Embedded newform 731.2.p.a.594.16

• Label: 731.2.p.a.594.16
• Level: $731$
• Weight: $2$
• Character: 731.594
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $384$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.p (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(384\)
Relative dimension:	\(64\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			594.16
Character	\(\chi\)	\(=\)	731.594
Dual form			731.2.p.a.16.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.359164 + 1.57360i) q^{2} +(1.14363 - 0.261026i) q^{3} +(-0.545279 - 0.262592i) q^{4} +(0.961694 + 0.766926i) q^{5} +1.89337i q^{6} +0.0287002i q^{7} +(-1.40365 + 1.76012i) q^{8} +(-1.46315 + 0.704615i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q - 12 q^{2} - 72 q^{4} - 8 q^{8} + 62 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/731\mathbb{Z}\right)^\times\).

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.359164	+	1.57360i	−0.253967	+	1.11270i	0.673615	+	0.739083i	\(0.264741\pi\)
							−0.927582	+	0.373620i	\(0.878116\pi\)
\(3\)	1.14363	−	0.261026i	0.660276	−	0.150704i	0.120764	−	0.992681i	\(-0.461466\pi\)
							0.539513	+	0.841978i	\(0.318609\pi\)
\(5\)	0.961694	+	0.766926i	0.430083	+	0.342980i	0.814483	−	0.580187i	\(-0.197020\pi\)
							−0.384401	+	0.923166i	\(0.625592\pi\)
\(7\)			0.0287002i			0.0108476i	0.999985	+	0.00542382i	\(0.00172646\pi\)
							−0.999985	+	0.00542382i	\(0.998274\pi\)
\(11\)	−0.135381	−	0.281121i	−0.0408188	−	0.0847611i	0.879562	−	0.475784i	\(-0.157836\pi\)
							−0.920381	+	0.391023i	\(0.872121\pi\)
\(13\)	−2.17511	+	2.72751i	−0.603268	+	0.756474i	−0.985883	−	0.167434i	\(-0.946452\pi\)
							0.382615	+	0.923908i	\(0.375023\pi\)
\(17\)	−2.03399	+	3.58649i	−0.493314	+	0.869851i
							\(19\)	0.616817	+	0.297043i	0.141508	+	0.0681464i	0.503298	−	0.864113i	\(-0.332120\pi\)
−0.361790	+	0.932260i	\(0.617834\pi\)
\(23\)	1.43200	+	2.97358i	0.298593	+	0.620034i	0.995248	−	0.0973721i	\(-0.0310437\pi\)
							−0.696656	+	0.717406i	\(0.745329\pi\)
\(29\)	6.25439	+	1.42752i	1.16141	+	0.265085i	0.759446	−	0.650570i	\(-0.225470\pi\)
							0.401965	+	0.915655i	\(0.368327\pi\)
\(31\)	3.42590	+	0.781939i	0.615310	+	0.140440i	0.518806	−	0.854892i	\(-0.326377\pi\)
							0.0965040	+	0.995333i	\(0.469234\pi\)
\(37\)			6.89643i			1.13377i	0.823798	+	0.566883i	\(0.191851\pi\)
							−0.823798	+	0.566883i	\(0.808149\pi\)
\(41\)	8.44800	+	1.92820i	1.31936	+	0.301134i	0.823522	−	0.567284i	\(-0.192006\pi\)
							0.495833	+	0.868418i	\(0.334863\pi\)
\(43\)	4.54781	−	4.72413i	0.693534	−	0.720424i
							\(47\)	−6.97044	−	3.35679i	−1.01674	−	0.489638i	−0.150156	−	0.988662i	\(-0.547978\pi\)
−0.866588	+	0.499024i	\(0.833692\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.p.a.594.16	yes	384
17.16	even	2	inner	731.2.p.a.594.15	yes	384
43.16	even	7	inner	731.2.p.a.16.15	✓	384
731.16	even	14	inner	731.2.p.a.16.16	yes	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.p.a.16.15	✓	384	43.16	even	7	inner
731.2.p.a.16.16	yes	384	731.16	even	14	inner
731.2.p.a.594.15	yes	384	17.16	even	2	inner
731.2.p.a.594.16	yes	384	1.1	even	1	trivial