Space of modular forms of level 600, weight 8, and trivial character

• Label: 600.8.a
• Level: $600$
• Weight: $8$
• Character orbit: 600.a
• Rep. character: $\chi_{600}(1,\cdot)$
• Character field: $\Q$
• Dimension: $67$
• Newform subspaces: $23$
• Sturm bound: $960$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	600.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 23 \)
Sturm bound:			\(960\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(600))\).

	Total	New	Old
Modular forms	864	67	797
Cusp forms	816	67	749
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	Dim
\(+\)	\(+\)	\(+\)	$+$	\(7\)
\(+\)	\(+\)	\(-\)	$-$	\(9\)
\(+\)	\(-\)	\(+\)	$-$	\(8\)
\(+\)	\(-\)	\(-\)	$+$	\(9\)
\(-\)	\(+\)	\(+\)	$-$	\(7\)
\(-\)	\(+\)	\(-\)	$+$	\(10\)
\(-\)	\(-\)	\(+\)	$+$	\(9\)
\(-\)	\(-\)	\(-\)	$-$	\(8\)
Plus space			\(+\)	\(35\)
Minus space			\(-\)	\(32\)

Trace form

\( 67 q + 27 q^{3} + 332 q^{7} + 48843 q^{9} + O(q^{10}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(600))\) 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
600.8.a.a	$600$	$8$	600.a	1.a	$1$	$1$	$1$	$187.431$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-27\)	\(0\)	\(-504\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}-504q^{7}+3^{6}q^{9}+3812q^{11}+\cdots\)
600.8.a.b	$600$	$8$	600.a	1.a	$1$	$1$	$1$	$187.431$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-27\)	\(0\)	\(-120\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}-120q^{7}+3^{6}q^{9}-7196q^{11}+\cdots\)
600.8.a.c	$600$	$8$	600.a	1.a	$1$	$1$	$1$	$187.431$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-27\)	\(0\)	\(540\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+540q^{7}+3^{6}q^{9}+3584q^{11}+\cdots\)
600.8.a.d	$600$	$8$	600.a	1.a	$1$	$1$	$1$	$187.431$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-27\)	\(0\)	\(776\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+776q^{7}+3^{6}q^{9}+612q^{11}+\cdots\)
600.8.a.e	$600$	$8$	600.a	1.a	$1$	$1$	$1$	$187.431$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(27\)	\(0\)	\(-1056\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}-1056q^{7}+3^{6}q^{9}+6412q^{11}+\cdots\)
600.8.a.f	$600$	$8$	600.a	1.a	$1$	$2$	$2$	$187.431$	\(\Q(\sqrt{7849}) \)	None	✓	$1$	$1$	\(0\)	\(-54\)	\(0\)	\(-952\)		$-$	$+$	$+$	$-$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+(-476-\beta )q^{7}+3^{6}q^{9}+\cdots\)
600.8.a.g	$600$	$8$	600.a	1.a	$1$	$2$	$2$	$187.431$	\(\Q(\sqrt{10761}) \)	None	✓	$1$	$0$	\(0\)	\(-54\)	\(0\)	\(896\)		$+$	$+$	$+$	$+$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+(448-\beta )q^{7}+3^{6}q^{9}+(-2576+\cdots)q^{11}+\cdots\)
600.8.a.h	$600$	$8$	600.a	1.a	$1$	$2$	$2$	$187.431$	\(\Q(\sqrt{6946}) \)	None	✓	$1$	$1$	\(0\)	\(54\)	\(0\)	\(-448\)		$+$	$-$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(-224+\beta )q^{7}+3^{6}q^{9}+\cdots\)
600.8.a.i	$600$	$8$	600.a	1.a	$1$	$2$	$2$	$187.431$	\(\Q(\sqrt{114}) \)	None	✓	$1$	$1$	\(0\)	\(54\)	\(0\)	\(-328\)		$+$	$-$	$+$	$-$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(-164+\beta )q^{7}+3^{6}q^{9}+\cdots\)
600.8.a.j	$600$	$8$	600.a	1.a	$1$	$2$	$2$	$187.431$	\(\Q(\sqrt{106}) \)	None	✓	$1$	$0$	\(0\)	\(54\)	\(0\)	\(704\)		$-$	$-$	$+$	$+$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(352+\beta )q^{7}+3^{6}q^{9}+(-1036+\cdots)q^{11}+\cdots\)
600.8.a.k	$600$	$8$	600.a	1.a	$1$	$2$	$2$	$187.431$	\(\Q(\sqrt{106}) \)	None	✓	$1$	$0$	\(0\)	\(54\)	\(0\)	\(824\)		$-$	$-$	$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(412+7\beta )q^{7}+3^{6}q^{9}+(-1736+\cdots)q^{11}+\cdots\)
600.8.a.l	$600$	$8$	600.a	1.a	$1$	$3$	$3$	$187.431$	3.3.1008564.1	None	✓	$1$	$0$	\(0\)	\(-81\)	\(0\)	\(-743\)		$+$	$+$	$+$	$+$	$2^{5}\cdot 3\cdot 5\cdot 7$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+(-248+\beta _{1})q^{7}+3^{6}q^{9}+\cdots\)
600.8.a.m	$600$	$8$	600.a	1.a	$1$	$3$	$3$	$187.431$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-81\)	\(0\)	\(113\)		$-$	$+$	$+$	$-$	$2^{5}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+(38+\beta _{2})q^{7}+3^{6}q^{9}+(885+\cdots)q^{11}+\cdots\)
600.8.a.n	$600$	$8$	600.a	1.a	$1$	$3$	$3$	$187.431$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(81\)	\(0\)	\(-113\)		$+$	$-$	$-$	$+$	$2^{5}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(-38-\beta _{2})q^{7}+3^{6}q^{9}+\cdots\)
600.8.a.o	$600$	$8$	600.a	1.a	$1$	$3$	$3$	$187.431$	3.3.1008564.1	None	✓	$1$	$1$	\(0\)	\(81\)	\(0\)	\(743\)		$-$	$-$	$-$	$-$	$2^{5}\cdot 3\cdot 5\cdot 7$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(248-\beta _{1})q^{7}+3^{6}q^{9}+(-503+\cdots)q^{11}+\cdots\)
600.8.a.p	$600$	$8$	600.a	1.a	$1$	$4$	$4$	$187.431$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-108\)	\(0\)	\(-172\)		$+$	$+$	$-$	$-$	$2^{10}\cdot 3\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+(-43-\beta _{1})q^{7}+3^{6}q^{9}+\cdots\)
600.8.a.q	$600$	$8$	600.a	1.a	$1$	$4$	$4$	$187.431$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-108\)	\(0\)	\(548\)		$-$	$+$	$-$	$+$	$2^{10}\cdot 3\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+(137-\beta _{1})q^{7}+3^{6}q^{9}+(-792+\cdots)q^{11}+\cdots\)
600.8.a.r	$600$	$8$	600.a	1.a	$1$	$4$	$4$	$187.431$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(108\)	\(0\)	\(-548\)		$+$	$-$	$+$	$-$	$2^{10}\cdot 3\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(-137+\beta _{1})q^{7}+3^{6}q^{9}+\cdots\)
600.8.a.s	$600$	$8$	600.a	1.a	$1$	$4$	$4$	$187.431$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(108\)	\(0\)	\(172\)		$-$	$-$	$+$	$+$	$2^{10}\cdot 3\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(43+\beta _{1})q^{7}+3^{6}q^{9}+(-252+\cdots)q^{11}+\cdots\)
600.8.a.t	$600$	$8$	600.a	1.a	$1$	$5$	$5$	$187.431$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-135\)	\(0\)	\(-62\)		$+$	$+$	$-$	$-$	$2^{11}\cdot 3^{2}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+(-12+\beta _{1})q^{7}+3^{6}q^{9}+\cdots\)
600.8.a.u	$600$	$8$	600.a	1.a	$1$	$5$	$5$	$187.431$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(135\)	\(0\)	\(62\)		$-$	$-$	$-$	$-$	$2^{11}\cdot 3^{2}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(12-\beta _{1})q^{7}+3^{6}q^{9}+(-3^{3}+\cdots)q^{11}+\cdots\)
600.8.a.v	$600$	$8$	600.a	1.a	$1$	$6$	$6$	$187.431$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-162\)	\(0\)	\(-624\)		$-$	$+$	$-$	$+$	$2^{15}\cdot 3^{3}\cdot 5^{5}$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+(-104-\beta _{2})q^{7}+3^{6}q^{9}+\cdots\)
600.8.a.w	$600$	$8$	600.a	1.a	$1$	$6$	$6$	$187.431$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(162\)	\(0\)	\(624\)		$+$	$-$	$-$	$+$	$2^{15}\cdot 3^{3}\cdot 5^{5}$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(104+\beta _{2})q^{7}+3^{6}q^{9}+(276+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(600))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(600)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 18}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 12}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 12}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 16}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 9}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 12}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 2}\)