Space of modular forms of level 208, weight 7, and Character 208.t

• Label: 208.7.t
• Level: $208$
• Weight: $7$
• Character orbit: 208.t
• Rep. character: $\chi_{208}(161,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $82$
• Newform subspaces: $6$
• Sturm bound: $196$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	208.t (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(196\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(208, [\chi])\).

	Total	New	Old
Modular forms	348	86	262
Cusp forms	324	82	242
Eisenstein series	24	4	20

Trace form

\( 82 q + 4 q^{3} - 46 q^{5} + 722 q^{7} + 18950 q^{9} + O(q^{10}) \)

Decomposition of \(S_{7}^{\mathrm{new}}(208, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
208.7.t.a	$208$	$7$	208.t	13.d	$4$	$6$	$3$	$47.851$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None			26.7.d.a	$2$	$0$	\(0\)	\(0\)	\(-150\)	\(-150\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{3}q^{3}+(-5^{2}-5^{2}\beta _{2}+5\beta _{4})q^{5}+\cdots\)
208.7.t.b	$208$	$7$	208.t	13.d	$4$	$8$	$4$	$47.851$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None			26.7.d.b	$2$	$0$	\(0\)	\(0\)	\(84\)	\(230\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{2}q^{3}+(10-\beta _{1}-\beta _{2}+10\beta _{3}+\beta _{5}+\cdots)q^{5}+\cdots\)
208.7.t.c	$208$	$7$	208.t	13.d	$4$	$12$	$6$	$47.851$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None			13.7.d.a	$2$	$0$	\(0\)	\(4\)	\(108\)	\(-398\)	$2^{9}\cdot 5^{2}\cdot 13^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{6}q^{3}+(9-9\beta _{2}-\beta _{6}+\beta _{7}-\beta _{11})q^{5}+\cdots\)
208.7.t.d	$208$	$7$	208.t	13.d	$4$	$14$	$7$	$47.851$	\(\mathbb{Q}[x]/(x^{14} + \cdots)\)	None			52.7.g.a	$2$	$0$	\(0\)	\(0\)	\(-66\)	\(320\)	$2^{19}\cdot 3^{2}\cdot 13^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{2}q^{3}+(-5-5\beta _{5}-\beta _{6})q^{5}+(23+\cdots)q^{7}+\cdots\)
208.7.t.e	$208$	$7$	208.t	13.d	$4$	$20$	$10$	$47.851$	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)	None			104.7.l.a	$2$	$0$	\(0\)	\(0\)	\(-128\)	\(170\)	$2^{33}\cdot 3^{4}\cdot 13^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{1}q^{3}+(-6+\beta _{1}-6\beta _{4}+\beta _{5}-\beta _{6}+\cdots)q^{5}+\cdots\)
208.7.t.f	$208$	$7$	208.t	13.d	$4$	$22$	$11$	$47.851$		None			104.7.l.b	$2$	$0$	\(0\)	\(0\)	\(106\)	\(550\)		$\mathrm{SU}(2)[C_{4}]$

Decomposition of \(S_{7}^{\mathrm{old}}(208, [\chi])\) into lower level spaces

\( S_{7}^{\mathrm{old}}(208, [\chi]) \simeq \) \(S_{7}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(52, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(104, [\chi])\)\(^{\oplus 2}\)