Space of modular forms of level 38, weight 2, and trivial character

• Label: 38.2.a
• Level: $38$
• Weight: $2$
• Character orbit: 38.a
• Rep. character: $\chi_{38}(1,\cdot)$
• Character field: $\Q$
• Dimension: $2$
• Newform subspaces: $2$
• Sturm bound: $10$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	38.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(10\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(38))\).

	Total	New	Old
Modular forms	7	2	5
Cusp forms	4	2	2
Eisenstein series	3	0	3

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

\(2\)	\(19\)	Fricke	Dim
\(+\)	\(-\)	\(-\)	\(1\)
\(-\)	\(+\)	\(-\)	\(1\)
Plus space		\(+\)	\(0\)
Minus space		\(-\)	\(2\)

Trace form

2 * q + 2 * q^4 - 4 * q^5 - 2 * q^6 + 2 * q^7 - 4 * q^9 - 4 * q^10 - 4 * q^11 + 4 * q^13 + 4 * q^14 + 4 * q^15 + 2 * q^16 + 6 * q^17 - 4 * q^20 - 4 * q^21 + 8 * q^22 + 2 * q^23 - 2 * q^24 + 6 * q^25 - 6 * q^26 + 2 * q^28 + 4 * q^29 + 4 * q^30 - 12 * q^31 - 8 * q^33 - 12 * q^35 - 4 * q^36 - 2 * q^38 + 6 * q^39 - 4 * q^40 - 8 * q^41 - 2 * q^42 + 12 * q^43 - 4 * q^44 + 8 * q^45 - 4 * q^46 + 8 * q^47 - 4 * q^49 + 16 * q^50 + 4 * q^52 - 4 * q^53 + 10 * q^54 - 8 * q^55 + 4 * q^56 + 2 * q^57 - 14 * q^58 + 24 * q^59 + 4 * q^60 - 8 * q^61 - 4 * q^62 - 4 * q^63 + 2 * q^64 + 4 * q^65 + 4 * q^66 + 8 * q^67 + 6 * q^68 + 4 * q^69 - 12 * q^70 - 4 * q^71 + 2 * q^73 - 4 * q^74 - 16 * q^75 + 12 * q^77 - 4 * q^78 - 20 * q^79 - 4 * q^80 + 2 * q^81 - 8 * q^82 - 12 * q^83 - 4 * q^84 - 12 * q^85 - 4 * q^86 + 14 * q^87 + 8 * q^88 - 12 * q^89 + 8 * q^90 - 8 * q^91 + 2 * q^92 + 4 * q^93 + 8 * q^94 + 4 * q^95 - 2 * q^96 - 12 * q^97 + 8 * q^98 + 8 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(38))\) 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	19
38.2.a.a	$38$	$2$	38.a	1.a	$1$	$1$	$1$	$0.303$	\(\Q\)	None	✓	✓	✓	✓	38.2.a.a	$1$	$0$	\(-1\)	\(1\)	\(0\)	\(-1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{7}-q^{8}+\cdots\)
38.2.a.b	$38$	$2$	38.a	1.a	$1$	$1$	$1$	$0.303$	\(\Q\)	None	✓	✓	✓	✓	38.2.a.b	$1$	$0$	\(1\)	\(-1\)	\(-4\)	\(3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-4q^{5}-q^{6}+3q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(38))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(38)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 2}\)