Properties

Label 420.2.q
Level $420$
Weight $2$
Character orbit 420.q
Rep. character $\chi_{420}(121,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $12$
Newform subspaces $4$
Sturm bound $192$
Trace bound $5$

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Defining parameters

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

Dimensions

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

Total New Old
Modular forms 216 12 204
Cusp forms 168 12 156
Eisenstein series 48 0 48

Trace form

\( 12q - 2q^{3} - 2q^{7} - 6q^{9} + O(q^{10}) \) \( 12q - 2q^{3} - 2q^{7} - 6q^{9} + 4q^{11} + 12q^{13} + 8q^{17} + 6q^{19} + 4q^{21} - 6q^{25} + 4q^{27} - 8q^{29} - 2q^{31} - 8q^{33} + 18q^{37} + 10q^{39} - 24q^{41} - 4q^{43} - 8q^{47} + 18q^{49} - 12q^{51} + 8q^{55} - 20q^{57} + 4q^{59} + 12q^{61} - 2q^{63} + 4q^{65} - 6q^{67} - 8q^{69} + 48q^{71} - 26q^{73} - 2q^{75} - 36q^{77} - 14q^{79} - 6q^{81} - 56q^{83} + 8q^{85} + 16q^{89} + 22q^{91} + 2q^{93} - 16q^{95} + 64q^{97} - 8q^{99} + O(q^{100}) \)

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

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

\( S_{2}^{\mathrm{old}}(420, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(140, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 2}\)