Space of modular forms of level 429, weight 2, and Character 429.i

• Label: 429.2.i
• Level: $429$
• Weight: $2$
• Character orbit: 429.i
• Rep. character: $\chi_{429}(100,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $44$
• Newform subspaces: $6$
• Sturm bound: $112$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 429 = 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	429.i (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(112\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(429, [\chi])\).

	Total	New	Old
Modular forms	120	44	76
Cusp forms	104	44	60
Eisenstein series	16	0	16

Trace form

\( 44 q + 4 q^{2} + 2 q^{3} - 20 q^{4} + 16 q^{5} - 2 q^{7} - 24 q^{8} - 22 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(429, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
429.2.i.a	$429$	$2$	429.i	13.c	$3$	$2$	$1$	$3.426$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(1\)	\(4\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{3}+2\zeta_{6}q^{4}+2q^{5}-\zeta_{6}q^{7}+\cdots\)
429.2.i.b	$429$	$2$	429.i	13.c	$3$	$2$	$1$	$3.426$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(1\)	\(4\)	\(3\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{3}+2\zeta_{6}q^{4}+2q^{5}+3\zeta_{6}q^{7}+\cdots\)
429.2.i.c	$429$	$2$	429.i	13.c	$3$	$10$	$5$	$3.426$	10.0.\(\cdots\).1	None		$2$	$0$	\(-2\)	\(-5\)	\(16\)	\(-9\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{4}q^{2}+\beta _{3}q^{3}+(-\beta _{1}+\beta _{6})q^{4}+\cdots\)
429.2.i.d	$429$	$2$	429.i	13.c	$3$	$10$	$5$	$3.426$	10.0.\(\cdots\).1	None		$2$	$0$	\(0\)	\(5\)	\(-4\)	\(-7\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{2}+\beta _{7}-\beta _{8}+\beta _{9})q^{2}+(1-\beta _{4}+\cdots)q^{3}+\cdots\)
429.2.i.e	$429$	$2$	429.i	13.c	$3$	$10$	$5$	$3.426$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None		$2$	$0$	\(2\)	\(5\)	\(4\)	\(9\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{2}+(1+\beta _{5})q^{3}+(\beta _{1}-\beta _{3}+\beta _{5}+\cdots)q^{4}+\cdots\)
429.2.i.f	$429$	$2$	429.i	13.c	$3$	$10$	$5$	$3.426$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None		$2$	$0$	\(4\)	\(-5\)	\(-8\)	\(3\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\beta _{1}-\beta _{6})q^{2}+(-1+\beta _{6})q^{3}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(429, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(429, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(143, [\chi])\)\(^{\oplus 2}\)