Space of modular forms of level 736, weight 4, and trivial character

• Label: 736.4.a
• Level: $736$
• Weight: $4$
• Character orbit: 736.a
• Rep. character: $\chi_{736}(1,\cdot)$
• Character field: $\Q$
• Dimension: $66$
• Newform subspaces: $10$
• Sturm bound: $384$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 736 = 2^{5} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	736.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(384\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(736))\).

	Total	New	Old
Modular forms	296	66	230
Cusp forms	280	66	214
Eisenstein series	16	0	16

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
\(+\)	\(+\)	\(+\)		\(79\)	\(17\)	\(62\)		\(75\)	\(17\)	\(58\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)		\(69\)	\(15\)	\(54\)		\(65\)	\(15\)	\(50\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)		\(75\)	\(16\)	\(59\)		\(71\)	\(16\)	\(55\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)		\(73\)	\(18\)	\(55\)		\(69\)	\(18\)	\(51\)		\(4\)	\(0\)	\(4\)
Plus space		\(+\)		\(152\)	\(35\)	\(117\)		\(144\)	\(35\)	\(109\)		\(8\)	\(0\)	\(8\)
Minus space		\(-\)		\(144\)	\(31\)	\(113\)		\(136\)	\(31\)	\(105\)		\(8\)	\(0\)	\(8\)

Trace form

66 * q - 4 * q^5 + 554 * q^9 + 236 * q^13 + 308 * q^17 - 240 * q^21 + 1182 * q^25 + 284 * q^29 + 1280 * q^33 - 1668 * q^37 - 588 * q^41 - 180 * q^45 + 4770 * q^49 - 212 * q^53 + 416 * q^57 - 1780 * q^61 - 840 * q^65 + 324 * q^73 + 368 * q^77 + 1506 * q^81 + 3176 * q^85 + 1748 * q^89 - 3696 * q^93 + 228 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(736))\) 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
736.4.a.a	$736$	$4$	736.a	1.a	$1$	$3$	$3$	$43.425$	3.3.11032.1	None	✓	✓			736.4.a.a	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(16\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(-\beta _{1}-\beta _{2})q^{5}+(4+\cdots)q^{7}+\cdots\)
736.4.a.b	$736$	$4$	736.a	1.a	$1$	$3$	$3$	$43.425$	3.3.11032.1	None	✓	✓			736.4.a.a	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-16\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(-\beta _{1}-\beta _{2})q^{5}+(-4+\cdots)q^{7}+\cdots\)
736.4.a.c	$736$	$4$	736.a	1.a	$1$	$4$	$4$	$43.425$	4.4.310848.1	None	✓	✓			736.4.a.c	$1$	$1$	\(0\)	\(0\)	\(-20\)	\(-44\)		$+$	$-$	$-$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+(-5+\beta _{1}-\beta _{2}+\beta _{3})q^{5}+\cdots\)
736.4.a.d	$736$	$4$	736.a	1.a	$1$	$4$	$4$	$43.425$	4.4.310848.1	None	✓	✓			736.4.a.c	$1$	$1$	\(0\)	\(0\)	\(-20\)	\(44\)		$-$	$+$	$-$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+(-5+\beta _{1}-\beta _{2}+\beta _{3})q^{5}+\cdots\)
736.4.a.e	$736$	$4$	736.a	1.a	$1$	$8$	$8$	$43.425$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓		736.4.a.e	$1$	$1$	\(0\)	\(-12\)	\(-12\)	\(14\)		$+$	$-$	$-$	$2^{8}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{3}+(-1-\beta _{3})q^{5}+(2+\cdots)q^{7}+\cdots\)
736.4.a.f	$736$	$4$	736.a	1.a	$1$	$8$	$8$	$43.425$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓			736.4.a.e	$1$	$0$	\(0\)	\(12\)	\(-12\)	\(-14\)		$+$	$+$	$+$	$2^{8}$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{3}+(-1-\beta _{3})q^{5}+(-2+\cdots)q^{7}+\cdots\)
736.4.a.g	$736$	$4$	736.a	1.a	$1$	$9$	$9$	$43.425$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓	✓		736.4.a.g	$1$	$0$	\(0\)	\(-14\)	\(30\)	\(28\)		$+$	$+$	$+$	$2^{10}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{3}+(3-\beta _{4})q^{5}+(3-\beta _{6}+\cdots)q^{7}+\cdots\)
736.4.a.h	$736$	$4$	736.a	1.a	$1$	$9$	$9$	$43.425$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓	✓		736.4.a.h	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-42\)		$-$	$+$	$-$	$2^{10}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}-\beta _{5}q^{5}+(-5+\beta _{3})q^{7}+(8+\cdots)q^{9}+\cdots\)
736.4.a.i	$736$	$4$	736.a	1.a	$1$	$9$	$9$	$43.425$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓			736.4.a.h	$1$	$0$	\(0\)	\(0\)	\(0\)	\(42\)		$-$	$-$	$+$	$2^{10}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}-\beta _{5}q^{5}+(5-\beta _{3})q^{7}+(8-\beta _{3}+\cdots)q^{9}+\cdots\)
736.4.a.j	$736$	$4$	736.a	1.a	$1$	$9$	$9$	$43.425$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓			736.4.a.g	$1$	$0$	\(0\)	\(14\)	\(30\)	\(-28\)		$-$	$-$	$+$	$2^{10}$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{3}+(3-\beta _{4})q^{5}+(-3+\beta _{6}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(736))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(736)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(92))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(184))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(368))\)\(^{\oplus 2}\)