Space of modular forms of level 208, weight 6, and trivial character

• Label: 208.6.a
• Level: $208$
• Weight: $6$
• Character orbit: 208.a
• Rep. character: $\chi_{208}(1,\cdot)$
• Character field: $\Q$
• Dimension: $30$
• Newform subspaces: $13$
• Sturm bound: $168$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	208.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(168\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(208))\).

	Total	New	Old
Modular forms	146	30	116
Cusp forms	134	30	104
Eisenstein series	12	0	12

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(13\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(7\)
\(+\)	\(-\)	\(-\)	\(8\)
\(-\)	\(+\)	\(-\)	\(8\)
\(-\)	\(-\)	\(+\)	\(7\)
Plus space		\(+\)	\(14\)
Minus space		\(-\)	\(16\)

Trace form

\( 30 q - 98 q^{7} + 2474 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(208))\) 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					A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	13
208.6.a.a	$208$	$6$	208.a	1.a	$1$	$1$	$1$	$33.360$	\(\Q\)	None	✓				52.6.a.b	$1$	$1$	\(0\)	\(-17\)	\(-91\)	\(233\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-17q^{3}-91q^{5}+233q^{7}+46q^{9}+\cdots\)
208.6.a.b	$208$	$6$	208.a	1.a	$1$	$1$	$1$	$33.360$	\(\Q\)	None	✓				26.6.a.a	$1$	$0$	\(0\)	\(0\)	\(-14\)	\(170\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-14q^{5}+170q^{7}-3^{5}q^{9}+250q^{11}+\cdots\)
208.6.a.c	$208$	$6$	208.a	1.a	$1$	$1$	$1$	$33.360$	\(\Q\)	None	✓				52.6.a.a	$1$	$1$	\(0\)	\(5\)	\(-3\)	\(-53\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5q^{3}-3q^{5}-53q^{7}-218q^{9}+702q^{11}+\cdots\)
208.6.a.d	$208$	$6$	208.a	1.a	$1$	$1$	$1$	$33.360$	\(\Q\)	None	✓				104.6.a.a	$1$	$1$	\(0\)	\(23\)	\(-9\)	\(-165\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+23q^{3}-9q^{5}-165q^{7}+286q^{9}+\cdots\)
208.6.a.e	$208$	$6$	208.a	1.a	$1$	$2$	$2$	$33.360$	\(\Q(\sqrt{337}) \)	None	✓				104.6.a.b	$1$	$1$	\(0\)	\(-23\)	\(-47\)	\(95\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-11-\beta )q^{3}+(-21-5\beta )q^{5}+(7^{2}+\cdots)q^{7}+\cdots\)
208.6.a.f	$208$	$6$	208.a	1.a	$1$	$2$	$2$	$33.360$	\(\Q(\sqrt{2785}) \)	None	✓				26.6.a.b	$1$	$1$	\(0\)	\(-9\)	\(-37\)	\(-327\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-4-\beta )q^{3}+(-18-\beta )q^{5}+(-164+\cdots)q^{7}+\cdots\)
208.6.a.g	$208$	$6$	208.a	1.a	$1$	$2$	$2$	$33.360$	\(\Q(\sqrt{849}) \)	None	✓				26.6.a.c	$1$	$0$	\(0\)	\(-9\)	\(73\)	\(-155\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-4-\beta )q^{3}+(38-3\beta )q^{5}+(-82+\cdots)q^{7}+\cdots\)
208.6.a.h	$208$	$6$	208.a	1.a	$1$	$2$	$2$	$33.360$	\(\Q(\sqrt{17}) \)	None	✓				13.6.a.a	$1$	$0$	\(0\)	\(28\)	\(-42\)	\(36\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(14-3\beta )q^{3}+(-21-20\beta )q^{5}+(18+\cdots)q^{7}+\cdots\)
208.6.a.i	$208$	$6$	208.a	1.a	$1$	$3$	$3$	$33.360$	3.3.203961.1	None	✓		✓		52.6.a.c	$1$	$0$	\(0\)	\(-12\)	\(8\)	\(-138\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-5-2\beta _{1}+\beta _{2})q^{3}+(6+5\beta _{1}-5\beta _{2})q^{5}+\cdots\)
208.6.a.j	$208$	$6$	208.a	1.a	$1$	$3$	$3$	$33.360$	3.3.168897.1	None	✓		✓		13.6.a.b	$1$	$1$	\(0\)	\(-8\)	\(56\)	\(60\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-3+\beta _{1})q^{3}+(21-4\beta _{1}+3\beta _{2})q^{5}+\cdots\)
208.6.a.k	$208$	$6$	208.a	1.a	$1$	$3$	$3$	$33.360$	3.3.1848689.1	None	✓				104.6.a.c	$1$	$0$	\(0\)	\(20\)	\(56\)	\(76\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(7-\beta _{1})q^{3}+(19+\beta _{2})q^{5}+(5^{2}+5\beta _{1}+\cdots)q^{7}+\cdots\)
208.6.a.l	$208$	$6$	208.a	1.a	$1$	$4$	$4$	$33.360$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		104.6.a.d	$1$	$1$	\(0\)	\(11\)	\(31\)	\(-39\)		$+$	$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(3-\beta _{2})q^{3}+(7+\beta _{1}+3\beta _{2}+\beta _{3})q^{5}+\cdots\)
208.6.a.m	$208$	$6$	208.a	1.a	$1$	$5$	$5$	$33.360$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓		✓		104.6.a.e	$1$	$0$	\(0\)	\(-9\)	\(19\)	\(109\)		$+$	$-$	$-$	$2^{6}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{3}+(4-\beta _{1}+\beta _{3})q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(208))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(208)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 5}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 2}\)