Properties

Label 6.22.a
Level $6$
Weight $22$
Character orbit 6.a
Rep. character $\chi_{6}(1,\cdot)$
Character field $\Q$
Dimension $3$
Newform subspaces $3$
Sturm bound $22$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 6 = 2 \cdot 3 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 6.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(22\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{22}(\Gamma_0(6))\).

Total New Old
Modular forms 23 3 20
Cusp forms 19 3 16
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

\( 3 q + 1024 q^{2} + 59049 q^{3} + 3145728 q^{4} + 16153674 q^{5} - 60466176 q^{6} - 1636375008 q^{7} + 1073741824 q^{8} + 10460353203 q^{9} + O(q^{10}) \) \( 3 q + 1024 q^{2} + 59049 q^{3} + 3145728 q^{4} + 16153674 q^{5} - 60466176 q^{6} - 1636375008 q^{7} + 1073741824 q^{8} + 10460353203 q^{9} - 37617076224 q^{10} - 98395423908 q^{11} + 61917364224 q^{12} + 1099574688426 q^{13} - 2015853297664 q^{14} - 576003745026 q^{15} + 3298534883328 q^{16} + 23877926268102 q^{17} + 3570467226624 q^{18} - 11567072167068 q^{19} + 16938354868224 q^{20} - 39996769291200 q^{21} + 114034790043648 q^{22} + 340893203414856 q^{23} - 63403380965376 q^{24} - 23054276382099 q^{25} + 438673445795840 q^{26} + 205891132094649 q^{27} - 1715863560388608 q^{28} + 4030132752182658 q^{29} - 3787829463988224 q^{30} - 7075669005317352 q^{31} + 1125899906842624 q^{32} - 19469104089550092 q^{33} - 5441907977385984 q^{34} + 28933847452893504 q^{35} + 10968475320188928 q^{36} - 5837601526121118 q^{37} - 19155079287869440 q^{38} + 30012138157854078 q^{39} - 39444363318657024 q^{40} + 64202623305887310 q^{41} - 61045473892220928 q^{42} + 425104435306891356 q^{43} - 103175080019755008 q^{44} + 56324378522039274 q^{45} - 106337564428640256 q^{46} - 557743059082102848 q^{47} + 64925062108545024 q^{48} + 332159879552591115 q^{49} - 479240611205989376 q^{50} + 627334178863920210 q^{51} + 1152987628490981376 q^{52} - 2728003506188961654 q^{53} - 210832519264920576 q^{54} + 1262544774270568200 q^{55} - 2113775387451326464 q^{56} - 4057814241887276340 q^{57} - 366680730067200000 q^{58} + 5653138491641034780 q^{59} - 603983702944382976 q^{60} - 408089737914454038 q^{61} + 10620503418739417088 q^{62} - 5705686852080650208 q^{63} + 3458764513820540928 q^{64} + 1818329229000884748 q^{65} - 7253127250796187648 q^{66} + 30566343163574783652 q^{67} + 25037820414501322752 q^{68} - 19478918202179686056 q^{69} + 20631684004056170496 q^{70} - 69525201378432548808 q^{71} + 3743906242624487424 q^{72} - 33536250911384785794 q^{73} - 82718090089237129216 q^{74} + 35134083667603341351 q^{75} - 12128954264655495168 q^{76} + 71553913985152541184 q^{77} - 9851418861070546944 q^{78} + 66909092022784482216 q^{79} + 17761152394302849024 q^{80} + 36472996377170786403 q^{81} - 111647600565496584192 q^{82} - 384466770624937646364 q^{83} - 41939652356289331200 q^{84} + 410124662544577652244 q^{85} + 166165071545072095232 q^{86} + 320552290165541526630 q^{87} + 119574144004808245248 q^{88} + 130769230072423245966 q^{89} - 131162634589071181824 q^{90} - 705608677189118001984 q^{91} + 357452431663936045056 q^{92} - 6244696289600779032 q^{93} - 1265485229912200937472 q^{94} + 1480681379169368573304 q^{95} - 66483263599150104576 q^{96} - 11286510703838866938 q^{97} + 1427520150706407564288 q^{98} - 343083629212196859108 q^{99} + O(q^{100}) \)

Decomposition of \(S_{22}^{\mathrm{new}}(\Gamma_0(6))\) 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
6.22.a.a 6.a 1.a $1$ $16.769$ \(\Q\) None \(-1024\) \(59049\) \(26444550\) \(166115864\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2^{10}q^{2}+3^{10}q^{3}+2^{20}q^{4}+26444550q^{5}+\cdots\)
6.22.a.b 6.a 1.a $1$ $16.769$ \(\Q\) None \(1024\) \(-59049\) \(12954174\) \(-479513104\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2^{10}q^{2}-3^{10}q^{3}+2^{20}q^{4}+12954174q^{5}+\cdots\)
6.22.a.c 6.a 1.a $1$ $16.769$ \(\Q\) None \(1024\) \(59049\) \(-23245050\) \(-1322977768\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2^{10}q^{2}+3^{10}q^{3}+2^{20}q^{4}-23245050q^{5}+\cdots\)

Decomposition of \(S_{22}^{\mathrm{old}}(\Gamma_0(6))\) into lower level spaces

\( S_{22}^{\mathrm{old}}(\Gamma_0(6)) \cong \) \(S_{22}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 2}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)