Space of modular forms of level 30, weight 8, and trivial character

• Label: 30.8.a
• Level: $30$
• Weight: $8$
• Character orbit: 30.a
• Rep. character: $\chi_{30}(1,\cdot)$
• Character field: $\Q$
• Dimension: $6$
• Newform subspaces: $6$
• Sturm bound: $48$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 30 = 2 \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	30.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(48\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(30))\).

	Total	New	Old
Modular forms	46	6	40
Cusp forms	38	6	32
Eisenstein series	8	0	8

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
\(+\)	\(+\)	\(+\)	$+$	\(1\)
\(+\)	\(+\)	\(-\)	$-$	\(1\)
\(+\)	\(-\)	\(-\)	$+$	\(1\)
\(-\)	\(+\)	\(+\)	$-$	\(1\)
\(-\)	\(+\)	\(-\)	$+$	\(1\)
\(-\)	\(-\)	\(+\)	$+$	\(1\)
Plus space			\(+\)	\(4\)
Minus space			\(-\)	\(2\)

Trace form

\( 6 q - 54 q^{3} + 384 q^{4} + 228 q^{7} + 4374 q^{9} + O(q^{10}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(30))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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
30.8.a.a	$30$	$8$	30.a	1.a	$1$	$1$	$1$	$9.372$	\(\Q\)	None	✓	$1$	$0$	\(-8\)	\(-27\)	\(-125\)	\(-1084\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}-3^{3}q^{3}+2^{6}q^{4}-5^{3}q^{5}+6^{3}q^{6}+\cdots\)
30.8.a.b	$30$	$8$	30.a	1.a	$1$	$1$	$1$	$9.372$	\(\Q\)	None	✓	$1$	$1$	\(-8\)	\(-27\)	\(125\)	\(416\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}-3^{3}q^{3}+2^{6}q^{4}+5^{3}q^{5}+6^{3}q^{6}+\cdots\)
30.8.a.c	$30$	$8$	30.a	1.a	$1$	$1$	$1$	$9.372$	\(\Q\)	None	✓	$1$	$0$	\(-8\)	\(27\)	\(125\)	\(-232\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}+3^{3}q^{3}+2^{6}q^{4}+5^{3}q^{5}-6^{3}q^{6}+\cdots\)
30.8.a.d	$30$	$8$	30.a	1.a	$1$	$1$	$1$	$9.372$	\(\Q\)	None	✓	$1$	$1$	\(8\)	\(-27\)	\(-125\)	\(-988\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}-3^{3}q^{3}+2^{6}q^{4}-5^{3}q^{5}-6^{3}q^{6}+\cdots\)
30.8.a.e	$30$	$8$	30.a	1.a	$1$	$1$	$1$	$9.372$	\(\Q\)	None	✓	$1$	$0$	\(8\)	\(-27\)	\(125\)	\(512\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}-3^{3}q^{3}+2^{6}q^{4}+5^{3}q^{5}-6^{3}q^{6}+\cdots\)
30.8.a.f	$30$	$8$	30.a	1.a	$1$	$1$	$1$	$9.372$	\(\Q\)	None	✓	$1$	$0$	\(8\)	\(27\)	\(-125\)	\(1604\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}+3^{3}q^{3}+2^{6}q^{4}-5^{3}q^{5}+6^{3}q^{6}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(30))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(30)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 2}\)