Embedded newform 169.2.i.a.9.4

• Label: 169.2.i.a.9.4
• Level: $169$
• Weight: $2$
• Character: 169.9
• 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			9.4
Character	\(\chi\)	\(=\)	169.9
Dual form			169.2.i.a.94.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.431904 + 1.49111i) q^{2} +(2.71969 + 1.15875i) q^{3} +(-0.346475 - 0.219098i) q^{4} +(-2.18511 - 3.16567i) q^{5} +(-2.90247 + 3.55488i) q^{6} +(2.71121 + 0.905134i) q^{7} +(-1.84763 + 1.63686i) q^{8} +(3.97585 + 4.13930i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 26 q^{2} - 26 q^{3} - 12 q^{4} - 26 q^{5} - 26 q^{6} - 26 q^{7} - 26 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{23}{39}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.431904	+	1.49111i	−0.305402	+	1.05437i	0.650213	+	0.759752i	\(0.274680\pi\)
							−0.955615	+	0.294619i	\(0.904807\pi\)
\(3\)	2.71969	+	1.15875i	1.57022	+	0.669006i	0.989425	−	0.145046i	\(-0.0463331\pi\)
							0.580791	+	0.814053i	\(0.302743\pi\)
\(5\)	−2.18511	−	3.16567i	−0.977209	−	1.41573i	−0.908400	−	0.418102i	\(-0.862695\pi\)
							−0.0688094	−	0.997630i	\(-0.521920\pi\)
\(7\)	2.71121	+	0.905134i	1.02474	+	0.342109i	0.778864	−	0.627193i	\(-0.215796\pi\)
							0.245876	+	0.969301i	\(0.420924\pi\)
\(11\)	−2.40279	+	2.50157i	−0.724468	+	0.754252i	−0.976977	−	0.213343i	\(-0.931565\pi\)
							0.252509	+	0.967595i	\(0.418744\pi\)
\(13\)	−0.546128	−	3.56395i	−0.151469	−	0.988462i
							\(17\)	−1.63203	−	0.544851i	−0.395825	−	0.132146i	0.111786	−	0.993732i	\(-0.464343\pi\)
−0.507611	+	0.861587i	\(0.669471\pi\)
\(19\)	−1.32114	−	2.28827i	−0.303089	−	0.524966i	0.673745	−	0.738964i	\(-0.264685\pi\)
							−0.976834	+	0.213998i	\(0.931351\pi\)
\(23\)	3.10855	−	5.38416i	0.648177	−	1.12267i	−0.335381	−	0.942082i	\(-0.608865\pi\)
							0.983558	−	0.180592i	\(-0.0578015\pi\)
\(29\)	−0.178637	+	0.616727i	−0.0331721	+	0.114523i	−0.975396	−	0.220461i	\(-0.929244\pi\)
							0.942224	+	0.334984i	\(0.108731\pi\)
\(31\)	−0.552238	−	1.45613i	−0.0991849	−	0.261529i	0.876235	−	0.481885i	\(-0.160048\pi\)
							−0.975420	+	0.220355i	\(0.929278\pi\)
\(37\)	−0.805408	−	0.130891i	−0.132408	−	0.0215184i	0.0938460	−	0.995587i	\(-0.470084\pi\)
							−0.226254	+	0.974068i	\(0.572648\pi\)
\(41\)	6.07268	+	2.58733i	0.948394	+	0.404073i	0.810082	−	0.586316i	\(-0.199422\pi\)
							0.138311	+	0.990389i	\(0.455833\pi\)
\(43\)	0.297556	−	0.0483575i	0.0453768	−	0.00737445i	−0.137680	−	0.990477i	\(-0.543965\pi\)
							0.183057	+	0.983102i	\(0.441401\pi\)
\(47\)	1.03371	+	0.542534i	0.150782	+	0.0791367i	0.538427	−	0.842672i	\(-0.319019\pi\)
							−0.387644	+	0.921809i	\(0.626711\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.2.i.a.9.4	✓	336
169.94	even	39	inner	169.2.i.a.94.4	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.i.a.9.4	✓	336	1.1	even	1	trivial
169.2.i.a.94.4	yes	336	169.94	even	39	inner