Space of modular forms of level 1352, weight 2, and Character 1352.f

• Label: 1352.2.f
• Level: $1352$
• Weight: $2$
• Character orbit: 1352.f
• Rep. character: $\chi_{1352}(337,\cdot)$
• Character field: $\Q$
• Dimension: $38$
• Newform subspaces: $7$
• Sturm bound: $364$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 1352 = 2^{3} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1352.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(364\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(3\), \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1352, [\chi])\).

	Total	New	Old
Modular forms	210	38	172
Cusp forms	154	38	116
Eisenstein series	56	0	56

Trace form

\( 38 q + 2 q^{3} + 36 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1352, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1352.2.f.a	$1352$	$2$	1352.f	13.b	$2$	$2$	$2$	$10.796$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+q^{3}+2iq^{5}+iq^{7}-2q^{9}-iq^{11}+\cdots\)
1352.2.f.b	$1352$	$2$	1352.f	13.b	$2$	$2$	$2$	$10.796$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+q^{3}-iq^{5}-5iq^{7}-2q^{9}+2iq^{11}+\cdots\)
1352.2.f.c	$1352$	$2$	1352.f	13.b	$2$	$4$	$4$	$10.796$	\(\Q(i, \sqrt{17})\)	None		$2$	$0$	\(0\)	\(2\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1-\beta _{3})q^{3}+(-\beta _{1}-\beta _{2})q^{5}+\beta _{1}q^{7}+\cdots\)
1352.2.f.d	$1352$	$2$	1352.f	13.b	$2$	$4$	$4$	$10.796$	\(\Q(i, \sqrt{17})\)	None		$2$	$0$	\(0\)	\(2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1-\beta _{3})q^{3}+(\beta _{1}-\beta _{2})q^{5}-\beta _{1}q^{7}+\cdots\)
1352.2.f.e	$1352$	$2$	1352.f	13.b	$2$	$6$	$6$	$10.796$	6.0.153664.1	None		$2$	$0$	\(0\)	\(-8\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-2+\beta _{2}+\beta _{4})q^{3}+(\beta _{3}-\beta _{5})q^{5}+\cdots\)
1352.2.f.f	$1352$	$2$	1352.f	13.b	$2$	$8$	$8$	$10.796$	8.0.195105024.2	None		$2$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1+\beta _{1})q^{3}+(\beta _{3}+\beta _{6}-\beta _{7})q^{5}+\cdots\)
1352.2.f.g	$1352$	$2$	1352.f	13.b	$2$	$12$	$12$	$10.796$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		$2$	$0$	\(0\)	\(6\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1-\beta _{4}-\beta _{6})q^{3}+(-\beta _{1}-\beta _{7}-\beta _{8}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1352, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1352, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(52, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(104, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(169, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(338, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(676, [\chi])\)\(^{\oplus 2}\)