Embedded newform 169.4.i.a.3.9

• Label: 169.4.i.a.3.9
• 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			3.9
Character	\(\chi\)	\(=\)	169.3
Dual form			169.4.i.a.113.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.14814 + 2.36567i) q^{2} +(1.68249 - 8.24137i) q^{3} +(2.08864 - 7.21081i) q^{4} +(-7.14936 - 3.75227i) q^{5} +(14.1997 + 29.9252i) q^{6} +(-10.7212 - 1.74236i) q^{7} +(-0.688137 - 1.81447i) q^{8} +(-40.2500 - 17.1489i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1080 q - 27 q^{2} - 19 q^{3} + 155 q^{4} - 30 q^{5} - 56 q^{6} + 9 q^{7} + 4 q^{8} + 364 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.14814	+	2.36567i	−1.11304	+	0.836392i	−0.987799	−	0.155735i	\(-0.950226\pi\)
							−0.125237	+	0.992127i	\(0.539969\pi\)
\(3\)	1.68249	−	8.24137i	0.323795	−	1.58605i	−0.413514	−	0.910498i	\(-0.635699\pi\)
							0.737309	−	0.675555i	\(-0.236096\pi\)
\(5\)	−7.14936	−	3.75227i	−0.639458	−	0.335614i	0.113640	−	0.993522i	\(-0.463749\pi\)
							−0.753098	+	0.657908i	\(0.771441\pi\)
\(7\)	−10.7212	−	1.74236i	−0.578888	−	0.0940785i	−0.136085	−	0.990697i	\(-0.543452\pi\)
							−0.442803	+	0.896619i	\(0.646016\pi\)
\(11\)	4.51432	−	1.92337i	0.123738	−	0.0527199i	−0.329210	−	0.944257i	\(-0.606782\pi\)
							0.452948	+	0.891537i	\(0.350372\pi\)
\(13\)	−10.9456	−	45.5762i	−0.233521	−	0.972352i
							\(17\)	30.8591	+	5.01509i	0.440260	+	0.0715492i	0.376501	−	0.926416i	\(-0.377127\pi\)
0.0637589	+	0.997965i	\(0.479691\pi\)
\(19\)	−67.6991	+	117.258i	−0.817434	+	1.41584i	0.0901332	+	0.995930i	\(0.471271\pi\)
							−0.907567	+	0.419907i	\(0.862063\pi\)
\(23\)	87.5485	+	151.638i	0.793701	+	1.37473i	0.923661	+	0.383211i	\(0.125182\pi\)
							−0.129960	+	0.991519i	\(0.541485\pi\)
\(29\)	32.6745	−	24.5533i	0.209224	−	0.157222i	−0.490965	−	0.871180i	\(-0.663356\pi\)
							0.700189	+	0.713958i	\(0.253099\pi\)
\(31\)	−25.5621	+	37.0331i	−0.148100	+	0.214560i	−0.890012	−	0.455938i	\(-0.849304\pi\)
							0.741912	+	0.670497i	\(0.233919\pi\)
\(37\)	−253.552	−	20.4688i	−1.12658	−	0.0909473i	−0.496907	−	0.867804i	\(-0.665531\pi\)
							−0.629677	+	0.776857i	\(0.716813\pi\)
\(41\)	−59.8442	+	293.136i	−0.227953	+	1.11659i	0.691192	+	0.722671i	\(0.257086\pi\)
							−0.919145	+	0.393919i	\(0.871119\pi\)
\(43\)	110.368	−	8.90985i	0.391419	−	0.0315986i	0.116809	−	0.993154i	\(-0.462733\pi\)
							0.274610	+	0.961556i	\(0.411451\pi\)
\(47\)	−54.7645	−	13.4982i	−0.169962	−	0.0418919i	0.153416	−	0.988162i	\(-0.450972\pi\)
							−0.323378	+	0.946270i	\(0.604819\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.4.i.a.3.9	✓	1080
169.113	even	39	inner	169.4.i.a.113.9	yes	1080

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.4.i.a.3.9	✓	1080	1.1	even	1	trivial
169.4.i.a.113.9	yes	1080	169.113	even	39	inner