Space of modular forms of level 261, weight 12, and trivial character

• Label: 261.12.a
• Level: $261$
• Weight: $12$
• Character orbit: 261.a
• Rep. character: $\chi_{261}(1,\cdot)$
• Character field: $\Q$
• Dimension: $129$
• Newform subspaces: $8$
• Sturm bound: $360$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 261 = 3^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	261.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(360\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_0(261))\).

	Total	New	Old
Modular forms	334	129	205
Cusp forms	326	129	197
Eisenstein series	8	0	8

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	Dim
\(+\)	\(+\)	$+$	\(26\)
\(+\)	\(-\)	$-$	\(26\)
\(-\)	\(+\)	$-$	\(37\)
\(-\)	\(-\)	$+$	\(40\)
Plus space		\(+\)	\(66\)
Minus space		\(-\)	\(63\)

Trace form

\( 129 q + 32 q^{2} + 130188 q^{4} + 3104 q^{5} - 78272 q^{7} + 173142 q^{8} + O(q^{10}) \)

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(261))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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
261.12.a.a	$261$	$12$	261.a	1.a	$1$	$11$	$11$	$200.538$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓	$1$	$1$	\(32\)	\(0\)	\(2740\)	\(-49432\)		$-$	$+$	$-$	$2^{13}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+(3-\beta _{1})q^{2}+(832-6\beta _{1}+\beta _{2})q^{4}+\cdots\)
261.12.a.b	$261$	$12$	261.a	1.a	$1$	$11$	$11$	$200.538$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓	$1$	$1$	\(55\)	\(0\)	\(5656\)	\(-14784\)		$-$	$+$	$-$	$2^{12}\cdot 3^{4}\cdot 7$	$\mathrm{SU}(2)$	\(q+(5-\beta _{1})q^{2}+(704-8\beta _{1}+\beta _{2})q^{4}+\cdots\)
261.12.a.c	$261$	$12$	261.a	1.a	$1$	$12$	$12$	$200.538$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$1$	$0$	\(41\)	\(0\)	\(11906\)	\(-42544\)		$-$	$-$	$+$	$2^{13}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+(3+\beta _{1})q^{2}+(979+7\beta _{1}+\beta _{2})q^{4}+\cdots\)
261.12.a.d	$261$	$12$	261.a	1.a	$1$	$14$	$14$	$200.538$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	✓	$1$	$0$	\(-9\)	\(0\)	\(5656\)	\(52444\)		$-$	$-$	$+$	$2^{12}\cdot 3^{6}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(992+\beta _{1}+\beta _{2})q^{4}+\cdots\)
261.12.a.e	$261$	$12$	261.a	1.a	$1$	$14$	$14$	$200.538$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-9760\)	\(85024\)		$-$	$-$	$+$	$2^{20}\cdot 3^{7}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1312+3\beta _{1}+\beta _{2})q^{4}+(-698+\cdots)q^{5}+\cdots\)
261.12.a.f	$261$	$12$	261.a	1.a	$1$	$15$	$15$	$200.538$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	$1$	$1$	\(-87\)	\(0\)	\(-13094\)	\(24684\)		$-$	$+$	$-$	$2^{13}\cdot 3^{7}$	$\mathrm{SU}(2)$	\(q+(-6+\beta _{1})q^{2}+(1196-7\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
261.12.a.g	$261$	$12$	261.a	1.a	$1$	$26$	$26$	$200.538$		None	✓	$1$	$1$	\(-32\)	\(0\)	\(-12500\)	\(-66832\)		$+$	$-$	$-$		$\mathrm{SU}(2)$
261.12.a.h	$261$	$12$	261.a	1.a	$1$	$26$	$26$	$200.538$		None	✓	$1$	$0$	\(32\)	\(0\)	\(12500\)	\(-66832\)		$+$	$+$	$+$		$\mathrm{SU}(2)$

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(261))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_0(261)) \cong \) \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 6}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(29))\)\(^{\oplus 3}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(87))\)\(^{\oplus 2}\)