# Properties

 Label 18.4.c Level 18 Weight 4 Character orbit c Rep. character $$\chi_{18}(7,\cdot)$$ Character field $$\Q(\zeta_{3})$$ Dimension 6 Newform subspaces 2 Sturm bound 12 Trace bound 1

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## Defining parameters

 Level: $$N$$ $$=$$ $$18 = 2 \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 18.c (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$9$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$2$$ Sturm bound: $$12$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(18, [\chi])$$.

Total New Old
Modular forms 22 6 16
Cusp forms 14 6 8
Eisenstein series 8 0 8

## Trace form

 $$6q + 2q^{2} + 3q^{3} - 12q^{4} + 18q^{5} - 18q^{6} + 12q^{7} - 16q^{8} - 105q^{9} + O(q^{10})$$ $$6q + 2q^{2} + 3q^{3} - 12q^{4} + 18q^{5} - 18q^{6} + 12q^{7} - 16q^{8} - 105q^{9} + 39q^{11} + 24q^{12} - 24q^{13} + 100q^{14} + 90q^{15} - 48q^{16} - 78q^{17} + 156q^{18} + 210q^{19} + 72q^{20} - 36q^{21} - 18q^{22} - 264q^{23} - 24q^{24} - 219q^{25} - 392q^{26} - 96q^{28} - 348q^{29} - 288q^{30} - 6q^{31} + 32q^{32} + 765q^{33} + 90q^{34} + 1332q^{35} + 516q^{36} + 192q^{37} + 322q^{38} - 582q^{39} - 207q^{41} - 816q^{42} + 129q^{43} - 312q^{44} - 702q^{45} + 504q^{46} - 660q^{47} - 144q^{48} - 585q^{49} + 614q^{50} - 153q^{51} - 96q^{52} + 528q^{53} + 954q^{54} - 1404q^{55} + 400q^{56} + 987q^{57} + 252q^{58} - 327q^{59} - 936q^{60} + 858q^{61} - 1664q^{62} - 1794q^{63} + 384q^{64} + 414q^{65} + 288q^{66} + 1587q^{67} + 156q^{68} + 1494q^{69} + 216q^{70} - 312q^{71} - 24q^{72} - 258q^{73} + 856q^{74} + 3231q^{75} - 420q^{76} + 708q^{77} + 132q^{78} - 1482q^{79} - 576q^{80} + 315q^{81} - 2916q^{82} - 138q^{83} + 744q^{84} + 108q^{85} - 86q^{86} - 3204q^{87} - 72q^{88} - 3084q^{89} - 432q^{90} + 2508q^{91} - 1056q^{92} - 2634q^{93} + 612q^{94} - 2016q^{95} + 384q^{96} + 1029q^{97} + 2604q^{98} + 1152q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(18, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
18.4.c.a $$2$$ $$1.062$$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$0$$ $$9$$ $$31$$ $$q-2\zeta_{6}q^{2}+(3-6\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots$$
18.4.c.b $$4$$ $$1.062$$ $$\Q(\sqrt{-3}, \sqrt{-35})$$ None $$4$$ $$3$$ $$9$$ $$-19$$ $$q-2\beta _{1}q^{2}+(-\beta _{1}-\beta _{3})q^{3}+(-4-4\beta _{1}+\cdots)q^{4}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(18, [\chi])$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(18, [\chi]) \cong$$ $$S_{4}^{\mathrm{new}}(9, [\chi])$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 + 2 T + 4 T^{2}$$)($$( 1 - 2 T + 4 T^{2} )^{2}$$)
$3$ ($$1 + 27 T^{2}$$)($$1 - 3 T + 30 T^{2} - 81 T^{3} + 729 T^{4}$$)
$5$ ($$1 - 9 T - 44 T^{2} - 1125 T^{3} + 15625 T^{4}$$)($$1 - 9 T + 47 T^{2} + 1944 T^{3} - 24594 T^{4} + 243000 T^{5} + 734375 T^{6} - 17578125 T^{7} + 244140625 T^{8}$$)
$7$ ($$1 - 31 T + 618 T^{2} - 10633 T^{3} + 117649 T^{4}$$)($$1 + 19 T - 179 T^{2} - 2774 T^{3} + 50128 T^{4} - 951482 T^{5} - 21059171 T^{6} + 766718533 T^{7} + 13841287201 T^{8}$$)
$11$ ($$1 - 15 T - 1106 T^{2} - 19965 T^{3} + 1771561 T^{4}$$)($$1 - 24 T - 1285 T^{2} + 19224 T^{3} + 925104 T^{4} + 25587144 T^{5} - 2276455885 T^{6} - 56590744584 T^{7} + 3138428376721 T^{8}$$)
$13$ ($$1 - 37 T - 828 T^{2} - 81289 T^{3} + 4826809 T^{4}$$)($$1 + 61 T - 1367 T^{2} + 42334 T^{3} + 12885898 T^{4} + 93007798 T^{5} - 6598247903 T^{6} + 646874461753 T^{7} + 23298085122481 T^{8}$$)
$17$ ($$( 1 + 42 T + 4913 T^{2} )^{2}$$)($$( 1 - 3 T + 9592 T^{2} - 14739 T^{3} + 24137569 T^{4} )^{2}$$)
$19$ ($$( 1 + 28 T + 6859 T^{2} )^{2}$$)($$( 1 - 133 T + 12234 T^{2} - 912247 T^{3} + 47045881 T^{4} )^{2}$$)
$23$ ($$1 + 195 T + 25858 T^{2} + 2372565 T^{3} + 148035889 T^{4}$$)($$1 + 69 T - 20527 T^{2} + 65826 T^{3} + 433519968 T^{4} + 800904942 T^{5} - 3038732693503 T^{6} + 124279533640947 T^{7} + 21914624432020321 T^{8}$$)
$29$ ($$1 + 111 T - 12068 T^{2} + 2707179 T^{3} + 594823321 T^{4}$$)($$1 + 237 T + 4925 T^{2} + 584442 T^{3} + 661218474 T^{4} + 14253955938 T^{5} + 2929504855925 T^{6} + 3438193596280953 T^{7} + 353814783205469041 T^{8}$$)
$31$ ($$1 - 205 T + 12234 T^{2} - 6107155 T^{3} + 887503681 T^{4}$$)($$1 + 211 T - 24065 T^{2} + 1899844 T^{3} + 2490210604 T^{4} + 56598252604 T^{5} - 21357776083265 T^{6} + 5578760275901581 T^{7} + 787662783788549761 T^{8}$$)
$37$ ($$( 1 + 166 T + 50653 T^{2} )^{2}$$)($$( 1 - 262 T + 72162 T^{2} - 13271086 T^{3} + 2565726409 T^{4} )^{2}$$)
$41$ ($$1 - 261 T - 800 T^{2} - 17988381 T^{3} + 4750104241 T^{4}$$)($$1 + 468 T + 27371 T^{2} + 25183548 T^{3} + 16885415064 T^{4} + 1735675311708 T^{5} + 130015103180411 T^{6} + 153214745296373748 T^{7} + 22563490300366186081 T^{8}$$)
$43$ ($$1 - 43 T - 77658 T^{2} - 3418801 T^{3} + 6321363049 T^{4}$$)($$1 - 86 T - 17387 T^{2} + 11543866 T^{3} - 6295199732 T^{4} + 917818154062 T^{5} - 109909539332963 T^{6} - 43222964626568498 T^{7} + 39959630797262576401 T^{8}$$)
$47$ ($$1 + 177 T - 72494 T^{2} + 18376671 T^{3} + 10779215329 T^{4}$$)($$1 + 483 T + 20477 T^{2} + 2495178 T^{3} + 10288968168 T^{4} + 259056865494 T^{5} + 220725992291933 T^{6} + 540540018508636461 T^{7} +$$$$11\!\cdots\!41$$$$T^{8}$$)
$53$ ($$( 1 - 114 T + 148877 T^{2} )^{2}$$)($$( 1 - 150 T + 257074 T^{2} - 22331550 T^{3} + 22164361129 T^{4} )^{2}$$)
$59$ ($$1 + 159 T - 180098 T^{2} + 32655261 T^{3} + 42180533641 T^{4}$$)($$1 + 168 T - 388645 T^{2} + 1026648 T^{3} + 125802612624 T^{4} + 210851939592 T^{5} - 16393253496906445 T^{6} + 1455383297534029752 T^{7} +$$$$17\!\cdots\!81$$$$T^{8}$$)
$61$ ($$1 + 191 T - 190500 T^{2} + 43353371 T^{3} + 51520374361 T^{4}$$)($$1 - 1049 T + 424495 T^{2} - 232819256 T^{3} + 155558427094 T^{4} - 52845547546136 T^{5} + 21870141314372695 T^{6} - 12267159251383013909 T^{7} +$$$$26\!\cdots\!21$$$$T^{8}$$)
$67$ ($$1 - 421 T - 123522 T^{2} - 126621223 T^{3} + 90458382169 T^{4}$$)($$1 - 1166 T + 452161 T^{2} - 356643254 T^{3} + 324003162628 T^{4} - 107265095002802 T^{5} + 40901752539917209 T^{6} - 31722819106079908202 T^{7} +$$$$81\!\cdots\!61$$$$T^{8}$$)
$71$ ($$( 1 - 156 T + 357911 T^{2} )^{2}$$)($$( 1 + 312 T + 498238 T^{2} + 111668232 T^{3} + 128100283921 T^{4} )^{2}$$)
$73$ ($$( 1 - 182 T + 389017 T^{2} )^{2}$$)($$( 1 + 311 T + 698028 T^{2} + 120984287 T^{3} + 151334226289 T^{4} )^{2}$$)
$79$ ($$1 + 1133 T + 790650 T^{2} + 558613187 T^{3} + 243087455521 T^{4}$$)($$1 + 349 T - 854801 T^{2} - 3307124 T^{3} + 650611367644 T^{4} - 1630541109836 T^{5} - 207791400066806321 T^{6} + 41828206997933793331 T^{7} +$$$$59\!\cdots\!41$$$$T^{8}$$)
$83$ ($$1 - 1083 T + 601102 T^{2} - 619245321 T^{3} + 326940373369 T^{4}$$)($$1 + 1221 T + 78743 T^{2} + 327867804 T^{3} + 814636885368 T^{4} + 187470548045748 T^{5} + 25744265820195167 T^{6} +$$$$22\!\cdots\!63$$$$T^{7} +$$$$10\!\cdots\!61$$$$T^{8}$$)
$89$ ($$( 1 + 1050 T + 704969 T^{2} )^{2}$$)($$( 1 + 492 T + 1092454 T^{2} + 346844748 T^{3} + 496981290961 T^{4} )^{2}$$)
$97$ ($$1 - 901 T - 100872 T^{2} - 822318373 T^{3} + 832972004929 T^{4}$$)($$1 - 128 T - 1698713 T^{2} + 14111872 T^{3} + 2093632480048 T^{4} + 12879524553856 T^{5} - 1414980373408956377 T^{6} - 97309575507784347776 T^{7} +$$$$69\!\cdots\!41$$$$T^{8}$$)
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