Properties

Label 420.2.f
Level $420$
Weight $2$
Character orbit 420.f
Rep. character $\chi_{420}(209,\cdot)$
Character field $\Q$
Dimension $16$
Newform subspaces $2$
Sturm bound $192$
Trace bound $9$

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Defining parameters

Level: \( N \) \(=\) \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 420.f (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 105 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(192\)
Trace bound: \(9\)
Distinguishing \(T_p\): \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(420, [\chi])\).

Total New Old
Modular forms 108 16 92
Cusp forms 84 16 68
Eisenstein series 24 0 24

Trace form

\( 16 q + 6 q^{9} + O(q^{10}) \) \( 16 q + 6 q^{9} + 10 q^{15} + 2 q^{21} + 6 q^{39} + 32 q^{49} - 26 q^{51} - 92 q^{79} - 22 q^{81} - 20 q^{85} - 4 q^{91} - 38 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(420, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
420.2.f.a $8$ $3.354$ 8.0.\(\cdots\).2 \(\Q(\sqrt{-35}) \) \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{3}+\beta _{4}q^{5}+(-\beta _{1}-\beta _{6})q^{7}+\cdots\)
420.2.f.b $8$ $3.354$ 8.0.40960000.1 None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{6}q^{3}-\beta _{2}q^{5}+(\beta _{5}+\beta _{6})q^{7}+(1+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(420, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(420, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 2}\)