Embedded newform 289.3.e.p.40.2

• Label: 289.3.e.p.40.2
• Level: $289$
• Weight: $3$
• Character: 289.40
• Analytic conductor: $7.875$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $16$

Newspace parameters

Level:	\( N \)	\(=\)	\( 289 = 17^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	289.e (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.87467964001\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(4\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			40.2
Character	\(\chi\)	\(=\)	289.40
Dual form			289.3.e.p.224.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.65493 - 0.685496i) q^{2} +(0.448072 - 0.0891271i) q^{3} +(-0.559525 - 0.559525i) q^{4} +(-0.708463 + 1.06029i) q^{5} +(-0.802626 - 0.159652i) q^{6} +(2.73896 + 4.09915i) q^{7} +(3.28441 + 7.92927i) q^{8} +(-8.12209 + 3.36428i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/289\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{15}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.65493	−	0.685496i	−0.827467	−	0.342748i	−0.0715676	−	0.997436i	\(-0.522800\pi\)
							−0.755900	+	0.654688i	\(0.772800\pi\)
\(3\)	0.448072	−	0.0891271i	0.149357	−	0.0297090i	−0.119845	−	0.992793i	\(-0.538240\pi\)
							0.269203	+	0.963084i	\(0.413240\pi\)
\(5\)	−0.708463	+	1.06029i	−0.141693	+	0.212058i	−0.895528	−	0.445006i	\(-0.853202\pi\)
							0.753835	+	0.657064i	\(0.228202\pi\)
\(7\)	2.73896	+	4.09915i	0.391280	+	0.585592i	0.973851	−	0.227188i	\(-0.0729531\pi\)
							−0.582571	+	0.812780i	\(0.697953\pi\)
\(11\)	3.25662	−	16.3721i	0.296056	−	1.48837i	−0.490819	−	0.871261i	\(-0.663302\pi\)
							0.786875	−	0.617112i	\(-0.211698\pi\)
\(13\)	12.6971	−	12.6971i	0.976702	−	0.976702i	−0.0230332	−	0.999735i	\(-0.507332\pi\)
							0.999735	+	0.0230332i	\(0.00733233\pi\)
\(17\)		0			0
							\(19\)	−17.3609	−	7.19111i	−0.913731	−	0.378480i	−0.124247	−	0.992251i	\(-0.539652\pi\)
−0.789484	+	0.613772i	\(0.789652\pi\)
\(23\)	−19.1345	−	3.80610i	−0.831937	−	0.165482i	−0.239293	−	0.970947i	\(-0.576916\pi\)
							−0.592643	+	0.805465i	\(0.701916\pi\)
\(29\)	−19.3396	−	12.9223i	−0.666883	−	0.445597i	0.175497	−	0.984480i	\(-0.443847\pi\)
							−0.842381	+	0.538883i	\(0.818847\pi\)
\(31\)	−9.55439	−	48.0331i	−0.308206	−	1.54946i	−0.755548	−	0.655093i	\(-0.772629\pi\)
							0.447342	−	0.894363i	\(-0.352371\pi\)
\(37\)	−30.7649	+	6.11952i	−0.831484	+	0.165392i	−0.592438	−	0.805616i	\(-0.701835\pi\)
							−0.239046	+	0.971008i	\(0.576835\pi\)
\(41\)	−36.5886	−	54.7587i	−0.892405	−	1.33558i	−0.941581	−	0.336788i	\(-0.890659\pi\)
							0.0491757	−	0.998790i	\(-0.484341\pi\)
\(43\)	17.7784	−	7.36405i	0.413451	−	0.171257i	−0.166255	−	0.986083i	\(-0.553167\pi\)
							0.579706	+	0.814826i	\(0.303167\pi\)
\(47\)	−6.89268	+	6.89268i	−0.146653	+	0.146653i	−0.776621	−	0.629968i	\(-0.783068\pi\)
							0.629968	+	0.776621i	\(0.283068\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	289.3.e.p.40.2	yes	32
17.2	even	8	inner	289.3.e.p.75.4	yes	32
17.3	odd	16	inner	289.3.e.p.224.2	yes	32
17.4	even	4	inner	289.3.e.p.249.3	yes	32
17.5	odd	16	inner	289.3.e.p.65.3	yes	32
17.6	odd	16	inner	289.3.e.p.131.2	yes	32
17.7	odd	16	inner	289.3.e.p.158.3	yes	32
17.8	even	8	inner	289.3.e.p.214.1	yes	32
17.9	even	8	inner	289.3.e.p.214.2	yes	32
17.10	odd	16	inner	289.3.e.p.158.4	yes	32
17.11	odd	16	inner	289.3.e.p.131.1	yes	32
17.12	odd	16	inner	289.3.e.p.65.4	yes	32
17.13	even	4	inner	289.3.e.p.249.4	yes	32
17.14	odd	16	inner	289.3.e.p.224.1	yes	32
17.15	even	8	inner	289.3.e.p.75.3	yes	32
17.16	even	2	inner	289.3.e.p.40.1	✓	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
289.3.e.p.40.1	✓	32	17.16	even	2	inner
289.3.e.p.40.2	yes	32	1.1	even	1	trivial
289.3.e.p.65.3	yes	32	17.5	odd	16	inner
289.3.e.p.65.4	yes	32	17.12	odd	16	inner
289.3.e.p.75.3	yes	32	17.15	even	8	inner
289.3.e.p.75.4	yes	32	17.2	even	8	inner
289.3.e.p.131.1	yes	32	17.11	odd	16	inner
289.3.e.p.131.2	yes	32	17.6	odd	16	inner
289.3.e.p.158.3	yes	32	17.7	odd	16	inner
289.3.e.p.158.4	yes	32	17.10	odd	16	inner
289.3.e.p.214.1	yes	32	17.8	even	8	inner
289.3.e.p.214.2	yes	32	17.9	even	8	inner
289.3.e.p.224.1	yes	32	17.14	odd	16	inner
289.3.e.p.224.2	yes	32	17.3	odd	16	inner
289.3.e.p.249.3	yes	32	17.4	even	4	inner
289.3.e.p.249.4	yes	32	17.13	even	4	inner