Space of modular forms of level 208, weight 4, and Character 208.w

• Label: 208.4.w
• Level: $208$
• Weight: $4$
• Character orbit: 208.w
• Rep. character: $\chi_{208}(17,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $40$
• Newform subspaces: $5$
• Sturm bound: $112$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	208.w (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(112\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(208, [\chi])\).

	Total	New	Old
Modular forms	180	44	136
Cusp forms	156	40	116
Eisenstein series	24	4	20

Trace form

\( 40 q + 7 q^{3} + 57 q^{7} - 163 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(208, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
208.4.w.a	$208$	$4$	208.w	13.e	$6$	$2$	$1$	$12.272$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(-7\)	\(0\)	\(-39\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-7+7\zeta_{6})q^{3}+(-8+2^{4}\zeta_{6})q^{5}+\cdots\)
208.4.w.b	$208$	$4$	208.w	13.e	$6$	$2$	$1$	$12.272$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(2\)	\(0\)	\(24\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(2-2\zeta_{6})q^{3}+(1-2\zeta_{6})q^{5}+(2^{4}-8\zeta_{6})q^{7}+\cdots\)
208.4.w.c	$208$	$4$	208.w	13.e	$6$	$8$	$4$	$12.272$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(36\)	$2^{6}\cdot 3$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{5}q^{3}+(2-3\beta _{1}+\beta _{2}-\beta _{4})q^{5}+\cdots\)
208.4.w.d	$208$	$4$	208.w	13.e	$6$	$8$	$4$	$12.272$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		$2$	$0$	\(0\)	\(6\)	\(0\)	\(-18\)	$2^{4}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(1+\beta _{1}-\beta _{6})q^{3}+(2+4\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
208.4.w.e	$208$	$4$	208.w	13.e	$6$	$20$	$10$	$12.272$	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)	None		$2$	$0$	\(0\)	\(6\)	\(0\)	\(54\)	$2^{40}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(1-\beta _{1}+\beta _{2})q^{3}+(-1+\beta _{1}-\beta _{7}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(208, [\chi])\) into lower level spaces

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