Space of modular forms of level 329, weight 1, and Character 329.f

• Label: 329.1.f
• Level: $329$
• Weight: $1$
• Character orbit: 329.f
• Rep. character: $\chi_{329}(46,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $10$
• Newform subspaces: $2$
• Sturm bound: $32$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 329 = 7 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	329.f (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 329 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(32\)
Trace bound:			\(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(329, [\chi])\).

	Total	New	Old
Modular forms	14	14	0
Cusp forms	10	10	0
Eisenstein series	4	4	0

The following table gives the dimensions of subspaces with specified projective image type.

	\(D_n\)	\(A_4\)	\(S_4\)	\(A_5\)
Dimension	10	0	0	0

Trace form

10 * q - 5 * q^4 - 5 * q^9 + 5 * q^12 - 5 * q^14 - 5 * q^16 + 5 * q^18 - 10 * q^24 - 5 * q^25 + 5 * q^32 + 10 * q^36 - 5 * q^42 - 5 * q^47 - 10 * q^48 + 5 * q^51 + 5 * q^54 + 10 * q^56 + 10 * q^63 + 10 * q^64 + 5 * q^68 + 5 * q^72 - 5 * q^81 - 10 * q^83 - 5 * q^84 - 10 * q^96

Decomposition of \(S_{1}^{\mathrm{new}}(329, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	Image	CM	RM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																				$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
329.1.f.a	$329$	$1$	329.f	329.f	$6$	$2$	$1$	$0.164$	\(\Q(\sqrt{-3}) \)	$D_{3}$	\(\Q(\sqrt{-47}) \)	None		✓			329.1.f.a	$4$	$0$	\(1\)	\(-2\)	\(0\)	\(-1\)	$1$		\(q+\zeta_{6}q^{2}+\zeta_{6}^{2}q^{3}-2q^{6}+\zeta_{6}^{2}q^{7}+\cdots\)
329.1.f.b	$329$	$1$	329.f	329.f	$6$	$8$	$4$	$0.164$	\(\Q(\zeta_{15})\)	$D_{15}$	\(\Q(\sqrt{-47}) \)	None		✓	✓		329.1.f.b	$4$	$0$	\(-1\)	\(2\)	\(0\)	\(1\)	$1$		\(q+(\zeta_{30}^{8}+\zeta_{30}^{12})q^{2}+(-\zeta_{30}^{11}+\zeta_{30}^{14}+\cdots)q^{3}+\cdots\)