Embedded newform 169.4.i.a.3.10

• Label: 169.4.i.a.3.10
• Level: $169$
• Weight: $4$
• Character: 169.3
• Analytic conductor: $9.971$
• Analytic rank: $0$
• Dimension: $1080$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	169.i (of order \(39\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.97132279097\)
Analytic rank:	\(0\)
Dimension:	\(1080\)
Relative dimension:	\(45\) over \(\Q(\zeta_{39})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{39}]$

Embedding invariants

Embedding label			3.10
Character	\(\chi\)	\(=\)	169.3
Dual form			169.4.i.a.113.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.11755 + 2.34268i) q^{2} +(0.204779 - 1.00307i) q^{3} +(2.00519 - 6.92271i) q^{4} +(4.77311 + 2.50512i) q^{5} +(1.71148 + 3.60686i) q^{6} +(-12.1144 - 1.96879i) q^{7} +(-1.09622 - 2.89048i) q^{8} +(23.8752 + 10.1723i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1080 q - 27 q^{2} - 19 q^{3} + 155 q^{4} - 30 q^{5} - 56 q^{6} + 9 q^{7} + 4 q^{8} + 364 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.11755	+	2.34268i	−1.10222	+	0.828264i	−0.986295	−	0.164994i	\(-0.947240\pi\)
							−0.115925	+	0.993258i	\(0.536983\pi\)
\(3\)	0.204779	−	1.00307i	0.0394097	−	0.193042i	−0.955320	−	0.295574i	\(-0.904489\pi\)
							0.994730	+	0.102532i	\(0.0326944\pi\)
\(5\)	4.77311	+	2.50512i	0.426920	+	0.224065i	0.664477	−	0.747309i	\(-0.268654\pi\)
							−0.237557	+	0.971374i	\(0.576347\pi\)
\(7\)	−12.1144	−	1.96879i	−0.654117	−	0.106304i	−0.175709	−	0.984442i	\(-0.556222\pi\)
							−0.478408	+	0.878138i	\(0.658786\pi\)
\(11\)	−7.66899	+	3.26745i	−0.210208	+	0.0895613i	−0.494504	−	0.869175i	\(-0.664650\pi\)
							0.284296	+	0.958737i	\(0.408240\pi\)
\(13\)	−46.7902	−	2.77067i	−0.998251	−	0.0591113i
							\(17\)	−109.333	−	17.7683i	−1.55983	−	0.253497i	−0.681596	−	0.731729i	\(-0.738714\pi\)
−0.878234	+	0.478232i	\(0.841278\pi\)
\(19\)	13.4151	−	23.2356i	0.161981	−	0.280559i	−0.773598	−	0.633676i	\(-0.781545\pi\)
							0.935579	+	0.353118i	\(0.114878\pi\)
\(23\)	−30.8226	−	53.3862i	−0.279432	−	0.483991i	0.691811	−	0.722078i	\(-0.256813\pi\)
							−0.971244	+	0.238087i	\(0.923480\pi\)
\(29\)	186.056	−	139.812i	1.19137	−	0.895256i	0.195034	−	0.980796i	\(-0.437518\pi\)
							0.996335	+	0.0855408i	\(0.0272618\pi\)
\(31\)	28.9367	−	41.9220i	0.167651	−	0.242885i	−0.730149	−	0.683288i	\(-0.760549\pi\)
							0.897800	+	0.440404i	\(0.145165\pi\)
\(37\)	−20.3743	−	1.64478i	−0.0905273	−	0.00730812i	0.0351199	−	0.999383i	\(-0.488819\pi\)
							−0.125647	+	0.992075i	\(0.540101\pi\)
\(41\)	95.6882	−	468.712i	0.364488	−	1.78538i	−0.221608	−	0.975136i	\(-0.571130\pi\)
							0.586095	−	0.810242i	\(-0.300664\pi\)
\(43\)	480.435	−	38.7847i	1.70385	−	0.137549i	0.810153	−	0.586218i	\(-0.199384\pi\)
							0.893698	+	0.448669i	\(0.148102\pi\)
\(47\)	−431.842	−	106.440i	−1.34023	−	0.330336i	−0.496933	−	0.867789i	\(-0.665541\pi\)
							−0.843294	+	0.537452i	\(0.819387\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.4.i.a.3.10	✓	1080
169.113	even	39	inner	169.4.i.a.113.10	yes	1080

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.4.i.a.3.10	✓	1080	1.1	even	1	trivial
169.4.i.a.113.10	yes	1080	169.113	even	39	inner