Space of modular forms of level 38, weight 4, and trivial character

• Label: 38.4.a
• Level: $38$
• Weight: $4$
• Character orbit: 38.a
• Rep. character: $\chi_{38}(1,\cdot)$
• Character field: $\Q$
• Dimension: $5$
• Newform subspaces: $3$
• Sturm bound: $20$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	38.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(20\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(38))\).

	Total	New	Old
Modular forms	17	5	12
Cusp forms	13	5	8
Eisenstein series	4	0	4

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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(6\)	\(2\)	\(4\)		\(5\)	\(2\)	\(3\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(3\)	\(1\)	\(2\)		\(2\)	\(1\)	\(1\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(4\)	\(0\)	\(4\)		\(3\)	\(0\)	\(3\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(4\)	\(2\)	\(2\)		\(3\)	\(2\)	\(1\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(10\)	\(4\)	\(6\)		\(8\)	\(4\)	\(4\)		\(2\)	\(0\)	\(2\)
Minus space		\(-\)		\(7\)	\(1\)	\(6\)		\(5\)	\(1\)	\(4\)		\(2\)	\(0\)	\(2\)

Trace form

5 * q - 2 * q^2 + 8 * q^3 + 20 * q^4 - 8 * q^5 + 20 * q^6 + 8 * q^7 - 8 * q^8 + 35 * q^9 - 20 * q^10 + 50 * q^11 + 32 * q^12 - 22 * q^13 - 88 * q^14 - 304 * q^15 + 80 * q^16 - 62 * q^17 + 22 * q^18 + 19 * q^19 - 32 * q^20 - 48 * q^21 - 168 * q^22 - 154 * q^23 + 80 * q^24 + 229 * q^25 + 112 * q^26 + 368 * q^27 + 32 * q^28 - 226 * q^29 + 8 * q^30 + 228 * q^31 - 32 * q^32 + 28 * q^33 - 196 * q^34 + 906 * q^35 + 140 * q^36 + 346 * q^37 + 114 * q^38 + 42 * q^39 - 80 * q^40 - 1098 * q^41 - 812 * q^42 + 906 * q^43 + 200 * q^44 - 884 * q^45 + 600 * q^46 - 10 * q^47 + 128 * q^48 + 1705 * q^49 + 18 * q^50 - 1924 * q^51 - 88 * q^52 - 1254 * q^53 + 20 * q^54 - 850 * q^55 - 352 * q^56 + 114 * q^57 + 464 * q^58 - 416 * q^59 - 1216 * q^60 - 380 * q^61 + 392 * q^62 + 278 * q^63 + 320 * q^64 + 272 * q^65 - 216 * q^66 - 572 * q^67 - 248 * q^68 + 2020 * q^69 + 264 * q^70 + 1248 * q^71 + 88 * q^72 + 14 * q^73 + 76 * q^74 - 84 * q^75 + 76 * q^76 - 1298 * q^77 + 2120 * q^78 + 756 * q^79 - 128 * q^80 - 139 * q^81 + 1996 * q^82 + 1544 * q^83 - 192 * q^84 + 2358 * q^85 + 896 * q^86 + 2074 * q^87 - 672 * q^88 - 1706 * q^89 - 2588 * q^90 - 688 * q^91 - 616 * q^92 - 2140 * q^93 - 1536 * q^94 - 532 * q^95 + 320 * q^96 + 854 * q^97 - 3170 * q^98 - 826 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(38))\) 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
38.4.a.a	$38$	$4$	38.a	1.a	$1$	$1$	$1$	$2.242$	\(\Q\)	None	✓	✓	✓	✓	38.4.a.a	$1$	$1$	\(-2\)	\(-2\)	\(-9\)	\(-31\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-2q^{3}+4q^{4}-9q^{5}+4q^{6}+\cdots\)
38.4.a.b	$38$	$4$	38.a	1.a	$1$	$2$	$2$	$2.242$	\(\Q(\sqrt{177}) \)	None	✓	✓	✓	✓	38.4.a.b	$1$	$0$	\(-4\)	\(1\)	\(10\)	\(57\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+(1-\beta )q^{3}+4q^{4}+(4+2\beta )q^{5}+\cdots\)
38.4.a.c	$38$	$4$	38.a	1.a	$1$	$2$	$2$	$2.242$	\(\Q(\sqrt{73}) \)	None	✓	✓	✓	✓	38.4.a.c	$1$	$0$	\(4\)	\(9\)	\(-9\)	\(-18\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+(5-\beta )q^{3}+4q^{4}+(-6+3\beta )q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(38))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(38)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 2}\)