Embedded newform 349.2.i.a.9.7

• Label: 349.2.i.a.9.7
• Level: $349$
• Weight: $2$
• Character: 349.9
• Analytic conductor: $2.787$
• Analytic rank: $0$
• Dimension: $1568$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 349 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	349.i (of order \(87\), degree \(56\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.78677903054\)
Analytic rank:	\(0\)
Dimension:	\(1568\)
Relative dimension:	\(28\) over \(\Q(\zeta_{87})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{87}]$

Embedding invariants

Embedding label			9.7
Character	\(\chi\)	\(=\)	349.9
Dual form			349.2.i.a.194.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.76537 - 0.895491i) q^{2} +(-1.48767 + 0.561835i) q^{3} +(1.13323 + 1.54797i) q^{4} +(-3.26214 + 0.842600i) q^{5} +(3.12940 + 0.340343i) q^{6} +(1.24531 + 0.0449880i) q^{7} +(0.0261182 + 0.159314i) q^{8} +(-0.353559 + 0.311477i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1568 q - 58 q^{2} - 58 q^{3} - 28 q^{4} - 62 q^{5} - 72 q^{6} - 60 q^{7} + 23 q^{8} - 30 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/349\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{13}{87}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.76537	−	0.895491i	−1.24831	−	0.633208i	−0.301027	−	0.953616i	\(-0.597329\pi\)
							−0.947280	+	0.320408i	\(0.896180\pi\)
\(3\)	−1.48767	+	0.561835i	−0.858904	+	0.324376i	−0.743331	−	0.668923i	\(-0.766755\pi\)
							−0.115573	+	0.993299i	\(0.536870\pi\)
\(5\)	−3.26214	+	0.842600i	−1.45887	+	0.376822i	−0.893892	−	0.448282i	\(-0.852036\pi\)
							−0.564981	+	0.825104i	\(0.691117\pi\)
\(7\)	1.24531	+	0.0449880i	0.470683	+	0.0170039i	0.270429	−	0.962740i	\(-0.412835\pi\)
							0.200254	+	0.979744i	\(0.435823\pi\)
\(11\)	−1.45923	−	0.675113i	−0.439975	−	0.203554i	0.187380	−	0.982288i	\(-0.440000\pi\)
							−0.627355	+	0.778733i	\(0.715863\pi\)
\(13\)	0.0840742	+	4.65602i	0.0233180	+	1.29135i	0.775059	+	0.631889i	\(0.217720\pi\)
							−0.751741	+	0.659459i	\(0.770785\pi\)
\(17\)	0.393328	−	2.39919i	0.0953960	−	0.581890i	−0.895264	−	0.445536i	\(-0.853013\pi\)
							0.990660	−	0.136354i	\(-0.0435385\pi\)
\(19\)	−1.58829	−	0.472034i	−0.364379	−	0.108292i	0.0974982	−	0.995236i	\(-0.468916\pi\)
							−0.461877	+	0.886944i	\(0.652824\pi\)
\(23\)	−0.0913699	−	5.06005i	−0.0190519	−	1.05509i	−0.858860	−	0.512211i	\(-0.828827\pi\)
							0.839808	−	0.542884i	\(-0.182668\pi\)
\(29\)	−0.781891	+	0.509775i	−0.145193	+	0.0946629i	−0.616694	−	0.787203i	\(-0.711529\pi\)
							0.471501	+	0.881866i	\(0.343712\pi\)
\(31\)	−0.991728	−	1.46269i	−0.178120	−	0.262707i	0.728140	−	0.685428i	\(-0.240385\pi\)
							−0.906259	+	0.422722i	\(0.861075\pi\)
\(37\)	6.64800	−	3.99997i	1.09292	−	0.657591i	0.149361	−	0.988783i	\(-0.452278\pi\)
							0.943563	+	0.331192i	\(0.107451\pi\)
\(41\)	−1.20631	−	7.35819i	−0.188395	−	1.14916i	−0.896498	−	0.443048i	\(-0.853897\pi\)
							0.708103	−	0.706109i	\(-0.249551\pi\)
\(43\)	0.201611	+	2.22721i	0.0307454	+	0.339646i	0.996715	+	0.0809904i	\(0.0258083\pi\)
							−0.965970	+	0.258656i	\(0.916720\pi\)
\(47\)	−0.714761	−	0.543347i	−0.104259	−	0.0792553i	0.551746	−	0.834012i	\(-0.313962\pi\)
							−0.656005	+	0.754757i	\(0.727755\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	349.2.i.a.9.7	✓	1568
349.194	even	87	inner	349.2.i.a.194.7	yes	1568

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
349.2.i.a.9.7	✓	1568	1.1	even	1	trivial
349.2.i.a.194.7	yes	1568	349.194	even	87	inner