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

• Label: 624.2.cn
• Level: $624$
• Weight: $2$
• Character orbit: 624.cn
• Rep. character: $\chi_{624}(305,\cdot)$
• Character field: $\Q(\zeta_{12})$
• Dimension: $104$
• 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.cn (of order \(12\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 39 \)
Character field:			\(\Q(\zeta_{12})\)
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	496	120	376
Cusp forms	400	104	296
Eisenstein series	96	16	80

Trace form

104 * q + 2 * q^3 + 4 * q^7 - 2 * q^9 - 8 * q^13 + 10 * q^15 + 4 * q^19 + 2 * q^21 + 8 * q^27 + 36 * q^31 - 10 * q^33 + 8 * q^37 - 26 * q^39 + 96 * q^43 - 14 * q^45 + 12 * q^49 + 44 * q^55 - 14 * q^57 - 20 * q^61 - 10 * q^63 + 40 * q^67 - 6 * q^69 - 16 * q^73 - 12 * q^75 + 16 * q^79 - 2 * q^81 - 4 * q^85 + 2 * q^87 - 44 * q^91 - 54 * q^93 - 66 * 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.cn.a	$624$	$2$	624.cn	39.k	$12$	$4$	$1$	$4.983$	\(\Q(\zeta_{12})\)	\(\Q(\sqrt{-3}) \)					156.2.u.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)	$1$	$\mathrm{U}(1)[D_{12}]$	\(q+(-\zeta_{12}-\zeta_{12}^{3})q^{3}+(-2-2\zeta_{12}+\cdots)q^{7}+\cdots\)
624.2.cn.b	$624$	$2$	624.cn	39.k	$12$	$4$	$1$	$4.983$	\(\Q(\zeta_{12})\)	\(\Q(\sqrt{-3}) \)					39.2.k.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(10\)	$1$	$\mathrm{U}(1)[D_{12}]$	\(q+(\zeta_{12}+\zeta_{12}^{3})q^{3}+(2+2\zeta_{12}+\zeta_{12}^{2}+\cdots)q^{7}+\cdots\)
624.2.cn.c	$624$	$2$	624.cn	39.k	$12$	$8$	$2$	$4.983$	8.0.56070144.2	None					39.2.k.b	$4$	$0$	\(0\)	\(2\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(\beta _{1}+\beta _{2}+\beta _{4}-2\beta _{5}-\beta _{6})q^{3}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
624.2.cn.d	$624$	$2$	624.cn	39.k	$12$	$16$	$4$	$4.983$	16.0.\(\cdots\).9	None					78.2.k.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{12}]$	\(q+(-\beta _{10}-\beta _{12})q^{3}+(\beta _{1}-\beta _{3}-\beta _{9}+\cdots)q^{5}+\cdots\)
624.2.cn.e	$624$	$2$	624.cn	39.k	$12$	$16$	$4$	$4.983$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None					156.2.u.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$3^{2}$	$\mathrm{SU}(2)[C_{12}]$	\(q+(\beta _{9}+\beta _{13}+\beta _{15})q^{3}+(-\beta _{3}-\beta _{9}+\cdots)q^{5}+\cdots\)
624.2.cn.f	$624$	$2$	624.cn	39.k	$12$	$56$	$14$	$4.983$		None			✓		312.2.bp.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)		$\mathrm{SU}(2)[C_{12}]$

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

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