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}$