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

• Label: 600.2.d
• Level: $600$
• Weight: $2$
• Character orbit: 600.d
• Rep. character: $\chi_{600}(349,\cdot)$
• Character field: $\Q$
• Dimension: $36$
• Newform subspaces: $8$
• Sturm bound: $240$
• Trace bound: $3$

Defining parameters

Level:	$N$	$=$	$600 = 2^{3} \cdot 3 \cdot 5^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	600.d (of order $2$ and degree $1$)
Character conductor:	$\operatorname{cond}(\chi)$	$=$	$40$
Character field:			$\Q$
Newform subspaces:			$8$
Sturm bound:			$240$
Trace bound:			$3$
Distinguishing $T_p$:			$7$, $11$, $13$, $37$

Dimensions

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

	Total	New	Old
Modular forms	132	36	96
Cusp forms	108	36	72
Eisenstein series	24	0	24

Trace form

36 * q - 4 * q^4 + 4 * q^6 + 36 * q^9 + 28 * q^14 + 4 * q^16 + 4 * q^24 + 36 * q^26 + 40 * q^31 - 32 * q^34 - 4 * q^36 + 16 * q^39 + 8 * q^41 - 32 * q^46 - 36 * q^49 + 4 * q^54 - 8 * q^56 + 20 * q^64 - 32 * q^66 - 64 * q^71 - 88 * q^74 - 16 * q^76 - 24 * q^79 + 36 * q^81 - 8 * q^84 - 100 * q^86 + 40 * q^89 - 32 * q^94 + 4 * q^96

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.d.a	$600$	$2$	600.d	40.f	$2$	$2$	$2$	$4.791$	$\Q(\sqrt{-1})$	None					120.2.k.a	$2$	$0$	$-2$	$-2$	$0$	$0$	$1$	$\mathrm{SU}(2)[C_{2}]$	$q+(i-1)q^{2}-q^{3}-2 i q^{4}+(-i+1)q^{6}+\cdots$
600.2.d.b	$600$	$2$	600.d	40.f	$2$	$2$	$2$	$4.791$	$\Q(\sqrt{-1})$	None					24.2.d.a	$2$	$0$	$-2$	$2$	$0$	$0$	$1$	$\mathrm{SU}(2)[C_{2}]$	$q+(i-1)q^{2}+q^{3}-2 i q^{4}+(i-1)q^{6}+\cdots$
600.2.d.c	$600$	$2$	600.d	40.f	$2$	$2$	$2$	$4.791$	$\Q(\sqrt{-1})$	None					24.2.d.a	$2$	$0$	$2$	$-2$	$0$	$0$	$1$	$\mathrm{SU}(2)[C_{2}]$	$q+(i+1)q^{2}-q^{3}+2 i q^{4}+(-i-1)q^{6}+\cdots$
600.2.d.d	$600$	$2$	600.d	40.f	$2$	$2$	$2$	$4.791$	$\Q(\sqrt{-1})$	None					120.2.k.a	$2$	$0$	$2$	$2$	$0$	$0$	$1$	$\mathrm{SU}(2)[C_{2}]$	$q+(i+1)q^{2}+q^{3}+2 i q^{4}+(i+1)q^{6}+\cdots$
600.2.d.e	$600$	$2$	600.d	40.f	$2$	$6$	$6$	$4.791$	6.0.399424.1	None					120.2.k.b	$2$	$0$	$0$	$-6$	$0$	$0$	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{3}q^{2}-q^{3}-\beta _{2}q^{4}-\beta _{3}q^{6}+(\beta _{1}+\cdots)q^{7}+\cdots$
600.2.d.f	$600$	$2$	600.d	40.f	$2$	$6$	$6$	$4.791$	6.0.399424.1	None					120.2.k.b	$2$	$0$	$0$	$6$	$0$	$0$	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	$q-\beta _{2}q^{2}+q^{3}-\beta _{1}q^{4}-\beta _{2}q^{6}+(\beta _{1}+\cdots)q^{7}+\cdots$
600.2.d.g	$600$	$2$	600.d	40.f	$2$	$8$	$8$	$4.791$	8.0.214798336.3	None		✓			600.2.k.d	$2$	$0$	$-2$	$-8$	$0$	$0$	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	$q-\beta _{6}q^{2}-q^{3}+(-1-\beta _{1}+\beta _{3}+\beta _{6}+\cdots)q^{4}+\cdots$
600.2.d.h	$600$	$2$	600.d	40.f	$2$	$8$	$8$	$4.791$	8.0.214798336.3	None		✓			600.2.k.d	$2$	$0$	$2$	$8$	$0$	$0$	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{6}q^{2}+q^{3}+(-1-\beta _{1}+\beta _{3}+\beta _{6}+\cdots)q^{4}+\cdots$

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

$S_{2}^{\mathrm{old}}(600, [\chi]) \simeq$ $S_{2}^{\mathrm{new}}(40, [\chi])$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(120, [\chi])$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(200, [\chi])$$^{\oplus 2}$