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

• Label: 208.4.i
• Level: $208$
• Weight: $4$
• Character orbit: 208.i
• Rep. character: $\chi_{208}(81,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $40$
• Newform subspaces: $8$
• Sturm bound: $112$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	208.i (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 8 \)
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

\( 40 q - 5 q^{3} - 2 q^{5} - 17 q^{7} - 163 q^{9} + O(q^{10}) \)

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.i.a	$208$	$4$	208.i	13.c	$3$	$2$	$1$	$12.272$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(-3\)	\(4\)	\(-5\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-3+3\zeta_{6})q^{3}+2q^{5}-5\zeta_{6}q^{7}+\cdots\)
208.4.i.b	$208$	$4$	208.i	13.c	$3$	$2$	$1$	$12.272$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(2\)	\(34\)	\(20\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2-2\zeta_{6})q^{3}+17q^{5}+20\zeta_{6}q^{7}+\cdots\)
208.4.i.c	$208$	$4$	208.i	13.c	$3$	$2$	$1$	$12.272$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(8\)	\(-18\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(8-8\zeta_{6})q^{3}-9q^{5}-4\zeta_{6}q^{7}-37\zeta_{6}q^{9}+\cdots\)
208.4.i.d	$208$	$4$	208.i	13.c	$3$	$4$	$2$	$12.272$	\(\Q(\sqrt{-3}, \sqrt{217})\)	None		$2$	$0$	\(0\)	\(-3\)	\(-14\)	\(-45\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2+\beta _{1}+\beta _{2}+\beta _{3})q^{3}+(-3-\beta _{3})q^{5}+\cdots\)
208.4.i.e	$208$	$4$	208.i	13.c	$3$	$4$	$2$	$12.272$	\(\Q(\sqrt{-3}, \sqrt{17})\)	None		$2$	$0$	\(0\)	\(5\)	\(-30\)	\(15\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(3\beta _{1}+\beta _{2})q^{3}+(-10-5\beta _{3})q^{5}+\cdots\)
208.4.i.f	$208$	$4$	208.i	13.c	$3$	$6$	$3$	$12.272$	6.0.6622206867.2	None		$2$	$0$	\(0\)	\(0\)	\(-10\)	\(16\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{1}+\beta _{5})q^{3}+(-2-\beta _{1}+\beta _{2}-\beta _{4}+\cdots)q^{5}+\cdots\)
208.4.i.g	$208$	$4$	208.i	13.c	$3$	$8$	$4$	$12.272$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$2$	$0$	\(0\)	\(-11\)	\(14\)	\(-15\)	$2^{6}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-3+\beta _{1}+3\beta _{2})q^{3}+(2-\beta _{4})q^{5}+\cdots\)
208.4.i.h	$208$	$4$	208.i	13.c	$3$	$12$	$6$	$12.272$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$2$	$0$	\(0\)	\(-3\)	\(18\)	\(1\)	$2^{12}\cdot 3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\beta _{1}-\beta _{2})q^{3}+(2+\beta _{3}-\beta _{4}+\cdots)q^{5}+\cdots\)

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

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