Space of modular forms of level 52, weight 4, and Character 52.h

• Label: 52.4.h
• Level: $52$
• Weight: $4$
• Character orbit: 52.h
• Rep. character: $\chi_{52}(17,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $8$
• Newform subspaces: $1$
• Sturm bound: $28$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 52 = 2^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	52.h (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(28\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(52, [\chi])\).

	Total	New	Old
Modular forms	48	8	40
Cusp forms	36	8	28
Eisenstein series	12	0	12

Trace form

8 * q - 36 * q^7 - 70 * q^9 + 72 * q^11 + 62 * q^13 + 96 * q^15 + 88 * q^17 - 144 * q^19 - 20 * q^23 - 84 * q^25 - 432 * q^27 - 484 * q^29 + 1038 * q^33 + 40 * q^35 + 996 * q^37 - 236 * q^39 + 156 * q^41 + 504 * q^43 - 1530 * q^45 + 922 * q^49 - 1808 * q^51 - 1164 * q^53 - 1128 * q^55 + 600 * q^59 - 1224 * q^61 + 6480 * q^63 + 670 * q^65 + 960 * q^67 + 1738 * q^69 - 2964 * q^71 + 1448 * q^75 - 3972 * q^77 - 3968 * q^79 - 4132 * q^81 + 3870 * q^85 - 1660 * q^87 + 5430 * q^89 - 1720 * q^91 + 3324 * q^93 + 2400 * q^95 - 3042 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(52, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
52.4.h.a	$52$	$4$	52.h	13.e	$6$	$8$	$4$	$3.068$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		✓	✓	✓	52.4.h.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-36\)	$2^{6}\cdot 3$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{5}q^{3}+(2-3\beta _{1}+\beta _{2}-\beta _{4})q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(52, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(52, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 2}\)