Space of modular forms of level 600, weight 6, and trivial character

• Label: 600.6.a
• Level: $600$
• Weight: $6$
• Character orbit: 600.a
• Rep. character: $\chi_{600}(1,\cdot)$
• Character field: $\Q$
• Dimension: $47$
• Newform subspaces: $23$
• Sturm bound: $720$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	600.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 23 \)
Sturm bound:			\(720\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(600))\).

	Total	New	Old
Modular forms	624	47	577
Cusp forms	576	47	529
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
\(+\)	\(+\)	\(+\)	\(+\)		\(75\)	\(6\)	\(69\)		\(69\)	\(6\)	\(63\)		\(6\)	\(0\)	\(6\)
\(+\)	\(+\)	\(-\)	\(-\)		\(80\)	\(6\)	\(74\)		\(74\)	\(6\)	\(68\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(+\)	\(-\)		\(78\)	\(5\)	\(73\)		\(72\)	\(5\)	\(67\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)		\(79\)	\(6\)	\(73\)		\(73\)	\(6\)	\(67\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(+\)	\(-\)		\(81\)	\(7\)	\(74\)		\(75\)	\(7\)	\(68\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)	\(+\)		\(76\)	\(5\)	\(71\)		\(70\)	\(5\)	\(65\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)	\(+\)		\(78\)	\(5\)	\(73\)		\(72\)	\(5\)	\(67\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(-\)	\(-\)		\(77\)	\(7\)	\(70\)		\(71\)	\(7\)	\(64\)		\(6\)	\(0\)	\(6\)
Plus space			\(+\)		\(308\)	\(22\)	\(286\)		\(284\)	\(22\)	\(262\)		\(24\)	\(0\)	\(24\)
Minus space			\(-\)		\(316\)	\(25\)	\(291\)		\(292\)	\(25\)	\(267\)		\(24\)	\(0\)	\(24\)

Trace form

47 * q - 9 * q^3 + 100 * q^7 + 3807 * q^9 - 20 * q^11 + 662 * q^13 - 1902 * q^17 + 2128 * q^19 + 180 * q^21 - 2752 * q^23 - 729 * q^27 - 526 * q^29 - 9564 * q^31 - 4896 * q^33 + 4854 * q^37 - 5238 * q^39 - 18946 * q^41 + 10916 * q^43 - 23360 * q^47 + 62883 * q^49 + 9162 * q^51 - 13858 * q^53 + 12276 * q^57 + 61100 * q^59 - 53978 * q^61 + 8100 * q^63 - 173276 * q^67 - 36288 * q^69 + 600 * q^71 - 36090 * q^73 + 272640 * q^77 + 66776 * q^79 + 308367 * q^81 + 206804 * q^83 - 169434 * q^87 - 175938 * q^89 + 138844 * q^91 - 73512 * q^93 + 331238 * q^97 - 1620 * q^99

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(600))\) 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
600.6.a.a	$600$	$6$	600.a	1.a	$1$	$1$	$1$	$96.230$	\(\Q\)	None	✓				24.6.a.c	$1$	$0$	\(0\)	\(-9\)	\(0\)	\(-120\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-9q^{3}-120q^{7}+3^{4}q^{9}+524q^{11}+\cdots\)
600.6.a.b	$600$	$6$	600.a	1.a	$1$	$1$	$1$	$96.230$	\(\Q\)	None	✓				120.6.a.e	$1$	$1$	\(0\)	\(-9\)	\(0\)	\(28\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-9q^{3}+28q^{7}+3^{4}q^{9}-208q^{11}+\cdots\)
600.6.a.c	$600$	$6$	600.a	1.a	$1$	$1$	$1$	$96.230$	\(\Q\)	None	✓				120.6.a.f	$1$	$0$	\(0\)	\(-9\)	\(0\)	\(160\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-9q^{3}+160q^{7}+3^{4}q^{9}-596q^{11}+\cdots\)
600.6.a.d	$600$	$6$	600.a	1.a	$1$	$1$	$1$	$96.230$	\(\Q\)	None	✓				24.6.a.b	$1$	$0$	\(0\)	\(9\)	\(0\)	\(-144\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+9q^{3}-12^{2}q^{7}+3^{4}q^{9}-380q^{11}+\cdots\)
600.6.a.e	$600$	$6$	600.a	1.a	$1$	$1$	$1$	$96.230$	\(\Q\)	None	✓				120.6.a.d	$1$	$1$	\(0\)	\(9\)	\(0\)	\(-128\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+9q^{3}-2^{7}q^{7}+3^{4}q^{9}-308q^{11}+\cdots\)
600.6.a.f	$600$	$6$	600.a	1.a	$1$	$1$	$1$	$96.230$	\(\Q\)	None	✓				120.6.a.b	$1$	$1$	\(0\)	\(9\)	\(0\)	\(-108\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+9q^{3}-108q^{7}+3^{4}q^{9}-8q^{11}+\cdots\)
600.6.a.g	$600$	$6$	600.a	1.a	$1$	$1$	$1$	$96.230$	\(\Q\)	None	✓				120.6.a.c	$1$	$0$	\(0\)	\(9\)	\(0\)	\(80\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+9q^{3}+80q^{7}+3^{4}q^{9}+684q^{11}+\cdots\)
600.6.a.h	$600$	$6$	600.a	1.a	$1$	$1$	$1$	$96.230$	\(\Q\)	None	✓				120.6.a.a	$1$	$0$	\(0\)	\(9\)	\(0\)	\(100\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+9q^{3}+10^{2}q^{7}+3^{4}q^{9}-136q^{11}+\cdots\)
600.6.a.i	$600$	$6$	600.a	1.a	$1$	$1$	$1$	$96.230$	\(\Q\)	None	✓				24.6.a.a	$1$	$1$	\(0\)	\(9\)	\(0\)	\(240\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+9q^{3}+240q^{7}+3^{4}q^{9}-124q^{11}+\cdots\)
600.6.a.j	$600$	$6$	600.a	1.a	$1$	$2$	$2$	$96.230$	\(\Q(\sqrt{6661}) \)	None	✓	✓			600.6.a.j	$1$	$1$	\(0\)	\(-18\)	\(0\)	\(-66\)		$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-9q^{3}+(-33-\beta )q^{7}+3^{4}q^{9}+(112+\cdots)q^{11}+\cdots\)
600.6.a.k	$600$	$6$	600.a	1.a	$1$	$2$	$2$	$96.230$	\(\Q(\sqrt{1489}) \)	None	✓				120.6.a.g	$1$	$0$	\(0\)	\(-18\)	\(0\)	\(-16\)		$-$	$+$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-9q^{3}+(-8-\beta )q^{7}+3^{4}q^{9}+(2^{5}+\cdots)q^{11}+\cdots\)
600.6.a.l	$600$	$6$	600.a	1.a	$1$	$2$	$2$	$96.230$	\(\Q(\sqrt{2161}) \)	None	✓				120.6.a.h	$1$	$1$	\(0\)	\(-18\)	\(0\)	\(8\)		$+$	$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-9q^{3}+(4+\beta )q^{7}+3^{4}q^{9}+(244+2\beta )q^{11}+\cdots\)
600.6.a.m	$600$	$6$	600.a	1.a	$1$	$2$	$2$	$96.230$	\(\Q(\sqrt{109}) \)	None	✓	✓			600.6.a.m	$1$	$0$	\(0\)	\(-18\)	\(0\)	\(38\)		$+$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-9q^{3}+(19-\beta )q^{7}+3^{4}q^{9}+(4-3^{3}\beta )q^{11}+\cdots\)
600.6.a.n	$600$	$6$	600.a	1.a	$1$	$2$	$2$	$96.230$	\(\Q(\sqrt{109}) \)	None	✓	✓	✓		600.6.a.m	$1$	$1$	\(0\)	\(18\)	\(0\)	\(-38\)		$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+9q^{3}+(-19-\beta )q^{7}+3^{4}q^{9}+(4+\cdots)q^{11}+\cdots\)
600.6.a.o	$600$	$6$	600.a	1.a	$1$	$2$	$2$	$96.230$	\(\Q(\sqrt{6661}) \)	None	✓	✓	✓		600.6.a.j	$1$	$0$	\(0\)	\(18\)	\(0\)	\(66\)		$+$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+9q^{3}+(33-\beta )q^{7}+3^{4}q^{9}+(112+\cdots)q^{11}+\cdots\)
600.6.a.p	$600$	$6$	600.a	1.a	$1$	$3$	$3$	$96.230$	3.3.246388.1	None	✓	✓	✓		600.6.a.p	$1$	$0$	\(0\)	\(-27\)	\(0\)	\(-81\)		$-$	$+$	$+$	$-$	$2^{5}\cdot 5$	$\mathrm{SU}(2)$	\(q-9q^{3}+(-3^{3}-\beta _{2})q^{7}+3^{4}q^{9}+(13+\cdots)q^{11}+\cdots\)
600.6.a.q	$600$	$6$	600.a	1.a	$1$	$3$	$3$	$96.230$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓	✓		600.6.a.q	$1$	$1$	\(0\)	\(-27\)	\(0\)	\(-9\)		$+$	$+$	$+$	$+$	$2^{5}\cdot 5$	$\mathrm{SU}(2)$	\(q-9q^{3}+(-3+\beta _{1})q^{7}+3^{4}q^{9}+(-97+\cdots)q^{11}+\cdots\)
600.6.a.r	$600$	$6$	600.a	1.a	$1$	$3$	$3$	$96.230$	3.3.20073.1	None	✓		✓		120.6.f.a	$1$	$1$	\(0\)	\(-27\)	\(0\)	\(104\)		$-$	$+$	$-$	$+$	$2^{3}\cdot 5$	$\mathrm{SU}(2)$	\(q-9q^{3}+(33-2\beta _{1}-3\beta _{2})q^{7}+3^{4}q^{9}+\cdots\)
600.6.a.s	$600$	$6$	600.a	1.a	$1$	$3$	$3$	$96.230$	3.3.20073.1	None	✓				120.6.f.a	$1$	$1$	\(0\)	\(27\)	\(0\)	\(-104\)		$+$	$-$	$-$	$+$	$2^{3}\cdot 5$	$\mathrm{SU}(2)$	\(q+9q^{3}+(-33+2\beta _{1}+3\beta _{2})q^{7}+3^{4}q^{9}+\cdots\)
600.6.a.t	$600$	$6$	600.a	1.a	$1$	$3$	$3$	$96.230$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓			600.6.a.q	$1$	$0$	\(0\)	\(27\)	\(0\)	\(9\)		$-$	$-$	$-$	$-$	$2^{5}\cdot 5$	$\mathrm{SU}(2)$	\(q+9q^{3}+(3-\beta _{1})q^{7}+3^{4}q^{9}+(-97+\cdots)q^{11}+\cdots\)
600.6.a.u	$600$	$6$	600.a	1.a	$1$	$3$	$3$	$96.230$	3.3.246388.1	None	✓	✓			600.6.a.p	$1$	$1$	\(0\)	\(27\)	\(0\)	\(81\)		$+$	$-$	$-$	$+$	$2^{5}\cdot 5$	$\mathrm{SU}(2)$	\(q+9q^{3}+(3^{3}+\beta _{2})q^{7}+3^{4}q^{9}+(13+\cdots)q^{11}+\cdots\)
600.6.a.v	$600$	$6$	600.a	1.a	$1$	$4$	$4$	$96.230$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		120.6.f.b	$1$	$0$	\(0\)	\(-36\)	\(0\)	\(-6\)		$+$	$+$	$-$	$-$	$2^{6}\cdot 3\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q-9q^{3}+(-2-\beta _{3})q^{7}+3^{4}q^{9}+(86+\cdots)q^{11}+\cdots\)
600.6.a.w	$600$	$6$	600.a	1.a	$1$	$4$	$4$	$96.230$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		120.6.f.b	$1$	$0$	\(0\)	\(36\)	\(0\)	\(6\)		$-$	$-$	$-$	$-$	$2^{6}\cdot 3\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+9q^{3}+(2+\beta _{3})q^{7}+3^{4}q^{9}+(86+2\beta _{1}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(600))\) into lower level spaces

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