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Properties

Label 30.2.a
Level $30$
Weight $2$
Character orbit 30.a
Rep. character $\chi_{30}(1,\cdot)$
Character field $\Q$
Dimension $1$
Newform subspaces $1$
Sturm bound $12$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 30 = 2 \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 30.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(12\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(30))\).

Total New Old
Modular forms 10 1 9
Cusp forms 3 1 2
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.

\(2\)\(3\)\(5\)FrickeDim
\(+\)\(-\)\(+\)$-$\(1\)
Plus space\(+\)\(0\)
Minus space\(-\)\(1\)

Trace form

\( q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - 4 q^{7} - q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - 4 q^{7} - q^{8} + q^{9} + q^{10} + q^{12} + 2 q^{13} + 4 q^{14} - q^{15} + q^{16} + 6 q^{17} - q^{18} - 4 q^{19} - q^{20} - 4 q^{21} - q^{24} + q^{25} - 2 q^{26} + q^{27} - 4 q^{28} - 6 q^{29} + q^{30} + 8 q^{31} - q^{32} - 6 q^{34} + 4 q^{35} + q^{36} + 2 q^{37} + 4 q^{38} + 2 q^{39} + q^{40} - 6 q^{41} + 4 q^{42} - 4 q^{43} - q^{45} + q^{48} + 9 q^{49} - q^{50} + 6 q^{51} + 2 q^{52} - 6 q^{53} - q^{54} + 4 q^{56} - 4 q^{57} + 6 q^{58} - q^{60} - 10 q^{61} - 8 q^{62} - 4 q^{63} + q^{64} - 2 q^{65} - 4 q^{67} + 6 q^{68} - 4 q^{70} - q^{72} + 2 q^{73} - 2 q^{74} + q^{75} - 4 q^{76} - 2 q^{78} + 8 q^{79} - q^{80} + q^{81} + 6 q^{82} + 12 q^{83} - 4 q^{84} - 6 q^{85} + 4 q^{86} - 6 q^{87} + 18 q^{89} + q^{90} - 8 q^{91} + 8 q^{93} + 4 q^{95} - q^{96} + 2 q^{97} - 9 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(30))\) 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
30.2.a.a 30.a 1.a $1$ $0.240$ \(\Q\) None \(-1\) \(1\) \(-1\) \(-4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}-4q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(30))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(30)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 2}\)