Space of modular forms of level 52, weight 10, and trivial character

• Label: 52.10.a
• Level: $52$
• Weight: $10$
• Character orbit: 52.a
• Rep. character: $\chi_{52}(1,\cdot)$
• Character field: $\Q$
• Dimension: $9$
• Newform subspaces: $2$
• Sturm bound: $70$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 52 = 2^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	52.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(70\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(52))\).

	Total	New	Old
Modular forms	66	9	57
Cusp forms	60	9	51
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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(16\)	\(0\)	\(16\)		\(14\)	\(0\)	\(14\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)		\(17\)	\(0\)	\(17\)		\(15\)	\(0\)	\(15\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)		\(17\)	\(5\)	\(12\)		\(16\)	\(5\)	\(11\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(16\)	\(4\)	\(12\)		\(15\)	\(4\)	\(11\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(32\)	\(4\)	\(28\)		\(29\)	\(4\)	\(25\)		\(3\)	\(0\)	\(3\)
Minus space		\(-\)		\(34\)	\(5\)	\(29\)		\(31\)	\(5\)	\(26\)		\(3\)	\(0\)	\(3\)

Trace form

9 * q - 294 * q^3 - 226 * q^5 + 13568 * q^7 + 18995 * q^9 + 28600 * q^11 - 28561 * q^13 - 57080 * q^15 - 842628 * q^17 - 144252 * q^19 + 2415220 * q^21 - 363340 * q^23 + 6065533 * q^25 - 4130982 * q^27 + 582602 * q^29 + 3373288 * q^31 + 7528516 * q^33 - 11376958 * q^35 - 2166498 * q^37 + 21542622 * q^41 + 48277646 * q^43 + 35470994 * q^45 + 33948760 * q^47 + 55212639 * q^49 - 73516946 * q^51 - 49638666 * q^53 + 57562152 * q^55 - 202784624 * q^57 + 1745828 * q^59 + 73828778 * q^61 + 154051944 * q^63 - 104761748 * q^65 + 192381976 * q^67 - 142349292 * q^69 - 425758088 * q^71 + 264294534 * q^73 + 396482084 * q^75 - 322255136 * q^77 - 571308732 * q^79 - 270380503 * q^81 - 337679688 * q^83 + 976478552 * q^85 - 1778771784 * q^87 - 876294114 * q^89 + 198041974 * q^91 + 820173408 * q^93 + 72164076 * q^95 + 321520930 * q^97 + 1457101388 * q^99

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(52))\) 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
52.10.a.a	$52$	$10$	52.a	1.a	$1$	$4$	$4$	$26.782$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	52.10.a.a	$1$	$1$	\(0\)	\(-147\)	\(-1947\)	\(10251\)		$-$	$-$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(-37-\beta _{1})q^{3}+(-486+2\beta _{1}+\beta _{3})q^{5}+\cdots\)
52.10.a.b	$52$	$10$	52.a	1.a	$1$	$5$	$5$	$26.782$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓	✓	52.10.a.b	$1$	$0$	\(0\)	\(-147\)	\(1721\)	\(3317\)		$-$	$+$	$-$	$2^{8}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+(-29+\beta _{1})q^{3}+(345+\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(52))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(52)) \simeq \) \(S_{10}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 2}\)