Space of modular forms of level 169, weight 10, and trivial character

• Label: 169.10.a
• Level: $169$
• Weight: $10$
• Character orbit: 169.a
• Rep. character: $\chi_{169}(1,\cdot)$
• Character field: $\Q$
• Dimension: $111$
• Newform subspaces: $8$
• Sturm bound: $151$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	169.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(151\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(169))\).

	Total	New	Old
Modular forms	143	122	21
Cusp forms	129	111	18
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
\(+\)	\(54\)
\(-\)	\(57\)

Trace form

\( 111 q + 18 q^{2} + 2 q^{3} + 27394 q^{4} - 2274 q^{5} - 1164 q^{6} + 1142 q^{7} + 22392 q^{8} + 677481 q^{9} + O(q^{10}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(169))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		13
169.10.a.a	$169$	$10$	169.a	1.a	$1$	$4$	$4$	$87.041$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				13.10.a.a	$1$	$1$	\(33\)	\(-163\)	\(-471\)	\(11241\)		$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(8+\beta _{1})q^{2}+(-41+2\beta _{1}-\beta _{3})q^{3}+\cdots\)
169.10.a.b	$169$	$10$	169.a	1.a	$1$	$5$	$5$	$87.041$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				13.10.a.b	$1$	$1$	\(-15\)	\(161\)	\(-1803\)	\(-10099\)		$+$	$+$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-3-\beta _{1})q^{2}+(2^{5}+2\beta _{1}-\beta _{2})q^{3}+\cdots\)
169.10.a.c	$169$	$10$	169.a	1.a	$1$	$9$	$9$	$87.041$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓				13.10.c.a	$1$	$1$	\(-15\)	\(-161\)	\(-1140\)	\(1939\)		$+$	$+$	$2^{8}\cdot 3\cdot 13^{2}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{2}+(-18+\beta _{3})q^{3}+(200+\cdots)q^{4}+\cdots\)
169.10.a.d	$169$	$10$	169.a	1.a	$1$	$9$	$9$	$87.041$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓				13.10.c.a	$1$	$1$	\(15\)	\(-161\)	\(1140\)	\(-1939\)		$+$	$+$	$2^{8}\cdot 3\cdot 13^{2}$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{2}+(-18+\beta _{3})q^{3}+(200+\cdots)q^{4}+\cdots\)
169.10.a.e	$169$	$10$	169.a	1.a	$1$	$10$	$10$	$87.041$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓				13.10.b.a	$2$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)		$-$	$-$	$2^{10}\cdot 3^{4}\cdot 13^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{3}q^{3}+(2^{8}+\beta _{2})q^{4}+(-6\beta _{1}+\cdots)q^{5}+\cdots\)
169.10.a.f	$169$	$10$	169.a	1.a	$1$	$20$	$20$	$87.041$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	✓				13.10.e.a	$2$	$0$	\(0\)	\(326\)	\(0\)	\(0\)		$-$	$-$	$2^{25}\cdot 3^{10}\cdot 13^{12}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2^{4}+\beta _{4})q^{3}+(2^{8}+\beta _{3}+\beta _{4}+\cdots)q^{4}+\cdots\)
169.10.a.g	$169$	$10$	169.a	1.a	$1$	$27$	$27$	$87.041$		None	✓	✓	✓		169.10.a.g	$1$	$1$	\(-65\)	\(1\)	\(-3238\)	\(-17378\)		$+$	$+$		$\mathrm{SU}(2)$
169.10.a.h	$169$	$10$	169.a	1.a	$1$	$27$	$27$	$87.041$		None	✓	✓	✓		169.10.a.g	$1$	$0$	\(65\)	\(1\)	\(3238\)	\(17378\)		$-$	$-$		$\mathrm{SU}(2)$

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(169))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(169)) \simeq \) \(S_{10}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 2}\)