Space of modular forms of level 450, weight 2, and Character 450.e

• Label: 450.2.e
• Level: $450$
• Weight: $2$
• Character orbit: 450.e
• Rep. character: $\chi_{450}(151,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $38$
• Newform subspaces: $14$
• Sturm bound: $180$
• Trace bound: $7$

Defining parameters

Level:	$N$	$=$	$450 = 2 \cdot 3^{2} \cdot 5^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	450.e (of order $3$ and degree $2$)
Character conductor:	$\operatorname{cond}(\chi)$	$=$	$9$
Character field:			$\Q(\zeta_{3})$
Newform subspaces:			$14$
Sturm bound:			$180$
Trace bound:			$7$
Distinguishing $T_p$:			$7$, $11$, $17$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(450, [\chi])$.

	Total	New	Old
Modular forms	204	38	166
Cusp forms	156	38	118
Eisenstein series	48	0	48

Trace form

38 * q - q^2 + q^3 - 19 * q^4 - 5 * q^6 - 2 * q^7 + 2 * q^8 - 5 * q^9 - q^11 + 4 * q^12 - 2 * q^13 + 2 * q^14 - 19 * q^16 + 18 * q^17 + 2 * q^18 - 2 * q^19 - 2 * q^21 + 3 * q^22 + 6 * q^23 + q^24 - 20 * q^26 + 16 * q^27 + 4 * q^28 + 18 * q^29 - 8 * q^31 - q^32 + 15 * q^33 + 3 * q^34 - 5 * q^36 + 16 * q^37 - 11 * q^38 - 24 * q^39 - 13 * q^41 - 20 * q^42 + q^43 + 2 * q^44 + 12 * q^46 - 18 * q^47 - 5 * q^48 - 27 * q^49 + 37 * q^51 - 2 * q^52 - 48 * q^53 - 17 * q^54 + 2 * q^56 + 23 * q^57 + 6 * q^58 + 43 * q^59 - 20 * q^61 + 16 * q^62 - 28 * q^63 + 38 * q^64 + 26 * q^66 - 17 * q^67 - 9 * q^68 - 8 * q^69 - 24 * q^71 - 13 * q^72 + 22 * q^73 + 8 * q^74 + q^76 - 42 * q^77 + 10 * q^78 - 8 * q^79 - 41 * q^81 - 30 * q^82 + 24 * q^83 - 14 * q^84 + 35 * q^86 - 54 * q^87 + 3 * q^88 + 92 * q^89 + 56 * q^91 + 6 * q^92 + 20 * q^93 - 6 * q^94 + 4 * q^96 + 7 * q^97 + 42 * q^98 - 88 * q^99

Decomposition of $S_{2}^{\mathrm{new}}(450, [\chi])$ 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
450.2.e.a	$450$	$2$	450.e	9.c	$3$	$2$	$1$	$3.593$	$\Q(\sqrt{-3})$	None		✓			450.2.e.a	$2$	$1$	$-1$	$-3$	$0$	$-4$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(-1+\zeta_{6})q^{2}+(-2+\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots$
450.2.e.b	$450$	$2$	450.e	9.c	$3$	$2$	$1$	$3.593$	$\Q(\sqrt{-3})$	None					90.2.i.a	$2$	$0$	$-1$	$-3$	$0$	$1$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(-1+\zeta_{6})q^{2}+(-2+\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots$
450.2.e.c	$450$	$2$	450.e	9.c	$3$	$2$	$1$	$3.593$	$\Q(\sqrt{-3})$	None		✓			450.2.e.c	$2$	$0$	$-1$	$0$	$0$	$-2$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(-1+\zeta_{6})q^{2}+(-1+2\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots$
450.2.e.d	$450$	$2$	450.e	9.c	$3$	$2$	$1$	$3.593$	$\Q(\sqrt{-3})$	None					90.2.e.b	$2$	$0$	$-1$	$3$	$0$	$-4$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(-1+\zeta_{6})q^{2}+(1+\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots$
450.2.e.e	$450$	$2$	450.e	9.c	$3$	$2$	$1$	$3.593$	$\Q(\sqrt{-3})$	None					90.2.e.a	$2$	$0$	$1$	$-3$	$0$	$-1$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(1-\zeta_{6})q^{2}+(-2+\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots$
450.2.e.f	$450$	$2$	450.e	9.c	$3$	$2$	$1$	$3.593$	$\Q(\sqrt{-3})$	None		✓			450.2.e.c	$2$	$0$	$1$	$0$	$0$	$2$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(1-\zeta_{6})q^{2}+(1-2\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots$
450.2.e.g	$450$	$2$	450.e	9.c	$3$	$2$	$1$	$3.593$	$\Q(\sqrt{-3})$	None					90.2.i.a	$2$	$0$	$1$	$3$	$0$	$-1$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(1-\zeta_{6})q^{2}+(2-\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots$
450.2.e.h	$450$	$2$	450.e	9.c	$3$	$2$	$1$	$3.593$	$\Q(\sqrt{-3})$	None		✓			450.2.e.a	$2$	$0$	$1$	$3$	$0$	$4$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(1-\zeta_{6})q^{2}+(2-\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots$
450.2.e.i	$450$	$2$	450.e	9.c	$3$	$2$	$1$	$3.593$	$\Q(\sqrt{-3})$	None					18.2.c.a	$2$	$0$	$1$	$3$	$0$	$2$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(1-\zeta_{6})q^{2}+(1+\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots$
450.2.e.j	$450$	$2$	450.e	9.c	$3$	$4$	$2$	$3.593$	$\Q(\sqrt{-3}, \sqrt{-11})$	None					90.2.e.c	$2$	$0$	$-2$	$-2$	$0$	$1$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(-1+\beta _{2})q^{2}+(-\beta _{1}+\beta _{3})q^{3}-\beta _{2}q^{4}+\cdots$
450.2.e.k	$450$	$2$	450.e	9.c	$3$	$4$	$2$	$3.593$	$\Q(\sqrt{-2}, \sqrt{-3})$	None					90.2.i.b	$2$	$0$	$-2$	$2$	$0$	$4$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q-\beta _{2}q^{2}+(-\beta _{1}+\beta _{2}+\beta _{3})q^{3}+(-1+\cdots)q^{4}+\cdots$
450.2.e.l	$450$	$2$	450.e	9.c	$3$	$4$	$2$	$3.593$	$\Q(\sqrt{-2}, \sqrt{-3})$	None		✓			450.2.e.l	$2$	$0$	$-2$	$4$	$0$	$4$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(-1+\beta _{2})q^{2}+(1-\beta _{3})q^{3}-\beta _{2}q^{4}+\cdots$
450.2.e.m	$450$	$2$	450.e	9.c	$3$	$4$	$2$	$3.593$	$\Q(\sqrt{-2}, \sqrt{-3})$	None		✓			450.2.e.l	$2$	$0$	$2$	$-4$	$0$	$-4$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(1-\beta _{2})q^{2}+(-1+\beta _{3})q^{3}-\beta _{2}q^{4}+\cdots$
450.2.e.n	$450$	$2$	450.e	9.c	$3$	$4$	$2$	$3.593$	$\Q(\sqrt{-2}, \sqrt{-3})$	None					90.2.i.b	$2$	$0$	$2$	$-2$	$0$	$-4$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+\beta _{2}q^{2}+(-\beta _{1}-\beta _{2}+\beta _{3})q^{3}+(-1+\cdots)q^{4}+\cdots$

Decomposition of $S_{2}^{\mathrm{old}}(450, [\chi])$ into lower level spaces

$S_{2}^{\mathrm{old}}(450, [\chi]) \simeq$ $S_{2}^{\mathrm{new}}(18, [\chi])$$^{\oplus 3}$$\oplus$$S_{2}^{\mathrm{new}}(45, [\chi])$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(90, [\chi])$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(225, [\chi])$$^{\oplus 2}$