Space of modular forms of level 1520, weight 2, and trivial character

• Label: 1520.2.a
• Level: $1520$
• Weight: $2$
• Character orbit: 1520.a
• Rep. character: $\chi_{1520}(1,\cdot)$
• Character field: $\Q$
• Dimension: $36$
• Newform subspaces: $20$
• Sturm bound: $480$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1520.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 20 \)
Sturm bound:			\(480\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1520))\).

	Total	New	Old
Modular forms	252	36	216
Cusp forms	229	36	193
Eisenstein series	23	0	23

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

\(2\)	\(5\)	\(19\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(5\)
\(+\)	\(+\)	\(-\)	$-$	\(5\)
\(+\)	\(-\)	\(+\)	$-$	\(4\)
\(+\)	\(-\)	\(-\)	$+$	\(4\)
\(-\)	\(+\)	\(+\)	$-$	\(6\)
\(-\)	\(+\)	\(-\)	$+$	\(2\)
\(-\)	\(-\)	\(+\)	$+$	\(3\)
\(-\)	\(-\)	\(-\)	$-$	\(7\)
Plus space			\(+\)	\(14\)
Minus space			\(-\)	\(22\)

Trace form

\( 36 q + 4 q^{7} + 36 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1520))\) 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	5	19
1520.2.a.a	$1520$	$2$	1520.a	1.a	$1$	$1$	$1$	$12.137$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-3\)	\(1\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+q^{5}+q^{7}+6q^{9}-4q^{11}+\cdots\)
1520.2.a.b	$1520$	$2$	1520.a	1.a	$1$	$1$	$1$	$12.137$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(-1\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-q^{5}-2q^{7}+q^{9}+6q^{13}+\cdots\)
1520.2.a.c	$1520$	$2$	1520.a	1.a	$1$	$1$	$1$	$12.137$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(1\)	\(-4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{5}-4q^{7}+q^{9}+4q^{11}+\cdots\)
1520.2.a.d	$1520$	$2$	1520.a	1.a	$1$	$1$	$1$	$12.137$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+q^{7}-2q^{9}-q^{13}-q^{15}+\cdots\)
1520.2.a.e	$1520$	$2$	1520.a	1.a	$1$	$1$	$1$	$12.137$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+2q^{7}-3q^{9}+4q^{11}-4q^{13}+\cdots\)
1520.2.a.f	$1520$	$2$	1520.a	1.a	$1$	$1$	$1$	$12.137$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(1\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-3q^{9}+4q^{11}-6q^{13}-6q^{17}+\cdots\)
1520.2.a.g	$1520$	$2$	1520.a	1.a	$1$	$1$	$1$	$12.137$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+q^{7}-2q^{9}-3q^{13}-q^{15}+\cdots\)
1520.2.a.h	$1520$	$2$	1520.a	1.a	$1$	$1$	$1$	$12.137$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(2\)	\(1\)	\(-4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{5}-4q^{7}+q^{9}-4q^{11}+\cdots\)
1520.2.a.i	$1520$	$2$	1520.a	1.a	$1$	$1$	$1$	$12.137$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(2\)	\(1\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{5}+q^{9}+4q^{11}+4q^{13}+\cdots\)
1520.2.a.j	$1520$	$2$	1520.a	1.a	$1$	$1$	$1$	$12.137$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(3\)	\(-1\)	\(5\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-q^{5}+5q^{7}+6q^{9}+4q^{11}+\cdots\)
1520.2.a.k	$1520$	$2$	1520.a	1.a	$1$	$2$	$2$	$12.137$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(-2\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}-q^{5}+(1-2\beta )q^{9}-2q^{11}+\cdots\)
1520.2.a.l	$1520$	$2$	1520.a	1.a	$1$	$2$	$2$	$12.137$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(2\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+q^{5}-2q^{7}+(1-2\beta )q^{9}+\cdots\)
1520.2.a.m	$1520$	$2$	1520.a	1.a	$1$	$2$	$2$	$12.137$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-q^{5}+2\beta q^{7}-q^{9}+(-2+2\beta )q^{11}+\cdots\)
1520.2.a.n	$1520$	$2$	1520.a	1.a	$1$	$2$	$2$	$12.137$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(0\)	\(1\)	\(2\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+q^{5}+\beta q^{7}+(1+\beta )q^{9}-4q^{11}+\cdots\)
1520.2.a.o	$1520$	$2$	1520.a	1.a	$1$	$2$	$2$	$12.137$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(0\)	\(4\)	\(2\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{3}+q^{5}+(2-2\beta )q^{7}+(3+4\beta )q^{9}+\cdots\)
1520.2.a.p	$1520$	$2$	1520.a	1.a	$1$	$3$	$3$	$12.137$	3.3.148.1	None	✓	$1$	$0$	\(0\)	\(-2\)	\(3\)	\(0\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+q^{5}+(-\beta _{1}+\beta _{2})q^{7}+\cdots\)
1520.2.a.q	$1520$	$2$	1520.a	1.a	$1$	$3$	$3$	$12.137$	3.3.316.1	None	✓	$1$	$1$	\(0\)	\(1\)	\(-3\)	\(1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-q^{5}+(1-2\beta _{1}+\beta _{2})q^{7}+\cdots\)
1520.2.a.r	$1520$	$2$	1520.a	1.a	$1$	$3$	$3$	$12.137$	3.3.568.1	None	✓	$1$	$0$	\(0\)	\(1\)	\(-3\)	\(5\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-q^{5}+(2+\beta _{2})q^{7}+(1+2\beta _{1}+\cdots)q^{9}+\cdots\)
1520.2.a.s	$1520$	$2$	1520.a	1.a	$1$	$3$	$3$	$12.137$	3.3.229.1	None	✓	$1$	$0$	\(0\)	\(1\)	\(3\)	\(1\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+q^{5}-\beta _{1}q^{7}+(1-\beta _{1})q^{9}+\cdots\)
1520.2.a.t	$1520$	$2$	1520.a	1.a	$1$	$4$	$4$	$12.137$	4.4.11344.1	None	✓	$1$	$0$	\(0\)	\(-2\)	\(-4\)	\(-4\)		$-$	$+$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}-q^{5}+(-1+\beta _{1})q^{7}+(2-\beta _{3})q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1520))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1520)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(95))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(152))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(190))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(304))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(380))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(760))\)\(^{\oplus 2}\)