Embedded newform 169.2.k.a.4.9

• Label: 169.2.k.a.4.9
• Level: $169$
• Weight: $2$
• Character: 169.4
• Analytic conductor: $1.349$
• Analytic rank: $0$
• Dimension: $360$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	169.k (of order \(78\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.34947179416\)
Analytic rank:	\(0\)
Dimension:	\(360\)
Relative dimension:	\(15\) over \(\Q(\zeta_{78})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{78}]$

Embedding invariants

Embedding label			4.9
Character	\(\chi\)	\(=\)	169.4
Dual form			169.2.k.a.127.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.397859 + 0.0160332i) q^{2} +(0.320025 - 1.10485i) q^{3} +(-1.83548 - 0.148175i) q^{4} +(-2.52701 - 0.958369i) q^{5} +(0.145039 - 0.434446i) q^{6} +(-1.61420 - 3.78868i) q^{7} +(-1.51844 - 0.184373i) q^{8} +(1.41728 + 0.896235i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 360 q - 23 q^{2} - 24 q^{3} - 41 q^{4} - 26 q^{5} - 32 q^{6} - 26 q^{7} - 26 q^{8} - 11 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{78}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.397859	+	0.0160332i	0.281329	+	0.0113372i	0.180529	−	0.983570i	\(-0.442219\pi\)
							0.100800	+	0.994907i	\(0.467860\pi\)
\(3\)	0.320025	−	1.10485i	0.184767	−	0.637888i	−0.813635	−	0.581376i	\(-0.802515\pi\)
							0.998402	−	0.0565126i	\(-0.0179981\pi\)
\(5\)	−2.52701	−	0.958369i	−1.13011	−	0.428595i	−0.282530	−	0.959258i	\(-0.591174\pi\)
							−0.847583	+	0.530663i	\(0.821943\pi\)
\(7\)	−1.61420	−	3.78868i	−0.610112	−	1.43199i	−0.882738	−	0.469865i	\(-0.844302\pi\)
							0.272626	−	0.962120i	\(-0.412108\pi\)
\(11\)	−0.145996	−	0.230874i	−0.0440194	−	0.0696111i	0.822463	−	0.568818i	\(-0.192599\pi\)
							−0.866483	+	0.499207i	\(0.833625\pi\)
\(13\)	3.49504	−	0.885841i	0.969349	−	0.245688i
							\(17\)	2.03601	−	0.867463i	0.493805	−	0.210391i	−0.130684	−	0.991424i	\(-0.541717\pi\)
0.624489	+	0.781033i	\(0.285307\pi\)
\(19\)	−1.19452	+	0.689657i	−0.274042	+	0.158218i	−0.630723	−	0.776008i	\(-0.717242\pi\)
							0.356681	+	0.934226i	\(0.383908\pi\)
\(23\)	−2.79485	+	4.84082i	−0.582766	+	1.00938i	0.412384	+	0.911010i	\(0.364696\pi\)
							−0.995150	+	0.0983704i	\(0.968637\pi\)
\(29\)	0.298762	−	7.41370i	0.0554787	−	1.37669i	−0.696525	−	0.717532i	\(-0.745272\pi\)
							0.752004	−	0.659158i	\(-0.229087\pi\)
\(31\)	−6.91875	−	7.80965i	−1.24264	−	1.40265i	−0.882871	−	0.469615i	\(-0.844393\pi\)
							−0.359772	−	0.933040i	\(-0.617146\pi\)
\(37\)	6.52253	−	1.33158i	1.07230	−	0.218911i	0.368727	−	0.929538i	\(-0.379794\pi\)
							0.703569	+	0.710627i	\(0.251588\pi\)
\(41\)	1.55996	+	0.451849i	0.243625	+	0.0705669i	0.397784	−	0.917479i	\(-0.369779\pi\)
							−0.154158	+	0.988046i	\(0.549267\pi\)
\(43\)	−0.598243	+	2.93039i	−0.0912311	+	0.446880i	0.908357	+	0.418195i	\(0.137337\pi\)
							−0.999589	+	0.0286848i	\(0.990868\pi\)
\(47\)	3.21346	−	2.21809i	0.468731	−	0.323542i	−0.310178	−	0.950679i	\(-0.600389\pi\)
							0.778909	+	0.627137i	\(0.215773\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.2.k.a.4.9	✓	360
169.127	even	78	inner	169.2.k.a.127.9	yes	360

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.k.a.4.9	✓	360	1.1	even	1	trivial
169.2.k.a.127.9	yes	360	169.127	even	78	inner