Embedded newform 169.3.j.a.5.7

• Label: 169.3.j.a.5.7
• Level: $169$
• Weight: $3$
• Character: 169.5
• Analytic conductor: $4.605$
• Analytic rank: $0$
• Dimension: $720$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	169.j (of order \(52\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.60491646769\)
Analytic rank:	\(0\)
Dimension:	\(720\)
Relative dimension:	\(30\) over \(\Q(\zeta_{52})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{52}]$

Embedding invariants

Embedding label			5.7
Character	\(\chi\)	\(=\)	169.5
Dual form			169.3.j.a.34.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.68961 - 0.492890i) q^{2} +(1.35223 + 0.709706i) q^{3} +(3.25102 + 1.23295i) q^{4} +(-0.320821 + 5.30380i) q^{5} +(-3.28717 - 2.57534i) q^{6} +(-11.1706 - 6.75288i) q^{7} +(1.22394 + 0.739895i) q^{8} +(-3.78774 - 5.48748i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 720 q - 22 q^{2} - 22 q^{3} - 26 q^{4} - 34 q^{5} - 10 q^{6} - 14 q^{7} - 62 q^{8} - 202 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{3}{52}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.68961	−	0.492890i	−1.34481	−	0.246445i	−0.540843	−	0.841123i	\(-0.681895\pi\)
							−0.803964	+	0.594678i	\(0.797279\pi\)
\(3\)	1.35223	+	0.709706i	0.450744	+	0.236569i	0.674790	−	0.738010i	\(-0.264234\pi\)
							−0.224046	+	0.974579i	\(0.571927\pi\)
\(5\)	−0.320821	+	5.30380i	−0.0641642	+	1.06076i	0.813525	+	0.581530i	\(0.197546\pi\)
							−0.877689	+	0.479230i	\(0.840916\pi\)
\(7\)	−11.1706	−	6.75288i	−1.59580	−	0.964697i	−0.984927	−	0.172973i	\(-0.944663\pi\)
							−0.610878	−	0.791725i	\(-0.709183\pi\)
\(11\)	9.13501	−	1.67405i	0.830456	−	0.152187i	0.251821	−	0.967774i	\(-0.418970\pi\)
							0.578634	+	0.815587i	\(0.303586\pi\)
\(13\)	11.2097	+	6.58357i	0.862282	+	0.506429i
							\(17\)	−5.85636	−	23.7602i	−0.344492	−	1.39766i	−0.846313	−	0.532687i	\(-0.821182\pi\)
0.501821	−	0.864972i	\(-0.332664\pi\)
\(19\)	0.617774	−	0.617774i	0.0325144	−	0.0325144i	−0.690663	−	0.723177i	\(-0.742681\pi\)
							0.723177	+	0.690663i	\(0.242681\pi\)
\(23\)		−	38.3762i		−	1.66853i	−0.551363	−	0.834265i	\(-0.685892\pi\)
							0.551363	−	0.834265i	\(-0.314108\pi\)
\(29\)	−6.35242	−	9.20307i	−0.219049	−	0.317347i	0.698018	−	0.716080i	\(-0.254065\pi\)
							−0.917067	+	0.398733i	\(0.869450\pi\)
\(31\)	−8.05741	−	6.31258i	−0.259916	−	0.203631i	0.478298	−	0.878198i	\(-0.341254\pi\)
							−0.738214	+	0.674566i	\(0.764331\pi\)
\(37\)	6.14528	−	7.84387i	0.166089	−	0.211996i	−0.696970	−	0.717100i	\(-0.745469\pi\)
							0.863059	+	0.505104i	\(0.168546\pi\)
\(41\)	−47.3369	+	14.7508i	−1.15456	+	0.359775i	−0.814236	−	0.580534i	\(-0.802844\pi\)
							−0.340322	+	0.940309i	\(0.610536\pi\)
\(43\)	−79.9921	−	9.71280i	−1.86028	−	0.225879i	−0.888025	−	0.459796i	\(-0.847923\pi\)
							−0.972257	+	0.233917i	\(0.924846\pi\)
\(47\)	10.6649	−	23.6965i	0.226914	−	0.504182i	−0.762906	−	0.646510i	\(-0.776228\pi\)
							0.989820	+	0.142328i	\(0.0454588\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.3.j.a.5.7	✓	720
169.34	odd	52	inner	169.3.j.a.34.7	yes	720

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.3.j.a.5.7	✓	720	1.1	even	1	trivial
169.3.j.a.34.7	yes	720	169.34	odd	52	inner