Space of modular forms of level 261, weight 2, and trivial character

• Label: 261.2.a
• Level: $261$
• Weight: $2$
• Character orbit: 261.a
• Rep. character: $\chi_{261}(1,\cdot)$
• Character field: $\Q$
• Dimension: $11$
• Newform subspaces: $5$
• Sturm bound: $60$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 261 = 3^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	261.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(60\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(261))\).

	Total	New	Old
Modular forms	34	11	23
Cusp forms	27	11	16
Eisenstein series	7	0	7

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

\(3\)	\(29\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(5\)	\(2\)	\(3\)		\(4\)	\(2\)	\(2\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(11\)	\(2\)	\(9\)		\(9\)	\(2\)	\(7\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)		\(9\)	\(5\)	\(4\)		\(7\)	\(5\)	\(2\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)		\(9\)	\(2\)	\(7\)		\(7\)	\(2\)	\(5\)		\(2\)	\(0\)	\(2\)
Plus space		\(+\)		\(14\)	\(4\)	\(10\)		\(11\)	\(4\)	\(7\)		\(3\)	\(0\)	\(3\)
Minus space		\(-\)		\(20\)	\(7\)	\(13\)		\(16\)	\(7\)	\(9\)		\(4\)	\(0\)	\(4\)

Trace form

11 * q - q^2 + 5 * q^4 + 3 * q^8 - 4 * q^10 + 2 * q^11 - 4 * q^13 + 4 * q^14 - 7 * q^16 - 6 * q^17 + 24 * q^20 - 6 * q^22 + 7 * q^25 - 8 * q^26 + 4 * q^28 - 3 * q^29 + 6 * q^31 - 5 * q^32 + 22 * q^34 + 4 * q^35 - 10 * q^37 - 8 * q^38 - 16 * q^40 - 10 * q^41 - 2 * q^43 + 14 * q^44 - 12 * q^46 + 14 * q^47 - 5 * q^49 - 53 * q^50 - 28 * q^52 - 28 * q^53 + 14 * q^55 - 32 * q^56 + q^58 + 16 * q^59 - 14 * q^61 + 46 * q^62 - 7 * q^64 - 2 * q^65 + 20 * q^67 - 38 * q^68 - 8 * q^70 + 32 * q^71 + 10 * q^73 + 10 * q^74 + 24 * q^76 + 8 * q^77 - 2 * q^79 + 32 * q^80 - 18 * q^82 + 16 * q^83 - 24 * q^85 - 26 * q^86 + 10 * q^88 + 6 * q^89 - 32 * q^91 - 40 * q^92 - 26 * q^94 + 44 * q^95 + 34 * q^97 + 11 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(261))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	29
261.2.a.a	$261$	$2$	261.a	1.a	$1$	$2$	$2$	$2.084$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓	✓	261.2.a.a	$1$	$1$	\(-1\)	\(0\)	\(-4\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-1+\beta )q^{4}-2q^{5}+(-1+\cdots)q^{7}+\cdots\)
261.2.a.b	$261$	$2$	261.a	1.a	$1$	$2$	$2$	$2.084$	\(\Q(\sqrt{5}) \)	None	✓		✓	✓	87.2.a.a	$1$	$1$	\(-1\)	\(0\)	\(-2\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-1+\beta )q^{4}+(-2+2\beta )q^{5}+\cdots\)
261.2.a.c	$261$	$2$	261.a	1.a	$1$	$2$	$2$	$2.084$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓	✓	261.2.a.a	$1$	$0$	\(1\)	\(0\)	\(4\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-1+\beta )q^{4}+2q^{5}+(-1+\cdots)q^{7}+\cdots\)
261.2.a.d	$261$	$2$	261.a	1.a	$1$	$2$	$2$	$2.084$	\(\Q(\sqrt{2}) \)	None	✓				29.2.a.a	$1$	$0$	\(2\)	\(0\)	\(2\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(1+2\beta )q^{4}+q^{5}-2\beta q^{7}+\cdots\)
261.2.a.e	$261$	$2$	261.a	1.a	$1$	$3$	$3$	$2.084$	3.3.229.1	None	✓		✓		87.2.a.b	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{2}+(2+\beta _{1})q^{4}+2\beta _{1}q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(261))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(261)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(29))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(87))\)\(^{\oplus 2}\)