Space of modular forms of level 1320, weight 2, and trivial character

• Label: 1320.2.a
• Level: $1320$
• Weight: $2$
• Character orbit: 1320.a
• Rep. character: $\chi_{1320}(1,\cdot)$
• Character field: $\Q$
• Dimension: $20$
• Newform subspaces: $17$
• Sturm bound: $576$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 1320 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1320.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 17 \)
Sturm bound:			\(576\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(7\), \(13\), \(17\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1320))\).

	Total	New	Old
Modular forms	304	20	284
Cusp forms	273	20	253
Eisenstein series	31	0	31

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

\(2\)	\(3\)	\(5\)	\(11\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(+\)	\(1\)
\(+\)	\(+\)	\(+\)	\(-\)	\(-\)	\(2\)
\(+\)	\(+\)	\(-\)	\(+\)	\(-\)	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)	\(+\)	\(1\)
\(+\)	\(-\)	\(+\)	\(+\)	\(-\)	\(1\)
\(+\)	\(-\)	\(+\)	\(-\)	\(+\)	\(1\)
\(+\)	\(-\)	\(-\)	\(-\)	\(-\)	\(2\)
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)	\(+\)	\(2\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)	\(1\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)	\(1\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)	\(+\)	\(1\)
Plus space				\(+\)	\(8\)
Minus space				\(-\)	\(12\)

Trace form

\( 20 q - 4 q^{3} - 8 q^{7} + 20 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1320))\) 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	11
1320.2.a.a	$1320$	$2$	1320.a	1.a	$1$	$1$	$1$	$10.540$	\(\Q\)	None	✓	✓			1320.2.a.a	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(-2\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-2q^{7}+q^{9}-q^{11}-4q^{13}+\cdots\)
1320.2.a.b	$1320$	$2$	1320.a	1.a	$1$	$1$	$1$	$10.540$	\(\Q\)	None	✓	✓	✓	✓	1320.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(-2\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-2q^{7}+q^{9}+q^{11}+4q^{13}+\cdots\)
1320.2.a.c	$1320$	$2$	1320.a	1.a	$1$	$1$	$1$	$10.540$	\(\Q\)	None	✓	✓	✓	✓	1320.2.a.c	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(2\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+2q^{7}+q^{9}-q^{11}-4q^{13}+\cdots\)
1320.2.a.d	$1320$	$2$	1320.a	1.a	$1$	$1$	$1$	$10.540$	\(\Q\)	None	✓	✓			1320.2.a.d	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(4\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+4q^{7}+q^{9}-q^{11}+2q^{13}+\cdots\)
1320.2.a.e	$1320$	$2$	1320.a	1.a	$1$	$1$	$1$	$10.540$	\(\Q\)	None	✓	✓			1320.2.a.e	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(-4\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-4q^{7}+q^{9}-q^{11}+2q^{13}+\cdots\)
1320.2.a.f	$1320$	$2$	1320.a	1.a	$1$	$1$	$1$	$10.540$	\(\Q\)	None	✓	✓			1320.2.a.f	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(0\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+q^{9}-q^{11}-2q^{13}-q^{15}+\cdots\)
1320.2.a.g	$1320$	$2$	1320.a	1.a	$1$	$1$	$1$	$10.540$	\(\Q\)	None	✓	✓	✓	✓	1320.2.a.g	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(0\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+q^{9}+q^{11}-6q^{13}-q^{15}+\cdots\)
1320.2.a.h	$1320$	$2$	1320.a	1.a	$1$	$1$	$1$	$10.540$	\(\Q\)	None	✓	✓	✓	✓	1320.2.a.h	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(4\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+4q^{7}+q^{9}+q^{11}+2q^{13}+\cdots\)
1320.2.a.i	$1320$	$2$	1320.a	1.a	$1$	$1$	$1$	$10.540$	\(\Q\)	None	✓	✓	✓	✓	1320.2.a.i	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(-2\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-2q^{7}+q^{9}-q^{11}-4q^{13}+\cdots\)
1320.2.a.j	$1320$	$2$	1320.a	1.a	$1$	$1$	$1$	$10.540$	\(\Q\)	None	✓	✓	✓	✓	1320.2.a.j	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(-2\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-2q^{7}+q^{9}-q^{11}-q^{15}+\cdots\)
1320.2.a.k	$1320$	$2$	1320.a	1.a	$1$	$1$	$1$	$10.540$	\(\Q\)	None	✓	✓	✓	✓	1320.2.a.k	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(-2\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-2q^{7}+q^{9}+q^{11}-q^{15}+\cdots\)
1320.2.a.l	$1320$	$2$	1320.a	1.a	$1$	$1$	$1$	$10.540$	\(\Q\)	None	✓	✓	✓	✓	1320.2.a.l	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(2\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+2q^{7}+q^{9}+q^{11}-q^{15}+\cdots\)
1320.2.a.m	$1320$	$2$	1320.a	1.a	$1$	$1$	$1$	$10.540$	\(\Q\)	None	✓	✓	✓	✓	1320.2.a.m	$1$	$1$	\(0\)	\(1\)	\(1\)	\(-4\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-4q^{7}+q^{9}+q^{11}-6q^{13}+\cdots\)
1320.2.a.n	$1320$	$2$	1320.a	1.a	$1$	$1$	$1$	$10.540$	\(\Q\)	None	✓	✓	✓	✓	1320.2.a.n	$1$	$0$	\(0\)	\(1\)	\(1\)	\(0\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+q^{9}-q^{11}-2q^{13}+q^{15}+\cdots\)
1320.2.a.o	$1320$	$2$	1320.a	1.a	$1$	$2$	$2$	$10.540$	\(\Q(\sqrt{17}) \)	None	✓	✓	✓	✓	1320.2.a.o	$1$	$0$	\(0\)	\(-2\)	\(-2\)	\(-2\)		$+$	$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+(-1-\beta )q^{7}+q^{9}+q^{11}+\cdots\)
1320.2.a.p	$1320$	$2$	1320.a	1.a	$1$	$2$	$2$	$10.540$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓	✓	1320.2.a.p	$1$	$0$	\(0\)	\(-2\)	\(2\)	\(0\)		$+$	$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+\beta q^{7}+q^{9}-q^{11}+(2+\cdots)q^{13}+\cdots\)
1320.2.a.q	$1320$	$2$	1320.a	1.a	$1$	$2$	$2$	$10.540$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓	✓	1320.2.a.q	$1$	$0$	\(0\)	\(2\)	\(2\)	\(0\)		$+$	$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+\beta q^{7}+q^{9}+q^{11}+(2+\cdots)q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1320))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1320)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(66))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(88))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(110))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(132))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(165))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(220))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(264))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(330))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(440))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(660))\)\(^{\oplus 2}\)