Space of modular forms of level 600, weight 2, and Character 600.r

• Label: 600.2.r
• Level: $600$
• Weight: $2$
• Character orbit: 600.r
• Rep. character: $\chi_{600}(257,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $36$
• Newform subspaces: $6$
• 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.r (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 15 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(240\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(600, [\chi])\).

	Total	New	Old
Modular forms	288	36	252
Cusp forms	192	36	156
Eisenstein series	96	0	96

Trace form

\( 36 q - 4 q^{7} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(600, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
600.2.r.a	$600$	$2$	600.r	15.e	$4$	$4$	$2$	$4.791$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(-4\)	\(0\)	\(-8\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1-\zeta_{8}^{2}-\zeta_{8}^{3})q^{3}+(-2+2\zeta_{8}^{2}+\cdots)q^{7}+\cdots\)
600.2.r.b	$600$	$2$	600.r	15.e	$4$	$4$	$2$	$4.791$	\(\Q(\zeta_{8})\)	None		$2$	$0$	\(0\)	\(-4\)	\(0\)	\(4\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1+\zeta_{8}^{2})q^{3}+(1+\zeta_{8}-\zeta_{8}^{2}-\zeta_{8}^{3})q^{7}+\cdots\)
600.2.r.c	$600$	$2$	600.r	15.e	$4$	$4$	$2$	$4.791$	\(\Q(\zeta_{8})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(\zeta_{8}-\zeta_{8}^{3})q^{3}+(1+\zeta_{8}-\zeta_{8}^{2}-\zeta_{8}^{3})q^{7}+\cdots\)
600.2.r.d	$600$	$2$	600.r	15.e	$4$	$4$	$2$	$4.791$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(4\)	\(0\)	\(-12\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1+\zeta_{8}^{2}-\zeta_{8}^{3})q^{3}+(-3+3\zeta_{8}^{2}+\cdots)q^{7}+\cdots\)
600.2.r.e	$600$	$2$	600.r	15.e	$4$	$4$	$2$	$4.791$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(4\)	\(0\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1+\zeta_{8}^{2}-\zeta_{8}^{3})q^{3}+(2-2\zeta_{8}^{2})q^{7}+\cdots\)
600.2.r.f	$600$	$2$	600.r	15.e	$4$	$16$	$8$	$4.791$	16.0.\(\cdots\).1	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{12}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{10}q^{3}+(\beta _{3}-\beta _{4}-\beta _{5})q^{7}+(\beta _{7}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(600, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(600, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(300, [\chi])\)\(^{\oplus 2}\)