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Properties

Label 270.2.k
Level $270$
Weight $2$
Character orbit 270.k
Rep. character $\chi_{270}(31,\cdot)$
Character field $\Q(\zeta_{9})$
Dimension $72$
Newform subspaces $5$
Sturm bound $108$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 270.k (of order \(9\) and degree \(6\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 27 \)
Character field: \(\Q(\zeta_{9})\)
Newform subspaces: \( 5 \)
Sturm bound: \(108\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(270, [\chi])\).

Total New Old
Modular forms 348 72 276
Cusp forms 300 72 228
Eisenstein series 48 0 48

Trace form

\( 72 q + 6 q^{6} + 6 q^{8} + 12 q^{9} + O(q^{10}) \) \( 72 q + 6 q^{6} + 6 q^{8} + 12 q^{9} + 18 q^{11} + 6 q^{12} + 6 q^{14} + 24 q^{17} - 12 q^{18} + 24 q^{21} + 18 q^{22} - 72 q^{23} - 66 q^{27} - 66 q^{29} - 66 q^{33} + 18 q^{34} + 12 q^{35} + 6 q^{36} - 12 q^{38} + 12 q^{39} + 18 q^{41} + 18 q^{43} + 6 q^{45} - 72 q^{47} + 18 q^{49} + 48 q^{51} - 72 q^{53} + 36 q^{54} + 6 q^{56} - 18 q^{57} + 48 q^{59} + 18 q^{61} + 48 q^{62} - 72 q^{63} - 36 q^{64} - 24 q^{65} + 18 q^{67} - 12 q^{68} + 24 q^{69} - 36 q^{70} - 48 q^{71} + 36 q^{73} - 12 q^{74} - 36 q^{76} + 60 q^{77} - 96 q^{78} - 72 q^{79} - 12 q^{80} - 60 q^{81} - 60 q^{83} - 72 q^{85} - 90 q^{86} + 60 q^{87} - 36 q^{88} - 6 q^{89} - 48 q^{90} + 36 q^{91} + 36 q^{92} - 36 q^{93} - 36 q^{94} - 48 q^{95} - 12 q^{96} + 18 q^{97} + 30 q^{98} - 12 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(270, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
270.2.k.a 270.k 27.e $6$ $2.156$ \(\Q(\zeta_{18})\) None \(0\) \(0\) \(0\) \(6\) $\mathrm{SU}(2)[C_{9}]$ \(q+(-\zeta_{18}+\zeta_{18}^{4})q^{2}+(\zeta_{18}-2\zeta_{18}^{4}+\cdots)q^{3}+\cdots\)
270.2.k.b 270.k 27.e $12$ $2.156$ 12.0.\(\cdots\).1 None \(0\) \(3\) \(0\) \(-12\) $\mathrm{SU}(2)[C_{9}]$ \(q-\beta _{7}q^{2}+(1-\beta _{5}-\beta _{7}-\beta _{11})q^{3}+\cdots\)
270.2.k.c 270.k 27.e $12$ $2.156$ 12.0.\(\cdots\).1 None \(0\) \(3\) \(0\) \(6\) $\mathrm{SU}(2)[C_{9}]$ \(q+(-\beta _{7}-\beta _{10})q^{2}+(-\beta _{1}+\beta _{4}-\beta _{7}+\cdots)q^{3}+\cdots\)
270.2.k.d 270.k 27.e $18$ $2.156$ \(\mathbb{Q}[x]/(x^{18} - \cdots)\) None \(0\) \(-3\) \(0\) \(-3\) $\mathrm{SU}(2)[C_{9}]$ \(q+\beta _{2}q^{2}-\beta _{7}q^{3}+\beta _{3}q^{4}+(\beta _{3}-\beta _{8}+\cdots)q^{5}+\cdots\)
270.2.k.e 270.k 27.e $24$ $2.156$ None \(0\) \(-3\) \(0\) \(3\) $\mathrm{SU}(2)[C_{9}]$

Decomposition of \(S_{2}^{\mathrm{old}}(270, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(270, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(135, [\chi])\)\(^{\oplus 2}\)