Space of modular forms of level 736, weight 2, and trivial character

• Label: 736.2.a
• Level: $736$
• Weight: $2$
• Character orbit: 736.a
• Rep. character: $\chi_{736}(1,\cdot)$
• Character field: $\Q$
• Dimension: $22$
• Newform subspaces: $8$
• Sturm bound: $192$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 736 = 2^{5} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	736.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(192\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(736))\).

	Total	New	Old
Modular forms	104	22	82
Cusp forms	89	22	67
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.

\(2\)	\(23\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(21\)	\(5\)	\(16\)		\(18\)	\(5\)	\(13\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)		\(31\)	\(7\)	\(24\)		\(27\)	\(7\)	\(20\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)		\(25\)	\(6\)	\(19\)		\(21\)	\(6\)	\(15\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)		\(27\)	\(4\)	\(23\)		\(23\)	\(4\)	\(19\)		\(4\)	\(0\)	\(4\)
Plus space		\(+\)		\(48\)	\(9\)	\(39\)		\(41\)	\(9\)	\(32\)		\(7\)	\(0\)	\(7\)
Minus space		\(-\)		\(56\)	\(13\)	\(43\)		\(48\)	\(13\)	\(35\)		\(8\)	\(0\)	\(8\)

Trace form

22 * q + 4 * q^5 + 30 * q^9 - 12 * q^13 - 4 * q^17 - 16 * q^21 + 26 * q^25 + 4 * q^29 + 4 * q^37 - 36 * q^41 + 20 * q^45 + 22 * q^49 + 20 * q^53 - 32 * q^57 + 20 * q^61 - 24 * q^65 + 12 * q^73 + 16 * q^77 + 22 * q^81 + 24 * q^85 - 4 * q^89 - 16 * q^93 - 52 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(736))\) 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	23
736.2.a.a	$736$	$2$	736.a	1.a	$1$	$2$	$2$	$5.877$	\(\Q(\sqrt{2}) \)	None	✓	✓			736.2.a.a	$1$	$1$	\(0\)	\(-2\)	\(-4\)	\(4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+(-2+\beta )q^{5}+(2-\beta )q^{7}+\cdots\)
736.2.a.b	$736$	$2$	736.a	1.a	$1$	$2$	$2$	$5.877$	\(\Q(\sqrt{3}) \)	None	✓	✓			736.2.a.b	$1$	$1$	\(0\)	\(0\)	\(2\)	\(-6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(1-\beta )q^{5}+(-3-\beta )q^{7}+(-3+\cdots)q^{11}+\cdots\)
736.2.a.c	$736$	$2$	736.a	1.a	$1$	$2$	$2$	$5.877$	\(\Q(\sqrt{3}) \)	None	✓	✓			736.2.a.b	$1$	$0$	\(0\)	\(0\)	\(2\)	\(6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(1+\beta )q^{5}+(3-\beta )q^{7}+(3+\beta )q^{11}+\cdots\)
736.2.a.d	$736$	$2$	736.a	1.a	$1$	$2$	$2$	$5.877$	\(\Q(\sqrt{2}) \)	None	✓	✓			736.2.a.a	$1$	$1$	\(0\)	\(2\)	\(-4\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(-2-\beta )q^{5}+(-2-\beta )q^{7}+\cdots\)
736.2.a.e	$736$	$2$	736.a	1.a	$1$	$3$	$3$	$5.877$	3.3.316.1	None	✓	✓	✓		736.2.a.e	$1$	$1$	\(0\)	\(-4\)	\(-2\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{3}+(-1+\beta _{1}-\beta _{2})q^{5}+\cdots\)
736.2.a.f	$736$	$2$	736.a	1.a	$1$	$3$	$3$	$5.877$	3.3.316.1	None	✓	✓			736.2.a.e	$1$	$0$	\(0\)	\(4\)	\(-2\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{3}+(-1+\beta _{1}-\beta _{2})q^{5}+\cdots\)
736.2.a.g	$736$	$2$	736.a	1.a	$1$	$4$	$4$	$5.877$	4.4.13768.1	None	✓	✓	✓		736.2.a.g	$1$	$0$	\(0\)	\(-2\)	\(6\)	\(4\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+(2-\beta _{3})q^{5}+(1-\beta _{1})q^{7}+\cdots\)
736.2.a.h	$736$	$2$	736.a	1.a	$1$	$4$	$4$	$5.877$	4.4.13768.1	None	✓	✓	✓		736.2.a.g	$1$	$0$	\(0\)	\(2\)	\(6\)	\(-4\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+(2-\beta _{3})q^{5}+(-1+\beta _{1})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(736))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(736)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(92))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(184))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(368))\)\(^{\oplus 2}\)