Space of modular forms of level 1296, weight 4, and trivial character

• Label: 1296.4.a
• Level: $1296$
• Weight: $4$
• Character orbit: 1296.a
• Rep. character: $\chi_{1296}(1,\cdot)$
• Character field: $\Q$
• Dimension: $70$
• Newform subspaces: $30$
• Sturm bound: $864$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1296.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 30 \)
Sturm bound:			\(864\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1296))\).

	Total	New	Old
Modular forms	684	74	610
Cusp forms	612	70	542
Eisenstein series	72	4	68

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

\(2\)	\(3\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(174\)	\(19\)	\(155\)		\(156\)	\(19\)	\(137\)		\(18\)	\(0\)	\(18\)
\(+\)	\(-\)	\(-\)		\(168\)	\(17\)	\(151\)		\(150\)	\(17\)	\(133\)		\(18\)	\(0\)	\(18\)
\(-\)	\(+\)	\(-\)		\(168\)	\(18\)	\(150\)		\(150\)	\(16\)	\(134\)		\(18\)	\(2\)	\(16\)
\(-\)	\(-\)	\(+\)		\(174\)	\(20\)	\(154\)		\(156\)	\(18\)	\(138\)		\(18\)	\(2\)	\(16\)
Plus space		\(+\)		\(348\)	\(39\)	\(309\)		\(312\)	\(37\)	\(275\)		\(36\)	\(2\)	\(34\)
Minus space		\(-\)		\(336\)	\(35\)	\(301\)		\(300\)	\(33\)	\(267\)		\(36\)	\(2\)	\(34\)

Trace form

\( 70 q - 2 q^{7} + 2 q^{13} + 4 q^{19} + 1552 q^{25} - 182 q^{31} - 400 q^{37} + 250 q^{43} + 2844 q^{49} - 246 q^{55} + 470 q^{61} - 1226 q^{67} + 824 q^{73} - 362 q^{79} + 1584 q^{85} + 1298 q^{91} - 34 q^{97}+O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1296))\) 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
1296.4.a.a	$1296$	$4$	1296.a	1.a	$1$	$1$	$1$	$76.466$	\(\Q\)	None	✓				162.4.a.b	$1$	$1$	\(0\)	\(0\)	\(-21\)	\(-8\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-21q^{5}-8q^{7}+6^{2}q^{11}-7^{2}q^{13}+\cdots\)
1296.4.a.b	$1296$	$4$	1296.a	1.a	$1$	$1$	$1$	$76.466$	\(\Q\)	None	✓				18.4.c.a	$1$	$1$	\(0\)	\(0\)	\(-9\)	\(31\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-9q^{5}+31q^{7}+15q^{11}-37q^{13}+\cdots\)
1296.4.a.c	$1296$	$4$	1296.a	1.a	$1$	$1$	$1$	$76.466$	\(\Q\)	None	✓				648.4.a.a	$1$	$2$	\(0\)	\(0\)	\(-5\)	\(-36\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5q^{5}-6^{2}q^{7}-2^{6}q^{11}-65q^{13}+\cdots\)
1296.4.a.d	$1296$	$4$	1296.a	1.a	$1$	$1$	$1$	$76.466$	\(\Q\)	None	✓				324.4.a.a	$1$	$1$	\(0\)	\(0\)	\(-3\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{5}+4q^{7}-24q^{11}-5^{2}q^{13}+\cdots\)
1296.4.a.e	$1296$	$4$	1296.a	1.a	$1$	$1$	$1$	$76.466$	\(\Q\)	None	✓				324.4.a.a	$1$	$1$	\(0\)	\(0\)	\(3\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}+4q^{7}+24q^{11}-5^{2}q^{13}+\cdots\)
1296.4.a.f	$1296$	$4$	1296.a	1.a	$1$	$1$	$1$	$76.466$	\(\Q\)	None	✓				648.4.a.a	$1$	$0$	\(0\)	\(0\)	\(5\)	\(-36\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5q^{5}-6^{2}q^{7}+2^{6}q^{11}-65q^{13}+\cdots\)
1296.4.a.g	$1296$	$4$	1296.a	1.a	$1$	$1$	$1$	$76.466$	\(\Q\)	None	✓				18.4.c.a	$1$	$0$	\(0\)	\(0\)	\(9\)	\(31\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+9q^{5}+31q^{7}-15q^{11}-37q^{13}+\cdots\)
1296.4.a.h	$1296$	$4$	1296.a	1.a	$1$	$1$	$1$	$76.466$	\(\Q\)	None	✓				162.4.a.b	$1$	$1$	\(0\)	\(0\)	\(21\)	\(-8\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+21q^{5}-8q^{7}-6^{2}q^{11}-7^{2}q^{13}+\cdots\)
1296.4.a.i	$1296$	$4$	1296.a	1.a	$1$	$2$	$2$	$76.466$	\(\Q(\sqrt{33}) \)	None	✓				9.4.c.a	$1$	$0$	\(0\)	\(0\)	\(-15\)	\(-7\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-7-\beta )q^{5}+(-5+3\beta )q^{7}+(29+\cdots)q^{11}+\cdots\)
1296.4.a.j	$1296$	$4$	1296.a	1.a	$1$	$2$	$2$	$76.466$	\(\Q(\sqrt{3}) \)	None	✓				162.4.a.e	$1$	$0$	\(0\)	\(0\)	\(-12\)	\(-16\)		$-$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q+(-6+\beta )q^{5}+(-8+2\beta )q^{7}+(18+\cdots)q^{11}+\cdots\)
1296.4.a.k	$1296$	$4$	1296.a	1.a	$1$	$2$	$2$	$76.466$	\(\Q(\sqrt{57}) \)	None	✓				81.4.a.b	$1$	$1$	\(0\)	\(0\)	\(-12\)	\(-10\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-6-\beta )q^{5}+(-5-3\beta )q^{7}+(21+\cdots)q^{11}+\cdots\)
1296.4.a.l	$1296$	$4$	1296.a	1.a	$1$	$2$	$2$	$76.466$	\(\Q(\sqrt{105}) \)	None	✓				18.4.c.b	$1$	$1$	\(0\)	\(0\)	\(-9\)	\(-19\)		$-$	$+$	$-$	$3$	$\mathrm{SU}(2)$	\(q+(-5-\beta )q^{5}+(-9+\beta )q^{7}+(11-2\beta )q^{11}+\cdots\)
1296.4.a.m	$1296$	$4$	1296.a	1.a	$1$	$2$	$2$	$76.466$	\(\Q(\sqrt{201}) \)	None	✓				648.4.a.c	$1$	$0$	\(0\)	\(0\)	\(-8\)	\(30\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-4-\beta )q^{5}+(15-\beta )q^{7}+(-23+\cdots)q^{11}+\cdots\)
1296.4.a.n	$1296$	$4$	1296.a	1.a	$1$	$2$	$2$	$76.466$	\(\Q(\sqrt{129}) \)	None	✓				648.4.a.d	$1$	$0$	\(0\)	\(0\)	\(-4\)	\(6\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-2-\beta )q^{5}+(3+\beta )q^{7}+(5+3\beta )q^{11}+\cdots\)
1296.4.a.o	$1296$	$4$	1296.a	1.a	$1$	$2$	$2$	$76.466$	\(\Q(\sqrt{3}) \)	None	✓				81.4.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(44\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+7\beta q^{5}+22q^{7}+34\beta q^{11}-7^{2}q^{13}+\cdots\)
1296.4.a.p	$1296$	$4$	1296.a	1.a	$1$	$2$	$2$	$76.466$	\(\Q(\sqrt{129}) \)	None	✓				648.4.a.d	$1$	$0$	\(0\)	\(0\)	\(4\)	\(6\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{5}+(3+\beta )q^{7}+(-5-3\beta )q^{11}+\cdots\)
1296.4.a.q	$1296$	$4$	1296.a	1.a	$1$	$2$	$2$	$76.466$	\(\Q(\sqrt{201}) \)	None	✓				648.4.a.c	$1$	$0$	\(0\)	\(0\)	\(8\)	\(30\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(4+\beta )q^{5}+(15-\beta )q^{7}+(23+3\beta )q^{11}+\cdots\)
1296.4.a.r	$1296$	$4$	1296.a	1.a	$1$	$2$	$2$	$76.466$	\(\Q(\sqrt{105}) \)	None	✓				18.4.c.b	$1$	$0$	\(0\)	\(0\)	\(9\)	\(-19\)		$-$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q+(5+\beta )q^{5}+(-9+\beta )q^{7}+(-11+2\beta )q^{11}+\cdots\)
1296.4.a.s	$1296$	$4$	1296.a	1.a	$1$	$2$	$2$	$76.466$	\(\Q(\sqrt{3}) \)	None	✓				162.4.a.e	$1$	$0$	\(0\)	\(0\)	\(12\)	\(-16\)		$-$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q+(6+\beta )q^{5}+(-8-2\beta )q^{7}+(-18+\cdots)q^{11}+\cdots\)
1296.4.a.t	$1296$	$4$	1296.a	1.a	$1$	$2$	$2$	$76.466$	\(\Q(\sqrt{57}) \)	None	✓				81.4.a.b	$1$	$1$	\(0\)	\(0\)	\(12\)	\(-10\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(6-\beta )q^{5}+(-5+3\beta )q^{7}+(-21+\cdots)q^{11}+\cdots\)
1296.4.a.u	$1296$	$4$	1296.a	1.a	$1$	$2$	$2$	$76.466$	\(\Q(\sqrt{33}) \)	None	✓				9.4.c.a	$1$	$1$	\(0\)	\(0\)	\(15\)	\(-7\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(8-\beta )q^{5}+(-2-3\beta )q^{7}+(-37+\cdots)q^{11}+\cdots\)
1296.4.a.v	$1296$	$4$	1296.a	1.a	$1$	$3$	$3$	$76.466$	3.3.1509.1	None	✓		✓		36.4.e.a	$1$	$1$	\(0\)	\(0\)	\(-6\)	\(-6\)		$-$	$+$	$-$	$2\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{2})q^{5}+(-2+\beta _{1}-\beta _{2})q^{7}+\cdots\)
1296.4.a.w	$1296$	$4$	1296.a	1.a	$1$	$3$	$3$	$76.466$	3.3.1509.1	None	✓				36.4.e.a	$1$	$0$	\(0\)	\(0\)	\(6\)	\(-6\)		$-$	$-$	$+$	$2\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(2+\beta _{2})q^{5}+(-2+\beta _{1}-\beta _{2})q^{7}+\cdots\)
1296.4.a.x	$1296$	$4$	1296.a	1.a	$1$	$4$	$4$	$76.466$	4.4.29952.1	None	✓				648.4.a.g	$1$	$1$	\(0\)	\(0\)	\(-8\)	\(0\)		$+$	$-$	$-$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{1})q^{5}+(-\beta _{1}+\beta _{2})q^{7}+(2+\cdots)q^{11}+\cdots\)
1296.4.a.y	$1296$	$4$	1296.a	1.a	$1$	$4$	$4$	$76.466$	4.4.72153.1	None	✓				72.4.i.a	$1$	$1$	\(0\)	\(0\)	\(-5\)	\(3\)		$+$	$-$	$-$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{5}+(1+\beta _{3})q^{7}+(-4+\cdots)q^{11}+\cdots\)
1296.4.a.z	$1296$	$4$	1296.a	1.a	$1$	$4$	$4$	$76.466$	\(\Q(\sqrt{3}, \sqrt{7})\)	None	✓		✓		324.4.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(16\)		$-$	$-$	$+$	$2^{4}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{5}+(4-\beta _{3})q^{7}+(\beta _{1}-\beta _{2})q^{11}+\cdots\)
1296.4.a.ba	$1296$	$4$	1296.a	1.a	$1$	$4$	$4$	$76.466$	4.4.72153.1	None	✓				72.4.i.a	$1$	$0$	\(0\)	\(0\)	\(5\)	\(3\)		$+$	$+$	$+$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{5}+(1+\beta _{3})q^{7}+(4-\beta _{1}+\cdots)q^{11}+\cdots\)
1296.4.a.bb	$1296$	$4$	1296.a	1.a	$1$	$4$	$4$	$76.466$	4.4.29952.1	None	✓				648.4.a.g	$1$	$1$	\(0\)	\(0\)	\(8\)	\(0\)		$+$	$-$	$-$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{5}+(\beta _{1}-\beta _{2})q^{7}+(-2+\beta _{1}+\cdots)q^{11}+\cdots\)
1296.4.a.bc	$1296$	$4$	1296.a	1.a	$1$	$5$	$5$	$76.466$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓		✓		72.4.i.b	$1$	$1$	\(0\)	\(0\)	\(-5\)	\(-3\)		$+$	$-$	$-$	$2^{4}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{5}+(-1-\beta _{1})q^{7}+(-5+\cdots)q^{11}+\cdots\)
1296.4.a.bd	$1296$	$4$	1296.a	1.a	$1$	$5$	$5$	$76.466$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓		✓		72.4.i.b	$1$	$0$	\(0\)	\(0\)	\(5\)	\(-3\)		$+$	$+$	$+$	$2^{4}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{5}+(-1-\beta _{1})q^{7}+(5-2\beta _{2}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1296))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1296)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 15}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(162))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(216))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(324))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(432))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(648))\)\(^{\oplus 2}\)