Space of modular forms of level 3520, weight 2, and Character 3520.f

• Label: 3520.2.f
• Level: $3520$
• Weight: $2$
• Character orbit: 3520.f
• Rep. character: $\chi_{3520}(2111,\cdot)$
• Character field: $\Q$
• Dimension: $96$
• Newform subspaces: $13$
• Sturm bound: $1152$
• Trace bound: $33$

Defining parameters

Level:	\( N \)	\(=\)	\( 3520 = 2^{6} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3520.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 44 \)
Character field:			\(\Q\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(1152\)
Trace bound:			\(33\)
Distinguishing \(T_p\):			\(3\), \(7\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(3520, [\chi])\).

	Total	New	Old
Modular forms	600	96	504
Cusp forms	552	96	456
Eisenstein series	48	0	48

Trace form

\( 96 q - 96 q^{9} + 96 q^{25} + 16 q^{33} + 32 q^{37} + 96 q^{49} + 32 q^{53} - 16 q^{77} + 64 q^{81}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(3520, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
3520.2.f.a	$3520$	$2$	3520.f	44.c	$2$	$2$	$2$	$28.107$	\(\Q(\sqrt{-2}) \)	None					220.2.d.a	$2$	$0$	\(0\)	\(0\)	\(2\)	\(-8\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{3}+q^{5}-4q^{7}+q^{9}+(3+\beta )q^{11}+\cdots\)
3520.2.f.b	$3520$	$2$	3520.f	44.c	$2$	$2$	$2$	$28.107$	\(\Q(\sqrt{-2}) \)	None					880.2.f.a	$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{3}+q^{5}+q^{9}+(-3-\beta )q^{11}+\cdots\)
3520.2.f.c	$3520$	$2$	3520.f	44.c	$2$	$2$	$2$	$28.107$	\(\Q(\sqrt{-2}) \)	None					880.2.f.a	$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{3}+q^{5}+q^{9}+(3-\beta )q^{11}+3\beta q^{13}+\cdots\)
3520.2.f.d	$3520$	$2$	3520.f	44.c	$2$	$2$	$2$	$28.107$	\(\Q(\sqrt{-2}) \)	None					220.2.d.a	$2$	$0$	\(0\)	\(0\)	\(2\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{3}+q^{5}+4q^{7}+q^{9}+(-3+\beta )q^{11}+\cdots\)
3520.2.f.e	$3520$	$2$	3520.f	44.c	$2$	$4$	$4$	$28.107$	\(\Q(\zeta_{8})\)	None					1760.2.f.a	$4$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta_{2}-\beta_1)q^{3}-q^{5}+(\beta_{2}+\beta_1)q^{7}+\cdots\)
3520.2.f.f	$3520$	$2$	3520.f	44.c	$2$	$4$	$4$	$28.107$	\(\Q(\zeta_{8})\)	None					1760.2.f.b	$4$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta_{2}-\beta_1)q^{3}-q^{5}-q^{9}+(-2\beta_{2}+\beta_1)q^{11}+\cdots\)
3520.2.f.g	$3520$	$2$	3520.f	44.c	$2$	$4$	$4$	$28.107$	\(\Q(\sqrt{-2}, \sqrt{3})\)	None					880.2.f.c	$4$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}-q^{5}-2\beta _{2}q^{7}+q^{9}+(-2\beta _{1}+\cdots)q^{11}+\cdots\)
3520.2.f.h	$3520$	$2$	3520.f	44.c	$2$	$8$	$8$	$28.107$	8.0.170772624.1	None					880.2.f.e	$4$	$0$	\(0\)	\(0\)	\(-8\)	\(0\)	$2^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{3}-q^{5}+\beta _{2}q^{7}+(-1-\beta _{4}+\cdots)q^{9}+\cdots\)
3520.2.f.i	$3520$	$2$	3520.f	44.c	$2$	$8$	$8$	$28.107$	8.0.\(\cdots\).7	None					220.2.d.c	$4$	$0$	\(0\)	\(0\)	\(8\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{4}q^{3}+q^{5}-\beta _{3}q^{7}+(-3+\beta _{7})q^{9}+\cdots\)
3520.2.f.j	$3520$	$2$	3520.f	44.c	$2$	$8$	$8$	$28.107$	8.0.170772624.1	None					880.2.f.d	$4$	$0$	\(0\)	\(0\)	\(8\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{3}+q^{5}-\beta _{1}q^{7}+(-1-\beta _{4}+\cdots)q^{9}+\cdots\)
3520.2.f.k	$3520$	$2$	3520.f	44.c	$2$	$12$	$12$	$28.107$	12.0.\(\cdots\).1	None					220.2.d.d	$4$	$0$	\(0\)	\(0\)	\(-12\)	\(0\)	$2^{13}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}-q^{5}-\beta _{7}q^{7}+(-1+\beta _{1}+\cdots)q^{9}+\cdots\)
3520.2.f.l	$3520$	$2$	3520.f	44.c	$2$	$16$	$16$	$28.107$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None					1760.2.f.c	$4$	$0$	\(0\)	\(0\)	\(-16\)	\(0\)	$2^{22}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{4}q^{3}-q^{5}-\beta _{9}q^{7}+(-2-\beta _{1}+\cdots)q^{9}+\cdots\)
3520.2.f.m	$3520$	$2$	3520.f	44.c	$2$	$24$	$24$	$28.107$		None			✓		1760.2.f.d	$4$	$0$	\(0\)	\(0\)	\(24\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{2}^{\mathrm{old}}(3520, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(3520, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(44, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(176, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(220, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(352, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(704, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(880, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1760, [\chi])\)\(^{\oplus 2}\)