Space of modular forms of level 780, weight 2, and Character 780.z

• Label: 780.2.z
• Level: $780$
• Weight: $2$
• Character orbit: 780.z
• Rep. character: $\chi_{780}(629,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $56$
• Newform subspaces: $1$
• Sturm bound: $336$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 780 = 2^{2} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	780.z (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 195 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(336\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(780, [\chi])\).

	Total	New	Old
Modular forms	360	56	304
Cusp forms	312	56	256
Eisenstein series	48	0	48

Trace form

\( 56 q + 8 q^{15} - 16 q^{19} - 4 q^{21} - 16 q^{31} - 12 q^{45} - 16 q^{55} + 24 q^{61} + 56 q^{79} - 8 q^{81} - 24 q^{85} - 24 q^{91} + 44 q^{99}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(780, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
780.2.z.a	$780$	$2$	780.z	195.n	$4$	$56$	$28$	$6.228$		None		✓	✓	✓	780.2.z.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{4}]$

Decomposition of \(S_{2}^{\mathrm{old}}(780, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(780, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(195, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(390, [\chi])\)\(^{\oplus 2}\)