Space of modular forms of level 52, weight 2, and Character 52.f

• Label: 52.2.f
• Level: $52$
• Weight: $2$
• Character orbit: 52.f
• Rep. character: $\chi_{52}(31,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $10$
• Newform subspaces: $2$
• Sturm bound: $14$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 52 = 2^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	52.f (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 52 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(14\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(52, [\chi])\).

	Total	New	Old
Modular forms	18	18	0
Cusp forms	10	10	0
Eisenstein series	8	8	0

Trace form

10 * q - 2 * q^2 - 6 * q^5 + 8 * q^6 - 8 * q^8 - 10 * q^9 - 4 * q^13 - 20 * q^14 - 12 * q^16 + 6 * q^18 + 4 * q^20 + 16 * q^21 + 24 * q^22 + 32 * q^24 - 2 * q^26 + 12 * q^28 - 8 * q^29 - 12 * q^32 + 8 * q^33 + 8 * q^34 - 22 * q^37 + 28 * q^40 - 2 * q^41 + 4 * q^42 - 28 * q^44 - 2 * q^45 - 20 * q^46 - 44 * q^48 - 42 * q^50 - 4 * q^52 + 4 * q^53 - 32 * q^54 + 48 * q^57 + 20 * q^58 - 12 * q^60 + 4 * q^61 + 14 * q^65 - 32 * q^66 + 52 * q^68 + 8 * q^70 + 4 * q^72 - 54 * q^73 + 64 * q^74 + 32 * q^76 + 40 * q^78 + 40 * q^80 - 6 * q^81 - 48 * q^84 + 4 * q^85 + 32 * q^86 + 22 * q^89 + 24 * q^92 + 32 * q^93 + 20 * q^94 - 24 * q^96 + 2 * q^97 - 2 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(52, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
52.2.f.a	$52$	$2$	52.f	52.f	$4$	$2$	$1$	$0.415$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		✓			52.2.f.a	$4$	$0$	\(2\)	\(0\)	\(-6\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(i+1)q^{2}+2 i q^{4}+(-3 i-3)q^{5}+\cdots\)
52.2.f.b	$52$	$2$	52.f	52.f	$4$	$8$	$4$	$0.415$	8.0.18939904.2	None		✓	✓		52.2.f.b	$4$	$0$	\(-4\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1+\beta _{2}-\beta _{5}+\beta _{7})q^{2}+(1-2\beta _{1}+\cdots)q^{3}+\cdots\)