Embedded newform 169.2.k.a.4.14

• Label: 169.2.k.a.4.14
• 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			4.14
Character	\(\chi\)	\(=\)	169.4
Dual form			169.2.k.a.127.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.99581 + 0.0804286i) q^{2} +(-0.751408 + 2.59416i) q^{3} +(1.98329 + 0.160108i) q^{4} +(1.41011 + 0.534785i) q^{5} +(-1.70832 + 5.11703i) q^{6} +(-1.63799 - 3.84451i) q^{7} +(-0.0203394 - 0.00246965i) q^{8} +(-3.62949 - 2.29515i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 360 q - 23 q^{2} - 24 q^{3} - 41 q^{4} - 26 q^{5} - 32 q^{6} - 26 q^{7} - 26 q^{8} - 11 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.99581	+	0.0804286i	1.41125	+	0.0568716i	0.734136	−	0.679003i	\(-0.237588\pi\)
							0.677118	+	0.735874i	\(0.263229\pi\)
\(3\)	−0.751408	+	2.59416i	−0.433826	+	1.49774i	0.384576	+	0.923093i	\(0.374348\pi\)
							−0.818402	+	0.574647i	\(0.805139\pi\)
\(5\)	1.41011	+	0.534785i	0.630621	+	0.239163i	0.649144	−	0.760666i	\(-0.275127\pi\)
							−0.0185232	+	0.999828i	\(0.505896\pi\)
\(7\)	−1.63799	−	3.84451i	−0.619103	−	1.45309i	−0.873905	−	0.486096i	\(-0.838421\pi\)
							0.254802	−	0.966993i	\(-0.417990\pi\)
\(11\)	3.00913	+	4.75856i	0.907288	+	1.43476i	0.900535	+	0.434784i	\(0.143175\pi\)
							0.00675289	+	0.999977i	\(0.497850\pi\)
\(13\)	−0.967139	−	3.47342i	−0.268236	−	0.963353i
							\(17\)	6.45100	−	2.74851i	1.56460	−	0.666613i	0.576035	−	0.817425i	\(-0.304599\pi\)
0.988562	+	0.150812i	\(0.0481889\pi\)
\(19\)	−1.84691	+	1.06631i	−0.423709	+	0.244629i	−0.696663	−	0.717399i	\(-0.745333\pi\)
							0.272954	+	0.962027i	\(0.411999\pi\)
\(23\)	0.789020	−	1.36662i	0.164522	−	0.284960i	−0.771963	−	0.635667i	\(-0.780725\pi\)
							0.936485	+	0.350707i	\(0.114059\pi\)
\(29\)	−0.179760	+	4.46070i	−0.0333806	+	0.828332i	0.892620	+	0.450810i	\(0.148865\pi\)
							−0.926001	+	0.377522i	\(0.876776\pi\)
\(31\)	−2.69431	−	3.04125i	−0.483913	−	0.546225i	0.454949	−	0.890517i	\(-0.349657\pi\)
							−0.938863	+	0.344292i	\(0.888119\pi\)
\(37\)	3.77702	−	0.771085i	0.620939	−	0.126766i	0.120751	−	0.992683i	\(-0.461470\pi\)
							0.500187	+	0.865917i	\(0.333265\pi\)
\(41\)	−4.49274	−	1.30134i	−0.701648	−	0.203235i	−0.0918280	−	0.995775i	\(-0.529271\pi\)
							−0.609820	+	0.792540i	\(0.708758\pi\)
\(43\)	−0.625951	+	3.06611i	−0.0954566	+	0.467577i	0.903761	+	0.428037i	\(0.140795\pi\)
							−0.999218	+	0.0395406i	\(0.987411\pi\)
\(47\)	−6.46667	+	4.46362i	−0.943260	+	0.651085i	−0.937161	−	0.348896i	\(-0.886557\pi\)
							−0.00609861	+	0.999981i	\(0.501941\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.2.k.a.4.14	✓	360
169.127	even	78	inner	169.2.k.a.127.14	yes	360

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.k.a.4.14	✓	360	1.1	even	1	trivial
169.2.k.a.127.14	yes	360	169.127	even	78	inner