Space of modular forms of level 52, weight 3, and Character 52.b

• Label: 52.3.b
• Level: $52$
• Weight: $3$
• Character orbit: 52.b
• Rep. character: $\chi_{52}(51,\cdot)$
• Character field: $\Q$
• Dimension: $12$
• Newform subspaces: $4$
• Sturm bound: $21$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 52 = 2^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	52.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 52 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(21\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(3\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(52, [\chi])\).

	Total	New	Old
Modular forms	16	16	0
Cusp forms	12	12	0
Eisenstein series	4	4	0

Trace form

12 * q - 2 * q^4 - 28 * q^9 + 2 * q^10 + 34 * q^12 - 4 * q^13 - 2 * q^14 - 46 * q^16 + 16 * q^17 - 80 * q^22 + 36 * q^25 + 14 * q^26 - 24 * q^29 + 102 * q^30 - 52 * q^36 - 100 * q^38 + 162 * q^40 + 170 * q^42 + 238 * q^48 - 100 * q^49 - 140 * q^52 - 24 * q^53 - 314 * q^56 + 184 * q^61 + 40 * q^62 + 70 * q^64 + 56 * q^65 - 544 * q^66 - 246 * q^68 - 266 * q^74 - 336 * q^77 + 442 * q^78 + 156 * q^81 + 592 * q^82 - 48 * q^88 - 220 * q^90 + 884 * q^92 + 238 * q^94

Decomposition of \(S_{3}^{\mathrm{new}}(52, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
52.3.b.a	$52$	$3$	52.b	52.b	$2$	$1$	$1$	$1.417$	\(\Q\)	\(\Q(\sqrt{-13}) \)	✓	✓			52.3.b.a	$2$	$0$	\(-2\)	\(0\)	\(0\)	\(12\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-2q^{2}+4q^{4}+12q^{7}-8q^{8}+9q^{9}+\cdots\)
52.3.b.b	$52$	$3$	52.b	52.b	$2$	$1$	$1$	$1.417$	\(\Q\)	\(\Q(\sqrt{-13}) \)	✓	✓			52.3.b.a	$2$	$0$	\(2\)	\(0\)	\(0\)	\(-12\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+2q^{2}+4q^{4}-12q^{7}+8q^{8}+9q^{9}+\cdots\)
52.3.b.c	$52$	$3$	52.b	52.b	$2$	$2$	$2$	$1.417$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		✓			52.3.b.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta q^{2}-4 q^{4}+4\beta q^{5}-4\beta q^{8}+\cdots\)
52.3.b.d	$52$	$3$	52.b	52.b	$2$	$8$	$8$	$1.417$	8.0.\(\cdots\).3	None		✓	✓		52.3.b.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}-\beta _{6}q^{3}+\beta _{2}q^{4}+(-\beta _{1}-\beta _{7})q^{5}+\cdots\)