Space of modular forms of level 208, weight 8, and Character 208.w

• Label: 208.8.w
• Level: $208$
• Weight: $8$
• Character orbit: 208.w
• Rep. character: $\chi_{208}(17,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $96$
• Newform subspaces: $4$
• Sturm bound: $224$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	208.w (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(224\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(208, [\chi])\).

	Total	New	Old
Modular forms	404	100	304
Cusp forms	380	96	284
Eisenstein series	24	4	20

Trace form

96 * q + 55 * q^3 - 3015 * q^7 - 33535 * q^9 + 3 * q^11 - 3279 * q^13 - 6558 * q^15 + 726 * q^17 + 3 * q^19 + 73003 * q^23 - 1366942 * q^25 - 81254 * q^27 - 30212 * q^29 - 3 * q^33 + 436376 * q^35 + 310392 * q^37 - 158893 * q^39 + 330960 * q^41 - 196721 * q^43 + 1754337 * q^45 + 4676369 * q^49 + 1168126 * q^51 - 1692706 * q^53 - 3149528 * q^55 - 4709631 * q^59 - 1134430 * q^61 + 8793930 * q^63 + 3637379 * q^65 + 2765913 * q^67 + 1761981 * q^69 - 13600185 * q^71 - 8797315 * q^75 - 9603258 * q^77 + 5940128 * q^79 - 24101100 * q^81 + 8720469 * q^85 - 14323943 * q^87 - 5202435 * q^89 + 15401729 * q^91 + 17647590 * q^93 + 24189314 * q^95 - 4945755 * q^97

Decomposition of \(S_{8}^{\mathrm{new}}(208, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
208.8.w.a	$208$	$8$	208.w	13.e	$6$	$14$	$7$	$64.976$	\(\mathbb{Q}[x]/(x^{14} + \cdots)\)	None					13.8.e.a	$2$	$0$	\(0\)	\(-26\)	\(0\)	\(2772\)	$2^{22}\cdot 3^{3}\cdot 13^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-4\beta _{1}-\beta _{4}+\beta _{6})q^{3}+(-2^{5}+2^{6}\beta _{1}+\cdots)q^{5}+\cdots\)
208.8.w.b	$208$	$8$	208.w	13.e	$6$	$16$	$8$	$64.976$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None					26.8.e.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-2520\)	$2^{36}\cdot 3^{4}\cdot 13^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{3}q^{3}+(11+21\beta _{1}+\beta _{2}+\beta _{3}+\beta _{7}+\cdots)q^{5}+\cdots\)
208.8.w.c	$208$	$8$	208.w	13.e	$6$	$18$	$9$	$64.976$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None					52.8.h.a	$2$	$0$	\(0\)	\(27\)	\(0\)	\(-249\)	$2^{32}\cdot 3^{9}\cdot 13^{5}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(3-3\beta _{2}+\beta _{3})q^{3}+(-9-\beta _{1}+19\beta _{2}+\cdots)q^{5}+\cdots\)
208.8.w.d	$208$	$8$	208.w	13.e	$6$	$48$	$24$	$64.976$		None			✓		104.8.o.a	$2$	$0$	\(0\)	\(54\)	\(0\)	\(-3018\)		$\mathrm{SU}(2)[C_{6}]$

Decomposition of \(S_{8}^{\mathrm{old}}(208, [\chi])\) into lower level spaces

\( S_{8}^{\mathrm{old}}(208, [\chi]) \simeq \) \(S_{8}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(52, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(104, [\chi])\)\(^{\oplus 2}\)