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

• Label: 52.3.k
• Level: $52$
• Weight: $3$
• Character orbit: 52.k
• Rep. character: $\chi_{52}(33,\cdot)$
• Character field: $\Q(\zeta_{12})$
• Dimension: $8$
• Newform subspaces: $1$
• Sturm bound: $21$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 52 = 2^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	52.k (of order \(12\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(\zeta_{12})\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(21\)
Trace bound:			\(0\)

Dimensions

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

	Total	New	Old
Modular forms	68	8	60
Cusp forms	44	8	36
Eisenstein series	24	0	24

Trace form

8 * q + 6 * q^5 + 4 * q^7 - 6 * q^9 + 24 * q^11 + 18 * q^13 - 60 * q^15 - 54 * q^17 - 50 * q^19 - 54 * q^21 - 24 * q^23 + 36 * q^27 + 108 * q^29 + 176 * q^31 + 114 * q^33 - 30 * q^35 + 104 * q^37 + 120 * q^39 - 168 * q^41 - 198 * q^43 + 6 * q^45 - 96 * q^47 - 162 * q^49 - 72 * q^53 + 126 * q^55 + 318 * q^57 - 18 * q^61 + 180 * q^63 - 72 * q^65 - 326 * q^67 - 498 * q^69 - 162 * q^71 - 98 * q^73 - 240 * q^75 + 192 * q^79 + 240 * q^81 + 408 * q^83 + 318 * q^85 + 36 * q^87 + 54 * q^89 + 280 * q^91 - 66 * q^93 + 312 * q^95 + 182 * q^97

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.k.a	$52$	$3$	52.k	13.f	$12$	$8$	$2$	$1.417$	8.0.\(\cdots\).1	None		✓	✓	✓	52.3.k.a	$2$	$0$	\(0\)	\(0\)	\(6\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(\beta _{1}-\beta _{4}+\beta _{5})q^{3}+(-\beta _{2}+\beta _{4}-\beta _{5}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(52, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(52, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 2}\)