Space of modular forms of level 1296, weight 2, and trivial character

• Label: 1296.2.a
• Level: $1296$
• Weight: $2$
• Character orbit: 1296.a
• Rep. character: $\chi_{1296}(1,\cdot)$
• Character field: $\Q$
• Dimension: $22$
• Newform subspaces: $17$
• Sturm bound: $432$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1296.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 17 \)
Sturm bound:			\(432\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1296))\).

	Total	New	Old
Modular forms	252	26	226
Cusp forms	181	22	159
Eisenstein series	71	4	67

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
\(+\)	\(+\)	\(+\)		\(60\)	\(5\)	\(55\)		\(43\)	\(5\)	\(38\)		\(17\)	\(0\)	\(17\)
\(+\)	\(-\)	\(-\)		\(66\)	\(7\)	\(59\)		\(48\)	\(7\)	\(41\)		\(18\)	\(0\)	\(18\)
\(-\)	\(+\)	\(-\)		\(66\)	\(8\)	\(58\)		\(48\)	\(6\)	\(42\)		\(18\)	\(2\)	\(16\)
\(-\)	\(-\)	\(+\)		\(60\)	\(6\)	\(54\)		\(42\)	\(4\)	\(38\)		\(18\)	\(2\)	\(16\)
Plus space		\(+\)		\(120\)	\(11\)	\(109\)		\(85\)	\(9\)	\(76\)		\(35\)	\(2\)	\(33\)
Minus space		\(-\)		\(132\)	\(15\)	\(117\)		\(96\)	\(13\)	\(83\)		\(36\)	\(2\)	\(34\)

Trace form

\( 22 q - 2 q^{7} + 2 q^{13} + 4 q^{19} + 16 q^{25} + 10 q^{31} + 8 q^{37} + 10 q^{43} + 12 q^{49} - 6 q^{55} + 14 q^{61} + 22 q^{67} - 16 q^{73} + 22 q^{79} + 24 q^{85} + 2 q^{91} - 10 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1296))\) 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
1296.2.a.a	$1296$	$2$	1296.a	1.a	$1$	$1$	$1$	$10.349$	\(\Q\)	None	✓				324.2.a.b	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{5}-2q^{7}-6q^{11}+5q^{13}+3q^{17}+\cdots\)
1296.2.a.b	$1296$	$2$	1296.a	1.a	$1$	$1$	$1$	$10.349$	\(\Q\)	None	✓				36.2.e.a	$1$	$1$	\(0\)	\(0\)	\(-3\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{5}+q^{7}+3q^{11}-q^{13}-6q^{17}+\cdots\)
1296.2.a.c	$1296$	$2$	1296.a	1.a	$1$	$1$	$1$	$10.349$	\(\Q\)	None	✓				162.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{5}+4q^{7}-q^{13}-3q^{17}+4q^{19}+\cdots\)
1296.2.a.d	$1296$	$2$	1296.a	1.a	$1$	$1$	$1$	$10.349$	\(\Q\)	None	✓				648.2.a.b	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+4q^{11}-5q^{13}-5q^{17}-8q^{19}+\cdots\)
1296.2.a.e	$1296$	$2$	1296.a	1.a	$1$	$1$	$1$	$10.349$	\(\Q\)	None	✓				72.2.i.a	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+3q^{7}-5q^{11}-5q^{13}-2q^{17}+\cdots\)
1296.2.a.f	$1296$	$2$	1296.a	1.a	$1$	$1$	$1$	$10.349$	\(\Q\)	None	✓				18.2.c.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}-3q^{11}+2q^{13}+3q^{17}+q^{19}+\cdots\)
1296.2.a.g	$1296$	$2$	1296.a	1.a	$1$	$1$	$1$	$10.349$	\(\Q\)	None	✓				18.2.c.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}+3q^{11}+2q^{13}-3q^{17}+q^{19}+\cdots\)
1296.2.a.h	$1296$	$2$	1296.a	1.a	$1$	$1$	$1$	$10.349$	\(\Q\)	None	✓				648.2.a.b	$1$	$1$	\(0\)	\(0\)	\(1\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-4q^{11}-5q^{13}+5q^{17}-8q^{19}+\cdots\)
1296.2.a.i	$1296$	$2$	1296.a	1.a	$1$	$1$	$1$	$10.349$	\(\Q\)	None	✓				72.2.i.a	$1$	$0$	\(0\)	\(0\)	\(1\)	\(3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+3q^{7}+5q^{11}-5q^{13}+2q^{17}+\cdots\)
1296.2.a.j	$1296$	$2$	1296.a	1.a	$1$	$1$	$1$	$10.349$	\(\Q\)	None	✓				324.2.a.b	$1$	$0$	\(0\)	\(0\)	\(3\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}-2q^{7}+6q^{11}+5q^{13}-3q^{17}+\cdots\)
1296.2.a.k	$1296$	$2$	1296.a	1.a	$1$	$1$	$1$	$10.349$	\(\Q\)	None	✓				36.2.e.a	$1$	$0$	\(0\)	\(0\)	\(3\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}+q^{7}-3q^{11}-q^{13}+6q^{17}+\cdots\)
1296.2.a.l	$1296$	$2$	1296.a	1.a	$1$	$1$	$1$	$10.349$	\(\Q\)	None	✓				162.2.a.a	$1$	$0$	\(0\)	\(0\)	\(3\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}+4q^{7}-q^{13}+3q^{17}+4q^{19}+\cdots\)
1296.2.a.m	$1296$	$2$	1296.a	1.a	$1$	$2$	$2$	$10.349$	\(\Q(\sqrt{3}) \)	None	✓				648.2.a.e	$1$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-2+\beta )q^{5}+2\beta q^{7}+2q^{11}+(1+\cdots)q^{13}+\cdots\)
1296.2.a.n	$1296$	$2$	1296.a	1.a	$1$	$2$	$2$	$10.349$	\(\Q(\sqrt{33}) \)	None	✓		✓		72.2.i.b	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{5}+(-2+\beta )q^{7}-q^{11}+(2+\beta )q^{13}+\cdots\)
1296.2.a.o	$1296$	$2$	1296.a	1.a	$1$	$2$	$2$	$10.349$	\(\Q(\sqrt{3}) \)	None	✓		✓		81.2.a.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{5}-2q^{7}-2\beta q^{11}-q^{13}-3\beta q^{17}+\cdots\)
1296.2.a.p	$1296$	$2$	1296.a	1.a	$1$	$2$	$2$	$10.349$	\(\Q(\sqrt{33}) \)	None	✓				72.2.i.b	$1$	$0$	\(0\)	\(0\)	\(1\)	\(-3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+(-2+\beta )q^{7}+q^{11}+(2+\beta )q^{13}+\cdots\)
1296.2.a.q	$1296$	$2$	1296.a	1.a	$1$	$2$	$2$	$10.349$	\(\Q(\sqrt{3}) \)	None	✓				648.2.a.e	$1$	$0$	\(0\)	\(0\)	\(4\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{5}-2\beta q^{7}-2q^{11}+(1-2\beta )q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1296))\) into lower level spaces

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