Properties

Label 6.20.a
Level $6$
Weight $20$
Character orbit 6.a
Rep. character $\chi_{6}(1,\cdot)$
Character field $\Q$
Dimension $3$
Newform subspaces $3$
Sturm bound $20$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 6 = 2 \cdot 3 \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 6.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(20\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{20}(\Gamma_0(6))\).

Total New Old
Modular forms 21 3 18
Cusp forms 17 3 14
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\(+\)\(2\)
Minus space\(-\)\(1\)

Trace form

\( 3 q - 512 q^{2} + 19683 q^{3} + 786432 q^{4} - 7628574 q^{5} + 10077696 q^{6} + 64347072 q^{7} - 134217728 q^{8} + 1162261467 q^{9} + O(q^{10}) \) \( 3 q - 512 q^{2} + 19683 q^{3} + 786432 q^{4} - 7628574 q^{5} + 10077696 q^{6} + 64347072 q^{7} - 134217728 q^{8} + 1162261467 q^{9} + 5906101248 q^{10} - 12855640236 q^{11} + 5159780352 q^{12} + 20525875026 q^{13} + 8514805760 q^{14} - 3220650558 q^{15} + 206158430208 q^{16} + 1034935995942 q^{17} - 198359290368 q^{18} + 1057940622924 q^{19} - 1999784902656 q^{20} + 7158557194272 q^{21} + 8544952903680 q^{22} - 28175760231240 q^{23} + 2641807540224 q^{24} - 5256831071499 q^{25} - 7301588306944 q^{26} + 7625597484987 q^{27} + 16868198842368 q^{28} - 171676134994614 q^{29} + 41020314264576 q^{30} + 121025813245512 q^{31} - 35184372088832 q^{32} + 40619893311540 q^{33} + 92356574837760 q^{34} - 377327901232896 q^{35} + 304679870005248 q^{36} + 125711159560314 q^{37} + 2026025774077952 q^{38} - 1927970413420590 q^{39} + 1548249005555712 q^{40} + 4814212681852638 q^{41} - 2849114131587072 q^{42} - 3877441185382572 q^{43} - 3370028954025984 q^{44} - 2955465869452686 q^{45} + 511015718596608 q^{46} - 12704048498601600 q^{47} + 1352605460594688 q^{48} + 19957550681979387 q^{49} - 12932442419482112 q^{50} - 222119977442778 q^{51} + 5380734982815744 q^{52} - 11554685772118398 q^{53} + 3904305912313344 q^{54} + 74363725456988472 q^{55} + 2232105241149440 q^{56} - 17341971938234388 q^{57} - 9514864659151872 q^{58} - 36975948398764140 q^{59} - 844274219876352 q^{60} + 199406327145194994 q^{61} - 193171537527033856 q^{62} + 24929374099958208 q^{63} + 54043195528445952 q^{64} + 29785233633771468 q^{65} + 17837688443848704 q^{66} - 642672834346919028 q^{67} + 271302261720219648 q^{68} - 500716074932355528 q^{69} + 274180424465498112 q^{70} + 1076825911627850184 q^{71} - 51998697814228992 q^{72} - 1202308545775776114 q^{73} - 378661798622362624 q^{74} + 98954653995482517 q^{75} + 277332786655789056 q^{76} - 74881769901720960 q^{77} + 1050257217671783424 q^{78} + 1304200218769695672 q^{79} - 524231613521854464 q^{80} + 450283905890997363 q^{81} - 490084912219051008 q^{82} + 1157924915196617676 q^{83} + 1876572817135239168 q^{84} - 204916816651584636 q^{85} - 4196099495418370048 q^{86} + 504075658446394218 q^{87} + 2240008133982289920 q^{88} + 3275559485230199742 q^{89} + 2288144633583670272 q^{90} - 16000961310728191104 q^{91} - 7386106490058178560 q^{92} - 5292328778139637656 q^{93} + 4246141011142754304 q^{94} + 15429846493999760520 q^{95} + 692533995824480256 q^{96} + 10720227564511910214 q^{97} - 20212049453449892352 q^{98} - 4980538426639195404 q^{99} + O(q^{100}) \)

Decomposition of \(S_{20}^{\mathrm{new}}(\Gamma_0(6))\) 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
6.20.a.a 6.a 1.a $1$ $13.729$ \(\Q\) None \(-512\) \(-19683\) \(-3732474\) \(-149672656\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2^{9}q^{2}-3^{9}q^{3}+2^{18}q^{4}-3732474q^{5}+\cdots\)
6.20.a.b 6.a 1.a $1$ $13.729$ \(\Q\) None \(-512\) \(19683\) \(-5849490\) \(173530952\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2^{9}q^{2}+3^{9}q^{3}+2^{18}q^{4}-5849490q^{5}+\cdots\)
6.20.a.c 6.a 1.a $1$ $13.729$ \(\Q\) None \(512\) \(19683\) \(1953390\) \(40488776\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2^{9}q^{2}+3^{9}q^{3}+2^{18}q^{4}+1953390q^{5}+\cdots\)

Decomposition of \(S_{20}^{\mathrm{old}}(\Gamma_0(6))\) into lower level spaces

\( S_{20}^{\mathrm{old}}(\Gamma_0(6)) \cong \) \(S_{20}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{20}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 2}\)\(\oplus\)\(S_{20}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)