Space of modular forms of level 600, weight 4, and trivial character

• Label: 600.4.a
• Level: $600$
• Weight: $4$
• Character orbit: 600.a
• Rep. character: $\chi_{600}(1,\cdot)$
• Character field: $\Q$
• Dimension: $29$
• Newform subspaces: $23$
• Sturm bound: $480$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	600.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 23 \)
Sturm bound:			\(480\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(600))\).

	Total	New	Old
Modular forms	384	29	355
Cusp forms	336	29	307
Eisenstein series	48	0	48

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
\(+\)	\(+\)	\(+\)	\(+\)		\(51\)	\(3\)	\(48\)		\(45\)	\(3\)	\(42\)		\(6\)	\(0\)	\(6\)
\(+\)	\(+\)	\(-\)	\(-\)		\(46\)	\(4\)	\(42\)		\(40\)	\(4\)	\(36\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(+\)	\(-\)		\(48\)	\(4\)	\(44\)		\(42\)	\(4\)	\(38\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)		\(47\)	\(4\)	\(43\)		\(41\)	\(4\)	\(37\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(+\)	\(-\)		\(45\)	\(2\)	\(43\)		\(39\)	\(2\)	\(37\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)	\(+\)		\(50\)	\(5\)	\(45\)		\(44\)	\(5\)	\(39\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)	\(+\)		\(48\)	\(4\)	\(44\)		\(42\)	\(4\)	\(38\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(-\)	\(-\)		\(49\)	\(3\)	\(46\)		\(43\)	\(3\)	\(40\)		\(6\)	\(0\)	\(6\)
Plus space			\(+\)		\(196\)	\(16\)	\(180\)		\(172\)	\(16\)	\(156\)		\(24\)	\(0\)	\(24\)
Minus space			\(-\)		\(188\)	\(13\)	\(175\)		\(164\)	\(13\)	\(151\)		\(24\)	\(0\)	\(24\)

Trace form

29 * q + 3 * q^3 - 12 * q^7 + 261 * q^9 + 28 * q^11 + 58 * q^13 - 90 * q^17 - 272 * q^19 + 12 * q^21 - 160 * q^23 + 27 * q^27 + 158 * q^29 + 20 * q^31 - 48 * q^33 - 598 * q^37 - 198 * q^39 - 22 * q^41 + 116 * q^43 + 928 * q^47 + 2297 * q^49 - 366 * q^51 + 1394 * q^53 - 420 * q^57 - 1156 * q^59 + 1578 * q^61 - 108 * q^63 + 500 * q^67 + 576 * q^69 + 1464 * q^71 + 802 * q^73 - 768 * q^77 + 392 * q^79 + 2349 * q^81 + 836 * q^83 + 6 * q^87 + 5226 * q^89 - 1396 * q^91 - 1128 * q^93 + 50 * q^97 + 252 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(600))\) 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
600.4.a.a	$600$	$4$	600.a	1.a	$1$	$1$	$1$	$35.401$	\(\Q\)	None	✓				120.4.a.e	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(-20\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-20q^{7}+9q^{9}+2^{4}q^{11}-58q^{13}+\cdots\)
600.4.a.b	$600$	$4$	600.a	1.a	$1$	$1$	$1$	$35.401$	\(\Q\)	None	✓				120.4.f.a	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(-10\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-10q^{7}+9q^{9}-46q^{11}+34q^{13}+\cdots\)
600.4.a.c	$600$	$4$	600.a	1.a	$1$	$1$	$1$	$35.401$	\(\Q\)	None	✓				120.4.f.b	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-10\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-10q^{7}+9q^{9}-14q^{11}+82q^{13}+\cdots\)
600.4.a.d	$600$	$4$	600.a	1.a	$1$	$1$	$1$	$35.401$	\(\Q\)	None	✓				120.4.a.f	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-8\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-8q^{7}+9q^{9}+20q^{11}-22q^{13}+\cdots\)
600.4.a.e	$600$	$4$	600.a	1.a	$1$	$1$	$1$	$35.401$	\(\Q\)	None	✓	✓			600.4.a.e	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-5\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-5q^{7}+9q^{9}+14q^{11}-q^{13}+\cdots\)
600.4.a.f	$600$	$4$	600.a	1.a	$1$	$1$	$1$	$35.401$	\(\Q\)	None	✓				120.4.f.c	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+4q^{7}+9q^{9}-28q^{11}-2^{4}q^{13}+\cdots\)
600.4.a.g	$600$	$4$	600.a	1.a	$1$	$1$	$1$	$35.401$	\(\Q\)	None	✓	✓			600.4.a.g	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(19\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+19q^{7}+9q^{9}+22q^{11}-q^{13}+\cdots\)
600.4.a.h	$600$	$4$	600.a	1.a	$1$	$1$	$1$	$35.401$	\(\Q\)	None	✓				24.4.a.a	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(24\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+24q^{7}+9q^{9}-28q^{11}+74q^{13}+\cdots\)
600.4.a.i	$600$	$4$	600.a	1.a	$1$	$1$	$1$	$35.401$	\(\Q\)	None	✓				120.4.a.b	$1$	$1$	\(0\)	\(3\)	\(0\)	\(-20\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-20q^{7}+9q^{9}-56q^{11}+86q^{13}+\cdots\)
600.4.a.j	$600$	$4$	600.a	1.a	$1$	$1$	$1$	$35.401$	\(\Q\)	None	✓	✓			600.4.a.g	$1$	$1$	\(0\)	\(3\)	\(0\)	\(-19\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-19q^{7}+9q^{9}+22q^{11}+q^{13}+\cdots\)
600.4.a.k	$600$	$4$	600.a	1.a	$1$	$1$	$1$	$35.401$	\(\Q\)	None	✓				120.4.f.c	$1$	$1$	\(0\)	\(3\)	\(0\)	\(-4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-4q^{7}+9q^{9}-28q^{11}+2^{4}q^{13}+\cdots\)
600.4.a.l	$600$	$4$	600.a	1.a	$1$	$1$	$1$	$35.401$	\(\Q\)	None	✓				120.4.a.a	$1$	$0$	\(0\)	\(3\)	\(0\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-4q^{7}+9q^{9}+72q^{11}+6q^{13}+\cdots\)
600.4.a.m	$600$	$4$	600.a	1.a	$1$	$1$	$1$	$35.401$	\(\Q\)	None	✓				120.4.a.d	$1$	$1$	\(0\)	\(3\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+9q^{9}+4q^{11}-54q^{13}-114q^{17}+\cdots\)
600.4.a.n	$600$	$4$	600.a	1.a	$1$	$1$	$1$	$35.401$	\(\Q\)	None	✓	✓			600.4.a.e	$1$	$0$	\(0\)	\(3\)	\(0\)	\(5\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+5q^{7}+9q^{9}+14q^{11}+q^{13}+\cdots\)
600.4.a.o	$600$	$4$	600.a	1.a	$1$	$1$	$1$	$35.401$	\(\Q\)	None	✓				120.4.f.a	$1$	$0$	\(0\)	\(3\)	\(0\)	\(10\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+10q^{7}+9q^{9}-46q^{11}-34q^{13}+\cdots\)
600.4.a.p	$600$	$4$	600.a	1.a	$1$	$1$	$1$	$35.401$	\(\Q\)	None	✓				120.4.f.b	$1$	$1$	\(0\)	\(3\)	\(0\)	\(10\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+10q^{7}+9q^{9}-14q^{11}-82q^{13}+\cdots\)
600.4.a.q	$600$	$4$	600.a	1.a	$1$	$1$	$1$	$35.401$	\(\Q\)	None	✓				120.4.a.c	$1$	$0$	\(0\)	\(3\)	\(0\)	\(16\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+2^{4}q^{7}+9q^{9}-28q^{11}+26q^{13}+\cdots\)
600.4.a.r	$600$	$4$	600.a	1.a	$1$	$2$	$2$	$35.401$	\(\Q(\sqrt{181}) \)	None	✓	✓	✓		600.4.a.r	$1$	$1$	\(0\)	\(-6\)	\(0\)	\(-6\)		$+$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(-3-\beta )q^{7}+9q^{9}+(4+\beta )q^{11}+\cdots\)
600.4.a.s	$600$	$4$	600.a	1.a	$1$	$2$	$2$	$35.401$	\(\Q(\sqrt{109}) \)	None	✓	✓			600.4.a.s	$1$	$0$	\(0\)	\(-6\)	\(0\)	\(2\)		$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(1+\beta )q^{7}+9q^{9}+(-8+3\beta )q^{11}+\cdots\)
600.4.a.t	$600$	$4$	600.a	1.a	$1$	$2$	$2$	$35.401$	\(\Q(\sqrt{129}) \)	None	✓				120.4.f.d	$1$	$0$	\(0\)	\(-6\)	\(0\)	\(2\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-3q^{3}+(1+3\beta )q^{7}+9q^{9}+(37-\beta )q^{11}+\cdots\)
600.4.a.u	$600$	$4$	600.a	1.a	$1$	$2$	$2$	$35.401$	\(\Q(\sqrt{109}) \)	None	✓	✓	✓		600.4.a.s	$1$	$1$	\(0\)	\(6\)	\(0\)	\(-2\)		$+$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-1-\beta )q^{7}+9q^{9}+(-8+\cdots)q^{11}+\cdots\)
600.4.a.v	$600$	$4$	600.a	1.a	$1$	$2$	$2$	$35.401$	\(\Q(\sqrt{129}) \)	None	✓		✓		120.4.f.d	$1$	$0$	\(0\)	\(6\)	\(0\)	\(-2\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-1-3\beta )q^{7}+9q^{9}+(37+\cdots)q^{11}+\cdots\)
600.4.a.w	$600$	$4$	600.a	1.a	$1$	$2$	$2$	$35.401$	\(\Q(\sqrt{181}) \)	None	✓	✓	✓		600.4.a.r	$1$	$0$	\(0\)	\(6\)	\(0\)	\(6\)		$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(3+\beta )q^{7}+9q^{9}+(4+\beta )q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(600))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(600)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 2}\)