↑

Properties

Label 45.2.e
Level $45$
Weight $2$
Character orbit 45.e
Rep. character $\chi_{45}(16,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $8$
Newform subspaces $2$
Sturm bound $12$
Trace bound $1$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 45.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 2 \)
Sturm bound: \(12\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(45, [\chi])\).

Total New Old
Modular forms 16 8 8
Cusp forms 8 8 0
Eisenstein series 8 0 8

Trace form

\( 8 q - 2 q^{2} - 2 q^{3} - 4 q^{4} - 2 q^{5} - 4 q^{6} - 2 q^{7} - 4 q^{9} + O(q^{10}) \) \( 8 q - 2 q^{2} - 2 q^{3} - 4 q^{4} - 2 q^{5} - 4 q^{6} - 2 q^{7} - 4 q^{9} + 4 q^{11} + 14 q^{12} - 2 q^{13} + 12 q^{14} - 2 q^{15} - 4 q^{16} + 4 q^{17} - 20 q^{18} - 8 q^{19} - 6 q^{20} + 6 q^{22} - 6 q^{23} + 24 q^{24} - 4 q^{25} - 8 q^{26} - 2 q^{27} + 16 q^{28} + 8 q^{29} + 14 q^{30} - 8 q^{31} - 22 q^{32} + 20 q^{33} + 16 q^{35} + 16 q^{36} + 4 q^{37} + 10 q^{38} - 20 q^{39} - 6 q^{40} + 8 q^{41} - 42 q^{42} - 2 q^{43} - 40 q^{44} + 8 q^{45} - 20 q^{47} - 10 q^{48} - 2 q^{50} - 16 q^{51} + 10 q^{52} - 8 q^{53} - 4 q^{54} + 22 q^{57} + 18 q^{58} + 16 q^{59} - 22 q^{60} - 8 q^{61} + 84 q^{62} + 24 q^{63} - 16 q^{64} - 6 q^{65} - 8 q^{66} - 8 q^{67} + 26 q^{68} + 48 q^{69} + 6 q^{70} - 16 q^{71} + 6 q^{72} - 8 q^{73} + 20 q^{74} - 2 q^{75} + 4 q^{76} - 6 q^{77} + 2 q^{78} + 4 q^{79} + 12 q^{80} - 28 q^{81} - 48 q^{82} + 6 q^{83} - 36 q^{84} + 6 q^{85} - 20 q^{86} - 26 q^{87} + 18 q^{88} - 48 q^{89} - 2 q^{90} + 32 q^{91} - 36 q^{92} - 42 q^{93} + 24 q^{94} - 12 q^{95} + 28 q^{96} + 16 q^{97} - 76 q^{98} + 16 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(45, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
45.2.e.a 45.e 9.c $2$ $0.359$ \(\Q(\sqrt{-3}) \) None \(-1\) \(-3\) \(1\) \(3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{2}+(-2+\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
45.2.e.b 45.e 9.c $6$ $0.359$ 6.0.954288.1 None \(-1\) \(1\) \(-3\) \(-5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\beta _{1}+\beta _{2}-\beta _{4}-\beta _{5})q^{2}+\beta _{4}q^{3}+\cdots\)