Space of modular forms of level 169, weight 8, and trivial character

• Label: 169.8.a
• Level: $169$
• Weight: $8$
• Character orbit: 169.a
• Rep. character: $\chi_{169}(1,\cdot)$
• Character field: $\Q$
• Dimension: $85$
• Newform subspaces: $9$
• Sturm bound: $121$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	169.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(121\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(169))\).

	Total	New	Old
Modular forms	113	96	17
Cusp forms	99	85	14
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
\(+\)		\(58\)	\(49\)	\(9\)		\(51\)	\(44\)	\(7\)		\(7\)	\(5\)	\(2\)
\(-\)		\(55\)	\(47\)	\(8\)		\(48\)	\(41\)	\(7\)		\(7\)	\(6\)	\(1\)

Trace form

85 * q - 6 * q^2 - 52 * q^3 + 5186 * q^4 + 390 * q^5 + 84 * q^6 - 1056 * q^7 + 1320 * q^8 + 53529 * q^9 + 442 * q^10 + 7620 * q^11 - 2418 * q^12 - 11218 * q^14 + 21704 * q^15 + 330514 * q^16 + 28028 * q^17 - 23462 * q^18 - 46580 * q^19 + 73740 * q^20 + 70952 * q^21 - 168740 * q^22 - 146926 * q^23 - 53196 * q^24 + 925201 * q^25 - 59110 * q^27 + 273952 * q^28 - 47138 * q^29 + 129946 * q^30 + 316488 * q^31 + 250704 * q^32 - 169144 * q^33 - 56716 * q^34 + 23816 * q^35 + 2401578 * q^36 - 684986 * q^37 + 1006746 * q^38 - 1781402 * q^40 - 843054 * q^41 + 790660 * q^42 + 251238 * q^43 + 590604 * q^44 + 2624614 * q^45 + 496316 * q^46 + 1610664 * q^47 + 197928 * q^48 + 5220543 * q^49 - 5871426 * q^50 + 2316106 * q^51 - 3402970 * q^53 - 1043412 * q^54 + 1614548 * q^55 + 13960 * q^56 + 2563680 * q^57 + 4502440 * q^58 - 399996 * q^59 + 11818512 * q^60 - 2834780 * q^61 - 5325814 * q^62 - 10585472 * q^63 + 22397540 * q^64 - 22140858 * q^66 + 6391324 * q^67 + 5841060 * q^68 - 2660684 * q^69 + 13184060 * q^70 + 4184352 * q^71 + 4021200 * q^72 + 17699298 * q^73 - 12920752 * q^74 - 26285870 * q^75 - 27841480 * q^76 + 15513160 * q^77 - 21382074 * q^79 - 4873068 * q^80 + 40743805 * q^81 + 4407050 * q^82 + 13691388 * q^83 + 21686852 * q^84 + 6734372 * q^85 + 31170660 * q^86 - 23713440 * q^87 - 40221480 * q^88 - 33292110 * q^89 + 42918260 * q^90 - 50255880 * q^92 + 9696336 * q^93 + 3280706 * q^94 + 4142658 * q^95 + 36116084 * q^96 + 25966114 * q^97 + 12157482 * q^98 + 30302100 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(169))\) 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
169.8.a.a	$169$	$8$	169.a	1.a	$1$	$1$	$1$	$52.793$	\(\Q\)	None	✓				13.8.a.a	$1$	$0$	\(-10\)	\(-73\)	\(295\)	\(-1373\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-10q^{2}-73q^{3}-28q^{4}+295q^{5}+\cdots\)
169.8.a.b	$169$	$8$	169.a	1.a	$1$	$2$	$2$	$52.793$	\(\Q(\sqrt{337}) \)	None	✓				13.8.a.b	$1$	$0$	\(19\)	\(45\)	\(353\)	\(2009\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(10-\beta )q^{2}+(24-3\beta )q^{3}+(56-19\beta )q^{4}+\cdots\)
169.8.a.c	$169$	$8$	169.a	1.a	$1$	$4$	$4$	$52.793$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				13.8.a.c	$1$	$0$	\(-15\)	\(80\)	\(-258\)	\(-1692\)		$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-4+\beta _{1})q^{2}+(21-2\beta _{1}+\beta _{2})q^{3}+\cdots\)
169.8.a.d	$169$	$8$	169.a	1.a	$1$	$6$	$6$	$52.793$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				13.8.b.a	$2$	$1$	\(0\)	\(-56\)	\(0\)	\(0\)		$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-9-\beta _{2})q^{3}+(22+\beta _{4}+\cdots)q^{4}+\cdots\)
169.8.a.e	$169$	$8$	169.a	1.a	$1$	$8$	$8$	$52.793$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓				13.8.c.a	$1$	$0$	\(-9\)	\(28\)	\(-192\)	\(196\)		$+$	$+$	$2^{6}\cdot 13$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{2}+(4+\beta _{3})q^{3}+(72+2\beta _{1}+\cdots)q^{4}+\cdots\)
169.8.a.f	$169$	$8$	169.a	1.a	$1$	$8$	$8$	$52.793$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓				13.8.c.a	$1$	$0$	\(9\)	\(28\)	\(192\)	\(-196\)		$+$	$+$	$2^{6}\cdot 13$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{2}+(4+\beta _{3})q^{3}+(72+2\beta _{1}+\cdots)q^{4}+\cdots\)
169.8.a.g	$169$	$8$	169.a	1.a	$1$	$14$	$14$	$52.793$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	✓				13.8.e.a	$2$	$1$	\(0\)	\(-52\)	\(0\)	\(0\)		$-$	$-$	$2^{13}\cdot 3^{3}\cdot 13^{6}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-4-\beta _{5})q^{3}+(55+\beta _{4}+\cdots)q^{4}+\cdots\)
169.8.a.h	$169$	$8$	169.a	1.a	$1$	$21$	$21$	$52.793$		None	✓	✓	✓		169.8.a.h	$1$	$1$	\(-31\)	\(-26\)	\(-680\)	\(-2929\)		$-$	$-$		$\mathrm{SU}(2)$
169.8.a.i	$169$	$8$	169.a	1.a	$1$	$21$	$21$	$52.793$		None	✓	✓	✓		169.8.a.h	$1$	$0$	\(31\)	\(-26\)	\(680\)	\(2929\)		$+$	$+$		$\mathrm{SU}(2)$

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(169))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(169)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 2}\)