Space of modular forms of level 169, weight 3, and Character 169.f

• Label: 169.3.f
• Level: $169$
• Weight: $3$
• Character orbit: 169.f
• Rep. character: $\chi_{169}(19,\cdot)$
• Character field: $\Q(\zeta_{12})$
• Dimension: $84$
• Newform subspaces: $7$
• Sturm bound: $45$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	169.f (of order \(12\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(\zeta_{12})\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(45\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(169, [\chi])\).

	Total	New	Old
Modular forms	148	124	24
Cusp forms	92	84	8
Eisenstein series	56	40	16

Trace form

84 * q + 2 * q^2 + 2 * q^3 + 6 * q^4 + 14 * q^5 - 10 * q^6 - 16 * q^7 + 6 * q^8 - 70 * q^9 - 24 * q^10 - 4 * q^11 + 8 * q^14 + 14 * q^15 - 6 * q^16 + 12 * q^17 + 2 * q^18 - 10 * q^19 - 26 * q^20 - 40 * q^21 + 36 * q^22 - 18 * q^23 - 42 * q^24 + 32 * q^27 + 44 * q^28 + 10 * q^29 + 54 * q^30 + 20 * q^31 + 20 * q^32 + 32 * q^33 + 18 * q^34 - 76 * q^35 + 54 * q^36 + 68 * q^37 - 232 * q^40 - 100 * q^41 - 48 * q^42 - 180 * q^43 - 88 * q^44 + 2 * q^45 + 30 * q^46 + 68 * q^47 + 310 * q^48 + 72 * q^49 + 46 * q^50 - 216 * q^53 - 16 * q^54 + 188 * q^55 + 84 * q^56 + 20 * q^57 + 40 * q^58 + 164 * q^59 - 8 * q^60 + 108 * q^61 - 6 * q^62 - 52 * q^63 - 264 * q^66 - 118 * q^67 - 420 * q^68 - 72 * q^69 - 164 * q^70 + 86 * q^71 - 72 * q^72 - 58 * q^73 - 80 * q^74 - 48 * q^75 - 14 * q^76 - 32 * q^79 + 140 * q^80 + 518 * q^81 - 24 * q^82 + 188 * q^83 - 4 * q^84 - 96 * q^85 + 180 * q^86 + 242 * q^87 + 204 * q^88 + 110 * q^89 + 508 * q^92 + 20 * q^93 - 194 * q^94 + 78 * q^95 - 40 * q^96 - 178 * q^97 + 14 * q^98 - 208 * q^99

Decomposition of \(S_{3}^{\mathrm{new}}(169, [\chi])\) 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
169.3.f.a	$169$	$3$	169.f	13.f	$12$	$4$	$1$	$4.605$	\(\Q(\zeta_{12})\)	None					13.3.f.a	$2$	$0$	\(-4\)	\(-2\)	\(14\)	\(20\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(-1-\zeta_{12})q^{2}+(-\zeta_{12}-\zeta_{12}^{2}+\cdots)q^{3}+\cdots\)
169.3.f.b	$169$	$3$	169.f	13.f	$12$	$4$	$1$	$4.605$	\(\Q(\zeta_{12})\)	None					13.3.f.a	$2$	$0$	\(2\)	\(-2\)	\(14\)	\(-16\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(1-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+(\zeta_{12}-\zeta_{12}^{2}+\cdots)q^{3}+\cdots\)
169.3.f.c	$169$	$3$	169.f	13.f	$12$	$4$	$1$	$4.605$	\(\Q(\zeta_{12})\)	None					13.3.f.a	$2$	$0$	\(4\)	\(-2\)	\(-14\)	\(-20\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(1+\zeta_{12})q^{2}+(-\zeta_{12}-\zeta_{12}^{2}-\zeta_{12}^{3})q^{3}+\cdots\)
169.3.f.d	$169$	$3$	169.f	13.f	$12$	$8$	$2$	$4.605$	8.0.3317760000.2	None					13.3.d.a	$4$	$0$	\(-4\)	\(4\)	\(-16\)	\(-12\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(-1-\beta _{2}-\beta _{3}+\beta _{4}+\beta _{7})q^{2}+(1+\cdots)q^{3}+\cdots\)
169.3.f.e	$169$	$3$	169.f	13.f	$12$	$8$	$2$	$4.605$	8.0.\(\cdots\).9	None		✓			169.3.d.b	$8$	$0$	\(0\)	\(12\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+\beta _{1}q^{2}+3\beta _{4}q^{3}+7\beta _{2}q^{4}+(\beta _{1}-\beta _{5}+\cdots)q^{5}+\cdots\)
169.3.f.f	$169$	$3$	169.f	13.f	$12$	$8$	$2$	$4.605$	8.0.3317760000.2	None					13.3.d.a	$4$	$0$	\(4\)	\(4\)	\(16\)	\(12\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(1+\beta _{2}-\beta _{3}-\beta _{4}+\beta _{7})q^{2}+(1+\beta _{3}+\cdots)q^{3}+\cdots\)
169.3.f.g	$169$	$3$	169.f	13.f	$12$	$48$	$12$	$4.605$		None		✓	✓		169.3.d.e	$8$	$0$	\(0\)	\(-12\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{12}]$

Decomposition of \(S_{3}^{\mathrm{old}}(169, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(169, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 2}\)