Properties

Label 3.6.a
Level $3$
Weight $6$
Character orbit 3.a
Rep. character $\chi_{3}(1,\cdot)$
Character field $\Q$
Dimension $1$
Newform subspaces $1$
Sturm bound $2$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(2\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(3))\).

Total New Old
Modular forms 3 1 2
Cusp forms 1 1 0
Eisenstein series 2 0 2

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

\(3\)Dim.
\(-\)\(1\)

Trace form

\( q - 6 q^{2} + 9 q^{3} + 4 q^{4} + 6 q^{5} - 54 q^{6} - 40 q^{7} + 168 q^{8} + 81 q^{9} + O(q^{10}) \) \( q - 6 q^{2} + 9 q^{3} + 4 q^{4} + 6 q^{5} - 54 q^{6} - 40 q^{7} + 168 q^{8} + 81 q^{9} - 36 q^{10} - 564 q^{11} + 36 q^{12} + 638 q^{13} + 240 q^{14} + 54 q^{15} - 1136 q^{16} + 882 q^{17} - 486 q^{18} - 556 q^{19} + 24 q^{20} - 360 q^{21} + 3384 q^{22} - 840 q^{23} + 1512 q^{24} - 3089 q^{25} - 3828 q^{26} + 729 q^{27} - 160 q^{28} + 4638 q^{29} - 324 q^{30} + 4400 q^{31} + 1440 q^{32} - 5076 q^{33} - 5292 q^{34} - 240 q^{35} + 324 q^{36} - 2410 q^{37} + 3336 q^{38} + 5742 q^{39} + 1008 q^{40} - 6870 q^{41} + 2160 q^{42} + 9644 q^{43} - 2256 q^{44} + 486 q^{45} + 5040 q^{46} - 18672 q^{47} - 10224 q^{48} - 15207 q^{49} + 18534 q^{50} + 7938 q^{51} + 2552 q^{52} + 33750 q^{53} - 4374 q^{54} - 3384 q^{55} - 6720 q^{56} - 5004 q^{57} - 27828 q^{58} - 18084 q^{59} + 216 q^{60} + 39758 q^{61} - 26400 q^{62} - 3240 q^{63} + 27712 q^{64} + 3828 q^{65} + 30456 q^{66} - 23068 q^{67} + 3528 q^{68} - 7560 q^{69} + 1440 q^{70} - 4248 q^{71} + 13608 q^{72} - 41110 q^{73} + 14460 q^{74} - 27801 q^{75} - 2224 q^{76} + 22560 q^{77} - 34452 q^{78} + 21920 q^{79} - 6816 q^{80} + 6561 q^{81} + 41220 q^{82} + 82452 q^{83} - 1440 q^{84} + 5292 q^{85} - 57864 q^{86} + 41742 q^{87} - 94752 q^{88} - 94086 q^{89} - 2916 q^{90} - 25520 q^{91} - 3360 q^{92} + 39600 q^{93} + 112032 q^{94} - 3336 q^{95} + 12960 q^{96} + 49442 q^{97} + 91242 q^{98} - 45684 q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3
3.6.a.a 3.a 1.a $1$ $0.481$ \(\Q\) None \(-6\) \(9\) \(6\) \(-40\) $-$ $\mathrm{SU}(2)$ \(q-6q^{2}+9q^{3}+4q^{4}+6q^{5}-54q^{6}+\cdots\)