Space of modular forms of level 936, weight 4, and trivial character

• Label: 936.4.a
• Level: $936$
• Weight: $4$
• Character orbit: 936.a
• Rep. character: $\chi_{936}(1,\cdot)$
• Character field: $\Q$
• Dimension: $45$
• Newform subspaces: $17$
• Sturm bound: $672$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 936 = 2^{3} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	936.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 17 \)
Sturm bound:			\(672\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(936))\).

	Total	New	Old
Modular forms	520	45	475
Cusp forms	488	45	443
Eisenstein series	32	0	32

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

\(2\)	\(3\)	\(13\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(68\)	\(5\)	\(63\)		\(64\)	\(5\)	\(59\)		\(4\)	\(0\)	\(4\)
\(+\)	\(+\)	\(-\)	\(-\)		\(62\)	\(4\)	\(58\)		\(58\)	\(4\)	\(54\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(+\)	\(-\)		\(65\)	\(6\)	\(59\)		\(61\)	\(6\)	\(55\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)		\(65\)	\(8\)	\(57\)		\(61\)	\(8\)	\(53\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(+\)	\(-\)		\(62\)	\(5\)	\(57\)		\(58\)	\(5\)	\(53\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)	\(+\)		\(68\)	\(4\)	\(64\)		\(64\)	\(4\)	\(60\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)	\(+\)		\(65\)	\(7\)	\(58\)		\(61\)	\(7\)	\(54\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(-\)	\(-\)		\(65\)	\(6\)	\(59\)		\(61\)	\(6\)	\(55\)		\(4\)	\(0\)	\(4\)
Plus space			\(+\)		\(266\)	\(24\)	\(242\)		\(250\)	\(24\)	\(226\)		\(16\)	\(0\)	\(16\)
Minus space			\(-\)		\(254\)	\(21\)	\(233\)		\(238\)	\(21\)	\(217\)		\(16\)	\(0\)	\(16\)

Trace form

45 * q + 10 * q^5 + 12 * q^7 - 116 * q^11 - 13 * q^13 + 112 * q^17 + 24 * q^19 + 84 * q^23 + 945 * q^25 + 154 * q^29 - 276 * q^31 - 602 * q^35 - 122 * q^37 - 214 * q^41 + 286 * q^43 + 820 * q^47 + 2491 * q^49 + 134 * q^53 - 2192 * q^55 - 632 * q^59 + 546 * q^61 + 260 * q^65 + 1052 * q^67 + 356 * q^71 + 2782 * q^73 - 464 * q^77 + 124 * q^79 - 2852 * q^83 - 3064 * q^85 + 1058 * q^89 + 546 * q^91 + 4740 * q^95 - 2182 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(936))\) 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	3	13
936.4.a.a	$936$	$4$	936.a	1.a	$1$	$1$	$1$	$55.226$	\(\Q\)	None	✓				104.4.a.b	$1$	$0$	\(0\)	\(0\)	\(-19\)	\(-3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-19q^{5}-3q^{7}+2q^{11}-13q^{13}+\cdots\)
936.4.a.b	$936$	$4$	936.a	1.a	$1$	$1$	$1$	$55.226$	\(\Q\)	None	✓				104.4.a.a	$1$	$1$	\(0\)	\(0\)	\(7\)	\(-21\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+7q^{5}-21q^{7}-6q^{11}+13q^{13}+\cdots\)
936.4.a.c	$936$	$4$	936.a	1.a	$1$	$2$	$2$	$55.226$	\(\Q(\sqrt{43}) \)	None	✓				312.4.a.f	$1$	$1$	\(0\)	\(0\)	\(-12\)	\(44\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-6+\beta )q^{5}+(22-\beta )q^{7}-26q^{11}+\cdots\)
936.4.a.d	$936$	$4$	936.a	1.a	$1$	$2$	$2$	$55.226$	\(\Q(\sqrt{113}) \)	None	✓				312.4.a.c	$1$	$1$	\(0\)	\(0\)	\(-6\)	\(-10\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-3-\beta )q^{5}+(-5-\beta )q^{7}+(8+4\beta )q^{11}+\cdots\)
936.4.a.e	$936$	$4$	936.a	1.a	$1$	$2$	$2$	$55.226$	\(\Q(\sqrt{73}) \)	None	✓				104.4.a.c	$1$	$1$	\(0\)	\(0\)	\(3\)	\(-25\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3\beta q^{5}+(-12-\beta )q^{7}+(30-4\beta )q^{11}+\cdots\)
936.4.a.f	$936$	$4$	936.a	1.a	$1$	$2$	$2$	$55.226$	\(\Q(\sqrt{7}) \)	None	✓				312.4.a.e	$1$	$0$	\(0\)	\(0\)	\(4\)	\(-20\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{5}+(-10+3\beta )q^{7}+(30+4\beta )q^{11}+\cdots\)
936.4.a.g	$936$	$4$	936.a	1.a	$1$	$2$	$2$	$55.226$	\(\Q(\sqrt{3}) \)	None	✓				312.4.a.a	$1$	$1$	\(0\)	\(0\)	\(4\)	\(4\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{5}+(2+3\beta )q^{7}+(-14+6\beta )q^{11}+\cdots\)
936.4.a.h	$936$	$4$	936.a	1.a	$1$	$2$	$2$	$55.226$	\(\Q(\sqrt{55}) \)	None	✓				312.4.a.b	$1$	$0$	\(0\)	\(0\)	\(4\)	\(20\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{5}+(10-\beta )q^{7}+(10+2\beta )q^{11}+\cdots\)
936.4.a.i	$936$	$4$	936.a	1.a	$1$	$2$	$2$	$55.226$	\(\Q(\sqrt{321}) \)	None	✓				104.4.a.d	$1$	$0$	\(0\)	\(0\)	\(11\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(6-\beta )q^{5}+(2-3\beta )q^{7}-58q^{11}+\cdots\)
936.4.a.j	$936$	$4$	936.a	1.a	$1$	$2$	$2$	$55.226$	\(\Q(\sqrt{17}) \)	None	✓				312.4.a.d	$1$	$0$	\(0\)	\(0\)	\(18\)	\(-10\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(9-\beta )q^{5}+(-5-5\beta )q^{7}+(-2-14\beta )q^{11}+\cdots\)
936.4.a.k	$936$	$4$	936.a	1.a	$1$	$3$	$3$	$55.226$	3.3.36248.1	None	✓				312.4.a.g	$1$	$0$	\(0\)	\(0\)	\(-16\)	\(-22\)		$+$	$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-5-\beta _{2})q^{5}+(-8+\beta _{1}+\beta _{2})q^{7}+\cdots\)
936.4.a.l	$936$	$4$	936.a	1.a	$1$	$3$	$3$	$55.226$	3.3.13916.1	None	✓		✓		312.4.a.h	$1$	$1$	\(0\)	\(0\)	\(4\)	\(6\)		$-$	$-$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{2})q^{5}+(2+\beta _{1}+\beta _{2})q^{7}+(-11+\cdots)q^{11}+\cdots\)
936.4.a.m	$936$	$4$	936.a	1.a	$1$	$3$	$3$	$55.226$	3.3.18257.1	None	✓				104.4.a.e	$1$	$0$	\(0\)	\(0\)	\(8\)	\(36\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(3+2\beta _{1}+\beta _{2})q^{5}+(12+\beta _{1})q^{7}+\cdots\)
936.4.a.n	$936$	$4$	936.a	1.a	$1$	$4$	$4$	$55.226$	4.4.6390848.1	None	✓	✓	✓	✓	936.4.a.n	$1$	$1$	\(0\)	\(0\)	\(-8\)	\(24\)		$+$	$+$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{1})q^{5}+(6+\beta _{2})q^{7}+(-2^{4}+\cdots)q^{11}+\cdots\)
936.4.a.o	$936$	$4$	936.a	1.a	$1$	$4$	$4$	$55.226$	4.4.6390848.1	None	✓	✓	✓	✓	936.4.a.n	$1$	$0$	\(0\)	\(0\)	\(8\)	\(24\)		$-$	$+$	$-$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(2+\beta _{1})q^{5}+(6+\beta _{2})q^{7}+(2^{4}-\beta _{1}+\cdots)q^{11}+\cdots\)
936.4.a.p	$936$	$4$	936.a	1.a	$1$	$5$	$5$	$55.226$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓	✓	936.4.a.p	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(-18\)		$-$	$+$	$+$	$-$	$2^{8}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{5}+(-4-\beta _{1})q^{7}+(5+\beta _{1}+2\beta _{2}+\cdots)q^{11}+\cdots\)
936.4.a.q	$936$	$4$	936.a	1.a	$1$	$5$	$5$	$55.226$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓	✓	936.4.a.p	$1$	$0$	\(0\)	\(0\)	\(2\)	\(-18\)		$+$	$+$	$+$	$+$	$2^{8}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{5}+(-4-\beta _{1})q^{7}+(-5-\beta _{1}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(936))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(936)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(117))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(156))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(234))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(312))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(468))\)\(^{\oplus 2}\)