Embedded newform 169.2.g.a.14.6

• Label: 169.2.g.a.14.6
• Level: $169$
• Weight: $2$
• Character: 169.14
• Analytic conductor: $1.349$
• Analytic rank: $0$
• Dimension: $156$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	169.g (of order \(13\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.34947179416\)
Analytic rank:	\(0\)
Dimension:	\(156\)
Relative dimension:	\(13\) over \(\Q(\zeta_{13})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{13}]$

Embedding invariants

Embedding label			14.6
Character	\(\chi\)	\(=\)	169.14
Dual form			169.2.g.a.157.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.479816 + 0.695133i) q^{2} +(-2.71090 + 1.42279i) q^{3} +(0.456223 + 1.20296i) q^{4} +(-1.82534 - 1.61711i) q^{5} +(0.311704 - 2.56712i) q^{6} +(-0.220084 - 0.0542458i) q^{7} +(-2.69533 - 0.664340i) q^{8} +(3.62047 - 5.24515i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 10 q^{2} - 9 q^{3} - 20 q^{4} - 7 q^{5} - q^{6} - 5 q^{7} + 2 q^{8} - 14 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{9}{13}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.479816	+	0.695133i	−0.339281	+	0.491533i	−0.955071	−	0.296377i	\(-0.904222\pi\)
							0.615790	+	0.787910i	\(0.288837\pi\)
\(3\)	−2.71090	+	1.42279i	−1.56514	+	0.821449i	−0.999994	−	0.00354112i	\(-0.998873\pi\)
							−0.565147	+	0.824990i	\(0.691181\pi\)
\(5\)	−1.82534	−	1.61711i	−0.816317	−	0.723194i	0.147701	−	0.989032i	\(-0.452813\pi\)
							−0.964018	+	0.265838i	\(0.914351\pi\)
\(7\)	−0.220084	−	0.0542458i	−0.0831839	−	0.0205030i	0.197504	−	0.980302i	\(-0.436716\pi\)
							−0.280688	+	0.959799i	\(0.590563\pi\)
\(11\)	0.946194	+	1.37080i	0.285288	+	0.413312i	0.939259	−	0.343208i	\(-0.111514\pi\)
							−0.653971	+	0.756520i	\(0.726898\pi\)
\(13\)	1.80673	−	3.12021i	0.501098	−	0.865391i
							\(17\)	−6.94760	−	1.71243i	−1.68504	−	0.415325i	−0.723574	−	0.690247i	\(-0.757502\pi\)
−0.961467	+	0.274922i	\(0.911348\pi\)
\(19\)	−2.98564			−0.684952			−0.342476	−	0.939527i	\(-0.611266\pi\)
							−0.342476	+	0.939527i	\(0.611266\pi\)
\(23\)	−3.28291			−0.684533			−0.342267	−	0.939603i	\(-0.611195\pi\)
							−0.342267	+	0.939603i	\(0.611195\pi\)
\(29\)	−5.74770	+	8.32698i	−1.06732	+	1.54628i	−0.251604	+	0.967830i	\(0.580958\pi\)
							−0.815717	+	0.578451i	\(0.803657\pi\)
\(31\)	−0.249708	+	2.05653i	−0.0448488	+	0.369364i	0.952954	+	0.303114i	\(0.0980264\pi\)
							−0.997803	+	0.0662493i	\(0.978897\pi\)
\(37\)	−0.411599	+	3.38982i	−0.0676664	+	0.557283i	0.918968	+	0.394332i	\(0.129024\pi\)
							−0.986634	+	0.162950i	\(0.947899\pi\)
\(41\)	8.41285	−	4.41540i	1.31387	−	0.689570i	0.345339	−	0.938478i	\(-0.387764\pi\)
							0.968527	+	0.248908i	\(0.0800716\pi\)
\(43\)	0.371090	+	3.05620i	0.0565907	+	0.466067i	0.993265	+	0.115865i	\(0.0369639\pi\)
							−0.936674	+	0.350202i	\(0.886113\pi\)
\(47\)	1.34366	−	3.54295i	0.195993	−	0.516792i	−0.800606	−	0.599191i	\(-0.795489\pi\)
							0.996599	+	0.0823991i	\(0.0262582\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.2.g.a.14.6	✓	156
169.157	even	13	inner	169.2.g.a.157.6	yes	156

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.g.a.14.6	✓	156	1.1	even	1	trivial
169.2.g.a.157.6	yes	156	169.157	even	13	inner