Space of modular forms of level 612, weight 4, and Character 612.b

• Label: 612.4.b
• Level: $612$
• Weight: $4$
• Character orbit: 612.b
• Rep. character: $\chi_{612}(577,\cdot)$
• Character field: $\Q$
• Dimension: $22$
• Newform subspaces: $4$
• Sturm bound: $432$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 612 = 2^{2} \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	612.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 17 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(432\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(612, [\chi])\).

	Total	New	Old
Modular forms	336	22	314
Cusp forms	312	22	290
Eisenstein series	24	0	24

Trace form

\( 22 q + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(612, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
612.4.b.a	$612$	$4$	612.b	17.b	$2$	$4$	$4$	$36.109$	\(\Q(i, \sqrt{17})\)	\(\Q(\sqrt{-51}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{4}$	$\mathrm{U}(1)[D_{2}]$	\(q+(-\beta _{1}-2\beta _{2})q^{5}+(8\beta _{1}+3\beta _{2})q^{11}+\cdots\)
612.4.b.b	$612$	$4$	612.b	17.b	$2$	$4$	$4$	$36.109$	4.0.1499912.1	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{5}+(\beta _{1}-\beta _{2})q^{7}+(-\beta _{1}+2\beta _{2}+\cdots)q^{11}+\cdots\)
612.4.b.c	$612$	$4$	612.b	17.b	$2$	$4$	$4$	$36.109$	\(\Q(\sqrt{-34}, \sqrt{-562})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{5}-\beta _{2}q^{7}-6\beta _{1}q^{11}-6q^{13}+\cdots\)
612.4.b.d	$612$	$4$	612.b	17.b	$2$	$10$	$10$	$36.109$	\(\mathbb{Q}[x]/(x^{10} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{15}\cdot 3^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{5}+(-\beta _{1}+\beta _{2})q^{7}+(\beta _{4}+\beta _{5}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(612, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(612, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(17, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(34, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(51, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(153, [\chi])\)\(^{\oplus 3}\)