Space of modular forms of level 169, weight 4, and Character 169.c

• Label: 169.4.c
• Level: $169$
• Weight: $4$
• Character orbit: 169.c
• Rep. character: $\chi_{169}(22,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $66$
• Newform subspaces: $12$
• Sturm bound: $60$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	169.c (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(60\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(169, [\chi])\).

	Total	New	Old
Modular forms	106	86	20
Cusp forms	78	66	12
Eisenstein series	28	20	8

Trace form

66 * q - q^2 + 7 * q^3 - 107 * q^4 - 4 * q^5 - 30 * q^6 + 35 * q^7 + 30 * q^8 - 204 * q^9 + 99 * q^10 - 15 * q^11 - 336 * q^12 - 132 * q^14 + 124 * q^15 - 331 * q^16 + 59 * q^17 + 614 * q^18 - 111 * q^19 + 311 * q^20 - 206 * q^21 - 434 * q^22 + 267 * q^23 - 216 * q^24 + 58 * q^25 - 770 * q^27 + 240 * q^28 + 169 * q^29 - 28 * q^30 + 428 * q^31 - 151 * q^32 + 361 * q^33 + 182 * q^34 + 236 * q^35 + 675 * q^36 - 417 * q^37 - 1192 * q^38 + 418 * q^40 + 373 * q^41 - 172 * q^42 + 439 * q^43 - 848 * q^44 - 16 * q^45 + 230 * q^46 + 204 * q^47 + 42 * q^48 - 506 * q^49 + 1106 * q^50 + 770 * q^51 - 1616 * q^53 - 1314 * q^54 + 14 * q^55 - 86 * q^56 + 330 * q^57 + 193 * q^58 - 1673 * q^59 - 144 * q^60 + 845 * q^61 - 1190 * q^62 - 70 * q^63 + 2714 * q^64 + 5064 * q^66 + 387 * q^67 + 441 * q^68 + 53 * q^69 - 2560 * q^70 + 781 * q^71 - 1155 * q^72 - 1600 * q^73 + 395 * q^74 - 2083 * q^75 + 100 * q^76 + 674 * q^77 - 32 * q^79 - 2153 * q^80 + 2099 * q^81 - 2637 * q^82 - 1776 * q^83 + 1540 * q^84 - 1426 * q^85 + 5172 * q^86 + 2841 * q^87 - 828 * q^88 + 655 * q^89 + 1138 * q^90 + 2408 * q^92 + 2052 * q^93 + 1156 * q^94 + 2136 * q^95 - 2816 * q^96 - 177 * q^97 + 1513 * q^98 - 5892 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(169, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
169.4.c.a	$169$	$4$	169.c	13.c	$3$	$2$	$1$	$9.971$	\(\Q(\sqrt{-3}) \)	None					13.4.a.a	$2$	$0$	\(-5\)	\(7\)	\(14\)	\(-13\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-5+5\zeta_{6})q^{2}+(7-7\zeta_{6})q^{3}-17\zeta_{6}q^{4}+\cdots\)
169.4.c.b	$169$	$4$	169.c	13.c	$3$	$2$	$1$	$9.971$	\(\Q(\sqrt{-3}) \)	None					13.4.b.a	$2$	$0$	\(-3\)	\(1\)	\(-18\)	\(-15\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-3+3\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
169.4.c.c	$169$	$4$	169.c	13.c	$3$	$2$	$1$	$9.971$	\(\Q(\sqrt{-3}) \)	None					13.4.b.a	$2$	$0$	\(3\)	\(1\)	\(18\)	\(15\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(3-3\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
169.4.c.d	$169$	$4$	169.c	13.c	$3$	$2$	$1$	$9.971$	\(\Q(\sqrt{-3}) \)	None					13.4.c.a	$2$	$0$	\(4\)	\(-2\)	\(-34\)	\(20\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(4-4\zeta_{6})q^{2}+(-2+2\zeta_{6})q^{3}-8\zeta_{6}q^{4}+\cdots\)
169.4.c.e	$169$	$4$	169.c	13.c	$3$	$2$	$1$	$9.971$	\(\Q(\sqrt{-3}) \)	None					13.4.a.a	$2$	$0$	\(5\)	\(7\)	\(-14\)	\(13\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(5-5\zeta_{6})q^{2}+(7-7\zeta_{6})q^{3}-17\zeta_{6}q^{4}+\cdots\)
169.4.c.f	$169$	$4$	169.c	13.c	$3$	$4$	$2$	$9.971$	\(\Q(\sqrt{-3}, \sqrt{17})\)	None					13.4.c.b	$2$	$0$	\(-5\)	\(-5\)	\(30\)	\(15\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2+\beta _{1}+2\beta _{2}+\beta _{3})q^{2}+(-1+\cdots)q^{3}+\cdots\)
169.4.c.g	$169$	$4$	169.c	13.c	$3$	$4$	$2$	$9.971$	\(\Q(\sqrt{-3}, \sqrt{17})\)	None					13.4.a.b	$2$	$0$	\(-1\)	\(-5\)	\(-6\)	\(9\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{1}q^{2}+(3\beta _{1}-4\beta _{2})q^{3}+(4+\beta _{1}+\cdots)q^{4}+\cdots\)
169.4.c.h	$169$	$4$	169.c	13.c	$3$	$4$	$2$	$9.971$	\(\Q(\zeta_{12})\)	None					13.4.e.b	$4$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta_{2} q^{2}-2\beta_1 q^{3}+(-5\beta_1+5)q^{4}+\cdots\)
169.4.c.i	$169$	$4$	169.c	13.c	$3$	$4$	$2$	$9.971$	\(\Q(\zeta_{12})\)	None					13.4.e.a	$4$	$0$	\(0\)	\(14\)	\(0\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q-2\beta_{2} q^{2}+7\beta_1 q^{3}+(4\beta_1-4)q^{4}+\cdots\)
169.4.c.j	$169$	$4$	169.c	13.c	$3$	$4$	$2$	$9.971$	\(\Q(\sqrt{-3}, \sqrt{17})\)	None					13.4.a.b	$2$	$0$	\(1\)	\(-5\)	\(6\)	\(-9\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{2}+(3\beta _{1}-4\beta _{2})q^{3}+(4+\beta _{1}+\cdots)q^{4}+\cdots\)
169.4.c.k	$169$	$4$	169.c	13.c	$3$	$18$	$9$	$9.971$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None		✓			169.4.a.k	$2$	$0$	\(-5\)	\(-1\)	\(60\)	\(-38\)	$13^{4}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{1}+\beta _{2}-\beta _{3})q^{2}+(\beta _{11}-\beta _{14}+\cdots)q^{3}+\cdots\)
169.4.c.l	$169$	$4$	169.c	13.c	$3$	$18$	$9$	$9.971$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None		✓			169.4.a.k	$2$	$0$	\(5\)	\(-1\)	\(-60\)	\(38\)	$13^{4}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{1}-\beta _{2}+\beta _{3})q^{2}+(\beta _{11}-\beta _{14})q^{3}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(169, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(169, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 2}\)