Space of modular forms of level 361, weight 2, and trivial character

• Label: 361.2.a
• Level: $361$
• Weight: $2$
• Character orbit: 361.a
• Rep. character: $\chi_{361}(1,\cdot)$
• Character field: $\Q$
• Dimension: $20$
• Newform subspaces: $9$
• Sturm bound: $63$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 361 = 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	361.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(63\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(361))\).

	Total	New	Old
Modular forms	41	37	4
Cusp forms	22	20	2
Eisenstein series	19	17	2

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

\(19\)	Dim
\(+\)	\(8\)
\(-\)	\(12\)

Trace form

\( 20 q + 2 q^{3} + 14 q^{4} - 2 q^{5} + 2 q^{7} + 2 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(361))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		19
361.2.a.a	$361$	$2$	361.a	1.a	$1$	$1$	$1$	$2.883$	\(\Q\)	\(\Q(\sqrt{-19}) \)	✓	$2$	$1$	\(0\)	\(0\)	\(-1\)	\(3\)		$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-2q^{4}-q^{5}+3q^{7}-3q^{9}-5q^{11}+\cdots\)
361.2.a.b	$361$	$2$	361.a	1.a	$1$	$1$	$1$	$2.883$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(2\)	\(3\)	\(-1\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-2q^{4}+3q^{5}-q^{7}+q^{9}+3q^{11}+\cdots\)
361.2.a.c	$361$	$2$	361.a	1.a	$1$	$2$	$2$	$2.883$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(-1\)	\(-3\)	\(2\)	\(6\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-2+\beta )q^{3}+(-1+\beta )q^{4}+\cdots\)
361.2.a.d	$361$	$2$	361.a	1.a	$1$	$2$	$2$	$2.883$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(0\)	\(-4\)	\(1\)	\(-2\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-2\beta )q^{2}-2q^{3}+3q^{4}+(1-\beta )q^{5}+\cdots\)
361.2.a.e	$361$	$2$	361.a	1.a	$1$	$2$	$2$	$2.883$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(0\)	\(4\)	\(1\)	\(-2\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-2\beta )q^{2}+2q^{3}+3q^{4}+\beta q^{5}+\cdots\)
361.2.a.f	$361$	$2$	361.a	1.a	$1$	$2$	$2$	$2.883$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(1\)	\(3\)	\(2\)	\(6\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(2-\beta )q^{3}+(-1+\beta )q^{4}+2\beta q^{5}+\cdots\)
361.2.a.g	$361$	$2$	361.a	1.a	$1$	$3$	$3$	$2.883$	\(\Q(\zeta_{18})^+\)	None	✓	$1$	$1$	\(-3\)	\(-3\)	\(-3\)	\(0\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(-1+\beta _{2})q^{3}+(1+\cdots)q^{4}+\cdots\)
361.2.a.h	$361$	$2$	361.a	1.a	$1$	$3$	$3$	$2.883$	\(\Q(\zeta_{18})^+\)	None	✓	$1$	$0$	\(3\)	\(3\)	\(-3\)	\(0\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(1-\beta _{2})q^{3}+(1-2\beta _{1}+\cdots)q^{4}+\cdots\)
361.2.a.i	$361$	$2$	361.a	1.a	$1$	$4$	$4$	$2.883$	\(\Q(\zeta_{20})^+\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(-4\)	\(-8\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-\beta _{1}q^{3}+(1+\beta _{2})q^{4}+(-2+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(361))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(361)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 2}\)