Space of modular forms of level 4, weight 9, and Character 4.b

• Label: 4.9.b
• Level: $4$
• Weight: $9$
• Character orbit: 4.b
• Rep. character: $\chi_{4}(3,\cdot)$
• Character field: $\Q$
• Dimension: $3$
• Newform subspaces: $2$
• Sturm bound: $4$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 4 = 2^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	4.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 4 \)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(4\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(4, [\chi])\).

	Total	New	Old
Modular forms	5	5	0
Cusp forms	3	3	0
Eisenstein series	2	2	0

Trace form

3 * q - 4 * q^2 + 144 * q^4 + 166 * q^5 - 2496 * q^6 + 11456 * q^8 - 285 * q^9 - 29064 * q^10 + 49920 * q^12 - 11418 * q^13 - 34944 * q^14 - 52992 * q^16 + 82822 * q^17 + 173436 * q^18 - 338144 * q^20 - 279552 * q^21 + 461760 * q^22 - 359424 * q^24 + 683241 * q^25 + 101752 * q^26 + 698880 * q^28 - 1664282 * q^29 - 1522560 * q^30 + 1534976 * q^32 + 3694080 * q^33 - 2475528 * q^34 + 2062992 * q^36 - 6018138 * q^37 + 486720 * q^38 + 172416 * q^40 + 7871686 * q^41 + 2795520 * q^42 - 9235200 * q^44 - 11091354 * q^45 + 5925504 * q^46 - 5591040 * q^48 + 13380675 * q^49 + 11895156 * q^50 + 490272 * q^52 - 7972058 * q^53 - 7832448 * q^54 - 5031936 * q^56 + 3893760 * q^57 - 19960968 * q^58 + 30451200 * q^60 - 8770074 * q^61 + 1697280 * q^62 + 37392384 * q^64 - 6169588 * q^65 - 36940800 * q^66 - 24405728 * q^68 + 47404032 * q^69 - 21315840 * q^70 + 1680576 * q^72 - 66168378 * q^73 + 84255352 * q^74 - 9734400 * q^76 + 51717120 * q^77 + 13653120 * q^78 - 141377024 * q^80 - 64529469 * q^81 + 14309112 * q^82 + 15654912 * q^84 + 155949132 * q^85 + 148085184 * q^86 + 66493440 * q^88 - 196914362 * q^89 - 68884104 * q^90 - 118510080 * q^92 + 13578240 * q^93 - 190504704 * q^94 + 203833344 * q^96 + 67858182 * q^97 + 16078076 * q^98

Decomposition of \(S_{9}^{\mathrm{new}}(4, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
4.9.b.a	$4$	$9$	4.b	4.b	$2$	$1$	$1$	$1.630$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓	✓			4.9.b.a	$2$	$0$	\(16\)	\(0\)	\(-1054\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+2^{4}q^{2}+2^{8}q^{4}-1054q^{5}+2^{12}q^{8}+\cdots\)
4.9.b.b	$4$	$9$	4.b	4.b	$2$	$2$	$2$	$1.630$	\(\Q(\sqrt{-39}) \)	None		✓	✓		4.9.b.b	$2$	$0$	\(-20\)	\(0\)	\(1220\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-10-\beta )q^{2}-8\beta q^{3}+(-56+20\beta )q^{4}+\cdots\)