Space of modular forms of level 308, weight 2, and trivial character

• Label: 308.2.a
• Level: $308$
• Weight: $2$
• Character orbit: 308.a
• Rep. character: $\chi_{308}(1,\cdot)$
• Character field: $\Q$
• Dimension: $6$
• Newform subspaces: $3$
• Sturm bound: $96$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 308 = 2^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	308.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(96\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(308))\).

	Total	New	Old
Modular forms	54	6	48
Cusp forms	43	6	37
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.

\(2\)	\(7\)	\(11\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(5\)	\(0\)	\(5\)		\(4\)	\(0\)	\(4\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)		\(7\)	\(0\)	\(7\)		\(5\)	\(0\)	\(5\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(9\)	\(0\)	\(9\)		\(7\)	\(0\)	\(7\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(7\)	\(0\)	\(7\)		\(5\)	\(0\)	\(5\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(7\)	\(2\)	\(5\)		\(6\)	\(2\)	\(4\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)		\(5\)	\(1\)	\(4\)		\(4\)	\(1\)	\(3\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)		\(6\)	\(0\)	\(6\)		\(5\)	\(0\)	\(5\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)		\(8\)	\(3\)	\(5\)		\(7\)	\(3\)	\(4\)		\(1\)	\(0\)	\(1\)
Plus space			\(+\)		\(23\)	\(1\)	\(22\)		\(18\)	\(1\)	\(17\)		\(5\)	\(0\)	\(5\)
Minus space			\(-\)		\(31\)	\(5\)	\(26\)		\(25\)	\(5\)	\(20\)		\(6\)	\(0\)	\(6\)

Trace form

6 * q - 2 * q^3 + 2 * q^5 + 8 * q^9 + 2 * q^11 + 12 * q^13 + 10 * q^15 - 4 * q^19 + 2 * q^23 + 12 * q^25 - 2 * q^27 + 4 * q^29 - 2 * q^31 - 2 * q^33 - 4 * q^35 + 6 * q^37 - 8 * q^39 - 20 * q^45 - 20 * q^47 + 6 * q^49 - 36 * q^51 + 12 * q^53 - 6 * q^55 + 4 * q^57 - 22 * q^59 + 8 * q^61 - 16 * q^65 + 6 * q^67 - 6 * q^69 - 10 * q^71 - 8 * q^73 - 44 * q^75 + 4 * q^77 - 12 * q^79 - 18 * q^81 - 16 * q^83 + 20 * q^85 + 12 * q^87 + 6 * q^89 + 12 * q^91 - 22 * q^93 + 20 * q^95 + 26 * q^97 - 4 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(308))\) 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	7	11
308.2.a.a	$308$	$2$	308.a	1.a	$1$	$1$	$1$	$2.459$	\(\Q\)	None	✓	✓	✓	✓	308.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-q^{7}-2q^{9}+q^{11}-4q^{13}+\cdots\)
308.2.a.b	$308$	$2$	308.a	1.a	$1$	$2$	$2$	$2.459$	\(\Q(\sqrt{6}) \)	None	✓	✓	✓	✓	308.2.a.b	$1$	$0$	\(0\)	\(0\)	\(4\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+2q^{5}-q^{7}+3q^{9}-q^{11}+\cdots\)
308.2.a.c	$308$	$2$	308.a	1.a	$1$	$3$	$3$	$2.459$	3.3.1016.1	None	✓	✓	✓	✓	308.2.a.c	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-\beta _{1}-\beta _{2})q^{5}+q^{7}+(1+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(308))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(308)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(154))\)\(^{\oplus 2}\)