Space of modular forms of level 252, weight 4, and Character 252.b

• Label: 252.4.b
• Level: $252$
• Weight: $4$
• Character orbit: 252.b
• Rep. character: $\chi_{252}(55,\cdot)$
• Character field: $\Q$
• Dimension: $58$
• Newform subspaces: $7$
• Sturm bound: $192$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	252.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 28 \)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(192\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(11\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(252, [\chi])\).

	Total	New	Old
Modular forms	152	62	90
Cusp forms	136	58	78
Eisenstein series	16	4	12

Trace form

58 * q + q^2 - 7 * q^4 - 23 * q^8 - 67 * q^14 - 43 * q^16 - 182 * q^22 - 1086 * q^25 - 199 * q^28 - 140 * q^29 - 559 * q^32 - 516 * q^37 + 454 * q^44 - 250 * q^46 - 694 * q^49 + 1229 * q^50 + 20 * q^53 + 133 * q^56 + 66 * q^58 + 2153 * q^64 - 1008 * q^65 - 848 * q^70 - 498 * q^74 - 1324 * q^77 + 2320 * q^85 - 1138 * q^86 + 346 * q^88 - 1918 * q^92 + 1625 * q^98

Decomposition of \(S_{4}^{\mathrm{new}}(252, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
252.4.b.a	$252$	$4$	252.b	28.d	$2$	$2$	$2$	$14.868$	\(\Q(\sqrt{-7}) \)	\(\Q(\sqrt{-7}) \)					28.4.d.a	$4$	$0$	\(-5\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+(-3+\beta )q^{2}+(7-5\beta )q^{4}+(-7+14\beta )q^{7}+\cdots\)
252.4.b.b	$252$	$4$	252.b	28.d	$2$	$4$	$4$	$14.868$	\(\Q(\sqrt{-2}, \sqrt{7})\)	\(\Q(\sqrt{-21}) \)		✓			252.4.b.b	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q-2\beta _{1}q^{2}-8q^{4}+\beta _{2}q^{5}-7\beta _{3}q^{7}+\cdots\)
252.4.b.c	$252$	$4$	252.b	28.d	$2$	$4$	$4$	$14.868$	\(\Q(i, \sqrt{7})\)	\(\Q(\sqrt{-7}) \)		✓			252.4.b.c	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta _{3}q^{2}+(-2-5\beta _{2})q^{4}+(-7+14\beta _{2}+\cdots)q^{7}+\cdots\)
252.4.b.d	$252$	$4$	252.b	28.d	$2$	$8$	$8$	$14.868$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					28.4.d.b	$4$	$0$	\(8\)	\(0\)	\(0\)	\(0\)	$2^{12}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1-\beta _{2})q^{2}+(-2-\beta _{2}+\beta _{4}+\beta _{5}+\cdots)q^{4}+\cdots\)
252.4.b.e	$252$	$4$	252.b	28.d	$2$	$12$	$12$	$14.868$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None					84.4.b.a	$2$	$0$	\(-1\)	\(0\)	\(0\)	\(-10\)	$2^{13}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}+\beta _{2}q^{4}+(-\beta _{1}-\beta _{8})q^{5}+\cdots\)
252.4.b.f	$252$	$4$	252.b	28.d	$2$	$12$	$12$	$14.868$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None					84.4.b.a	$2$	$0$	\(-1\)	\(0\)	\(0\)	\(10\)	$2^{13}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{2}+\beta _{3}q^{4}+(\beta _{2}-\beta _{8})q^{5}+(1+\cdots)q^{7}+\cdots\)
252.4.b.g	$252$	$4$	252.b	28.d	$2$	$16$	$16$	$14.868$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		✓	✓		252.4.b.g	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{26}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}+(3+\beta _{5})q^{4}+\beta _{11}q^{5}+(1+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(252, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(252, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 2}\)