Space of modular forms of level 424, weight 2, and trivial character

• Label: 424.2.a
• Level: $424$
• Weight: $2$
• Character orbit: 424.a
• Rep. character: $\chi_{424}(1,\cdot)$
• Character field: $\Q$
• Dimension: $13$
• Newform subspaces: $4$
• Sturm bound: $108$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 424 = 2^{3} \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	424.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(108\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(424))\).

	Total	New	Old
Modular forms	58	13	45
Cusp forms	51	13	38
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.

\(2\)	\(53\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(13\)	\(2\)	\(11\)		\(12\)	\(2\)	\(10\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(16\)	\(5\)	\(11\)		\(14\)	\(5\)	\(9\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)		\(16\)	\(3\)	\(13\)		\(14\)	\(3\)	\(11\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)		\(13\)	\(3\)	\(10\)		\(11\)	\(3\)	\(8\)		\(2\)	\(0\)	\(2\)
Plus space		\(+\)		\(26\)	\(5\)	\(21\)		\(23\)	\(5\)	\(18\)		\(3\)	\(0\)	\(3\)
Minus space		\(-\)		\(32\)	\(8\)	\(24\)		\(28\)	\(8\)	\(20\)		\(4\)	\(0\)	\(4\)

Trace form

13 * q + 2 * q^5 - 4 * q^7 + 11 * q^9 + 2 * q^11 - 4 * q^13 + 8 * q^15 - 6 * q^17 - 4 * q^19 - 8 * q^21 + 21 * q^25 + 12 * q^27 - 12 * q^29 - 4 * q^31 - 8 * q^33 - 4 * q^35 + 12 * q^37 - 12 * q^39 + 6 * q^41 - 2 * q^43 + 10 * q^45 + 4 * q^47 + 29 * q^49 - 8 * q^51 + 3 * q^53 - 28 * q^55 - 10 * q^57 + 26 * q^59 + 14 * q^61 - 8 * q^63 - 4 * q^65 - 24 * q^67 + 22 * q^69 - 32 * q^71 - 18 * q^73 + 32 * q^75 + 24 * q^77 - 20 * q^79 - 19 * q^81 - 8 * q^83 - 24 * q^85 - 40 * q^87 - 8 * q^89 - 44 * q^91 - 28 * q^93 + 44 * q^95 - 14 * q^97 - 2 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(424))\) 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					A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	53
424.2.a.a	$424$	$2$	424.a	1.a	$1$	$2$	$2$	$3.386$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓	✓	424.2.a.a	$1$	$1$	\(0\)	\(-2\)	\(-4\)	\(4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}-2q^{5}+(2-2\beta )q^{7}+\cdots\)
424.2.a.b	$424$	$2$	424.a	1.a	$1$	$3$	$3$	$3.386$	3.3.148.1	None	✓	✓	✓	✓	424.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(-2\)	\(-8\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1+\beta _{1}-\beta _{2})q^{5}+(-3+\cdots)q^{7}+\cdots\)
424.2.a.c	$424$	$2$	424.a	1.a	$1$	$3$	$3$	$3.386$	3.3.316.1	None	✓	✓	✓	✓	424.2.a.c	$1$	$0$	\(0\)	\(2\)	\(3\)	\(6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(1-\beta _{2})q^{5}+2q^{7}+(1+\cdots)q^{9}+\cdots\)
424.2.a.d	$424$	$2$	424.a	1.a	$1$	$5$	$5$	$3.386$	5.5.4321108.1	None	✓	✓	✓	✓	424.2.a.d	$1$	$0$	\(0\)	\(1\)	\(5\)	\(-6\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1+\beta _{3})q^{5}+(-1-\beta _{2}-\beta _{4})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(424))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(424)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(53))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(106))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(212))\)\(^{\oplus 2}\)