Space of modular forms of level 30, weight 2, and trivial character

• Label: 30.2.a
• Level: $30$
• Weight: $2$
• Character orbit: 30.a
• Rep. character: $\chi_{30}(1,\cdot)$
• Character field: $\Q$
• Dimension: $1$
• Newform subspaces: $1$
• Sturm bound: $12$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 30 = 2 \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	30.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(12\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(30))\).

	Total	New	Old
Modular forms	10	1	9
Cusp forms	3	1	2
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.

\(2\)	\(3\)	\(5\)	Fricke	Dim
\(+\)	\(-\)	\(+\)	\(-\)	\(1\)
Plus space			\(+\)	\(0\)
Minus space			\(-\)	\(1\)

Trace form

q - q^2 + q^3 + q^4 - q^5 - q^6 - 4 * q^7 - q^8 + q^9 + q^10 + q^12 + 2 * q^13 + 4 * q^14 - q^15 + q^16 + 6 * q^17 - q^18 - 4 * q^19 - q^20 - 4 * q^21 - q^24 + q^25 - 2 * q^26 + q^27 - 4 * q^28 - 6 * q^29 + q^30 + 8 * q^31 - q^32 - 6 * q^34 + 4 * q^35 + q^36 + 2 * q^37 + 4 * q^38 + 2 * q^39 + q^40 - 6 * q^41 + 4 * q^42 - 4 * q^43 - q^45 + q^48 + 9 * q^49 - q^50 + 6 * q^51 + 2 * q^52 - 6 * q^53 - q^54 + 4 * q^56 - 4 * q^57 + 6 * q^58 - q^60 - 10 * q^61 - 8 * q^62 - 4 * q^63 + q^64 - 2 * q^65 - 4 * q^67 + 6 * q^68 - 4 * q^70 - q^72 + 2 * q^73 - 2 * q^74 + q^75 - 4 * q^76 - 2 * q^78 + 8 * q^79 - q^80 + q^81 + 6 * q^82 + 12 * q^83 - 4 * q^84 - 6 * q^85 + 4 * q^86 - 6 * q^87 + 18 * q^89 + q^90 - 8 * q^91 + 8 * q^93 + 4 * q^95 - q^96 + 2 * q^97 - 9 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(30))\) 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	3	5
30.2.a.a	$30$	$2$	30.a	1.a	$1$	$1$	$1$	$0.240$	\(\Q\)	None	✓	✓	✓	✓	30.2.a.a	$1$	$0$	\(-1\)	\(1\)	\(-1\)	\(-4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}-4q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(30))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(30)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 2}\)