Space of modular forms of level 208, weight 10, and Character 208.f

• Label: 208.10.f
• Level: $208$
• Weight: $10$
• Character orbit: 208.f
• Rep. character: $\chi_{208}(129,\cdot)$
• Character field: $\Q$
• Dimension: $62$
• Newform subspaces: $4$
• Sturm bound: $280$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	208.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(280\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(208, [\chi])\).

	Total	New	Old
Modular forms	258	64	194
Cusp forms	246	62	184
Eisenstein series	12	2	10

Trace form

62 * q + 164 * q^3 + 393658 * q^9 + 43078 * q^13 - 102000 * q^17 + 1679048 * q^23 - 23517134 * q^25 + 10615940 * q^27 + 1063436 * q^29 + 14101252 * q^35 + 1038736 * q^39 - 57696700 * q^43 - 301841610 * q^49 + 35636372 * q^51 + 44506132 * q^53 + 134671136 * q^55 + 125054108 * q^61 + 312493540 * q^65 - 170259288 * q^69 - 735481400 * q^75 - 633685704 * q^77 - 1485153656 * q^79 + 1864392030 * q^81 - 214111264 * q^87 + 508447412 * q^91 - 391914776 * q^95

Decomposition of \(S_{10}^{\mathrm{new}}(208, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
208.10.f.a	$208$	$10$	208.f	13.b	$2$	$10$	$10$	$107.127$	\(\mathbb{Q}[x]/(x^{10} + \cdots)\)	None					26.10.b.a	$2$	$0$	\(0\)	\(-162\)	\(0\)	\(0\)	$2^{22}\cdot 3^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-2^{4}+\beta _{1})q^{3}+(-5\beta _{3}+\beta _{5})q^{5}+\cdots\)
208.10.f.b	$208$	$10$	208.f	13.b	$2$	$10$	$10$	$107.127$	\(\mathbb{Q}[x]/(x^{10} + \cdots)\)	None					13.10.b.a	$2$	$0$	\(0\)	\(2\)	\(0\)	\(0\)	$2^{15}\cdot 3^{4}\cdot 13^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+(-\beta _{1}+\beta _{5})q^{5}+(2\beta _{1}+\beta _{5}+\cdots)q^{7}+\cdots\)
208.10.f.c	$208$	$10$	208.f	13.b	$2$	$10$	$10$	$107.127$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None					52.10.d.a	$2$	$0$	\(0\)	\(162\)	\(0\)	\(0\)	$2^{15}\cdot 3\cdot 13^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(2^{4}+\beta _{1})q^{3}+\beta _{2}q^{5}-\beta _{5}q^{7}+(5411+\cdots)q^{9}+\cdots\)
208.10.f.d	$208$	$10$	208.f	13.b	$2$	$32$	$32$	$107.127$		None			✓		104.10.f.a	$2$	$0$	\(0\)	\(162\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{10}^{\mathrm{old}}(208, [\chi])\) into lower level spaces

\( S_{10}^{\mathrm{old}}(208, [\chi]) \simeq \) \(S_{10}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(52, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(104, [\chi])\)\(^{\oplus 2}\)