Space of modular forms of level 423, weight 2, and trivial character

• Label: 423.2.a
• Level: $423$
• Weight: $2$
• Character orbit: 423.a
• Rep. character: $\chi_{423}(1,\cdot)$
• Character field: $\Q$
• Dimension: $19$
• Newform subspaces: $11$
• Sturm bound: $96$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 423 = 3^{2} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	423.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(96\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(423))\).

	Total	New	Old
Modular forms	52	19	33
Cusp forms	45	19	26
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.

\(3\)	\(47\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(4\)
\(+\)	\(-\)	$-$	\(4\)
\(-\)	\(+\)	$-$	\(8\)
\(-\)	\(-\)	$+$	\(3\)
Plus space		\(+\)	\(7\)
Minus space		\(-\)	\(12\)

Trace form

\( 19 q + 2 q^{2} + 20 q^{4} + 4 q^{5} - 4 q^{7} + 6 q^{8} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(423))\) 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	47
423.2.a.a	$423$	$2$	423.a	1.a	$1$	$1$	$1$	$3.378$	\(\Q\)	None	✓	$1$	$1$	\(-2\)	\(0\)	\(-3\)	\(1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}-3q^{5}+q^{7}+6q^{10}+\cdots\)
423.2.a.b	$423$	$2$	423.a	1.a	$1$	$1$	$1$	$3.378$	\(\Q\)	None	✓	$1$	$1$	\(-2\)	\(0\)	\(1\)	\(-3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}+q^{5}-3q^{7}-2q^{10}+\cdots\)
423.2.a.c	$423$	$2$	423.a	1.a	$1$	$1$	$1$	$3.378$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}+q^{5}-3q^{7}+3q^{11}-4q^{13}+\cdots\)
423.2.a.d	$423$	$2$	423.a	1.a	$1$	$1$	$1$	$3.378$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(0\)	\(-2\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-2q^{5}-3q^{8}-2q^{10}+\cdots\)
423.2.a.e	$423$	$2$	423.a	1.a	$1$	$1$	$1$	$3.378$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}+4q^{7}-3q^{8}+6q^{13}+\cdots\)
423.2.a.f	$423$	$2$	423.a	1.a	$1$	$1$	$1$	$3.378$	\(\Q\)	None	✓	$1$	$0$	\(2\)	\(0\)	\(3\)	\(-3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}+3q^{5}-3q^{7}+6q^{10}+\cdots\)
423.2.a.g	$423$	$2$	423.a	1.a	$1$	$1$	$1$	$3.378$	\(\Q\)	None	✓	$1$	$0$	\(2\)	\(0\)	\(3\)	\(1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}+3q^{5}+q^{7}+6q^{10}+\cdots\)
423.2.a.h	$423$	$2$	423.a	1.a	$1$	$2$	$2$	$3.378$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(1\)	\(0\)	\(-1\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(2+\beta )q^{4}+(-1+\beta )q^{5}+(1+\cdots)q^{7}+\cdots\)
423.2.a.i	$423$	$2$	423.a	1.a	$1$	$3$	$3$	$3.378$	3.3.316.1	None	✓	$1$	$1$	\(-2\)	\(0\)	\(-1\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(2-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
423.2.a.j	$423$	$2$	423.a	1.a	$1$	$3$	$3$	$3.378$	3.3.316.1	None	✓	$1$	$0$	\(2\)	\(0\)	\(1\)	\(-3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(2-2\beta _{1}+\beta _{2})q^{4}+\beta _{1}q^{5}+\cdots\)
423.2.a.k	$423$	$2$	423.a	1.a	$1$	$4$	$4$	$3.378$	4.4.1957.1	None	✓	$1$	$0$	\(-1\)	\(0\)	\(2\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{2}+(1-\beta _{1}+\beta _{2})q^{4}+(-2\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(423))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(423)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(47))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(141))\)\(^{\oplus 2}\)