Space of modular forms of level 624, weight 3, and Character 624.ba

• Label: 624.3.ba
• Level: $624$
• Weight: $3$
• Character orbit: 624.ba
• Rep. character: $\chi_{624}(385,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $56$
• Newform subspaces: $6$
• Sturm bound: $336$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	624.ba (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(336\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(624, [\chi])\).

	Total	New	Old
Modular forms	472	56	416
Cusp forms	424	56	368
Eisenstein series	48	0	48

Trace form

56 * q + 8 * q^5 + 16 * q^7 + 168 * q^9 - 32 * q^19 - 16 * q^31 + 8 * q^37 - 48 * q^39 - 72 * q^41 + 24 * q^45 - 96 * q^47 + 80 * q^53 + 320 * q^55 + 48 * q^57 + 64 * q^59 - 176 * q^61 + 48 * q^63 + 216 * q^65 + 256 * q^67 + 256 * q^71 - 40 * q^73 + 576 * q^79 + 504 * q^81 + 160 * q^83 + 16 * q^85 + 56 * q^89 + 544 * q^91 + 280 * q^97

Decomposition of \(S_{3}^{\mathrm{new}}(624, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
624.3.ba.a	$624$	$3$	624.ba	13.d	$4$	$4$	$2$	$17.003$	\(\Q(\zeta_{12})\)	None					78.3.f.a	$2$	$0$	\(0\)	\(0\)	\(12\)	\(4\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta_{3} q^{3}+(-\beta_{3}+\beta_{2}-3\beta_1+3)q^{5}+\cdots\)
624.3.ba.b	$624$	$3$	624.ba	13.d	$4$	$8$	$4$	$17.003$	8.0.\(\cdots\).10	None					39.3.g.a	$2$	$0$	\(0\)	\(0\)	\(-20\)	\(-8\)	$2^{5}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{2}q^{3}+(-3+\beta _{2}+3\beta _{5}+\beta _{6})q^{5}+\cdots\)
624.3.ba.c	$624$	$3$	624.ba	13.d	$4$	$8$	$4$	$17.003$	8.0.\(\cdots\).1	None					78.3.f.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$2^{6}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{1}q^{3}+(-\beta _{1}+\beta _{2}+\beta _{7})q^{5}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
624.3.ba.d	$624$	$3$	624.ba	13.d	$4$	$8$	$4$	$17.003$	8.0.\(\cdots\).10	None					156.3.j.a	$2$	$0$	\(0\)	\(0\)	\(12\)	\(8\)	$2^{5}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{2}q^{3}+(1-\beta _{1}+\beta _{4}+\beta _{5})q^{5}+(2+\cdots)q^{7}+\cdots\)
624.3.ba.e	$624$	$3$	624.ba	13.d	$4$	$12$	$6$	$17.003$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None					312.3.s.a	$2$	$0$	\(0\)	\(0\)	\(8\)	\(12\)	$2^{9}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{4}q^{3}+(1+\beta _{3}+\beta _{6})q^{5}+(1-\beta _{6}+\cdots)q^{7}+\cdots\)
624.3.ba.f	$624$	$3$	624.ba	13.d	$4$	$16$	$8$	$17.003$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None			✓		312.3.s.b	$2$	$0$	\(0\)	\(0\)	\(-4\)	\(4\)	$2^{14}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{3}q^{3}-\beta _{2}q^{5}+\beta _{11}q^{7}+3q^{9}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(624, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(624, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(52, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(78, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(104, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(156, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(208, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(312, [\chi])\)\(^{\oplus 2}\)