Space of modular forms of level 361, weight 6, and trivial character

• Label: 361.6.a
• Level: $361$
• Weight: $6$
• Character orbit: 361.a
• Rep. character: $\chi_{361}(1,\cdot)$
• Character field: $\Q$
• Dimension: $133$
• Newform subspaces: $13$
• Sturm bound: $190$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 361 = 19^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	361.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(190\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(361))\).

	Total	New	Old
Modular forms	169	150	19
Cusp forms	149	133	16
Eisenstein series	20	17	3

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
\(+\)	\(64\)
\(-\)	\(69\)

Trace form

\( 133 q + 6 q^{2} - 2 q^{3} + 1990 q^{4} + 24 q^{5} - 106 q^{6} - 22 q^{7} + 504 q^{8} + 9251 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(361))\) 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
361.6.a.a	$361$	$6$	361.a	1.a	$1$	$1$	$1$	$57.899$	\(\Q\)	\(\Q(\sqrt{-19}) \)	✓	$2$	$1$	\(0\)	\(0\)	\(-101\)	\(33\)		$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-2^{5}q^{4}-101q^{5}+33q^{7}-3^{5}q^{9}+\cdots\)
361.6.a.b	$361$	$6$	361.a	1.a	$1$	$1$	$1$	$57.899$	\(\Q\)	None	✓	$1$	$0$	\(2\)	\(1\)	\(-24\)	\(-167\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+q^{3}-28q^{4}-24q^{5}+2q^{6}+\cdots\)
361.6.a.c	$361$	$6$	361.a	1.a	$1$	$1$	$1$	$57.899$	\(\Q\)	None	✓	$1$	$0$	\(6\)	\(-4\)	\(54\)	\(248\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+6q^{2}-4q^{3}+4q^{4}+54q^{5}-24q^{6}+\cdots\)
361.6.a.d	$361$	$6$	361.a	1.a	$1$	$2$	$2$	$57.899$	\(\Q(\sqrt{177}) \)	None	✓	$1$	$0$	\(7\)	\(7\)	\(-133\)	\(72\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(4-\beta )q^{2}+(2+3\beta )q^{3}+(28-7\beta )q^{4}+\cdots\)
361.6.a.e	$361$	$6$	361.a	1.a	$1$	$4$	$4$	$57.899$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(-9\)	\(-6\)	\(90\)	\(-190\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-3+\beta _{1}-\beta _{2})q^{2}+(-3-3\beta _{2}+\cdots)q^{3}+\cdots\)
361.6.a.f	$361$	$6$	361.a	1.a	$1$	$6$	$6$	$57.899$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(22\)	\(-266\)		$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-\beta _{2}q^{3}+(5^{2}+\beta _{3}-\beta _{4})q^{4}+\cdots\)
361.6.a.g	$361$	$6$	361.a	1.a	$1$	$8$	$8$	$57.899$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$1$	\(-3\)	\(-28\)	\(-10\)	\(104\)		$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-4-\beta _{2})q^{3}+(19+\beta _{3}+\cdots)q^{4}+\cdots\)
361.6.a.h	$361$	$6$	361.a	1.a	$1$	$8$	$8$	$57.899$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(3\)	\(28\)	\(-10\)	\(104\)		$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(4+\beta _{2})q^{3}+(19+\beta _{3})q^{4}+\cdots\)
361.6.a.i	$361$	$6$	361.a	1.a	$1$	$16$	$16$	$57.899$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓	$1$	$0$	\(-7\)	\(-1\)	\(159\)	\(116\)		$-$	$-$	$2^{10}\cdot 19^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-\beta _{6}q^{3}+(19+\beta _{2})q^{4}+(10+\cdots)q^{5}+\cdots\)
361.6.a.j	$361$	$6$	361.a	1.a	$1$	$16$	$16$	$57.899$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓	$1$	$0$	\(7\)	\(1\)	\(159\)	\(116\)		$-$	$-$	$2^{10}\cdot 19^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{6}q^{3}+(19+\beta _{2})q^{4}+(10+\cdots)q^{5}+\cdots\)
361.6.a.k	$361$	$6$	361.a	1.a	$1$	$21$	$21$	$57.899$		None	✓	$1$	$1$	\(-12\)	\(-27\)	\(-33\)	\(180\)		$+$	$+$		$\mathrm{SU}(2)$
361.6.a.l	$361$	$6$	361.a	1.a	$1$	$21$	$21$	$57.899$		None	✓	$1$	$0$	\(12\)	\(27\)	\(-33\)	\(180\)		$-$	$-$		$\mathrm{SU}(2)$
361.6.a.m	$361$	$6$	361.a	1.a	$1$	$28$	$28$	$57.899$		None	✓	$2$	$1$	\(0\)	\(0\)	\(-116\)	\(-552\)		$+$	$+$		$\mathrm{SU}(2)$

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(361))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(361)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 2}\)