Space of modular forms of level 450, weight 4, and Character 450.f

• Label: 450.4.f
• Level: $450$
• Weight: $4$
• Character orbit: 450.f
• Rep. character: $\chi_{450}(107,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $36$
• Newform subspaces: $6$
• Sturm bound: $360$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	450.f (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 15 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(360\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(450, [\chi])\).

	Total	New	Old
Modular forms	588	36	552
Cusp forms	492	36	456
Eisenstein series	96	0	96

Trace form

36 * q + 24 * q^7 - 204 * q^13 - 576 * q^16 + 624 * q^22 + 96 * q^28 - 576 * q^31 + 564 * q^37 - 1920 * q^43 - 576 * q^46 + 816 * q^52 + 888 * q^58 - 960 * q^61 + 1392 * q^67 + 468 * q^73 + 1152 * q^76 - 264 * q^82 + 2496 * q^88 - 1392 * q^91 - 3996 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(450, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
450.4.f.a	$450$	$4$	450.f	15.e	$4$	$4$	$2$	$26.551$	\(\Q(\zeta_{8})\)	None		✓			450.4.f.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-36\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(\beta_{3}+\beta_{2})q^{2}+4\beta_1 q^{4}+(9\beta_1-9)q^{7}+\cdots\)
450.4.f.b	$450$	$4$	450.f	15.e	$4$	$4$	$2$	$26.551$	\(\Q(\zeta_{8})\)	None					90.4.f.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(16\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-\beta_{3}-\beta_{2})q^{2}+4\beta_1 q^{4}+(-4\beta_1+4)q^{7}+\cdots\)
450.4.f.c	$450$	$4$	450.f	15.e	$4$	$4$	$2$	$26.551$	\(\Q(\zeta_{8})\)	None		✓			450.4.f.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(36\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(\beta_{3}+\beta_{2})q^{2}+4\beta_1 q^{4}+(-9\beta_1+9)q^{7}+\cdots\)
450.4.f.d	$450$	$4$	450.f	15.e	$4$	$8$	$4$	$26.551$	\(\Q(\zeta_{24})\)	None		✓			450.4.f.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-48\)	$2^{6}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-\beta_{3}-\beta_{2})q^{2}-4\beta_1 q^{4}+(\beta_{4}-6\beta_1-6)q^{7}+\cdots\)
450.4.f.e	$450$	$4$	450.f	15.e	$4$	$8$	$4$	$26.551$	8.0.\(\cdots\).8	None					90.4.f.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$2^{7}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-\beta _{2}-\beta _{3})q^{2}-4\beta _{1}q^{4}+(1+\beta _{1}+\cdots)q^{7}+\cdots\)
450.4.f.f	$450$	$4$	450.f	15.e	$4$	$8$	$4$	$26.551$	\(\Q(\zeta_{24})\)	None		✓			450.4.f.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(48\)	$2^{6}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-\beta_{3}+\beta_{2})q^{2}+4\beta_1 q^{4}+(\beta_{5}-6\beta_1+6)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(450, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(450, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 2}\)