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

• Label: 600.2.m
• Level: $600$
• Weight: $2$
• Character orbit: 600.m
• Rep. character: $\chi_{600}(299,\cdot)$
• Character field: $\Q$
• Dimension: $68$
• Newform subspaces: $5$
• Sturm bound: $240$
• Trace bound: $6$

Defining parameters

Level:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	600.m (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 120 \)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(240\)
Trace bound:			\(6\)
Distinguishing \(T_p\):			\(7\), \(11\), \(29\)

Dimensions

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

	Total	New	Old
Modular forms	132	76	56
Cusp forms	108	68	40
Eisenstein series	24	8	16

Trace form

68 * q + 2 * q^6 + 4 * q^9 + 8 * q^16 + 24 * q^19 - 6 * q^24 + 4 * q^34 + 26 * q^36 - 64 * q^46 + 52 * q^49 + 8 * q^51 - 16 * q^54 - 24 * q^64 - 66 * q^66 - 124 * q^76 + 12 * q^81 - 44 * q^84 + 48 * q^91 + 104 * q^94 - 114 * q^96 - 96 * q^99

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.m.a	$600$	$2$	600.m	120.m	$2$	$4$	$4$	$4.791$	\(\Q(\zeta_{8})\)	\(\Q(\sqrt{-2}) \)					24.2.f.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta_{3} q^{2}+(-\beta_{3}-\beta_1)q^{3}+2 q^{4}+\cdots\)
600.2.m.b	$600$	$2$	600.m	120.m	$2$	$8$	$8$	$4.791$	\(\Q(\zeta_{24})\)	\(\Q(\sqrt{-2}) \)		✓			600.2.b.b	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta_{2} q^{2}-\beta_{3} q^{3}+2 q^{4}+(\beta_{5}-1)q^{6}+\cdots\)
600.2.m.c	$600$	$2$	600.m	120.m	$2$	$16$	$16$	$4.791$	16.0.\(\cdots\).1	None					120.2.b.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{11}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}-\beta _{7}q^{3}+\beta _{2}q^{4}+(-1+\beta _{11}+\cdots)q^{6}+\cdots\)
600.2.m.d	$600$	$2$	600.m	120.m	$2$	$16$	$16$	$4.791$	16.0.\(\cdots\).1	None					120.2.b.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{11}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+\beta _{6}q^{3}+\beta _{2}q^{4}-\beta _{14}q^{6}+\cdots\)
600.2.m.e	$600$	$2$	600.m	120.m	$2$	$24$	$24$	$4.791$		None		✓	✓		600.2.b.g	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

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

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