Space of modular forms of level 1530, weight 2, and trivial character

• Label: 1530.2.a
• Level: $1530$
• Weight: $2$
• Character orbit: 1530.a
• Rep. character: $\chi_{1530}(1,\cdot)$
• Character field: $\Q$
• Dimension: $24$
• Newform subspaces: $20$
• Sturm bound: $648$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 1530 = 2 \cdot 3^{2} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1530.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 20 \)
Sturm bound:			\(648\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1530))\).

	Total	New	Old
Modular forms	340	24	316
Cusp forms	309	24	285
Eisenstein series	31	0	31

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

\(2\)	\(3\)	\(5\)	\(17\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	$+$	\(2\)
\(+\)	\(+\)	\(+\)	\(-\)	$-$	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)	$+$	\(1\)
\(+\)	\(-\)	\(+\)	\(+\)	$-$	\(3\)
\(+\)	\(-\)	\(+\)	\(-\)	$+$	\(1\)
\(+\)	\(-\)	\(-\)	\(+\)	$+$	\(2\)
\(+\)	\(-\)	\(-\)	\(-\)	$-$	\(2\)
\(-\)	\(+\)	\(+\)	\(+\)	$-$	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)	$+$	\(1\)
\(-\)	\(+\)	\(-\)	\(-\)	$-$	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)	$+$	\(2\)
\(-\)	\(-\)	\(+\)	\(-\)	$-$	\(3\)
\(-\)	\(-\)	\(-\)	\(+\)	$-$	\(3\)
Plus space				\(+\)	\(9\)
Minus space				\(-\)	\(15\)

Trace form

\( 24 q + 24 q^{4} - 2 q^{5} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1530))\) 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	17
1530.2.a.a	$1530$	$2$	1530.a	1.a	$1$	$1$	$1$	$12.217$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(-1\)	\(-4\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{5}-4q^{7}-q^{8}+q^{10}+\cdots\)
1530.2.a.b	$1530$	$2$	1530.a	1.a	$1$	$1$	$1$	$12.217$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(-1\)	\(0\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{5}-q^{8}+q^{10}-4q^{11}+\cdots\)
1530.2.a.c	$1530$	$2$	1530.a	1.a	$1$	$1$	$1$	$12.217$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(-1\)	\(2\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{5}+2q^{7}-q^{8}+q^{10}+\cdots\)
1530.2.a.d	$1530$	$2$	1530.a	1.a	$1$	$1$	$1$	$12.217$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(1\)	\(-4\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{5}-4q^{7}-q^{8}-q^{10}+\cdots\)
1530.2.a.e	$1530$	$2$	1530.a	1.a	$1$	$1$	$1$	$12.217$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(1\)	\(-2\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{5}-2q^{7}-q^{8}-q^{10}+\cdots\)
1530.2.a.f	$1530$	$2$	1530.a	1.a	$1$	$1$	$1$	$12.217$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(1\)	\(0\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{5}-q^{8}-q^{10}-4q^{11}+\cdots\)
1530.2.a.g	$1530$	$2$	1530.a	1.a	$1$	$1$	$1$	$12.217$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(1\)	\(2\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{5}+2q^{7}-q^{8}-q^{10}+\cdots\)
1530.2.a.h	$1530$	$2$	1530.a	1.a	$1$	$1$	$1$	$12.217$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(1\)	\(2\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{5}+2q^{7}-q^{8}-q^{10}+\cdots\)
1530.2.a.i	$1530$	$2$	1530.a	1.a	$1$	$1$	$1$	$12.217$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(0\)	\(-1\)	\(-2\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{5}-2q^{7}+q^{8}-q^{10}+\cdots\)
1530.2.a.j	$1530$	$2$	1530.a	1.a	$1$	$1$	$1$	$12.217$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(0\)	\(-1\)	\(-2\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{5}-2q^{7}+q^{8}-q^{10}+\cdots\)
1530.2.a.k	$1530$	$2$	1530.a	1.a	$1$	$1$	$1$	$12.217$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(-1\)	\(-2\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{5}-2q^{7}+q^{8}-q^{10}+\cdots\)
1530.2.a.l	$1530$	$2$	1530.a	1.a	$1$	$1$	$1$	$12.217$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(-1\)	\(2\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{5}+2q^{7}+q^{8}-q^{10}+\cdots\)
1530.2.a.m	$1530$	$2$	1530.a	1.a	$1$	$1$	$1$	$12.217$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(0\)	\(1\)	\(-4\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{5}-4q^{7}+q^{8}+q^{10}+\cdots\)
1530.2.a.n	$1530$	$2$	1530.a	1.a	$1$	$1$	$1$	$12.217$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(1\)	\(2\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{5}+2q^{7}+q^{8}+q^{10}+\cdots\)
1530.2.a.o	$1530$	$2$	1530.a	1.a	$1$	$1$	$1$	$12.217$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(1\)	\(2\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{5}+2q^{7}+q^{8}+q^{10}+\cdots\)
1530.2.a.p	$1530$	$2$	1530.a	1.a	$1$	$1$	$1$	$12.217$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(1\)	\(2\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{5}+2q^{7}+q^{8}+q^{10}+\cdots\)
1530.2.a.q	$1530$	$2$	1530.a	1.a	$1$	$2$	$2$	$12.217$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(-2\)	\(0\)	\(-2\)	\(2\)		$+$	$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{5}+(1+\beta )q^{7}-q^{8}+\cdots\)
1530.2.a.r	$1530$	$2$	1530.a	1.a	$1$	$2$	$2$	$12.217$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(-2\)	\(0\)	\(-2\)	\(2\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{5}+2\beta q^{7}-q^{8}+q^{10}+\cdots\)
1530.2.a.s	$1530$	$2$	1530.a	1.a	$1$	$2$	$2$	$12.217$	\(\Q(\sqrt{6}) \)	None	✓	$1$	$0$	\(2\)	\(0\)	\(-2\)	\(0\)		$-$	$-$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{5}+\beta q^{7}+q^{8}-q^{10}+\cdots\)
1530.2.a.t	$1530$	$2$	1530.a	1.a	$1$	$2$	$2$	$12.217$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(2\)	\(0\)	\(2\)	\(2\)		$-$	$+$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{5}+(1+\beta )q^{7}+q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1530))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1530)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(51))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(85))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(102))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(153))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(170))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(255))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(306))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(510))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(765))\)\(^{\oplus 2}\)