Space of modular forms of level 1350, weight 2, and trivial character

• Label: 1350.2.a
• Level: $1350$
• Weight: $2$
• Character orbit: 1350.a
• Rep. character: $\chi_{1350}(1,\cdot)$
• Character field: $\Q$
• Dimension: $26$
• Newform subspaces: $24$
• Sturm bound: $540$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1350.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 24 \)
Sturm bound:			\(540\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1350))\).

	Total	New	Old
Modular forms	306	26	280
Cusp forms	235	26	209
Eisenstein series	71	0	71

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
\(+\)	\(+\)	\(+\)	\(+\)		\(33\)	\(3\)	\(30\)		\(25\)	\(3\)	\(22\)		\(8\)	\(0\)	\(8\)
\(+\)	\(+\)	\(-\)	\(-\)		\(42\)	\(4\)	\(38\)		\(33\)	\(4\)	\(29\)		\(9\)	\(0\)	\(9\)
\(+\)	\(-\)	\(+\)	\(-\)		\(42\)	\(3\)	\(39\)		\(33\)	\(3\)	\(30\)		\(9\)	\(0\)	\(9\)
\(+\)	\(-\)	\(-\)	\(+\)		\(36\)	\(3\)	\(33\)		\(27\)	\(3\)	\(24\)		\(9\)	\(0\)	\(9\)
\(-\)	\(+\)	\(+\)	\(-\)		\(39\)	\(4\)	\(35\)		\(30\)	\(4\)	\(26\)		\(9\)	\(0\)	\(9\)
\(-\)	\(+\)	\(-\)	\(+\)		\(39\)	\(2\)	\(37\)		\(30\)	\(2\)	\(28\)		\(9\)	\(0\)	\(9\)
\(-\)	\(-\)	\(+\)	\(+\)		\(39\)	\(2\)	\(37\)		\(30\)	\(2\)	\(28\)		\(9\)	\(0\)	\(9\)
\(-\)	\(-\)	\(-\)	\(-\)		\(36\)	\(5\)	\(31\)		\(27\)	\(5\)	\(22\)		\(9\)	\(0\)	\(9\)
Plus space			\(+\)		\(147\)	\(10\)	\(137\)		\(112\)	\(10\)	\(102\)		\(35\)	\(0\)	\(35\)
Minus space			\(-\)		\(159\)	\(16\)	\(143\)		\(123\)	\(16\)	\(107\)		\(36\)	\(0\)	\(36\)

Trace form

26 * q + 26 * q^4 - 6 * q^7 + 26 * q^16 - 6 * q^22 - 6 * q^28 + 2 * q^31 + 8 * q^34 + 12 * q^37 + 36 * q^43 - 4 * q^46 + 60 * q^49 + 36 * q^58 + 4 * q^61 + 26 * q^64 - 12 * q^67 + 30 * q^73 + 12 * q^79 + 12 * q^82 - 6 * q^88 + 24 * q^91 + 28 * q^94 - 30 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1350))\) 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
1350.2.a.a	$1350$	$2$	1350.a	1.a	$1$	$1$	$1$	$10.780$	\(\Q\)	None	✓	✓			1350.2.a.a	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(-4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-4q^{7}-q^{8}+3q^{11}-q^{13}+\cdots\)
1350.2.a.b	$1350$	$2$	1350.a	1.a	$1$	$1$	$1$	$10.780$	\(\Q\)	None	✓				270.2.c.a	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(-4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-4q^{7}-q^{8}+5q^{11}+3q^{13}+\cdots\)
1350.2.a.c	$1350$	$2$	1350.a	1.a	$1$	$1$	$1$	$10.780$	\(\Q\)	None	✓				270.2.a.a	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-2q^{7}-q^{8}-3q^{11}+q^{13}+\cdots\)
1350.2.a.d	$1350$	$2$	1350.a	1.a	$1$	$1$	$1$	$10.780$	\(\Q\)	None	✓				270.2.a.b	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(-2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-2q^{7}-q^{8}+3q^{11}-5q^{13}+\cdots\)
1350.2.a.e	$1350$	$2$	1350.a	1.a	$1$	$1$	$1$	$10.780$	\(\Q\)	None	✓	✓			1350.2.a.e	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-2q^{7}-q^{8}+3q^{11}-5q^{13}+\cdots\)
1350.2.a.f	$1350$	$2$	1350.a	1.a	$1$	$1$	$1$	$10.780$	\(\Q\)	None	✓	✓			1350.2.a.f	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(-1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{7}-q^{8}+2q^{13}+q^{14}+\cdots\)
1350.2.a.g	$1350$	$2$	1350.a	1.a	$1$	$1$	$1$	$10.780$	\(\Q\)	None	✓	✓			1350.2.a.f	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{7}-q^{8}-2q^{13}-q^{14}+\cdots\)
1350.2.a.h	$1350$	$2$	1350.a	1.a	$1$	$1$	$1$	$10.780$	\(\Q\)	None	✓				54.2.a.a	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{7}-q^{8}+3q^{11}+4q^{13}+\cdots\)
1350.2.a.i	$1350$	$2$	1350.a	1.a	$1$	$1$	$1$	$10.780$	\(\Q\)	None	✓	✓			1350.2.a.e	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+2q^{7}-q^{8}-3q^{11}+5q^{13}+\cdots\)
1350.2.a.j	$1350$	$2$	1350.a	1.a	$1$	$1$	$1$	$10.780$	\(\Q\)	None	✓				270.2.c.a	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+4q^{7}-q^{8}-5q^{11}-3q^{13}+\cdots\)
1350.2.a.k	$1350$	$2$	1350.a	1.a	$1$	$1$	$1$	$10.780$	\(\Q\)	None	✓	✓			1350.2.a.a	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+4q^{7}-q^{8}-3q^{11}+q^{13}+\cdots\)
1350.2.a.l	$1350$	$2$	1350.a	1.a	$1$	$1$	$1$	$10.780$	\(\Q\)	None	✓				270.2.c.a	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-4q^{7}+q^{8}-5q^{11}+3q^{13}+\cdots\)
1350.2.a.m	$1350$	$2$	1350.a	1.a	$1$	$1$	$1$	$10.780$	\(\Q\)	None	✓	✓			1350.2.a.a	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-4q^{7}+q^{8}-3q^{11}-q^{13}+\cdots\)
1350.2.a.n	$1350$	$2$	1350.a	1.a	$1$	$1$	$1$	$10.780$	\(\Q\)	None	✓	✓			1350.2.a.e	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-2q^{7}+q^{8}-3q^{11}-5q^{13}+\cdots\)
1350.2.a.o	$1350$	$2$	1350.a	1.a	$1$	$1$	$1$	$10.780$	\(\Q\)	None	✓				270.2.a.b	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-2q^{7}+q^{8}-3q^{11}-5q^{13}+\cdots\)
1350.2.a.p	$1350$	$2$	1350.a	1.a	$1$	$1$	$1$	$10.780$	\(\Q\)	None	✓				270.2.a.a	$1$	$0$	\(1\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-2q^{7}+q^{8}+3q^{11}+q^{13}+\cdots\)
1350.2.a.q	$1350$	$2$	1350.a	1.a	$1$	$1$	$1$	$10.780$	\(\Q\)	None	✓	✓			1350.2.a.f	$1$	$0$	\(1\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{7}+q^{8}+2q^{13}-q^{14}+\cdots\)
1350.2.a.r	$1350$	$2$	1350.a	1.a	$1$	$1$	$1$	$10.780$	\(\Q\)	None	✓				54.2.a.a	$1$	$0$	\(1\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{7}+q^{8}-3q^{11}+4q^{13}+\cdots\)
1350.2.a.s	$1350$	$2$	1350.a	1.a	$1$	$1$	$1$	$10.780$	\(\Q\)	None	✓	✓			1350.2.a.f	$1$	$0$	\(1\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{7}+q^{8}-2q^{13}+q^{14}+\cdots\)
1350.2.a.t	$1350$	$2$	1350.a	1.a	$1$	$1$	$1$	$10.780$	\(\Q\)	None	✓	✓			1350.2.a.e	$1$	$0$	\(1\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+2q^{7}+q^{8}+3q^{11}+5q^{13}+\cdots\)
1350.2.a.u	$1350$	$2$	1350.a	1.a	$1$	$1$	$1$	$10.780$	\(\Q\)	None	✓	✓			1350.2.a.a	$1$	$0$	\(1\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+4q^{7}+q^{8}+3q^{11}+q^{13}+\cdots\)
1350.2.a.v	$1350$	$2$	1350.a	1.a	$1$	$1$	$1$	$10.780$	\(\Q\)	None	✓				270.2.c.a	$1$	$0$	\(1\)	\(0\)	\(0\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+4q^{7}+q^{8}+5q^{11}-3q^{13}+\cdots\)
1350.2.a.w	$1350$	$2$	1350.a	1.a	$1$	$2$	$2$	$10.780$	\(\Q(\sqrt{19}) \)	None	✓		✓		270.2.c.c	$2$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+\beta q^{7}-q^{8}+\beta q^{11}-\beta q^{14}+\cdots\)
1350.2.a.x	$1350$	$2$	1350.a	1.a	$1$	$2$	$2$	$10.780$	\(\Q(\sqrt{19}) \)	None	✓		✓		270.2.c.c	$2$	$0$	\(2\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+\beta q^{7}+q^{8}-\beta q^{11}+\beta q^{14}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1350))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1350)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(135))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(270))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(450))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(675))\)\(^{\oplus 2}\)