Space of modular forms of level 361, weight 2, and Character 361.e

• Label: 361.2.e
• Level: $361$
• Weight: $2$
• Character orbit: 361.e
• Rep. character: $\chi_{361}(28,\cdot)$
• Character field: $\Q(\zeta_{9})$
• Dimension: $120$
• Newform subspaces: $13$
• Sturm bound: $63$
• Trace bound: $12$

Defining parameters

Level:	\( N \)	\(=\)	\( 361 = 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	361.e (of order \(9\) and degree \(6\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 19 \)
Character field:			\(\Q(\zeta_{9})\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(63\)
Trace bound:			\(12\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(361, [\chi])\).

	Total	New	Old
Modular forms	252	216	36
Cusp forms	132	120	12
Eisenstein series	120	96	24

Trace form

\( 120 q + 6 q^{2} + 3 q^{3} + 6 q^{5} - 3 q^{6} - 3 q^{7} - 6 q^{8} - 3 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(361, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
361.2.e.a	$361$	$2$	361.e	19.e	$9$	$6$	$1$	$2.883$	\(\Q(\zeta_{18})\)	None		$2$	$0$	\(-3\)	\(-6\)	\(3\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{9}]$	\(q+(\zeta_{18}+\zeta_{18}^{2}-\zeta_{18}^{3}-\zeta_{18}^{4})q^{2}+\cdots\)
361.2.e.b	$361$	$2$	361.e	19.e	$9$	$6$	$1$	$2.883$	\(\Q(\zeta_{18})\)	None		$2$	$0$	\(-3\)	\(3\)	\(3\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{9}]$	\(q+(-1+\zeta_{18}-\zeta_{18}^{2}+\zeta_{18}^{3}-\zeta_{18}^{4}+\cdots)q^{2}+\cdots\)
361.2.e.c	$361$	$2$	361.e	19.e	$9$	$6$	$1$	$2.883$	\(\Q(\zeta_{18})\)	\(\Q(\sqrt{-19}) \)		$12$	$0$	\(0\)	\(0\)	\(0\)	\(-9\)	$1$	$\mathrm{U}(1)[D_{9}]$	\(q+2\zeta_{18}^{5}q^{4}-\zeta_{18}^{4}q^{5}+(-3+3\zeta_{18}^{3}+\cdots)q^{7}+\cdots\)
361.2.e.d	$361$	$2$	361.e	19.e	$9$	$6$	$1$	$2.883$	\(\Q(\zeta_{18})\)	None		$6$	$0$	\(0\)	\(0\)	\(0\)	\(3\)	$1$	$\mathrm{SU}(2)[C_{9}]$	\(q+2\zeta_{18}q^{3}+2\zeta_{18}^{5}q^{4}+3\zeta_{18}^{4}q^{5}+\cdots\)
361.2.e.e	$361$	$2$	361.e	19.e	$9$	$6$	$1$	$2.883$	\(\Q(\zeta_{18})\)	None		$6$	$0$	\(0\)	\(0\)	\(0\)	\(3\)	$1$	$\mathrm{SU}(2)[C_{9}]$	\(q-2\zeta_{18}q^{3}+2\zeta_{18}^{5}q^{4}+3\zeta_{18}^{4}q^{5}+\cdots\)
361.2.e.f	$361$	$2$	361.e	19.e	$9$	$6$	$1$	$2.883$	\(\Q(\zeta_{18})\)	None		$2$	$0$	\(3\)	\(-3\)	\(3\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{9}]$	\(q+(1-\zeta_{18}+\zeta_{18}^{2}-\zeta_{18}^{3}+\zeta_{18}^{4}+\cdots)q^{2}+\cdots\)
361.2.e.g	$361$	$2$	361.e	19.e	$9$	$6$	$1$	$2.883$	\(\Q(\zeta_{18})\)	None		$2$	$0$	\(3\)	\(6\)	\(3\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{9}]$	\(q+(-\zeta_{18}-\zeta_{18}^{2}+\zeta_{18}^{3}+\zeta_{18}^{4}+\cdots)q^{2}+\cdots\)
361.2.e.h	$361$	$2$	361.e	19.e	$9$	$6$	$1$	$2.883$	\(\Q(\zeta_{18})\)	None		$2$	$0$	\(6\)	\(3\)	\(-6\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{9}]$	\(q+(1+\zeta_{18}-\zeta_{18}^{4}-\zeta_{18}^{5})q^{2}+(1+\cdots)q^{3}+\cdots\)
361.2.e.i	$361$	$2$	361.e	19.e	$9$	$12$	$2$	$2.883$	12.0.\(\cdots\).1	None		$6$	$0$	\(0\)	\(0\)	\(0\)	\(-18\)	$1$	$\mathrm{SU}(2)[C_{9}]$	\(q-\beta _{7}q^{2}+(-\beta _{1}-2\beta _{3})q^{3}+(\beta _{5}+\beta _{10}+\cdots)q^{4}+\cdots\)
361.2.e.j	$361$	$2$	361.e	19.e	$9$	$12$	$2$	$2.883$	12.0.\(\cdots\).1	None		$6$	$0$	\(0\)	\(0\)	\(0\)	\(-18\)	$1$	$\mathrm{SU}(2)[C_{9}]$	\(q-\beta _{1}q^{2}+(-\beta _{1}+\beta _{7}-2\beta _{9})q^{3}+(\beta _{4}+\cdots)q^{4}+\cdots\)
361.2.e.k	$361$	$2$	361.e	19.e	$9$	$12$	$2$	$2.883$	12.0.\(\cdots\).1	None		$6$	$0$	\(0\)	\(0\)	\(0\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{9}]$	\(q+(-\beta _{5}-2\beta _{10}+\beta _{11})q^{2}+2\beta _{5}q^{3}+\cdots\)
361.2.e.l	$361$	$2$	361.e	19.e	$9$	$12$	$2$	$2.883$	12.0.\(\cdots\).1	None		$6$	$0$	\(0\)	\(0\)	\(0\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{9}]$	\(q+(-\beta _{5}-2\beta _{10}+\beta _{11})q^{2}-2\beta _{5}q^{3}+\cdots\)
361.2.e.m	$361$	$2$	361.e	19.e	$9$	$24$	$4$	$2.883$		None		$12$	$0$	\(0\)	\(0\)	\(0\)	\(24\)		$\mathrm{SU}(2)[C_{9}]$

Decomposition of \(S_{2}^{\mathrm{old}}(361, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(361, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(19, [\chi])\)\(^{\oplus 2}\)