Space of modular forms of level 208, weight 2, and trivial character

• Label: 208.2.a
• Level: $208$
• Weight: $2$
• Character orbit: 208.a
• Rep. character: $\chi_{208}(1,\cdot)$
• Character field: $\Q$
• Dimension: $6$
• Newform subspaces: $5$
• Sturm bound: $56$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	208.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(56\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(208))\).

	Total	New	Old
Modular forms	34	6	28
Cusp forms	23	6	17
Eisenstein series	11	0	11

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
\(+\)	\(+\)	\(+\)	\(1\)
\(+\)	\(-\)	\(-\)	\(2\)
\(-\)	\(+\)	\(-\)	\(2\)
\(-\)	\(-\)	\(+\)	\(1\)
Plus space		\(+\)	\(2\)
Minus space		\(-\)	\(4\)

Trace form

\( 6 q - 2 q^{7} + 2 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(208))\) 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
208.2.a.a	$208$	$2$	208.a	1.a	$1$	$1$	$1$	$1.661$	\(\Q\)	None	✓		✓	✓	26.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(-3\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-3q^{5}+q^{7}-2q^{9}-6q^{11}+\cdots\)
208.2.a.b	$208$	$2$	208.a	1.a	$1$	$1$	$1$	$1.661$	\(\Q\)	None	✓		✓	✓	104.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(-5\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-5q^{7}-2q^{9}+2q^{11}+\cdots\)
208.2.a.c	$208$	$2$	208.a	1.a	$1$	$1$	$1$	$1.661$	\(\Q\)	None	✓				52.2.a.a	$1$	$0$	\(0\)	\(0\)	\(2\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+2q^{7}-3q^{9}+2q^{11}-q^{13}+\cdots\)
208.2.a.d	$208$	$2$	208.a	1.a	$1$	$1$	$1$	$1.661$	\(\Q\)	None	✓				26.2.a.b	$1$	$0$	\(0\)	\(3\)	\(-1\)	\(-1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-q^{5}-q^{7}+6q^{9}+2q^{11}+\cdots\)
208.2.a.e	$208$	$2$	208.a	1.a	$1$	$2$	$2$	$1.661$	\(\Q(\sqrt{17}) \)	None	✓		✓	✓	104.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(3\)	\(1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(2-\beta )q^{5}+\beta q^{7}+(1+\beta )q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(208))\) into lower level spaces

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