Properties

Label 42.2.f
Level $42$
Weight $2$
Character orbit 42.f
Rep. character $\chi_{42}(5,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $4$
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.f (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 21 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 1 \)
Sturm bound: \(16\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(42, [\chi])\).

Total New Old
Modular forms 24 4 20
Cusp forms 8 4 4
Eisenstein series 16 0 16

Trace form

\( 4 q + 2 q^{4} - 10 q^{7} - 6 q^{9} + O(q^{10}) \) \( 4 q + 2 q^{4} - 10 q^{7} - 6 q^{9} - 6 q^{10} + 12 q^{15} - 2 q^{16} + 12 q^{19} + 12 q^{22} + 6 q^{24} + 4 q^{25} - 2 q^{28} - 6 q^{31} - 18 q^{33} - 12 q^{36} + 4 q^{37} - 6 q^{40} - 6 q^{42} - 32 q^{43} - 12 q^{46} + 22 q^{49} + 12 q^{51} + 12 q^{52} + 18 q^{54} + 6 q^{58} + 6 q^{60} + 24 q^{63} - 4 q^{64} - 4 q^{67} + 18 q^{70} + 24 q^{73} - 24 q^{78} + 2 q^{79} - 18 q^{81} - 24 q^{82} - 24 q^{85} - 18 q^{87} + 6 q^{88} - 12 q^{91} - 24 q^{94} + 6 q^{96} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(42, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
42.2.f.a 42.f 21.g $4$ $0.335$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(-10\) $\mathrm{SU}(2)[C_{6}]$ \(q+\zeta_{12}q^{2}+(-\zeta_{12}-\zeta_{12}^{3})q^{3}+\zeta_{12}^{2}q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(42, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(42, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 2}\)