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Properties

Label 42.2.a
Level $42$
Weight $2$
Character orbit 42.a
Rep. character $\chi_{42}(1,\cdot)$
Character field $\Q$
Dimension $1$
Newform subspaces $1$
Sturm bound $16$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 42.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(16\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(42))\).

Total New Old
Modular forms 12 1 11
Cusp forms 5 1 4
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.

\(2\)\(3\)\(7\)FrickeDim
\(-\)\(+\)\(+\)$-$\(1\)
Plus space\(+\)\(0\)
Minus space\(-\)\(1\)

Trace form

\( q + q^{2} - q^{3} + q^{4} - 2 q^{5} - q^{6} - q^{7} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} - q^{3} + q^{4} - 2 q^{5} - q^{6} - q^{7} + q^{8} + q^{9} - 2 q^{10} - 4 q^{11} - q^{12} + 6 q^{13} - q^{14} + 2 q^{15} + q^{16} + 2 q^{17} + q^{18} - 4 q^{19} - 2 q^{20} + q^{21} - 4 q^{22} + 8 q^{23} - q^{24} - q^{25} + 6 q^{26} - q^{27} - q^{28} - 2 q^{29} + 2 q^{30} + q^{32} + 4 q^{33} + 2 q^{34} + 2 q^{35} + q^{36} - 10 q^{37} - 4 q^{38} - 6 q^{39} - 2 q^{40} - 6 q^{41} + q^{42} - 4 q^{43} - 4 q^{44} - 2 q^{45} + 8 q^{46} - q^{48} + q^{49} - q^{50} - 2 q^{51} + 6 q^{52} + 6 q^{53} - q^{54} + 8 q^{55} - q^{56} + 4 q^{57} - 2 q^{58} + 4 q^{59} + 2 q^{60} + 6 q^{61} - q^{63} + q^{64} - 12 q^{65} + 4 q^{66} + 4 q^{67} + 2 q^{68} - 8 q^{69} + 2 q^{70} + 8 q^{71} + q^{72} + 10 q^{73} - 10 q^{74} + q^{75} - 4 q^{76} + 4 q^{77} - 6 q^{78} - 2 q^{80} + q^{81} - 6 q^{82} - 4 q^{83} + q^{84} - 4 q^{85} - 4 q^{86} + 2 q^{87} - 4 q^{88} - 6 q^{89} - 2 q^{90} - 6 q^{91} + 8 q^{92} + 8 q^{95} - q^{96} - 14 q^{97} + q^{98} - 4 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(42))\) 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 7
42.2.a.a 42.a 1.a $1$ $0.335$ \(\Q\) None \(1\) \(-1\) \(-2\) \(-1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-2q^{5}-q^{6}-q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(42))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(42)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 2}\)