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

• Label: 600.2.w
• Level: $600$
• Weight: $2$
• Character orbit: 600.w
• Rep. character: $\chi_{600}(293,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $136$
• Newform subspaces: $11$
• 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.w (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 120 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(240\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\), \(11\), \(17\)

Dimensions

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

	Total	New	Old
Modular forms	264	152	112
Cusp forms	216	136	80
Eisenstein series	48	16	32

Trace form

136 * q - 4 * q^6 + 8 * q^7 + 8 * q^12 - 16 * q^16 + 20 * q^18 + 20 * q^22 - 28 * q^28 + 16 * q^31 + 16 * q^33 - 20 * q^36 + 32 * q^42 - 72 * q^46 - 44 * q^48 - 8 * q^52 + 16 * q^57 - 44 * q^58 - 24 * q^63 - 116 * q^66 - 32 * q^72 + 8 * q^73 + 56 * q^76 - 64 * q^78 + 24 * q^81 - 64 * q^82 - 64 * q^87 + 116 * q^88 + 92 * q^96 - 8 * q^97

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
600.2.w.a	$600$	$2$	600.w	120.w	$4$	$4$	$2$	$4.791$	\(\Q(i, \sqrt{6})\)	\(\Q(\sqrt{-30}) \)		✓			600.2.w.a	$8$	$0$	\(-4\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(-1+\beta _{2})q^{2}+\beta _{1}q^{3}-2\beta _{2}q^{4}+\cdots\)
600.2.w.b	$600$	$2$	600.w	120.w	$4$	$4$	$2$	$4.791$	\(\Q(i, \sqrt{6})\)	\(\Q(\sqrt{-6}) \)					120.2.w.a	$4$	$0$	\(-4\)	\(0\)	\(0\)	\(4\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(-1+\beta _{2})q^{2}+\beta _{1}q^{3}-2\beta _{2}q^{4}+\cdots\)
600.2.w.c	$600$	$2$	600.w	120.w	$4$	$4$	$2$	$4.791$	\(\Q(\zeta_{12})\)	None		✓			600.2.w.c	$4$	$1$	\(-2\)	\(0\)	\(0\)	\(-12\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(\beta_{2}-\beta_1)q^{2}+(\beta_{3}-\beta_1+1)q^{3}+\cdots\)
600.2.w.d	$600$	$2$	600.w	120.w	$4$	$4$	$2$	$4.791$	\(\Q(\zeta_{12})\)	None		✓			600.2.w.c	$4$	$0$	\(-2\)	\(0\)	\(0\)	\(12\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(\beta_{2}-\beta_1)q^{2}+(\beta_{3}-\beta_{2}+\beta_1)q^{3}+\cdots\)
600.2.w.e	$600$	$2$	600.w	120.w	$4$	$4$	$2$	$4.791$	\(\Q(\zeta_{12})\)	None		✓			600.2.w.c	$4$	$0$	\(2\)	\(0\)	\(0\)	\(-12\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-\beta_{2}+\beta_1)q^{2}+(-\beta_{3}+\beta_{2}-\beta_1)q^{3}+\cdots\)
600.2.w.f	$600$	$2$	600.w	120.w	$4$	$4$	$2$	$4.791$	\(\Q(\zeta_{12})\)	None		✓			600.2.w.c	$4$	$0$	\(2\)	\(0\)	\(0\)	\(12\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-\beta_{2}+\beta_1)q^{2}+(-\beta_{3}+\beta_1-1)q^{3}+\cdots\)
600.2.w.g	$600$	$2$	600.w	120.w	$4$	$4$	$2$	$4.791$	\(\Q(i, \sqrt{6})\)	\(\Q(\sqrt{-30}) \)		✓			600.2.w.a	$8$	$0$	\(4\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(1-\beta _{2})q^{2}+\beta _{1}q^{3}-2\beta _{2}q^{4}+(\beta _{1}+\cdots)q^{6}+\cdots\)
600.2.w.h	$600$	$2$	600.w	120.w	$4$	$4$	$2$	$4.791$	\(\Q(i, \sqrt{6})\)	\(\Q(\sqrt{-6}) \)					120.2.w.a	$4$	$0$	\(4\)	\(0\)	\(0\)	\(4\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(1-\beta _{2})q^{2}+\beta _{1}q^{3}-2\beta _{2}q^{4}+(\beta _{1}+\cdots)q^{6}+\cdots\)
600.2.w.i	$600$	$2$	600.w	120.w	$4$	$8$	$4$	$4.791$	8.0.3317760000.5	\(\Q(\sqrt{-15}) \)		✓			600.2.w.i	$16$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+\beta _{1}q^{2}+(\beta _{4}+\beta _{6})q^{3}+\beta _{2}q^{4}+(2+\cdots)q^{6}+\cdots\)
600.2.w.j	$600$	$2$	600.w	120.w	$4$	$32$	$16$	$4.791$		None					120.2.w.c	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{4}]$
600.2.w.k	$600$	$2$	600.w	120.w	$4$	$64$	$32$	$4.791$		None		✓	✓		600.2.w.k	$16$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{4}]$

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

\( S_{2}^{\mathrm{old}}(600, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 2}\)