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Properties

Label 165.2.a
Level $165$
Weight $2$
Character orbit 165.a
Rep. character $\chi_{165}(1,\cdot)$
Character field $\Q$
Dimension $7$
Newform subspaces $3$
Sturm bound $48$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 165.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(48\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(165))\).

Total New Old
Modular forms 28 7 21
Cusp forms 21 7 14
Eisenstein series 7 0 7

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

\(3\)\(5\)\(11\)FrickeDim
\(+\)\(+\)\(+\)\(+\)\(2\)
\(-\)\(+\)\(+\)\(-\)\(2\)
\(-\)\(-\)\(-\)\(-\)\(3\)
Plus space\(+\)\(2\)
Minus space\(-\)\(5\)

Trace form

\( 7 q - 3 q^{2} + 3 q^{3} + 9 q^{4} - q^{5} + q^{6} - 15 q^{8} + 7 q^{9} + O(q^{10}) \) \( 7 q - 3 q^{2} + 3 q^{3} + 9 q^{4} - q^{5} + q^{6} - 15 q^{8} + 7 q^{9} + q^{10} - q^{11} + 5 q^{12} + 2 q^{13} - 16 q^{14} + 3 q^{15} + 9 q^{16} - 10 q^{17} - 3 q^{18} + 4 q^{19} + q^{20} + 8 q^{21} + q^{22} - 8 q^{23} - 3 q^{24} + 7 q^{25} + 14 q^{26} + 3 q^{27} + 8 q^{28} - 14 q^{29} - 3 q^{30} - 23 q^{32} + 3 q^{33} - 30 q^{34} + 9 q^{36} + 10 q^{37} + 4 q^{38} + 2 q^{39} - 3 q^{40} - 10 q^{41} - 8 q^{42} - 4 q^{43} + q^{44} - q^{45} + 8 q^{46} - 16 q^{47} - 3 q^{48} + 15 q^{49} - 3 q^{50} + 6 q^{51} - 58 q^{52} - 22 q^{53} + q^{54} + 7 q^{55} + 20 q^{57} + 6 q^{58} + 4 q^{59} + 5 q^{60} - 14 q^{61} + 40 q^{62} + q^{64} - 6 q^{65} - 3 q^{66} + 12 q^{67} + 50 q^{68} + 8 q^{69} - 8 q^{70} + 24 q^{71} - 15 q^{72} - 10 q^{73} - 2 q^{74} + 3 q^{75} + 28 q^{76} - 18 q^{78} - 8 q^{79} + 17 q^{80} + 7 q^{81} + 2 q^{82} + 4 q^{83} - 16 q^{84} + 6 q^{85} + 36 q^{86} - 6 q^{87} - 3 q^{88} - 26 q^{89} + q^{90} - 16 q^{91} + 8 q^{92} + 16 q^{94} + 12 q^{95} - 35 q^{96} + 14 q^{97} + 61 q^{98} - q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(165))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 5 11
165.2.a.a 165.a 1.a $2$ $1.318$ \(\Q(\sqrt{2}) \) None 165.2.a.a \(-2\) \(-2\) \(-2\) \(-4\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}-q^{3}+(1-2\beta )q^{4}-q^{5}+\cdots\)
165.2.a.b 165.a 1.a $2$ $1.318$ \(\Q(\sqrt{3}) \) None 165.2.a.b \(0\) \(2\) \(-2\) \(4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+q^{3}+q^{4}-q^{5}+\beta q^{6}+2q^{7}+\cdots\)
165.2.a.c 165.a 1.a $3$ $1.318$ 3.3.148.1 None 165.2.a.c \(-1\) \(3\) \(3\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(165))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(165)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 2}\)