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

• Label: 624.2.bf
• Level: $624$
• Weight: $2$
• Character orbit: 624.bf
• Rep. character: $\chi_{624}(161,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $52$
• Newform subspaces: $7$
• Sturm bound: $224$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	624.bf (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 39 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 7 \)
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	60	188
Cusp forms	200	52	148
Eisenstein series	48	8	40

Trace form

52 * q + 4 * q^3 + 8 * q^7 - 4 * q^9 - 4 * q^13 - 4 * q^15 + 8 * q^19 - 8 * q^21 + 4 * q^27 - 24 * q^31 + 4 * q^33 - 20 * q^37 + 32 * q^39 + 8 * q^45 + 40 * q^55 + 8 * q^57 - 40 * q^61 + 16 * q^63 - 16 * q^67 + 4 * q^73 + 8 * q^79 - 4 * q^81 - 8 * q^85 - 32 * q^87 + 56 * q^91 - 12 * q^97 - 24 * q^99

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.bf.a	$624$	$2$	624.bf	39.f	$4$	$4$	$2$	$4.983$	\(\Q(\zeta_{8})\)	None					312.2.x.a	$4$	$1$	\(0\)	\(-4\)	\(0\)	\(-12\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(\beta_{2}-1)q^{3}+(-\beta_{3}+\beta_{2})q^{5}+(3\beta_1-3)q^{7}+\cdots\)
624.2.bf.b	$624$	$2$	624.bf	39.f	$4$	$4$	$2$	$4.983$	\(\Q(\zeta_{12})\)	\(\Q(\sqrt{-3}) \)					156.2.m.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$2^{2}$	$\mathrm{U}(1)[D_{4}]$	\(q+\beta_{3} q^{3}+(-\beta_{3}+\beta_{2}+2\beta_1-2)q^{7}+\cdots\)
624.2.bf.c	$624$	$2$	624.bf	39.f	$4$	$4$	$2$	$4.983$	\(\Q(\zeta_{12})\)	None					156.2.m.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta_{2} q^{3}+(-\beta_{3}+\beta_{2})q^{5}+(-\beta_1+1)q^{7}+\cdots\)
624.2.bf.d	$624$	$2$	624.bf	39.f	$4$	$4$	$2$	$4.983$	\(\Q(\zeta_{8})\)	None					39.2.f.a	$4$	$0$	\(0\)	\(4\)	\(0\)	\(-4\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(\beta_{2}+1)q^{3}+(\beta_{3}+\beta_{2})q^{5}+(-\beta_1-1)q^{7}+\cdots\)
624.2.bf.e	$624$	$2$	624.bf	39.f	$4$	$8$	$4$	$4.983$	\(\Q(\zeta_{24})\)	None					312.2.x.b	$4$	$0$	\(0\)	\(8\)	\(0\)	\(-8\)	$2^{9}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-\beta_{2}+1)q^{3}-\beta_{7} q^{5}+(-\beta_1-1)q^{7}+\cdots\)
624.2.bf.f	$624$	$2$	624.bf	39.f	$4$	$12$	$6$	$4.983$	12.0.\(\cdots\).52	None					78.2.g.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(12\)	$2\cdot 3^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{2}q^{3}+(-\beta _{5}+\beta _{6})q^{5}+(1+\beta _{4}+\cdots)q^{7}+\cdots\)
624.2.bf.g	$624$	$2$	624.bf	39.f	$4$	$16$	$8$	$4.983$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None			✓		312.2.x.c	$4$	$0$	\(0\)	\(-4\)	\(0\)	\(24\)	$2^{8}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{12}q^{3}+(\beta _{4}-\beta _{8}-\beta _{10}-\beta _{14}+\cdots)q^{5}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(624, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(78, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(156, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(312, [\chi])\)\(^{\oplus 2}\)