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Properties

Label 36.3.g
Level $36$
Weight $3$
Character orbit 36.g
Rep. character $\chi_{36}(5,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $4$
Newform subspaces $1$
Sturm bound $18$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 36.g (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 1 \)
Sturm bound: \(18\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(36, [\chi])\).

Total New Old
Modular forms 30 4 26
Cusp forms 18 4 14
Eisenstein series 12 0 12

Trace form

\( 4 q + 3 q^{3} + 9 q^{5} - q^{7} - 15 q^{9} + O(q^{10}) \) \( 4 q + 3 q^{3} + 9 q^{5} - q^{7} - 15 q^{9} - 36 q^{11} + 5 q^{13} - 45 q^{15} + 2 q^{19} + 99 q^{21} + 99 q^{23} + 13 q^{25} - 63 q^{29} - 7 q^{31} - 36 q^{33} - 64 q^{37} + 57 q^{39} - 18 q^{41} - 46 q^{43} - 99 q^{45} - 81 q^{47} - 51 q^{49} - 27 q^{51} + 90 q^{55} + 51 q^{57} + 126 q^{59} + 41 q^{61} + 141 q^{63} + 171 q^{65} + 116 q^{67} + 99 q^{69} + 86 q^{73} - 297 q^{75} - 279 q^{77} + 83 q^{79} - 63 q^{81} - 81 q^{83} + 18 q^{85} - 63 q^{87} - 302 q^{91} - 159 q^{93} - 144 q^{95} - 196 q^{97} + 171 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(36, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
36.3.g.a 36.g 9.d $4$ $0.981$ \(\Q(\sqrt{-3}, \sqrt{-11})\) None \(0\) \(3\) \(9\) \(-1\) $\mathrm{SU}(2)[C_{6}]$ \(q+(1-\beta _{1}+\beta _{3})q^{3}+(2-2\beta _{1}-2\beta _{2}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(36, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(36, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 2}\)