Space of modular forms of level 52, weight 6, and trivial character

• Label: 52.6.a
• Level: $52$
• Weight: $6$
• Character orbit: 52.a
• Rep. character: $\chi_{52}(1,\cdot)$
• Character field: $\Q$
• Dimension: $5$
• Newform subspaces: $3$
• Sturm bound: $42$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 52 = 2^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	52.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(42\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(52))\).

	Total	New	Old
Modular forms	38	5	33
Cusp forms	32	5	27
Eisenstein series	6	0	6

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(13\)	Fricke	Dim
\(-\)	\(+\)	\(-\)	\(3\)
\(-\)	\(-\)	\(+\)	\(2\)
Plus space		\(+\)	\(2\)
Minus space		\(-\)	\(3\)

Trace form

5 * q + 24 * q^3 - 86 * q^5 - 42 * q^7 + 869 * q^9 - 858 * q^11 - 169 * q^13 + 3280 * q^15 + 1370 * q^17 - 1758 * q^19 - 5384 * q^21 + 3992 * q^23 + 9903 * q^25 - 9576 * q^27 + 4802 * q^29 - 6782 * q^31 - 2984 * q^33 + 2452 * q^35 - 3678 * q^37 - 44230 * q^41 + 24684 * q^43 - 1846 * q^45 - 3498 * q^47 + 32277 * q^49 - 29672 * q^51 + 15950 * q^53 - 80948 * q^55 + 100480 * q^57 + 52206 * q^59 + 30414 * q^61 - 58482 * q^63 - 17238 * q^65 - 57182 * q^67 - 29160 * q^69 + 20402 * q^71 - 14414 * q^73 + 177344 * q^75 + 102036 * q^77 + 102748 * q^79 + 131981 * q^81 + 13554 * q^83 - 271868 * q^85 - 323520 * q^87 - 219718 * q^89 - 53742 * q^91 + 96480 * q^93 + 433556 * q^95 - 152790 * q^97 - 221314 * q^99

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(52))\) 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					A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	13
52.6.a.a	$52$	$6$	52.a	1.a	$1$	$1$	$1$	$8.340$	\(\Q\)	None	✓	✓			52.6.a.a	$1$	$1$	\(0\)	\(-5\)	\(-3\)	\(53\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5q^{3}-3q^{5}+53q^{7}-218q^{9}-702q^{11}+\cdots\)
52.6.a.b	$52$	$6$	52.a	1.a	$1$	$1$	$1$	$8.340$	\(\Q\)	None	✓	✓			52.6.a.b	$1$	$1$	\(0\)	\(17\)	\(-91\)	\(-233\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+17q^{3}-91q^{5}-233q^{7}+46q^{9}+\cdots\)
52.6.a.c	$52$	$6$	52.a	1.a	$1$	$3$	$3$	$8.340$	3.3.203961.1	None	✓	✓	✓	✓	52.6.a.c	$1$	$0$	\(0\)	\(12\)	\(8\)	\(138\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(5+2\beta _{1}-\beta _{2})q^{3}+(6+5\beta _{1}-5\beta _{2})q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(52))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(52)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 2}\)