T^{12} )^{2}$$)
$73$ ($$( 1 - 1098 T + 389017 T^{2} )( 1 + 592 T + 389017 T^{2} )$$)($$( 1 - 718 T + 257762 T^{2} - 319815230 T^{3} + 531457295055 T^{4} - 223706294377092 T^{5} + 74772514744761596 T^{6} - 87025551519693198564 T^{7} +$$$$80\!\cdots\!95$$$$T^{8} -$$$$18\!\cdots\!90$$$$T^{9} +$$$$59\!\cdots\!02$$$$T^{10} -$$$$63\!\cdots\!26$$$$T^{11} +$$$$34\!\cdots\!69$$$$T^{12} )^{2}$$)
$79$ ($$( 1 + 493039 T^{2} )^{2}$$)($$( 1 + 574874 T^{2} + 335775077615 T^{4} + 287028695918457900 T^{6} +$$$$81\!\cdots\!15$$$$T^{8} +$$$$33\!\cdots\!34$$$$T^{10} +$$$$14\!\cdots\!61$$$$T^{12} )^{2}$$)
$83$ ($$1 + 326940373369 T^{4}$$)($$1 - 1040598079986 T^{4} +$$$$60\!\cdots\!75$$$$T^{8} -$$$$23\!\cdots\!40$$$$T^{12} +$$$$64\!\cdots\!75$$$$T^{16} -$$$$11\!\cdots\!06$$$$T^{20} +$$$$12\!\cdots\!81$$$$T^{24}$$)
$89$ ($$( 1 - 1670 T + 704969 T^{2} )( 1 + 1670 T + 704969 T^{2} )$$)($$( 1 - 2798646 T^{2} + 3790036251455 T^{4} - 3254227731213032180 T^{6} +$$$$18\!\cdots\!55$$$$T^{8} -$$$$69\!\cdots\!66$$$$T^{10} +$$$$12\!\cdots\!81$$$$T^{12} )^{2}$$)
$97$ ($$( 1 - 594 T + 912673 T^{2} )( 1 + 1816 T + 912673 T^{2} )$$)($$( 1 + 2386 T + 2846498 T^{2} + 2507112050 T^{3} + 1418732317695 T^{4} + 602523284624124 T^{5} + 542007267886579004 T^{6} +$$$$54\!\cdots\!52$$$$T^{7} +$$$$11\!\cdots\!55$$$$T^{8} +$$$$19\!\cdots\!50$$$$T^{9} +$$$$19\!\cdots\!18$$$$T^{10} +$$$$15\!\cdots\!98$$$$T^{11} +$$$$57\!\cdots\!89$$$$T^{12} )^{2}$$)