Space of modular forms of level 624, weight 2, and Character 624.bu

• Label: 624.2.bu
• Level: $624$
• Weight: $2$
• Character orbit: 624.bu
• Rep. character: $\chi_{624}(191,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $56$
• Newform subspaces: $15$
• Sturm bound: $224$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	624.bu (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 156 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 15 \)
Sturm bound:			\(224\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(5\), \(7\), \(11\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(624, [\chi])\).

	Total	New	Old
Modular forms	248	56	192
Cusp forms	200	56	144
Eisenstein series	48	0	48

Trace form

\( 56 q + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(624, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
624.2.bu.a	$624$	$2$	624.bu	156.p	$6$	$2$	$1$	$4.983$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(-3\)	\(0\)	\(-3\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(-2+\zeta_{6})q^{3}+(-1-\zeta_{6})q^{7}+(3+\cdots)q^{9}+\cdots\)
624.2.bu.b	$624$	$2$	624.bu	156.p	$6$	$2$	$1$	$4.983$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(-3\)	\(0\)	\(3\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-1-\zeta_{6})q^{3}+(1+\zeta_{6})q^{7}+3\zeta_{6}q^{9}+\cdots\)
624.2.bu.c	$624$	$2$	624.bu	156.p	$6$	$2$	$1$	$4.983$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(-3\)	\(0\)	\(9\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(-2+\zeta_{6})q^{3}+(3+3\zeta_{6})q^{7}+(3-3\zeta_{6})q^{9}+\cdots\)
624.2.bu.d	$624$	$2$	624.bu	156.p	$6$	$2$	$1$	$4.983$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-3\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(1-2\zeta_{6})q^{3}+(-1-\zeta_{6})q^{7}-3q^{9}+\cdots\)
624.2.bu.e	$624$	$2$	624.bu	156.p	$6$	$2$	$1$	$4.983$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(3\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-1+2\zeta_{6})q^{3}+(1+\zeta_{6})q^{7}-3q^{9}+\cdots\)
624.2.bu.f	$624$	$2$	624.bu	156.p	$6$	$2$	$1$	$4.983$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(3\)	\(0\)	\(-9\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(2-\zeta_{6})q^{3}+(-3-3\zeta_{6})q^{7}+(3-3\zeta_{6})q^{9}+\cdots\)
624.2.bu.g	$624$	$2$	624.bu	156.p	$6$	$2$	$1$	$4.983$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(3\)	\(0\)	\(-3\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(1+\zeta_{6})q^{3}+(-1-\zeta_{6})q^{7}+3\zeta_{6}q^{9}+\cdots\)
624.2.bu.h	$624$	$2$	624.bu	156.p	$6$	$2$	$1$	$4.983$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(3\)	\(0\)	\(3\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(2-\zeta_{6})q^{3}+(1+\zeta_{6})q^{7}+(3-3\zeta_{6})q^{9}+\cdots\)
624.2.bu.i	$624$	$2$	624.bu	156.p	$6$	$4$	$2$	$4.983$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{6}]$	\(q-\zeta_{12}^{2}q^{3}+(2-4\zeta_{12})q^{5}+(\zeta_{12}^{2}+\cdots)q^{7}+\cdots\)
624.2.bu.j	$624$	$2$	624.bu	156.p	$6$	$4$	$2$	$4.983$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{6}]$	\(q-\zeta_{12}^{3}q^{3}+(-2+4\zeta_{12})q^{5}+(\zeta_{12}^{2}+\cdots)q^{7}+\cdots\)
624.2.bu.k	$624$	$2$	624.bu	156.p	$6$	$6$	$3$	$4.983$	6.0.954288.1	None		$2$	$0$	\(0\)	\(-1\)	\(0\)	\(-12\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{1}q^{3}+(1+\beta _{1}+2\beta _{3}+\beta _{4}+\beta _{5})q^{5}+\cdots\)
624.2.bu.l	$624$	$2$	624.bu	156.p	$6$	$6$	$3$	$4.983$	6.0.954288.1	None		$2$	$0$	\(0\)	\(-1\)	\(0\)	\(12\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{4}q^{3}+(1+\beta _{1}+2\beta _{3}+\beta _{4}+\beta _{5})q^{5}+\cdots\)
624.2.bu.m	$624$	$2$	624.bu	156.p	$6$	$6$	$3$	$4.983$	6.0.954288.1	None		$2$	$0$	\(0\)	\(1\)	\(0\)	\(-12\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{4}q^{3}+(1+\beta _{1}+2\beta _{3}+\beta _{4}+\beta _{5})q^{5}+\cdots\)
624.2.bu.n	$624$	$2$	624.bu	156.p	$6$	$6$	$3$	$4.983$	6.0.954288.1	None		$2$	$0$	\(0\)	\(1\)	\(0\)	\(12\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{2}-\beta _{5})q^{3}+(-1-\beta _{1}-2\beta _{3}-\beta _{4}+\cdots)q^{5}+\cdots\)
624.2.bu.o	$624$	$2$	624.bu	156.p	$6$	$8$	$4$	$4.983$	8.0.3317760000.8	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{7}q^{3}+(-\beta _{2}-\beta _{6})q^{5}+(\beta _{4}+\beta _{5}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(624, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(624, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(156, [\chi])\)\(^{\oplus 3}\)