Space of modular forms of level 608, weight 2, and trivial character

• Label: 608.2.a
• Level: $608$
• Weight: $2$
• Character orbit: 608.a
• Rep. character: $\chi_{608}(1,\cdot)$
• Character field: $\Q$
• Dimension: $18$
• Newform subspaces: $10$
• Sturm bound: $160$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 608 = 2^{5} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	608.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(160\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(608))\).

	Total	New	Old
Modular forms	88	18	70
Cusp forms	73	18	55
Eisenstein series	15	0	15

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

\(2\)	\(19\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(18\)	\(3\)	\(15\)		\(15\)	\(3\)	\(12\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)		\(24\)	\(6\)	\(18\)		\(20\)	\(6\)	\(14\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)		\(26\)	\(6\)	\(20\)		\(22\)	\(6\)	\(16\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)		\(20\)	\(3\)	\(17\)		\(16\)	\(3\)	\(13\)		\(4\)	\(0\)	\(4\)
Plus space		\(+\)		\(38\)	\(6\)	\(32\)		\(31\)	\(6\)	\(25\)		\(7\)	\(0\)	\(7\)
Minus space		\(-\)		\(50\)	\(12\)	\(38\)		\(42\)	\(12\)	\(30\)		\(8\)	\(0\)	\(8\)

Trace form

18 * q + 4 * q^5 + 18 * q^9 + 4 * q^13 - 12 * q^17 + 16 * q^21 + 22 * q^25 + 20 * q^29 + 16 * q^33 + 20 * q^37 - 12 * q^41 - 12 * q^45 + 34 * q^49 - 44 * q^53 + 20 * q^61 - 24 * q^65 + 4 * q^73 + 8 * q^77 - 30 * q^81 - 60 * q^89 - 16 * q^93 - 60 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(608))\) 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	19
608.2.a.a	$608$	$2$	608.a	1.a	$1$	$1$	$1$	$4.855$	\(\Q\)	None	✓	✓			608.2.a.a	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+q^{7}+6q^{9}-2q^{11}-q^{13}+\cdots\)
608.2.a.b	$608$	$2$	608.a	1.a	$1$	$1$	$1$	$4.855$	\(\Q\)	None	✓	✓			608.2.a.b	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-q^{7}-3q^{9}+3q^{11}-4q^{13}+\cdots\)
608.2.a.c	$608$	$2$	608.a	1.a	$1$	$1$	$1$	$4.855$	\(\Q\)	None	✓	✓			608.2.a.b	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+q^{7}-3q^{9}-3q^{11}-4q^{13}+\cdots\)
608.2.a.d	$608$	$2$	608.a	1.a	$1$	$1$	$1$	$4.855$	\(\Q\)	None	✓	✓			608.2.a.d	$1$	$1$	\(0\)	\(0\)	\(3\)	\(-5\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}-5q^{7}-3q^{9}-5q^{11}-4q^{13}+\cdots\)
608.2.a.e	$608$	$2$	608.a	1.a	$1$	$1$	$1$	$4.855$	\(\Q\)	None	✓	✓			608.2.a.d	$1$	$0$	\(0\)	\(0\)	\(3\)	\(5\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}+5q^{7}-3q^{9}+5q^{11}-4q^{13}+\cdots\)
608.2.a.f	$608$	$2$	608.a	1.a	$1$	$1$	$1$	$4.855$	\(\Q\)	None	✓	✓			608.2.a.a	$1$	$0$	\(0\)	\(3\)	\(0\)	\(-1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-q^{7}+6q^{9}+2q^{11}-q^{13}+\cdots\)
608.2.a.g	$608$	$2$	608.a	1.a	$1$	$2$	$2$	$4.855$	\(\Q(\sqrt{17}) \)	None	✓	✓	✓		608.2.a.g	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(-6\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(-1+\beta )q^{5}-3q^{7}+(1+\beta )q^{9}+\cdots\)
608.2.a.h	$608$	$2$	608.a	1.a	$1$	$2$	$2$	$4.855$	\(\Q(\sqrt{17}) \)	None	✓	✓			608.2.a.g	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(6\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(-1+\beta )q^{5}+3q^{7}+(1+\beta )q^{9}+\cdots\)
608.2.a.i	$608$	$2$	608.a	1.a	$1$	$4$	$4$	$4.855$	4.4.15317.1	None	✓	✓	✓		608.2.a.i	$1$	$0$	\(0\)	\(-2\)	\(1\)	\(1\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{3}-\beta _{1}q^{5}+(-\beta _{2}+\beta _{3})q^{7}+\cdots\)
608.2.a.j	$608$	$2$	608.a	1.a	$1$	$4$	$4$	$4.855$	4.4.15317.1	None	✓	✓	✓		608.2.a.i	$1$	$0$	\(0\)	\(2\)	\(1\)	\(-1\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{3}+(\beta _{1}-\beta _{2})q^{5}+\beta _{3}q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(608))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(608)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(152))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(304))\)\(^{\oplus 2}\)