Embedded newform 169.4.i.a.3.13

• Label: 169.4.i.a.3.13
• Level: $169$
• Weight: $4$
• Character: 169.3
• Analytic conductor: $9.971$
• Analytic rank: $0$
• Dimension: $1080$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	169.i (of order \(39\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.97132279097\)
Analytic rank:	\(0\)
Dimension:	\(1080\)
Relative dimension:	\(45\) over \(\Q(\zeta_{39})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{39}]$

Embedding invariants

Embedding label			3.13
Character	\(\chi\)	\(=\)	169.3
Dual form			169.4.i.a.113.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.15092 + 1.61631i) q^{2} +(1.23779 - 6.06311i) q^{3} +(-0.211743 + 0.731021i) q^{4} +(-2.40618 - 1.26286i) q^{5} +(7.13748 + 15.0419i) q^{6} +(23.6338 + 3.84087i) q^{7} +(-8.35871 - 22.0401i) q^{8} +(-10.3897 - 4.42663i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1080 q - 27 q^{2} - 19 q^{3} + 155 q^{4} - 30 q^{5} - 56 q^{6} + 9 q^{7} + 4 q^{8} + 364 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{31}{39}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.15092	+	1.61631i	−0.760466	+	0.571453i	−0.908676	−	0.417503i	\(-0.862905\pi\)
							0.148210	+	0.988956i	\(0.452649\pi\)
\(3\)	1.23779	−	6.06311i	0.238213	−	1.16685i	−0.668045	−	0.744121i	\(-0.732869\pi\)
							0.906258	−	0.422725i	\(-0.138926\pi\)
\(5\)	−2.40618	−	1.26286i	−0.215215	−	0.112953i	0.353671	−	0.935370i	\(-0.384933\pi\)
							−0.568886	+	0.822416i	\(0.692625\pi\)
\(7\)	23.6338	+	3.84087i	1.27611	+	0.207388i	0.760444	−	0.649404i	\(-0.224981\pi\)
							0.515663	+	0.856791i	\(0.327546\pi\)
\(11\)	−45.5748	+	19.4176i	−1.24921	+	0.532239i	−0.912387	−	0.409329i	\(-0.865763\pi\)
							−0.336823	+	0.941568i	\(0.609352\pi\)
\(13\)	45.8517	−	9.72749i	0.978228	−	0.207532i
							\(17\)	3.04314	+	0.494559i	0.0434159	+	0.00705577i	0.182080	−	0.983284i	\(-0.441717\pi\)
−0.138665	+	0.990339i	\(0.544281\pi\)
\(19\)	45.2635	−	78.3986i	0.546534	−	0.946625i	−0.451974	−	0.892031i	\(-0.649280\pi\)
							0.998509	−	0.0545941i	\(-0.0173865\pi\)
\(23\)	−21.0844	−	36.5192i	−0.191148	−	0.331078i	0.754483	−	0.656319i	\(-0.227888\pi\)
							−0.945631	+	0.325242i	\(0.894554\pi\)
\(29\)	81.6676	−	61.3692i	0.522941	−	0.392965i	−0.306232	−	0.951957i	\(-0.599068\pi\)
							0.829173	+	0.558992i	\(0.188812\pi\)
\(31\)	155.810	−	225.729i	0.902718	−	1.30781i	−0.0483503	−	0.998830i	\(-0.515396\pi\)
							0.951068	−	0.308982i	\(-0.0999882\pi\)
\(37\)	184.677	+	14.9087i	0.820559	+	0.0662424i	0.483613	−	0.875282i	\(-0.339324\pi\)
							0.336946	+	0.941524i	\(0.390606\pi\)
\(41\)	13.5418	−	66.3322i	0.0515824	−	0.252667i	−0.945821	−	0.324690i	\(-0.894740\pi\)
							0.997403	+	0.0720224i	\(0.0229453\pi\)
\(43\)	−90.1896	+	7.28085i	−0.319855	+	0.0258214i	−0.239352	−	0.970933i	\(-0.576935\pi\)
							−0.0805036	+	0.996754i	\(0.525653\pi\)
\(47\)	337.889	+	83.2821i	1.04864	+	0.258467i	0.725743	−	0.687966i	\(-0.241496\pi\)
							0.322899	+	0.946433i	\(0.395343\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.4.i.a.3.13	✓	1080
169.113	even	39	inner	169.4.i.a.113.13	yes	1080

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.4.i.a.3.13	✓	1080	1.1	even	1	trivial
169.4.i.a.113.13	yes	1080	169.113	even	39	inner