Properties

Label 225.2.e
Level $225$
Weight $2$
Character orbit 225.e
Rep. character $\chi_{225}(76,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $32$
Newform subspaces $5$
Sturm bound $60$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 225.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 5 \)
Sturm bound: \(60\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(225, [\chi])\).

Total New Old
Modular forms 72 44 28
Cusp forms 48 32 16
Eisenstein series 24 12 12

Trace form

\( 32 q + 2 q^{2} + 2 q^{3} - 12 q^{4} + 2 q^{7} + 6 q^{9} + O(q^{10}) \) \( 32 q + 2 q^{2} + 2 q^{3} - 12 q^{4} + 2 q^{7} + 6 q^{9} - 6 q^{11} - 14 q^{12} + 2 q^{13} + 6 q^{14} - 8 q^{16} - 4 q^{17} + 20 q^{18} + 8 q^{19} - 18 q^{21} - 6 q^{22} + 6 q^{23} - 54 q^{24} - 24 q^{26} + 2 q^{27} - 16 q^{28} - 6 q^{29} + 4 q^{31} + 22 q^{32} - 20 q^{33} - 14 q^{34} - 12 q^{36} - 4 q^{37} - 10 q^{38} + 42 q^{39} - 6 q^{41} + 42 q^{42} + 2 q^{43} + 72 q^{44} - 16 q^{46} + 20 q^{47} + 10 q^{48} + 10 q^{49} - 30 q^{51} - 10 q^{52} + 8 q^{53} + 48 q^{54} + 48 q^{56} - 22 q^{57} - 18 q^{58} - 42 q^{59} + 10 q^{61} - 84 q^{62} - 24 q^{63} + 28 q^{64} + 36 q^{66} + 8 q^{67} - 26 q^{68} - 18 q^{69} + 24 q^{71} - 6 q^{72} + 8 q^{73} - 48 q^{74} - 30 q^{76} + 6 q^{77} - 2 q^{78} + 2 q^{79} + 42 q^{81} + 48 q^{82} - 6 q^{83} + 54 q^{84} + 12 q^{86} + 26 q^{87} - 18 q^{88} - 60 q^{89} - 60 q^{91} + 36 q^{92} + 42 q^{93} - 2 q^{94} + 36 q^{96} - 16 q^{97} + 76 q^{98} + 18 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(225, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
225.2.e.a 225.e 9.c $2$ $1.797$ \(\Q(\sqrt{-3}) \) None \(1\) \(3\) \(0\) \(-3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{2}+(2-\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
225.2.e.b 225.e 9.c $6$ $1.797$ 6.0.954288.1 None \(1\) \(-1\) \(0\) \(5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{1}-\beta _{2}+\beta _{4}+\beta _{5})q^{2}-\beta _{4}q^{3}+\cdots\)
225.2.e.c 225.e 9.c $8$ $1.797$ 8.0.1223810289.2 None \(-2\) \(1\) \(0\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{1}q^{2}+(-\beta _{4}+\beta _{7})q^{3}+(-1+\beta _{1}+\cdots)q^{4}+\cdots\)
225.2.e.d 225.e 9.c $8$ $1.797$ \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{24}^{7}q^{2}+(\zeta_{24}^{6}-\zeta_{24}^{7})q^{3}-\zeta_{24}^{2}q^{4}+\cdots\)
225.2.e.e 225.e 9.c $8$ $1.797$ 8.0.1223810289.2 None \(2\) \(-1\) \(0\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{2}+(\beta _{4}-\beta _{7})q^{3}+(-1+\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(225, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(225, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 2}\)