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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(7\)	\(1\)	\(6\)		\(5\)	\(1\)	\(4\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)		\(9\)	\(2\)	\(7\)		\(6\)	\(2\)	\(4\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)		\(10\)	\(2\)	\(8\)		\(7\)	\(2\)	\(5\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)		\(8\)	\(1\)	\(7\)		\(5\)	\(1\)	\(4\)		\(3\)	\(0\)	\(3\)
Plus space		\(+\)		\(15\)	\(2\)	\(13\)		\(10\)	\(2\)	\(8\)		\(5\)	\(0\)	\(5\)
Minus space		\(-\)		\(19\)	\(4\)	\(15\)		\(13\)	\(4\)	\(9\)		\(6\)	\(0\)	\(6\)

Trace form

6 * q - 2 * q^7 + 2 * q^9 + 2 * q^11 + 8 * q^15 - 4 * q^17 - 2 * q^19 - 8 * q^21 + 8 * q^23 - 2 * q^25 + 12 * q^27 - 14 * q^31 - 8 * q^33 + 8 * q^37 - 4 * q^39 + 4 * q^41 - 12 * q^43 - 8 * q^45 - 10 * q^47 - 2 * q^49 - 28 * q^51 + 4 * q^53 + 4 * q^55 + 18 * q^59 + 12 * q^61 + 6 * q^63 - 26 * q^67 + 8 * q^69 - 6 * q^71 - 20 * q^73 + 12 * q^75 + 4 * q^77 - 28 * q^79 + 6 * q^81 + 22 * q^83 + 8 * q^87 + 4 * q^89 + 6 * q^91 - 16 * q^93 + 36 * q^95 - 4 * q^97 + 34 * q^99

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

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