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Properties

Label 6.18.a
Level $6$
Weight $18$
Character orbit 6.a
Rep. character $\chi_{6}(1,\cdot)$
Character field $\Q$
Dimension $3$
Newform subspaces $3$
Sturm bound $18$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 6 = 2 \cdot 3 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 6.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(18\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{18}(\Gamma_0(6))\).

Total New Old
Modular forms 19 3 16
Cusp forms 15 3 12
Eisenstein series 4 0 4

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(3\)FrickeDim
\(+\)\(+\)$+$\(1\)
\(+\)\(-\)$-$\(1\)
\(-\)\(+\)$-$\(1\)
Plus space\(+\)\(1\)
Minus space\(-\)\(2\)

Trace form

\( 3 q - 256 q^{2} - 6561 q^{3} + 196608 q^{4} + 373314 q^{5} - 1679616 q^{6} + 20293512 q^{7} - 16777216 q^{8} + 129140163 q^{9} + O(q^{10}) \) \( 3 q - 256 q^{2} - 6561 q^{3} + 196608 q^{4} + 373314 q^{5} - 1679616 q^{6} + 20293512 q^{7} - 16777216 q^{8} + 129140163 q^{9} - 197789184 q^{10} + 784792212 q^{11} - 429981696 q^{12} + 1602597066 q^{13} + 7584004096 q^{14} - 3396537846 q^{15} + 12884901888 q^{16} + 63064451862 q^{17} - 11019960576 q^{18} - 112465967988 q^{19} + 24465506304 q^{20} - 246522226680 q^{21} - 136621919232 q^{22} + 502204534536 q^{23} - 110075314176 q^{24} - 1827528895779 q^{25} + 1754059256320 q^{26} - 282429536481 q^{27} + 1329955602432 q^{28} - 3268241365782 q^{29} + 1540184357376 q^{30} + 4428127804848 q^{31} - 1099511627776 q^{32} + 10063371421188 q^{33} + 2058013011456 q^{34} - 1795311930576 q^{35} + 8463329722368 q^{36} - 34265998294158 q^{37} - 4158262154240 q^{38} + 26240907525138 q^{39} - 12962311962624 q^{40} - 49334969807970 q^{41} - 20734268295168 q^{42} - 78330110674764 q^{43} + 51432142405632 q^{44} + 16069943603394 q^{45} - 254423840495616 q^{46} + 609853163061552 q^{47} - 28179280429056 q^{48} + 15522206736555 q^{49} + 97630780039424 q^{50} + 18993259062990 q^{51} + 105027801317376 q^{52} - 201393768300174 q^{53} - 72301961339136 q^{54} - 431372192162760 q^{55} + 497025292435456 q^{56} + 813714677695260 q^{57} - 330891509767680 q^{58} - 3978739340006460 q^{59} - 222595504275456 q^{60} + 2545999496260602 q^{61} + 1093581265543168 q^{62} + 873569149174152 q^{63} + 844424930131968 q^{64} - 4546522505652132 q^{65} - 2997996227693568 q^{66} + 3306215142940092 q^{67} + 4132991917228032 q^{68} - 1075810753198296 q^{69} - 2091756079263744 q^{70} - 718407280036008 q^{71} - 722204136308736 q^{72} + 5920503858987486 q^{73} + 19406546444491264 q^{74} + 2047501942916481 q^{75} - 7370569678061568 q^{76} - 8870296929042336 q^{77} - 20917802780757504 q^{78} + 24057256930293696 q^{79} + 1603371421138944 q^{80} + 5559060566555523 q^{81} - 37354157530163712 q^{82} - 33050904413718324 q^{83} - 16156080647700480 q^{84} - 13005613799191836 q^{85} + 59034957775885312 q^{86} + 69098564553893010 q^{87} - 8953654098788352 q^{88} - 114944772828400434 q^{89} - 8514175820465664 q^{90} + 59742068442608496 q^{91} + 32912476375351296 q^{92} + 61239254144690448 q^{93} - 4534883594514432 q^{94} - 22335787055851896 q^{95} - 7213895789838336 q^{96} - 3753776454789018 q^{97} + 195877499937900288 q^{98} + 33782731392936852 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{18}^{\mathrm{new}}(\Gamma_0(6))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3
6.18.a.a 6.a 1.a $1$ $10.993$ \(\Q\) None \(-256\) \(-6561\) \(645150\) \(3974432\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2^{8}q^{2}-3^{8}q^{3}+2^{16}q^{4}+645150q^{5}+\cdots\)
6.18.a.b 6.a 1.a $1$ $10.993$ \(\Q\) None \(-256\) \(6561\) \(-72186\) \(-8640184\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2^{8}q^{2}+3^{8}q^{3}+2^{16}q^{4}-72186q^{5}+\cdots\)
6.18.a.c 6.a 1.a $1$ $10.993$ \(\Q\) None \(256\) \(-6561\) \(-199650\) \(24959264\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2^{8}q^{2}-3^{8}q^{3}+2^{16}q^{4}-199650q^{5}+\cdots\)

Decomposition of \(S_{18}^{\mathrm{old}}(\Gamma_0(6))\) into lower level spaces

\( S_{18}^{\mathrm{old}}(\Gamma_0(6)) \cong \) \(S_{18}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 2}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)