Embedded newform 169.2.i.a.3.12

• Label: 169.2.i.a.3.12
• Level: $169$
• Weight: $2$
• Character: 169.3
• Analytic conductor: $1.349$
• Analytic rank: $0$
• Dimension: $336$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.34947179416\)
Analytic rank:	\(0\)
Dimension:	\(336\)
Relative dimension:	\(14\) over \(\Q(\zeta_{39})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{39}]$

Embedding invariants

Embedding label			3.12
Character	\(\chi\)	\(=\)	169.3
Dual form			169.2.i.a.113.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.43066 - 1.07507i) q^{2} +(-0.532083 + 2.60631i) q^{3} +(0.334579 - 1.15510i) q^{4} +(-0.657900 - 0.345292i) q^{5} +(2.04075 + 4.30078i) q^{6} +(2.44433 + 0.397242i) q^{7} +(0.506036 + 1.33431i) q^{8} +(-3.74982 - 1.59765i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 26 q^{2} - 26 q^{3} - 12 q^{4} - 26 q^{5} - 26 q^{6} - 26 q^{7} - 26 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.43066	−	1.07507i	1.01163	−	0.760192i	0.0406081	−	0.999175i	\(-0.487070\pi\)
							0.971024	+	0.238983i	\(0.0768141\pi\)
\(3\)	−0.532083	+	2.60631i	−0.307198	+	1.50476i	0.474030	+	0.880509i	\(0.342799\pi\)
							−0.781228	+	0.624246i	\(0.785406\pi\)
\(5\)	−0.657900	−	0.345292i	−0.294222	−	0.154419i	0.311154	−	0.950360i	\(-0.399285\pi\)
							−0.605375	+	0.795940i	\(0.706977\pi\)
\(7\)	2.44433	+	0.397242i	0.923868	+	0.150143i	0.603695	−	0.797216i	\(-0.293695\pi\)
							0.320174	+	0.947359i	\(0.396259\pi\)
\(11\)	0.986578	−	0.420341i	0.297464	−	0.126738i	−0.238078	−	0.971246i	\(-0.576517\pi\)
							0.535542	+	0.844508i	\(0.320107\pi\)
\(13\)	−2.64177	−	2.45378i	−0.732695	−	0.680557i
							\(17\)	−0.931036	−	0.151308i	−0.225810	−	0.0366976i	0.0464536	−	0.998920i	\(-0.485208\pi\)
−0.272263	+	0.962223i	\(0.587772\pi\)
\(19\)	2.05942	−	3.56703i	0.472464	−	0.818332i	−0.527039	−	0.849841i	\(-0.676698\pi\)
							0.999503	+	0.0315087i	\(0.0100312\pi\)
\(23\)	−3.92310	−	6.79501i	−0.818023	−	1.41686i	−0.907137	−	0.420836i	\(-0.861737\pi\)
							0.0891139	−	0.996021i	\(-0.471596\pi\)
\(29\)	−1.29617	+	0.974010i	−0.240693	+	0.180869i	−0.714243	−	0.699898i	\(-0.753229\pi\)
							0.473550	+	0.880767i	\(0.342972\pi\)
\(31\)	−1.92189	+	2.78434i	−0.345182	+	0.500082i	−0.956671	−	0.291170i	\(-0.905955\pi\)
							0.611489	+	0.791253i	\(0.290571\pi\)
\(37\)	5.74077	+	0.463443i	0.943777	+	0.0761896i	0.542766	−	0.839884i	\(-0.317377\pi\)
							0.401011	+	0.916073i	\(0.368659\pi\)
\(41\)	−0.352113	+	1.72476i	−0.0549907	+	0.269363i	−0.997985	−	0.0634457i	\(-0.979791\pi\)
							0.942995	+	0.332808i	\(0.107996\pi\)
\(43\)	9.84148	−	0.794487i	1.50081	−	0.121158i	0.697443	−	0.716640i	\(-0.254321\pi\)
							0.803369	+	0.595482i	\(0.203039\pi\)
\(47\)	−0.593493	−	0.146283i	−0.0865698	−	0.0213375i	0.195793	−	0.980645i	\(-0.437272\pi\)
							−0.282363	+	0.959308i	\(0.591118\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.2.i.a.3.12	✓	336
169.113	even	39	inner	169.2.i.a.113.12	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.i.a.3.12	✓	336	1.1	even	1	trivial
169.2.i.a.113.12	yes	336	169.113	even	39	inner