Space of modular forms of level 10, weight 4, and trivial character

• Label: 10.4.a
• Level: $10$
• Weight: $4$
• Character orbit: 10.a
• Rep. character: $\chi_{10}(1,\cdot)$
• Character field: $\Q$
• Dimension: $1$
• Newform subspaces: $1$
• Sturm bound: $6$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 10 = 2 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	10.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(6\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(10))\).

	Total	New	Old
Modular forms	7	1	6
Cusp forms	3	1	2
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.

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

Trace form

q + 2 * q^2 - 8 * q^3 + 4 * q^4 + 5 * q^5 - 16 * q^6 - 4 * q^7 + 8 * q^8 + 37 * q^9 + 10 * q^10 + 12 * q^11 - 32 * q^12 - 58 * q^13 - 8 * q^14 - 40 * q^15 + 16 * q^16 + 66 * q^17 + 74 * q^18 - 100 * q^19 + 20 * q^20 + 32 * q^21 + 24 * q^22 + 132 * q^23 - 64 * q^24 + 25 * q^25 - 116 * q^26 - 80 * q^27 - 16 * q^28 - 90 * q^29 - 80 * q^30 + 152 * q^31 + 32 * q^32 - 96 * q^33 + 132 * q^34 - 20 * q^35 + 148 * q^36 - 34 * q^37 - 200 * q^38 + 464 * q^39 + 40 * q^40 - 438 * q^41 + 64 * q^42 + 32 * q^43 + 48 * q^44 + 185 * q^45 + 264 * q^46 - 204 * q^47 - 128 * q^48 - 327 * q^49 + 50 * q^50 - 528 * q^51 - 232 * q^52 + 222 * q^53 - 160 * q^54 + 60 * q^55 - 32 * q^56 + 800 * q^57 - 180 * q^58 + 420 * q^59 - 160 * q^60 + 902 * q^61 + 304 * q^62 - 148 * q^63 + 64 * q^64 - 290 * q^65 - 192 * q^66 - 1024 * q^67 + 264 * q^68 - 1056 * q^69 - 40 * q^70 + 432 * q^71 + 296 * q^72 + 362 * q^73 - 68 * q^74 - 200 * q^75 - 400 * q^76 - 48 * q^77 + 928 * q^78 - 160 * q^79 + 80 * q^80 - 359 * q^81 - 876 * q^82 + 72 * q^83 + 128 * q^84 + 330 * q^85 + 64 * q^86 + 720 * q^87 + 96 * q^88 + 810 * q^89 + 370 * q^90 + 232 * q^91 + 528 * q^92 - 1216 * q^93 - 408 * q^94 - 500 * q^95 - 256 * q^96 + 1106 * q^97 - 654 * q^98 + 444 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(10))\) 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	5
10.4.a.a	$10$	$4$	10.a	1.a	$1$	$1$	$1$	$0.590$	\(\Q\)	None	✓	✓	✓	✓	10.4.a.a	$1$	$0$	\(2\)	\(-8\)	\(5\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-8q^{3}+4q^{4}+5q^{5}-2^{4}q^{6}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(10))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(10)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)