Space of modular forms of level 600, weight 2, and trivial character

• Label: 600.2.a
• Level: $600$
• Weight: $2$
• Character orbit: 600.a
• Rep. character: $\chi_{600}(1,\cdot)$
• Character field: $\Q$
• Dimension: $9$
• Newform subspaces: $9$
• Sturm bound: $240$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	600.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(240\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(600))\).

	Total	New	Old
Modular forms	144	9	135
Cusp forms	97	9	88
Eisenstein series	47	0	47

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
\(+\)	\(+\)	\(+\)	\(+\)	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)	\(1\)
\(+\)	\(-\)	\(+\)	\(-\)	\(1\)
\(+\)	\(-\)	\(-\)	\(+\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)	\(2\)
Plus space			\(+\)	\(3\)
Minus space			\(-\)	\(6\)

Trace form

9 * q - q^3 - 4 * q^7 + 9 * q^9 - 4 * q^11 + 2 * q^13 + 6 * q^17 - 4 * q^21 + 16 * q^23 - q^27 + 22 * q^29 - 4 * q^31 + 8 * q^33 + 2 * q^37 - 6 * q^39 + 34 * q^41 - 12 * q^43 - 16 * q^47 + 29 * q^49 + 10 * q^51 - 14 * q^53 - 4 * q^57 - 20 * q^59 + 2 * q^61 - 4 * q^63 + 4 * q^67 - 24 * q^71 + 10 * q^73 - 48 * q^79 + 9 * q^81 + 4 * q^83 + 14 * q^87 + 18 * q^89 - 20 * q^91 + 16 * q^93 - 6 * q^97 - 4 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(600))\) 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
600.2.a.a	$600$	$2$	600.a	1.a	$1$	$1$	$1$	$4.791$	\(\Q\)	None	✓				120.2.a.a	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-4q^{7}+q^{9}+6q^{13}+2q^{17}+\cdots\)
600.2.a.b	$600$	$2$	600.a	1.a	$1$	$1$	$1$	$4.791$	\(\Q\)	None	✓	✓			600.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-3\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-3q^{7}+q^{9}+2q^{11}+3q^{13}+\cdots\)
600.2.a.c	$600$	$2$	600.a	1.a	$1$	$1$	$1$	$4.791$	\(\Q\)	None	✓				120.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{9}-4q^{11}-6q^{13}+6q^{17}+\cdots\)
600.2.a.d	$600$	$2$	600.a	1.a	$1$	$1$	$1$	$4.791$	\(\Q\)	None	✓		✓	✓	120.2.f.a	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{7}+q^{9}+2q^{11}-2q^{13}+\cdots\)
600.2.a.e	$600$	$2$	600.a	1.a	$1$	$1$	$1$	$4.791$	\(\Q\)	None	✓	✓			600.2.a.e	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(5\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+5q^{7}+q^{9}-6q^{11}+3q^{13}+\cdots\)
600.2.a.f	$600$	$2$	600.a	1.a	$1$	$1$	$1$	$4.791$	\(\Q\)	None	✓	✓	✓	✓	600.2.a.e	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-5\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-5q^{7}+q^{9}-6q^{11}-3q^{13}+\cdots\)
600.2.a.g	$600$	$2$	600.a	1.a	$1$	$1$	$1$	$4.791$	\(\Q\)	None	✓				120.2.f.a	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{7}+q^{9}+2q^{11}+2q^{13}+\cdots\)
600.2.a.h	$600$	$2$	600.a	1.a	$1$	$1$	$1$	$4.791$	\(\Q\)	None	✓		✓	✓	24.2.a.a	$1$	$0$	\(0\)	\(1\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{9}+4q^{11}+2q^{13}-2q^{17}+\cdots\)
600.2.a.i	$600$	$2$	600.a	1.a	$1$	$1$	$1$	$4.791$	\(\Q\)	None	✓	✓			600.2.a.b	$1$	$0$	\(0\)	\(1\)	\(0\)	\(3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+3q^{7}+q^{9}+2q^{11}-3q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(600))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(600)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 2}\)