Space of modular forms of level 3240, weight 2, and trivial character

• Label: 3240.2.a
• Level: $3240$
• Weight: $2$
• Character orbit: 3240.a
• Rep. character: $\chi_{3240}(1,\cdot)$
• Character field: $\Q$
• Dimension: $48$
• Newform subspaces: $22$
• Sturm bound: $1296$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 3240 = 2^{3} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3240.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 22 \)
Sturm bound:			\(1296\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(7\), \(11\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3240))\).

	Total	New	Old
Modular forms	696	48	648
Cusp forms	601	48	553
Eisenstein series	95	0	95

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

\(2\)	\(3\)	\(5\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(84\)	\(5\)	\(79\)		\(73\)	\(5\)	\(68\)		\(11\)	\(0\)	\(11\)
\(+\)	\(+\)	\(-\)	\(-\)		\(90\)	\(8\)	\(82\)		\(78\)	\(8\)	\(70\)		\(12\)	\(0\)	\(12\)
\(+\)	\(-\)	\(+\)	\(-\)		\(90\)	\(7\)	\(83\)		\(78\)	\(7\)	\(71\)		\(12\)	\(0\)	\(12\)
\(+\)	\(-\)	\(-\)	\(+\)		\(84\)	\(4\)	\(80\)		\(72\)	\(4\)	\(68\)		\(12\)	\(0\)	\(12\)
\(-\)	\(+\)	\(+\)	\(-\)		\(90\)	\(6\)	\(84\)		\(78\)	\(6\)	\(72\)		\(12\)	\(0\)	\(12\)
\(-\)	\(+\)	\(-\)	\(+\)		\(84\)	\(5\)	\(79\)		\(72\)	\(5\)	\(67\)		\(12\)	\(0\)	\(12\)
\(-\)	\(-\)	\(+\)	\(+\)		\(84\)	\(6\)	\(78\)		\(72\)	\(6\)	\(66\)		\(12\)	\(0\)	\(12\)
\(-\)	\(-\)	\(-\)	\(-\)		\(90\)	\(7\)	\(83\)		\(78\)	\(7\)	\(71\)		\(12\)	\(0\)	\(12\)
Plus space			\(+\)		\(336\)	\(20\)	\(316\)		\(289\)	\(20\)	\(269\)		\(47\)	\(0\)	\(47\)
Minus space			\(-\)		\(360\)	\(28\)	\(332\)		\(312\)	\(28\)	\(284\)		\(48\)	\(0\)	\(48\)

Trace form

\( 48 q - 12 q^{19} + 48 q^{25} + 24 q^{31} + 12 q^{43} + 36 q^{49} - 12 q^{61} - 12 q^{67} + 36 q^{73} + 48 q^{91} + 36 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3240))\) 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	5
3240.2.a.a	$3240$	$2$	3240.a	1.a	$1$	$1$	$1$	$25.872$	\(\Q\)	None	✓	✓			3240.2.a.a	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-2q^{7}+3q^{11}-2q^{17}+q^{19}+\cdots\)
3240.2.a.b	$3240$	$2$	3240.a	1.a	$1$	$1$	$1$	$25.872$	\(\Q\)	None	✓				360.2.q.a	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-5q^{11}+3q^{17}+5q^{19}+6q^{23}+\cdots\)
3240.2.a.c	$3240$	$2$	3240.a	1.a	$1$	$1$	$1$	$25.872$	\(\Q\)	None	✓	✓			3240.2.a.c	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+q^{11}-7q^{19}+6q^{23}+q^{25}+\cdots\)
3240.2.a.d	$3240$	$2$	3240.a	1.a	$1$	$1$	$1$	$25.872$	\(\Q\)	None	✓	✓			3240.2.a.a	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-2q^{7}-3q^{11}+2q^{17}+q^{19}+\cdots\)
3240.2.a.e	$3240$	$2$	3240.a	1.a	$1$	$1$	$1$	$25.872$	\(\Q\)	None	✓	✓			3240.2.a.c	$1$	$1$	\(0\)	\(0\)	\(1\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-q^{11}-7q^{19}-6q^{23}+q^{25}+\cdots\)
3240.2.a.f	$3240$	$2$	3240.a	1.a	$1$	$1$	$1$	$25.872$	\(\Q\)	None	✓				360.2.q.a	$1$	$0$	\(0\)	\(0\)	\(1\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+5q^{11}-3q^{17}+5q^{19}-6q^{23}+\cdots\)
3240.2.a.g	$3240$	$2$	3240.a	1.a	$1$	$2$	$2$	$25.872$	\(\Q(\sqrt{3}) \)	None	✓	✓			3240.2.a.g	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+(-1+\beta )q^{7}+(2+\beta )q^{11}+(-2+\cdots)q^{13}+\cdots\)
3240.2.a.h	$3240$	$2$	3240.a	1.a	$1$	$2$	$2$	$25.872$	\(\Q(\sqrt{57}) \)	None	✓	✓			3240.2.a.h	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+\beta q^{7}+(-3+\beta )q^{11}+(2+\beta )q^{13}+\cdots\)
3240.2.a.i	$3240$	$2$	3240.a	1.a	$1$	$2$	$2$	$25.872$	\(\Q(\sqrt{33}) \)	None	✓	✓			3240.2.a.i	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+\beta q^{7}+(1-\beta )q^{11}-\beta q^{13}+\cdots\)
3240.2.a.j	$3240$	$2$	3240.a	1.a	$1$	$2$	$2$	$25.872$	\(\Q(\sqrt{6}) \)	None	✓		✓		360.2.q.c	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+(1+\beta )q^{7}-2q^{17}+(-2-2\beta )q^{19}+\cdots\)
3240.2.a.k	$3240$	$2$	3240.a	1.a	$1$	$2$	$2$	$25.872$	\(\Q(\sqrt{3}) \)	None	✓				360.2.q.b	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+(2+\beta )q^{7}+(2+2\beta )q^{11}+(-2+\cdots)q^{13}+\cdots\)
3240.2.a.l	$3240$	$2$	3240.a	1.a	$1$	$2$	$2$	$25.872$	\(\Q(\sqrt{3}) \)	None	✓	✓			3240.2.a.g	$1$	$1$	\(0\)	\(0\)	\(2\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+(-1+\beta )q^{7}+(-2-\beta )q^{11}+\cdots\)
3240.2.a.m	$3240$	$2$	3240.a	1.a	$1$	$2$	$2$	$25.872$	\(\Q(\sqrt{33}) \)	None	✓	✓			3240.2.a.i	$1$	$0$	\(0\)	\(0\)	\(2\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+\beta q^{7}+(-1+\beta )q^{11}-\beta q^{13}+\cdots\)
3240.2.a.n	$3240$	$2$	3240.a	1.a	$1$	$2$	$2$	$25.872$	\(\Q(\sqrt{57}) \)	None	✓	✓			3240.2.a.h	$1$	$0$	\(0\)	\(0\)	\(2\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+\beta q^{7}+(3-\beta )q^{11}+(2+\beta )q^{13}+\cdots\)
3240.2.a.o	$3240$	$2$	3240.a	1.a	$1$	$2$	$2$	$25.872$	\(\Q(\sqrt{6}) \)	None	✓				360.2.q.c	$1$	$0$	\(0\)	\(0\)	\(2\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+(1+\beta )q^{7}+2q^{17}+(-2-2\beta )q^{19}+\cdots\)
3240.2.a.p	$3240$	$2$	3240.a	1.a	$1$	$2$	$2$	$25.872$	\(\Q(\sqrt{3}) \)	None	✓				360.2.q.b	$1$	$1$	\(0\)	\(0\)	\(2\)	\(4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+(2+\beta )q^{7}+(-2-2\beta )q^{11}+\cdots\)
3240.2.a.q	$3240$	$2$	3240.a	1.a	$1$	$3$	$3$	$25.872$	3.3.564.1	None	✓				360.2.q.d	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(-5\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+(-1-\beta _{1}+\beta _{2})q^{7}+(1+\beta _{2})q^{11}+\cdots\)
3240.2.a.r	$3240$	$2$	3240.a	1.a	$1$	$3$	$3$	$25.872$	3.3.564.1	None	✓		✓		360.2.q.d	$1$	$1$	\(0\)	\(0\)	\(3\)	\(-5\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+(-1-\beta _{1}+\beta _{2})q^{7}+(-1-\beta _{2})q^{11}+\cdots\)
3240.2.a.s	$3240$	$2$	3240.a	1.a	$1$	$4$	$4$	$25.872$	4.4.29268.1	None	✓		✓		360.2.q.e	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(-1\)		$-$	$-$	$+$	$+$	$3$	$\mathrm{SU}(2)$	\(q-q^{5}-\beta _{2}q^{7}+(\beta _{2}+\beta _{3})q^{11}+(1+\beta _{1}+\cdots)q^{13}+\cdots\)
3240.2.a.t	$3240$	$2$	3240.a	1.a	$1$	$4$	$4$	$25.872$	4.4.62352.1	None	✓	✓	✓		3240.2.a.t	$1$	$0$	\(0\)	\(0\)	\(-4\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+(\beta _{1}-\beta _{2})q^{7}+(-2+\beta _{1}-\beta _{3})q^{11}+\cdots\)
3240.2.a.u	$3240$	$2$	3240.a	1.a	$1$	$4$	$4$	$25.872$	4.4.29268.1	None	✓		✓		360.2.q.e	$1$	$0$	\(0\)	\(0\)	\(4\)	\(-1\)		$+$	$+$	$-$	$-$	$3$	$\mathrm{SU}(2)$	\(q+q^{5}-\beta _{2}q^{7}+(-\beta _{2}-\beta _{3})q^{11}+(1+\cdots)q^{13}+\cdots\)
3240.2.a.v	$3240$	$2$	3240.a	1.a	$1$	$4$	$4$	$25.872$	4.4.62352.1	None	✓	✓	✓		3240.2.a.t	$1$	$0$	\(0\)	\(0\)	\(4\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+(\beta _{1}-\beta _{2})q^{7}+(2-\beta _{1}+\beta _{3})q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3240))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3240)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(135))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(162))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(216))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(270))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(324))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(360))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(405))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(540))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(648))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(810))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1080))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1620))\)\(^{\oplus 2}\)