Properties

Label 18.3.d
Level 18
Weight 3
Character orbit d
Rep. character \(\chi_{18}(5,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 4
Newform subspaces 1
Sturm bound 9
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 18.d (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 1 \)
Sturm bound: \(9\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(18, [\chi])\).

Total New Old
Modular forms 16 4 12
Cusp forms 8 4 4
Eisenstein series 8 0 8

Trace form

\( 4q + 4q^{4} - 18q^{5} - 12q^{6} + 2q^{7} + 12q^{9} + O(q^{10}) \) \( 4q + 4q^{4} - 18q^{5} - 12q^{6} + 2q^{7} + 12q^{9} + 18q^{11} + 12q^{12} - 10q^{13} + 36q^{14} + 18q^{15} - 8q^{16} - 24q^{18} - 40q^{19} - 36q^{20} - 42q^{21} - 12q^{22} + 18q^{23} + 4q^{25} + 8q^{28} + 18q^{29} + 72q^{30} + 38q^{31} + 54q^{33} + 24q^{34} + 12q^{36} + 128q^{37} - 72q^{38} - 102q^{39} - 126q^{41} - 48q^{42} - 46q^{43} - 54q^{45} - 24q^{46} + 54q^{47} + 24q^{48} - 12q^{49} - 72q^{51} + 20q^{52} + 36q^{54} - 108q^{55} + 72q^{56} + 144q^{57} + 24q^{58} + 126q^{59} - 36q^{60} + 62q^{61} + 222q^{63} - 32q^{64} + 90q^{65} - 72q^{66} - 106q^{67} + 72q^{68} + 18q^{69} - 108q^{70} - 96q^{72} - 208q^{73} + 72q^{74} - 12q^{75} - 40q^{76} - 90q^{77} + 144q^{78} + 14q^{79} - 252q^{81} + 144q^{82} - 378q^{83} - 144q^{84} + 108q^{85} - 108q^{86} - 54q^{87} + 24q^{88} + 412q^{91} + 36q^{92} + 222q^{93} + 84q^{94} + 180q^{95} + 48q^{96} + 14q^{97} + 126q^{99} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(18, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
18.3.d.a \(4\) \(0.490\) \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(0\) \(-18\) \(2\) \(q+\beta _{1}q^{2}+(1-2\beta _{1}-2\beta _{2}+\beta _{3})q^{3}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(18, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(18, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T^{2} + 4 T^{4} \)
$3$ \( 1 - 6 T^{2} + 81 T^{4} \)
$5$ \( ( 1 + 9 T + 52 T^{2} + 225 T^{3} + 625 T^{4} )^{2} \)
$7$ \( 1 - 2 T - 41 T^{2} + 106 T^{3} - 572 T^{4} + 5194 T^{5} - 98441 T^{6} - 235298 T^{7} + 5764801 T^{8} \)
$11$ \( 1 - 18 T + 359 T^{2} - 4518 T^{3} + 61428 T^{4} - 546678 T^{5} + 5256119 T^{6} - 31888098 T^{7} + 214358881 T^{8} \)
$13$ \( 1 + 10 T - 47 T^{2} - 1910 T^{3} - 23852 T^{4} - 322790 T^{5} - 1342367 T^{6} + 48268090 T^{7} + 815730721 T^{8} \)
$17$ \( 1 - 796 T^{2} + 294342 T^{4} - 66482716 T^{6} + 6975757441 T^{8} \)
$19$ \( ( 1 + 20 T + 606 T^{2} + 7220 T^{3} + 130321 T^{4} )^{2} \)
$23$ \( 1 - 18 T + 1175 T^{2} - 19206 T^{3} + 915780 T^{4} - 10159974 T^{5} + 328813175 T^{6} - 2664646002 T^{7} + 78310985281 T^{8} \)
$29$ \( 1 - 18 T + 1745 T^{2} - 29466 T^{3} + 2063316 T^{4} - 24780906 T^{5} + 1234205345 T^{6} - 10706819778 T^{7} + 500246412961 T^{8} \)
$31$ \( 1 - 38 T - 353 T^{2} + 4750 T^{3} + 918004 T^{4} + 4564750 T^{5} - 326002913 T^{6} - 33725139878 T^{7} + 852891037441 T^{8} \)
$37$ \( ( 1 - 64 T + 3546 T^{2} - 87616 T^{3} + 1874161 T^{4} )^{2} \)
$41$ \( 1 + 126 T + 9329 T^{2} + 508662 T^{3} + 22367460 T^{4} + 855060822 T^{5} + 26361524369 T^{6} + 598513134366 T^{7} + 7984925229121 T^{8} \)
$43$ \( 1 + 46 T - 1625 T^{2} + 1978 T^{3} + 6663796 T^{4} + 3657322 T^{5} - 5555551625 T^{6} + 290782700254 T^{7} + 11688200277601 T^{8} \)
$47$ \( 1 - 54 T + 4751 T^{2} - 204066 T^{3} + 11548308 T^{4} - 450781794 T^{5} + 23183364431 T^{6} - 582077627766 T^{7} + 23811286661761 T^{8} \)
$53$ \( 1 - 2236 T^{2} - 2409114 T^{4} - 17643115516 T^{6} + 62259690411361 T^{8} \)
$59$ \( 1 - 126 T + 10535 T^{2} - 660618 T^{3} + 33793140 T^{4} - 2299611258 T^{5} + 127656398135 T^{6} - 5314747238766 T^{7} + 146830437604321 T^{8} \)
$61$ \( 1 - 62 T - 2615 T^{2} + 60946 T^{3} + 13569316 T^{4} + 226780066 T^{5} - 36206874215 T^{6} - 3194263210382 T^{7} + 191707312997281 T^{8} \)
$67$ \( 1 + 106 T - 65 T^{2} + 246238 T^{3} + 57123076 T^{4} + 1105362382 T^{5} - 1309822865 T^{6} + 9588588509914 T^{7} + 406067677556641 T^{8} \)
$71$ \( 1 - 12460 T^{2} + 77194662 T^{4} - 316629545260 T^{6} + 645753531245761 T^{8} \)
$73$ \( ( 1 + 104 T + 11418 T^{2} + 554216 T^{3} + 28398241 T^{4} )^{2} \)
$79$ \( 1 - 14 T - 10985 T^{2} + 18214 T^{3} + 84841444 T^{4} + 113673574 T^{5} - 427866639785 T^{6} - 3403224377294 T^{7} + 1517108809906561 T^{8} \)
$83$ \( 1 + 378 T + 72863 T^{2} + 9538830 T^{3} + 917456196 T^{4} + 65712999870 T^{5} + 3457955643023 T^{6} + 123583461133482 T^{7} + 2252292232139041 T^{8} \)
$89$ \( 1 - 8860 T^{2} + 51019782 T^{4} - 555896255260 T^{6} + 3936588805702081 T^{8} \)
$97$ \( 1 - 14 T - 8087 T^{2} + 147490 T^{3} - 21765356 T^{4} + 1387733410 T^{5} - 715936295447 T^{6} - 11661608069006 T^{7} + 7837433594376961 T^{8} \)
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