Space of modular forms of level 169, weight 12, and trivial character

• Label: 169.12.a
• Level: $169$
• Weight: $12$
• Character orbit: 169.a
• Rep. character: $\chi_{169}(1,\cdot)$
• Character field: $\Q$
• Dimension: $136$
• Newform subspaces: $9$
• Sturm bound: $182$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	169.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(182\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_0(169))\).

	Total	New	Old
Modular forms	175	147	28
Cusp forms	161	136	25
Eisenstein series	14	11	3

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

\(13\)	Dim
\(+\)	\(69\)
\(-\)	\(67\)

Trace form

\( 136 q + 10 q^{2} - 232 q^{3} + 128578 q^{4} - 5600 q^{5} + 28404 q^{6} + 57216 q^{7} - 42936 q^{8} + 7095444 q^{9} + O(q^{10}) \)

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(169))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		13
169.12.a.a	$169$	$12$	169.a	1.a	$1$	$1$	$1$	$129.850$	\(\Q\)	None	✓	$1$	$0$	\(24\)	\(252\)	\(-4830\)	\(16744\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+24q^{2}+252q^{3}-1472q^{4}-4830q^{5}+\cdots\)
169.12.a.b	$169$	$12$	169.a	1.a	$1$	$5$	$5$	$129.850$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(41\)	\(-496\)	\(2542\)	\(36296\)		$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(8+\beta _{1})q^{2}+(-99-2\beta _{1}-\beta _{4})q^{3}+\cdots\)
169.12.a.c	$169$	$12$	169.a	1.a	$1$	$6$	$6$	$129.850$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(-55\)	\(476\)	\(-3312\)	\(4176\)		$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-9-\beta _{1})q^{2}+(80-2\beta _{1}+\beta _{4})q^{3}+\cdots\)
169.12.a.d	$169$	$12$	169.a	1.a	$1$	$12$	$12$	$129.850$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$1$	$0$	\(-31\)	\(244\)	\(-5218\)	\(2928\)		$+$	$+$	$2^{16}\cdot 3\cdot 13^{4}$	$\mathrm{SU}(2)$	\(q+(-3+\beta _{1})q^{2}+(20+\beta _{3})q^{3}+(941+\cdots)q^{4}+\cdots\)
169.12.a.e	$169$	$12$	169.a	1.a	$1$	$12$	$12$	$129.850$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$2$	$1$	\(0\)	\(-488\)	\(0\)	\(0\)		$-$	$-$	$2^{13}\cdot 3^{4}\cdot 13^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-41-\beta _{2})q^{3}+(2^{10}+\beta _{2}+\cdots)q^{4}+\cdots\)
169.12.a.f	$169$	$12$	169.a	1.a	$1$	$12$	$12$	$129.850$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$1$	$0$	\(31\)	\(244\)	\(5218\)	\(-2928\)		$+$	$+$	$2^{16}\cdot 3\cdot 13^{4}$	$\mathrm{SU}(2)$	\(q+(3-\beta _{1})q^{2}+(20+\beta _{3})q^{3}+(941-7\beta _{1}+\cdots)q^{4}+\cdots\)
169.12.a.g	$169$	$12$	169.a	1.a	$1$	$22$	$22$	$129.850$		None	✓	$2$	$1$	\(0\)	\(-484\)	\(0\)	\(0\)		$-$	$-$		$\mathrm{SU}(2)$
169.12.a.h	$169$	$12$	169.a	1.a	$1$	$33$	$33$	$129.850$		None	✓	$1$	$1$	\(-119\)	\(10\)	\(-22260\)	\(-97413\)		$-$	$-$		$\mathrm{SU}(2)$
169.12.a.i	$169$	$12$	169.a	1.a	$1$	$33$	$33$	$129.850$		None	✓	$1$	$0$	\(119\)	\(10\)	\(22260\)	\(97413\)		$+$	$+$		$\mathrm{SU}(2)$

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(169))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_0(169)) \cong \) \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 2}\)