Space of modular forms of level 429, weight 2, and trivial character

• Label: 429.2.a
• Level: $429$
• Weight: $2$
• Character orbit: 429.a
• Rep. character: $\chi_{429}(1,\cdot)$
• Character field: $\Q$
• Dimension: $19$
• Newform subspaces: $8$
• Sturm bound: $112$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 429 = 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	429.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(112\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(429))\).

	Total	New	Old
Modular forms	60	19	41
Cusp forms	53	19	34
Eisenstein series	7	0	7

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)	\(11\)	\(13\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(2\)
\(+\)	\(+\)	\(-\)	$-$	\(4\)
\(+\)	\(-\)	\(+\)	$-$	\(3\)
\(+\)	\(-\)	\(-\)	$+$	\(1\)
\(-\)	\(+\)	\(+\)	$-$	\(3\)
\(-\)	\(+\)	\(-\)	$+$	\(1\)
\(-\)	\(-\)	\(+\)	$+$	\(2\)
\(-\)	\(-\)	\(-\)	$-$	\(3\)
Plus space			\(+\)	\(6\)
Minus space			\(-\)	\(13\)

Trace form

\( 19 q - 3 q^{2} - q^{3} + 21 q^{4} - 6 q^{5} + 5 q^{6} - 8 q^{7} + 9 q^{8} + 19 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(429))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces					A-L signs			Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	11	13
429.2.a.a	$429$	$2$	429.a	1.a	$1$	$1$	$1$	$3.426$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(-1\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}-q^{4}+q^{6}+3q^{8}+q^{9}+\cdots\)
429.2.a.b	$429$	$2$	429.a	1.a	$1$	$1$	$1$	$3.426$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(1\)	\(-2\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}-q^{4}-2q^{5}-q^{6}+3q^{8}+\cdots\)
429.2.a.c	$429$	$2$	429.a	1.a	$1$	$2$	$2$	$3.426$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(-2\)	\(2\)	\(-4\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+q^{3}+(1-2\beta )q^{4}+(-2+\cdots)q^{5}+\cdots\)
429.2.a.d	$429$	$2$	429.a	1.a	$1$	$2$	$2$	$3.426$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(-2\)	\(-4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-q^{3}+q^{4}+(-1-\beta )q^{5}-\beta q^{6}+\cdots\)
429.2.a.e	$429$	$2$	429.a	1.a	$1$	$3$	$3$	$3.426$	3.3.564.1	None	✓	$1$	$0$	\(-1\)	\(-3\)	\(-2\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-q^{3}+(2+\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
429.2.a.f	$429$	$2$	429.a	1.a	$1$	$3$	$3$	$3.426$	3.3.148.1	None	✓	$1$	$0$	\(1\)	\(3\)	\(0\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+q^{3}+(\beta _{1}+\beta _{2})q^{4}-\beta _{2}q^{5}+\cdots\)
429.2.a.g	$429$	$2$	429.a	1.a	$1$	$3$	$3$	$3.426$	3.3.148.1	None	✓	$1$	$0$	\(3\)	\(3\)	\(4\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{2}+q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
429.2.a.h	$429$	$2$	429.a	1.a	$1$	$4$	$4$	$3.426$	4.4.8468.1	None	✓	$1$	$0$	\(-2\)	\(-4\)	\(0\)	\(2\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}-q^{3}+(2-\beta _{1})q^{4}+\beta _{3}q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(429))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(429)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(143))\)\(^{\oplus 2}\)