Properties

Label 420.2.d
Level $420$
Weight $2$
Character orbit 420.d
Rep. character $\chi_{420}(41,\cdot)$
Character field $\Q$
Dimension $12$
Newform subspaces $4$
Sturm bound $192$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 420.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 21 \)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(192\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(11\), \(17\), \(41\)

Dimensions

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

Total New Old
Modular forms 108 12 96
Cusp forms 84 12 72
Eisenstein series 24 0 24

Trace form

\( 12q - 4q^{7} + 6q^{9} + O(q^{10}) \) \( 12q - 4q^{7} + 6q^{9} + 2q^{15} + 2q^{21} + 12q^{25} - 32q^{37} - 30q^{39} + 4q^{49} + 22q^{51} - 12q^{57} + 32q^{63} + 32q^{67} - 4q^{79} - 10q^{81} - 12q^{85} + 4q^{91} - 20q^{93} + 6q^{99} + O(q^{100}) \)

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.d.a \(2\) \(3.354\) \(\Q(\sqrt{-3}) \) None \(0\) \(-3\) \(-2\) \(4\) \(q+(-1-\zeta_{6})q^{3}-q^{5}+(1+2\zeta_{6})q^{7}+\cdots\)
420.2.d.b \(2\) \(3.354\) \(\Q(\sqrt{-3}) \) None \(0\) \(3\) \(2\) \(4\) \(q+(1+\zeta_{6})q^{3}+q^{5}+(3-2\zeta_{6})q^{7}+3\zeta_{6}q^{9}+\cdots\)
420.2.d.c \(4\) \(3.354\) \(\Q(i, \sqrt{5})\) None \(0\) \(-2\) \(4\) \(-6\) \(q+(-1+\beta _{3})q^{3}+q^{5}+(-1+\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
420.2.d.d \(4\) \(3.354\) \(\Q(i, \sqrt{5})\) None \(0\) \(2\) \(-4\) \(-6\) \(q+(1+\beta _{1})q^{3}-q^{5}+(-1+\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(420, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 2}\)