Space of modular forms of level 1620, weight 2, and trivial character

• Label: 1620.2.a
• Level: $1620$
• Weight: $2$
• Character orbit: 1620.a
• Rep. character: $\chi_{1620}(1,\cdot)$
• Character field: $\Q$
• Dimension: $16$
• Newform subspaces: $10$
• Sturm bound: $648$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1620.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(648\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\), \(11\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1620))\).

	Total	New	Old
Modular forms	360	16	344
Cusp forms	289	16	273
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	Dim
\(-\)	\(+\)	\(+\)	\(-\)	\(5\)
\(-\)	\(+\)	\(-\)	\(+\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)	\(5\)
Plus space			\(+\)	\(6\)
Minus space			\(-\)	\(10\)

Trace form

\( 16 q - 4 q^{7} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1620))\) 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
1620.2.a.a	$1620$	$2$	1620.a	1.a	$1$	$1$	$1$	$12.936$	\(\Q\)	None	✓	✓			1620.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-4q^{7}-3q^{11}-4q^{13}+5q^{19}+\cdots\)
1620.2.a.b	$1620$	$2$	1620.a	1.a	$1$	$1$	$1$	$12.936$	\(\Q\)	None	✓				180.2.i.a	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-q^{7}-4q^{13}+6q^{17}+2q^{19}+\cdots\)
1620.2.a.c	$1620$	$2$	1620.a	1.a	$1$	$1$	$1$	$12.936$	\(\Q\)	None	✓	✓			1620.2.a.c	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+2q^{7}+3q^{11}-4q^{13}+6q^{17}+\cdots\)
1620.2.a.d	$1620$	$2$	1620.a	1.a	$1$	$1$	$1$	$12.936$	\(\Q\)	None	✓	✓			1620.2.a.a	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-4q^{7}+3q^{11}-4q^{13}+5q^{19}+\cdots\)
1620.2.a.e	$1620$	$2$	1620.a	1.a	$1$	$1$	$1$	$12.936$	\(\Q\)	None	✓				180.2.i.a	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-q^{7}-4q^{13}-6q^{17}+2q^{19}+\cdots\)
1620.2.a.f	$1620$	$2$	1620.a	1.a	$1$	$1$	$1$	$12.936$	\(\Q\)	None	✓	✓			1620.2.a.c	$1$	$1$	\(0\)	\(0\)	\(1\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+2q^{7}-3q^{11}-4q^{13}-6q^{17}+\cdots\)
1620.2.a.g	$1620$	$2$	1620.a	1.a	$1$	$2$	$2$	$12.936$	\(\Q(\sqrt{3}) \)	None	✓	✓	✓		1620.2.a.g	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+(-1+\beta )q^{7}-\beta q^{11}+(2-2\beta )q^{13}+\cdots\)
1620.2.a.h	$1620$	$2$	1620.a	1.a	$1$	$2$	$2$	$12.936$	\(\Q(\sqrt{3}) \)	None	✓	✓			1620.2.a.g	$1$	$0$	\(0\)	\(0\)	\(2\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+(-1+\beta )q^{7}+\beta q^{11}+(2-2\beta )q^{13}+\cdots\)
1620.2.a.i	$1620$	$2$	1620.a	1.a	$1$	$3$	$3$	$12.936$	3.3.564.1	None	✓		✓		180.2.i.b	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(3\)		$-$	$+$	$+$	$-$	$3$	$\mathrm{SU}(2)$	\(q-q^{5}+(1-\beta _{1})q^{7}-\beta _{2}q^{11}+(2+\beta _{2})q^{13}+\cdots\)
1620.2.a.j	$1620$	$2$	1620.a	1.a	$1$	$3$	$3$	$12.936$	3.3.564.1	None	✓		✓		180.2.i.b	$1$	$0$	\(0\)	\(0\)	\(3\)	\(3\)		$-$	$-$	$-$	$-$	$3$	$\mathrm{SU}(2)$	\(q+q^{5}+(1-\beta _{1})q^{7}+\beta _{2}q^{11}+(2+\beta _{2})q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1620))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1620)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(135))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(162))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(270))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(324))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(405))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(540))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(810))\)\(^{\oplus 2}\)