Embedded newform 349.2.i.a.9.1

• Label: 349.2.i.a.9.1
• 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			9.1
Character	\(\chi\)	\(=\)	349.9
Dual form			349.2.i.a.194.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.46125 - 1.24848i) q^{2} +(0.387351 - 0.146288i) q^{3} +(3.31766 + 4.53187i) q^{4} +(1.25689 - 0.324652i) q^{5} +(-1.13601 - 0.123548i) q^{6} +(4.59338 + 0.165940i) q^{7} +(-1.61470 - 9.84922i) q^{8} +(-2.12241 + 1.86979i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1568 q - 58 q^{2} - 58 q^{3} - 28 q^{4} - 62 q^{5} - 72 q^{6} - 60 q^{7} + 23 q^{8} - 30 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.46125	−	1.24848i	−1.74037	−	0.882808i	−0.969236	−	0.246133i	\(-0.920840\pi\)
							−0.771133	−	0.636674i	\(-0.780310\pi\)
\(3\)	0.387351	−	0.146288i	0.223637	−	0.0844594i	−0.238954	−	0.971031i	\(-0.576805\pi\)
							0.462592	+	0.886571i	\(0.346920\pi\)
\(5\)	1.25689	−	0.324652i	0.562100	−	0.145189i	0.0417294	−	0.999129i	\(-0.486713\pi\)
							0.520371	+	0.853940i	\(0.325794\pi\)
\(7\)	4.59338	+	0.165940i	1.73614	+	0.0627196i	0.885955	−	0.463772i	\(-0.153504\pi\)
							0.850180	+	0.526491i	\(0.176493\pi\)
\(11\)	0.522225	+	0.241607i	0.157457	+	0.0728473i	0.497035	−	0.867731i	\(-0.334422\pi\)
							−0.339578	+	0.940578i	\(0.610284\pi\)
\(13\)	0.0766747	+	4.24624i	0.0212657	+	1.17769i	0.818676	+	0.574256i	\(0.194709\pi\)
							−0.797410	+	0.603438i	\(0.793797\pi\)
\(17\)	−0.537285	+	3.27729i	−0.130311	+	0.794861i	0.838103	+	0.545512i	\(0.183665\pi\)
							−0.968414	+	0.249349i	\(0.919783\pi\)
\(19\)	3.35680	+	0.997625i	0.770102	+	0.228871i	0.645937	−	0.763391i	\(-0.276467\pi\)
							0.124165	+	0.992262i	\(0.460375\pi\)
\(23\)	0.0383322	+	2.12283i	0.00799282	+	0.442641i	0.979030	+	0.203714i	\(0.0653014\pi\)
							−0.971038	+	0.238927i	\(0.923204\pi\)
\(29\)	−2.09713	+	1.36728i	−0.389427	+	0.253897i	−0.725907	−	0.687793i	\(-0.758580\pi\)
							0.336480	+	0.941691i	\(0.390763\pi\)
\(31\)	−5.76511	−	8.50290i	−1.03544	−	1.52717i	−0.837112	−	0.547031i	\(-0.815758\pi\)
							−0.198332	−	0.980135i	\(-0.563552\pi\)
\(37\)	6.84081	−	4.11598i	1.12462	−	0.676663i	0.173302	−	0.984869i	\(-0.444556\pi\)
							0.951320	+	0.308206i	\(0.0997286\pi\)
\(41\)	−0.442548	−	2.69942i	−0.0691144	−	0.421579i	−0.998608	−	0.0527415i	\(-0.983204\pi\)
							0.929494	−	0.368838i	\(-0.120244\pi\)
\(43\)	−0.292374	−	3.22987i	−0.0445866	−	0.492551i	−0.987291	−	0.158922i	\(-0.949198\pi\)
							0.942705	−	0.333629i	\(-0.108273\pi\)
\(47\)	9.00574	+	6.84598i	1.31362	+	0.998589i	0.998777	+	0.0494461i	\(0.0157456\pi\)
							0.314845	+	0.949143i	\(0.398048\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	349.2.i.a.9.1	✓	1568
349.194	even	87	inner	349.2.i.a.194.1	yes	1568

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
349.2.i.a.9.1	✓	1568	1.1	even	1	trivial
349.2.i.a.194.1	yes	1568	349.194	even	87	inner