Space of modular forms of level 540, weight 4, and trivial character

• Label: 540.4.a
• Level: $540$
• Weight: $4$
• Character orbit: 540.a
• Rep. character: $\chi_{540}(1,\cdot)$
• Character field: $\Q$
• Dimension: $16$
• Newform subspaces: $10$
• Sturm bound: $432$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 540 = 2^{2} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	540.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(432\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(540))\).

	Total	New	Old
Modular forms	342	16	326
Cusp forms	306	16	290
Eisenstein series	36	0	36

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
\(-\)	\(+\)	\(+\)	$-$	\(4\)
\(-\)	\(+\)	\(-\)	$+$	\(4\)
\(-\)	\(-\)	\(+\)	$+$	\(4\)
\(-\)	\(-\)	\(-\)	$-$	\(4\)
Plus space			\(+\)	\(8\)
Minus space			\(-\)	\(8\)

Trace form

\( 16 q + 32 q^{7} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(540))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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
540.4.a.a	$540$	$4$	540.a	1.a	$1$	$1$	$1$	$31.861$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-5\)	\(-22\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5q^{5}-22q^{7}-9q^{11}+17q^{13}+\cdots\)
540.4.a.b	$540$	$4$	540.a	1.a	$1$	$1$	$1$	$31.861$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-5\)	\(17\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5q^{5}+17q^{7}+30q^{11}-61q^{13}+\cdots\)
540.4.a.c	$540$	$4$	540.a	1.a	$1$	$1$	$1$	$31.861$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(5\)	\(-22\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{5}-22q^{7}+9q^{11}+17q^{13}+\cdots\)
540.4.a.d	$540$	$4$	540.a	1.a	$1$	$1$	$1$	$31.861$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(5\)	\(17\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{5}+17q^{7}-30q^{11}-61q^{13}+\cdots\)
540.4.a.e	$540$	$4$	540.a	1.a	$1$	$2$	$2$	$31.861$	\(\Q(\sqrt{21}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-10\)	\(-2\)		$-$	$+$	$+$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q-5q^{5}+(-1-\beta )q^{7}+(21-\beta )q^{11}+\cdots\)
540.4.a.f	$540$	$4$	540.a	1.a	$1$	$2$	$2$	$31.861$	\(\Q(\sqrt{69}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-10\)	\(10\)		$-$	$-$	$+$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q-5q^{5}+(5+\beta )q^{7}+(-9+\beta )q^{11}+\cdots\)
540.4.a.g	$540$	$4$	540.a	1.a	$1$	$2$	$2$	$31.861$	\(\Q(\sqrt{41}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-10\)	\(13\)		$-$	$+$	$+$	$-$	$3$	$\mathrm{SU}(2)$	\(q-5q^{5}+(6-\beta )q^{7}+(-5+5\beta )q^{11}+\cdots\)
540.4.a.h	$540$	$4$	540.a	1.a	$1$	$2$	$2$	$31.861$	\(\Q(\sqrt{21}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(10\)	\(-2\)		$-$	$-$	$-$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+5q^{5}+(-1-\beta )q^{7}+(-21+\beta )q^{11}+\cdots\)
540.4.a.i	$540$	$4$	540.a	1.a	$1$	$2$	$2$	$31.861$	\(\Q(\sqrt{69}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(10\)	\(10\)		$-$	$+$	$-$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+5q^{5}+(5+\beta )q^{7}+(9-\beta )q^{11}+(17+\cdots)q^{13}+\cdots\)
540.4.a.j	$540$	$4$	540.a	1.a	$1$	$2$	$2$	$31.861$	\(\Q(\sqrt{41}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(10\)	\(13\)		$-$	$+$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q+5q^{5}+(6-\beta )q^{7}+(5-5\beta )q^{11}+(3+\cdots)q^{13}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(540))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(540)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(135))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(270))\)\(^{\oplus 2}\)