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Properties

Label 330.2.a
Level $330$
Weight $2$
Character orbit 330.a
Rep. character $\chi_{330}(1,\cdot)$
Character field $\Q$
Dimension $5$
Newform subspaces $5$
Sturm bound $144$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 330 = 2 \cdot 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 330.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(144\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(7\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(330))\).

Total New Old
Modular forms 80 5 75
Cusp forms 65 5 60
Eisenstein series 15 0 15

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

\(2\)\(3\)\(5\)\(11\)FrickeDim
\(+\)\(+\)\(+\)\(-\)$-$\(1\)
\(+\)\(+\)\(-\)\(-\)$+$\(1\)
\(-\)\(+\)\(+\)\(+\)$-$\(1\)
\(-\)\(+\)\(-\)\(-\)$-$\(1\)
\(-\)\(-\)\(-\)\(+\)$-$\(1\)
Plus space\(+\)\(1\)
Minus space\(-\)\(4\)

Trace form

\( 5 q + q^{2} - 3 q^{3} + 5 q^{4} + q^{5} + q^{6} + q^{8} + 5 q^{9} + O(q^{10}) \) \( 5 q + q^{2} - 3 q^{3} + 5 q^{4} + q^{5} + q^{6} + q^{8} + 5 q^{9} + q^{10} + q^{11} - 3 q^{12} + 6 q^{13} + 8 q^{14} + q^{15} + 5 q^{16} + 2 q^{17} + q^{18} - 4 q^{19} + q^{20} - 3 q^{22} + q^{24} + 5 q^{25} + 6 q^{26} - 3 q^{27} - 2 q^{29} + q^{30} + q^{32} - 3 q^{33} + 10 q^{34} - 8 q^{35} + 5 q^{36} - 18 q^{37} - 4 q^{38} - 10 q^{39} + q^{40} - 6 q^{41} - 8 q^{42} - 12 q^{43} + q^{44} + q^{45} - 8 q^{46} - 8 q^{47} - 3 q^{48} - 3 q^{49} + q^{50} + 2 q^{51} + 6 q^{52} + 6 q^{53} + q^{54} + q^{55} + 8 q^{56} - 4 q^{57} - 10 q^{58} - 20 q^{59} + q^{60} + 22 q^{61} + 5 q^{64} - 2 q^{65} + q^{66} - 28 q^{67} + 2 q^{68} - 16 q^{71} + q^{72} + 18 q^{73} + 6 q^{74} - 3 q^{75} - 4 q^{76} - 8 q^{77} - 10 q^{78} - 8 q^{79} + q^{80} + 5 q^{81} + 2 q^{82} + 4 q^{83} + 2 q^{85} - 28 q^{86} - 2 q^{87} - 3 q^{88} + 2 q^{89} + q^{90} + 16 q^{91} - 28 q^{95} + q^{96} + 10 q^{97} - 7 q^{98} + q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(330))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 5 11
330.2.a.a 330.a 1.a $1$ $2.635$ \(\Q\) None \(-1\) \(-1\) \(-1\) \(0\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}-q^{8}+\cdots\)
330.2.a.b 330.a 1.a $1$ $2.635$ \(\Q\) None \(-1\) \(-1\) \(1\) \(-4\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-4q^{7}+\cdots\)
330.2.a.c 330.a 1.a $1$ $2.635$ \(\Q\) None \(1\) \(-1\) \(-1\) \(4\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+4q^{7}+\cdots\)
330.2.a.d 330.a 1.a $1$ $2.635$ \(\Q\) None \(1\) \(-1\) \(1\) \(0\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+q^{8}+\cdots\)
330.2.a.e 330.a 1.a $1$ $2.635$ \(\Q\) None \(1\) \(1\) \(1\) \(0\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}+q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(330))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(330)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(66))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(110))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(165))\)\(^{\oplus 2}\)