Space of modular forms of level 273, weight 12, and trivial character

• Label: 273.12.a
• Level: $273$
• Weight: $12$
• Character orbit: 273.a
• Rep. character: $\chi_{273}(1,\cdot)$
• Character field: $\Q$
• Dimension: $132$
• Newform subspaces: $8$
• Sturm bound: $448$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 273 = 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	273.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(448\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_0(273))\).

	Total	New	Old
Modular forms	416	132	284
Cusp forms	408	132	276
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.

\(3\)	\(7\)	\(13\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(15\)
\(+\)	\(+\)	\(-\)	$-$	\(18\)
\(+\)	\(-\)	\(+\)	$-$	\(16\)
\(+\)	\(-\)	\(-\)	$+$	\(17\)
\(-\)	\(+\)	\(+\)	$-$	\(17\)
\(-\)	\(+\)	\(-\)	$+$	\(18\)
\(-\)	\(-\)	\(+\)	$+$	\(18\)
\(-\)	\(-\)	\(-\)	$-$	\(13\)
Plus space			\(+\)	\(68\)
Minus space			\(-\)	\(64\)

Trace form

\( 132 q + 92 q^{2} + 149244 q^{4} - 44712 q^{6} - 67228 q^{7} - 36156 q^{8} + 7794468 q^{9} + O(q^{10}) \)

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(273))\) 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	7	13
273.12.a.a	$273$	$12$	273.a	1.a	$1$	$13$	$13$	$209.758$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓	$1$	$1$	\(-109\)	\(3159\)	\(-6095\)	\(218491\)		$-$	$-$	$-$	$-$	$2^{14}\cdot 3^{5}\cdot 7$	$\mathrm{SU}(2)$	\(q+(-8-\beta _{1})q^{2}+3^{5}q^{3}+(567+20\beta _{1}+\cdots)q^{4}+\cdots\)
273.12.a.b	$273$	$12$	273.a	1.a	$1$	$15$	$15$	$209.758$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	$1$	$0$	\(100\)	\(-3645\)	\(6095\)	\(-252105\)		$+$	$+$	$+$	$+$	$2^{16}\cdot 3^{6}\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q+(7-\beta _{1})q^{2}-3^{5}q^{3}+(1174-10\beta _{1}+\cdots)q^{4}+\cdots\)
273.12.a.c	$273$	$12$	273.a	1.a	$1$	$16$	$16$	$209.758$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓	$1$	$1$	\(-63\)	\(-3888\)	\(-3168\)	\(268912\)		$+$	$-$	$+$	$-$	$2^{18}\cdot 3^{12}\cdot 5\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q+(-4+\beta _{1})q^{2}-3^{5}q^{3}+(1298-4\beta _{1}+\cdots)q^{4}+\cdots\)
273.12.a.d	$273$	$12$	273.a	1.a	$1$	$17$	$17$	$209.758$	\(\mathbb{Q}[x]/(x^{17} - \cdots)\)	None	✓	$1$	$1$	\(-10\)	\(4131\)	\(-6405\)	\(-285719\)		$-$	$+$	$+$	$-$	$2^{17}\cdot 3^{9}\cdot 7^{5}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+3^{5}q^{3}+(1246-\beta _{1}+\cdots)q^{4}+\cdots\)
273.12.a.e	$273$	$12$	273.a	1.a	$1$	$17$	$17$	$209.758$	\(\mathbb{Q}[x]/(x^{17} - \cdots)\)	None	✓	$1$	$0$	\(65\)	\(-4131\)	\(6405\)	\(285719\)		$+$	$-$	$-$	$+$	$2^{20}\cdot 3^{10}\cdot 5^{2}\cdot 7^{5}$	$\mathrm{SU}(2)$	\(q+(4-\beta _{1})q^{2}-3^{5}q^{3}+(1071-4\beta _{1}+\cdots)q^{4}+\cdots\)
273.12.a.f	$273$	$12$	273.a	1.a	$1$	$18$	$18$	$209.758$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None	✓	$1$	$0$	\(19\)	\(4374\)	\(-3168\)	\(302526\)		$-$	$-$	$+$	$+$	$2^{18}\cdot 3^{14}\cdot 7^{5}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{2}+3^{5}q^{3}+(1354-\beta _{1}+\cdots)q^{4}+\cdots\)
273.12.a.g	$273$	$12$	273.a	1.a	$1$	$18$	$18$	$209.758$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None	✓	$1$	$1$	\(36\)	\(-4374\)	\(3168\)	\(-302526\)		$+$	$+$	$-$	$-$	$2^{18}\cdot 3^{12}\cdot 7^{5}$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{2}-3^{5}q^{3}+(1060-3\beta _{1}+\cdots)q^{4}+\cdots\)
273.12.a.h	$273$	$12$	273.a	1.a	$1$	$18$	$18$	$209.758$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None	✓	$1$	$0$	\(54\)	\(4374\)	\(3168\)	\(-302526\)		$-$	$+$	$-$	$+$	$2^{23}\cdot 3^{12}\cdot 5\cdot 7$	$\mathrm{SU}(2)$	\(q+(3-\beta _{1})q^{2}+3^{5}q^{3}+(1147-2\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(273))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_0(273)) \cong \) \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 8}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(91))\)\(^{\oplus 2}\)