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Properties

Label 630.2.z
Level $630$
Weight $2$
Character orbit 630.z
Rep. character $\chi_{630}(169,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $72$
Newform subspaces $3$
Sturm bound $288$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 630.z (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 45 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 3 \)
Sturm bound: \(288\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(630, [\chi])\).

Total New Old
Modular forms 304 72 232
Cusp forms 272 72 200
Eisenstein series 32 0 32

Trace form

\( 72 q + 36 q^{4} + 4 q^{5} + 16 q^{9} + O(q^{10}) \) \( 72 q + 36 q^{4} + 4 q^{5} + 16 q^{9} - 12 q^{11} + 8 q^{14} + 20 q^{15} - 36 q^{16} - 4 q^{20} + 4 q^{21} - 12 q^{25} + 16 q^{29} - 16 q^{30} + 24 q^{31} + 20 q^{36} - 4 q^{39} - 16 q^{41} - 24 q^{44} - 24 q^{45} + 36 q^{49} + 4 q^{50} - 112 q^{51} - 72 q^{54} + 24 q^{55} - 8 q^{56} - 16 q^{59} + 4 q^{60} - 72 q^{64} + 48 q^{65} + 40 q^{66} - 16 q^{69} + 88 q^{71} + 56 q^{74} + 12 q^{79} - 8 q^{80} + 32 q^{81} + 8 q^{84} + 24 q^{85} - 64 q^{86} + 160 q^{89} + 60 q^{90} + 24 q^{91} - 12 q^{95} - 108 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(630, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
630.2.z.a 630.z 45.j $4$ $5.031$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\zeta_{12}q^{2}+(-2\zeta_{12}+\zeta_{12}^{3})q^{3}+\zeta_{12}^{2}q^{4}+\cdots\)
630.2.z.b 630.z 45.j $24$ $5.031$ None \(0\) \(0\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{6}]$
630.2.z.c 630.z 45.j $44$ $5.031$ None \(0\) \(0\) \(2\) \(0\) $\mathrm{SU}(2)[C_{6}]$

Decomposition of \(S_{2}^{\mathrm{old}}(630, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(630, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(315, [\chi])\)\(^{\oplus 2}\)