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

• Label: 624.2.bc
• Level: $624$
• Weight: $2$
• Character orbit: 624.bc
• Rep. character: $\chi_{624}(31,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $28$
• Newform subspaces: $6$
• Sturm bound: $224$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	624.bc (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 52 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(224\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

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

	Total	New	Old
Modular forms	248	28	220
Cusp forms	200	28	172
Eisenstein series	48	0	48

Trace form

\( 28 q - 12 q^{5} - 28 q^{9} + 8 q^{21} - 20 q^{37} - 36 q^{41} + 12 q^{45} + 72 q^{53} + 8 q^{57} - 24 q^{61} + 60 q^{65} - 20 q^{73} + 28 q^{81} - 24 q^{85} + 12 q^{89} - 8 q^{93} - 52 q^{97}+O(q^{100}) \)

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
624.2.bc.a	$624$	$2$	624.bc	52.f	$4$	$2$	$1$	$4.983$	\(\Q(\sqrt{-1}) \)	None		✓			624.2.bc.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+i q^{3}+(-2 i-2)q^{7}-q^{9}+(-3 i-3)q^{11}+\cdots\)
624.2.bc.b	$624$	$2$	624.bc	52.f	$4$	$2$	$1$	$4.983$	\(\Q(\sqrt{-1}) \)	None		✓			624.2.bc.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-i q^{3}+(2 i+2)q^{7}-q^{9}+(3 i+3)q^{11}+\cdots\)
624.2.bc.c	$624$	$2$	624.bc	52.f	$4$	$4$	$2$	$4.983$	\(\Q(i, \sqrt{17})\)	None		✓			624.2.bc.c	$2$	$0$	\(0\)	\(0\)	\(-2\)	\(-8\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{1}q^{3}+(-1+\beta _{2})q^{5}+(-2-2\beta _{1}+\cdots)q^{7}+\cdots\)
624.2.bc.d	$624$	$2$	624.bc	52.f	$4$	$4$	$2$	$4.983$	\(\Q(i, \sqrt{17})\)	None		✓			624.2.bc.c	$2$	$0$	\(0\)	\(0\)	\(-2\)	\(8\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{1}q^{3}+(-1+\beta _{1}-\beta _{3})q^{5}+(2-2\beta _{1}+\cdots)q^{7}+\cdots\)
624.2.bc.e	$624$	$2$	624.bc	52.f	$4$	$8$	$4$	$4.983$	8.0.56070144.2	None		✓			624.2.bc.e	$2$	$0$	\(0\)	\(0\)	\(-4\)	\(-8\)	$2^{7}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{1}q^{3}+(-\beta _{1}+\beta _{5})q^{5}+(-1-\beta _{1}+\cdots)q^{7}+\cdots\)
624.2.bc.f	$624$	$2$	624.bc	52.f	$4$	$8$	$4$	$4.983$	8.0.56070144.2	None		✓			624.2.bc.e	$2$	$0$	\(0\)	\(0\)	\(-4\)	\(8\)	$2^{7}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{1}q^{3}+(-\beta _{1}+\beta _{5})q^{5}+(1+\beta _{1}+\cdots)q^{7}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(624, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(52, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(156, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(208, [\chi])\)\(^{\oplus 2}\)