Space of modular forms of level 210, weight 4, and Character 210.b

• Label: 210.4.b
• Level: $210$
• Weight: $4$
• Character orbit: 210.b
• Rep. character: $\chi_{210}(41,\cdot)$
• Character field: $\Q$
• Dimension: $32$
• Newform subspaces: $2$
• Sturm bound: $192$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	210.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 21 \)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(192\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(210, [\chi])\).

	Total	New	Old
Modular forms	152	32	120
Cusp forms	136	32	104
Eisenstein series	16	0	16

Trace form

32 * q - 128 * q^4 - 20 * q^7 + 60 * q^9 - 40 * q^15 + 512 * q^16 + 224 * q^18 + 332 * q^21 + 800 * q^25 + 80 * q^28 - 280 * q^30 - 240 * q^36 + 1400 * q^37 + 1260 * q^39 + 776 * q^42 + 704 * q^43 - 1248 * q^46 - 2056 * q^49 - 356 * q^51 - 1728 * q^57 + 624 * q^58 + 160 * q^60 + 1532 * q^63 - 2048 * q^64 + 1024 * q^67 - 360 * q^70 - 896 * q^72 - 736 * q^78 - 6632 * q^79 + 3956 * q^81 - 1328 * q^84 - 1440 * q^85 - 312 * q^91 + 5344 * q^93 - 7716 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(210, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
210.4.b.a	$210$	$4$	210.b	21.c	$2$	$16$	$16$	$12.390$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		✓			210.4.b.a	$2$	$0$	\(0\)	\(-4\)	\(80\)	\(-10\)	$2^{14}\cdot 3^{4}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{5}q^{2}-\beta _{2}q^{3}-4q^{4}+5q^{5}+(-2+\cdots)q^{6}+\cdots\)
210.4.b.b	$210$	$4$	210.b	21.c	$2$	$16$	$16$	$12.390$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		✓			210.4.b.a	$2$	$0$	\(0\)	\(4\)	\(-80\)	\(-10\)	$2^{14}\cdot 3^{4}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{5}q^{2}+\beta _{2}q^{3}-4q^{4}-5q^{5}+(2+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(210, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(210, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 2}\)