Space of modular forms of level 169, weight 4, and trivial character

• Label: 169.4.a
• Level: $169$
• Weight: $4$
• Character orbit: 169.a
• Rep. character: $\chi_{169}(1,\cdot)$
• Character field: $\Q$
• Dimension: $33$
• Newform subspaces: $12$
• Sturm bound: $60$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	169.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(60\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(169))\).

	Total	New	Old
Modular forms	53	44	9
Cusp forms	39	33	6
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.

\(13\)		Total				Cusp				Eisenstein
		All	New	Old		All	New	Old		All	New	Old
\(+\)		\(28\)	\(23\)	\(5\)		\(21\)	\(18\)	\(3\)		\(7\)	\(5\)	\(2\)
\(-\)		\(25\)	\(21\)	\(4\)		\(18\)	\(15\)	\(3\)		\(7\)	\(6\)	\(1\)

Trace form

33 * q + 4 * q^2 + 2 * q^3 + 110 * q^4 + 10 * q^5 - 12 * q^6 + 22 * q^7 + 48 * q^8 + 159 * q^9 - 78 * q^10 - 54 * q^11 + 150 * q^12 - 186 * q^14 - 16 * q^15 + 274 * q^16 - 98 * q^17 + 220 * q^18 + 210 * q^19 + 100 * q^20 + 212 * q^21 - 136 * q^22 - 144 * q^23 - 588 * q^24 + 581 * q^25 - 520 * q^27 + 96 * q^28 + 188 * q^29 + 226 * q^30 - 110 * q^31 + 208 * q^32 - 76 * q^33 + 520 * q^34 - 164 * q^35 - 258 * q^36 - 78 * q^37 - 482 * q^38 - 202 * q^40 - 106 * q^41 + 220 * q^42 - 226 * q^43 + 620 * q^44 + 334 * q^45 - 476 * q^46 - 330 * q^47 - 72 * q^48 - 211 * q^49 - 236 * q^50 + 184 * q^51 + 380 * q^53 + 84 * q^54 - 212 * q^55 + 344 * q^56 - 1488 * q^57 - 1204 * q^58 + 662 * q^59 - 1008 * q^60 + 916 * q^61 + 1706 * q^62 + 1846 * q^63 - 908 * q^64 + 2406 * q^66 - 546 * q^67 - 2148 * q^68 - 1844 * q^69 + 580 * q^70 + 122 * q^71 - 360 * q^72 - 554 * q^73 - 1256 * q^74 + 670 * q^75 + 2120 * q^76 - 1148 * q^77 - 148 * q^79 + 692 * q^80 - 839 * q^81 + 2070 * q^82 - 1650 * q^83 - 2956 * q^84 + 712 * q^85 - 2364 * q^86 + 1152 * q^87 - 120 * q^88 + 1910 * q^89 - 2740 * q^90 + 3208 * q^92 + 720 * q^93 - 478 * q^94 - 1092 * q^95 + 1268 * q^96 + 858 * q^97 - 220 * q^98 + 702 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(169))\) 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
169.4.a.a	$169$	$4$	169.a	1.a	$1$	$1$	$1$	$9.971$	\(\Q\)	None	✓				13.4.c.a	$1$	$0$	\(-4\)	\(2\)	\(-17\)	\(-20\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+2q^{3}+8q^{4}-17q^{5}-8q^{6}+\cdots\)
169.4.a.b	$169$	$4$	169.a	1.a	$1$	$1$	$1$	$9.971$	\(\Q\)	None	✓				13.4.b.a	$1$	$1$	\(-3\)	\(-1\)	\(9\)	\(-15\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}-q^{3}+q^{4}+9q^{5}+3q^{6}-15q^{7}+\cdots\)
169.4.a.c	$169$	$4$	169.a	1.a	$1$	$1$	$1$	$9.971$	\(\Q\)	None	✓				13.4.b.a	$1$	$1$	\(3\)	\(-1\)	\(-9\)	\(15\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}-q^{3}+q^{4}-9q^{5}-3q^{6}+15q^{7}+\cdots\)
169.4.a.d	$169$	$4$	169.a	1.a	$1$	$1$	$1$	$9.971$	\(\Q\)	None	✓				13.4.c.a	$1$	$0$	\(4\)	\(2\)	\(17\)	\(20\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+2q^{3}+8q^{4}+17q^{5}+8q^{6}+\cdots\)
169.4.a.e	$169$	$4$	169.a	1.a	$1$	$1$	$1$	$9.971$	\(\Q\)	None	✓				13.4.a.a	$1$	$0$	\(5\)	\(-7\)	\(7\)	\(13\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5q^{2}-7q^{3}+17q^{4}+7q^{5}-35q^{6}+\cdots\)
169.4.a.f	$169$	$4$	169.a	1.a	$1$	$2$	$2$	$9.971$	\(\Q(\sqrt{17}) \)	None	✓				13.4.c.b	$1$	$0$	\(-5\)	\(5\)	\(-15\)	\(15\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-2-\beta )q^{2}+(1+3\beta )q^{3}+5\beta q^{4}+\cdots\)
169.4.a.g	$169$	$4$	169.a	1.a	$1$	$2$	$2$	$9.971$	\(\Q(\sqrt{17}) \)	None	✓				13.4.a.b	$1$	$0$	\(-1\)	\(5\)	\(3\)	\(9\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(4-3\beta )q^{3}+(-4+\beta )q^{4}+\cdots\)
169.4.a.h	$169$	$4$	169.a	1.a	$1$	$2$	$2$	$9.971$	\(\Q(\sqrt{3}) \)	None	✓				13.4.e.a	$2$	$1$	\(0\)	\(-14\)	\(0\)	\(0\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2\beta q^{2}-7q^{3}+4q^{4}+8\beta q^{5}-14\beta q^{6}+\cdots\)
169.4.a.i	$169$	$4$	169.a	1.a	$1$	$2$	$2$	$9.971$	\(\Q(\sqrt{3}) \)	None	✓				13.4.e.b	$2$	$1$	\(0\)	\(4\)	\(0\)	\(0\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+2q^{3}-5q^{4}+\beta q^{5}+2\beta q^{6}+\cdots\)
169.4.a.j	$169$	$4$	169.a	1.a	$1$	$2$	$2$	$9.971$	\(\Q(\sqrt{17}) \)	None	✓				13.4.c.b	$1$	$0$	\(5\)	\(5\)	\(15\)	\(-15\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(3-\beta )q^{2}+(4-3\beta )q^{3}+(5-5\beta )q^{4}+\cdots\)
169.4.a.k	$169$	$4$	169.a	1.a	$1$	$9$	$9$	$9.971$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓	✓		169.4.a.k	$1$	$1$	\(-5\)	\(1\)	\(-30\)	\(-38\)		$-$	$-$	$13^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+\beta _{2}q^{3}+(5+\beta _{5}+\beta _{6}+\cdots)q^{4}+\cdots\)
169.4.a.l	$169$	$4$	169.a	1.a	$1$	$9$	$9$	$9.971$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓	✓		169.4.a.k	$1$	$0$	\(5\)	\(1\)	\(30\)	\(38\)		$+$	$+$	$13^{2}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+\beta _{2}q^{3}+(5+\beta _{5}+\beta _{6}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(169))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(169)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 2}\)