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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

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

Display 100 coefficients 10 coefficients

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

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
420.2.q.a 420.q 7.c $2$ $3.354$ \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(-1\) \(5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}-\zeta_{6}q^{5}+(2+\zeta_{6})q^{7}+\cdots\)
420.2.q.b 420.q 7.c $2$ $3.354$ \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(1\) \(-5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}+\zeta_{6}q^{5}+(-2-\zeta_{6})q^{7}+\cdots\)
420.2.q.c 420.q 7.c $4$ $3.354$ \(\Q(\sqrt{-3}, \sqrt{7})\) None \(0\) \(-2\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{2}q^{3}+(-1-\beta _{2})q^{5}+\beta _{1}q^{7}+(-1+\cdots)q^{9}+\cdots\)
420.2.q.d 420.q 7.c $4$ $3.354$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(-2\) \(2\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1-\beta _{1})q^{3}-\beta _{1}q^{5}+(\beta _{1}+\beta _{3})q^{7}+\cdots\)

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

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