Space of modular forms of level 224, weight 4, and Character 224.x

• Label: 224.4.x
• Level: $224$
• Weight: $4$
• Character orbit: 224.x
• Rep. character: $\chi_{224}(27,\cdot)$
• Character field: $\Q(\zeta_{8})$
• Dimension: $376$
• Newform subspaces: $2$
• Sturm bound: $128$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	224.x (of order \(8\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 224 \)
Character field:			\(\Q(\zeta_{8})\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(128\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(224, [\chi])\).

	Total	New	Old
Modular forms	392	392	0
Cusp forms	376	376	0
Eisenstein series	16	16	0

Trace form

376 * q - 8 * q^2 - 8 * q^4 - 4 * q^7 - 8 * q^8 - 8 * q^9 - 8 * q^11 + 204 * q^14 - 16 * q^15 - 308 * q^16 + 172 * q^18 - 4 * q^21 + 228 * q^22 - 336 * q^23 - 8 * q^25 + 376 * q^28 - 8 * q^29 - 1232 * q^30 - 8 * q^32 - 460 * q^35 - 448 * q^36 - 8 * q^37 - 8 * q^39 - 2264 * q^42 - 816 * q^43 + 3572 * q^44 - 8 * q^46 + 2848 * q^50 - 224 * q^51 + 744 * q^53 + 996 * q^56 - 8 * q^57 - 4040 * q^58 + 1928 * q^60 - 2192 * q^64 - 16 * q^65 + 2032 * q^67 - 400 * q^70 - 456 * q^71 - 8 * q^72 + 2988 * q^74 - 4 * q^77 - 4656 * q^78 - 16 * q^79 + 6552 * q^84 - 1008 * q^85 - 8 * q^86 + 608 * q^88 + 3596 * q^91 + 6612 * q^92 - 224 * q^93 - 7192 * q^98 + 208 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(224, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
224.4.x.a	$224$	$4$	224.x	224.x	$8$	$8$	$2$	$13.216$	8.0.157351936.1	\(\Q(\sqrt{-7}) \)		✓			224.4.x.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{8}]$	\(q+(-\beta _{2}-3\beta _{4})q^{2}+(-7\beta _{3}+5\beta _{5}+\cdots)q^{4}+\cdots\)
224.4.x.b	$224$	$4$	224.x	224.x	$8$	$368$	$92$	$13.216$		None		✓	✓		224.4.x.b	$4$	$0$	\(-8\)	\(0\)	\(0\)	\(-4\)		$\mathrm{SU}(2)[C_{8}]$