Embedded newform 349.2.i.a.9.6

• Label: 349.2.i.a.9.6
• 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			9.6
Character	\(\chi\)	\(=\)	349.9
Dual form			349.2.i.a.194.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.86652 - 0.946796i) q^{2} +(2.80322 - 1.05867i) q^{3} +(1.40605 + 1.92064i) q^{4} +(2.94974 - 0.761907i) q^{5} +(-6.23459 - 0.678053i) q^{6} +(-0.347829 - 0.0125657i) q^{7} +(-0.128769 - 0.785453i) q^{8} +(4.48620 - 3.95224i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1568 q - 58 q^{2} - 58 q^{3} - 28 q^{4} - 62 q^{5} - 72 q^{6} - 60 q^{7} + 23 q^{8} - 30 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.86652	−	0.946796i	−1.31983	−	0.669486i	−0.355617	−	0.934632i	\(-0.615729\pi\)
							−0.964208	+	0.265146i	\(0.914580\pi\)
\(3\)	2.80322	−	1.05867i	1.61844	−	0.611223i	0.631943	−	0.775015i	\(-0.282258\pi\)
							0.986495	+	0.163792i	\(0.0523725\pi\)
\(5\)	2.94974	−	0.761907i	1.31916	−	0.340735i	0.476499	−	0.879175i	\(-0.341906\pi\)
							0.842664	+	0.538440i	\(0.180986\pi\)
\(7\)	−0.347829	−	0.0125657i	−0.131467	−	0.00474938i	−0.0297093	−	0.999559i	\(-0.509458\pi\)
							−0.101758	+	0.994809i	\(0.532447\pi\)
\(11\)	−2.88602	−	1.33522i	−0.870169	−	0.402583i	−0.0665895	−	0.997780i	\(-0.521212\pi\)
							−0.803580	+	0.595197i	\(0.797074\pi\)
\(13\)	−0.0159367	−	0.882574i	−0.00442005	−	0.244782i	−0.994528	−	0.104473i	\(-0.966685\pi\)
							0.990108	−	0.140309i	\(-0.0448097\pi\)
\(17\)	−1.15911	+	7.07028i	−0.281126	+	1.71479i	0.348003	+	0.937493i	\(0.386860\pi\)
							−0.629129	+	0.777301i	\(0.716588\pi\)
\(19\)	2.58341	+	0.767778i	0.592675	+	0.176140i	0.567263	−	0.823537i	\(-0.308002\pi\)
							0.0254118	+	0.999677i	\(0.491910\pi\)
\(23\)	−0.0560579	−	3.10448i	−0.0116889	−	0.647330i	−0.951844	−	0.306583i	\(-0.900814\pi\)
							0.940155	−	0.340747i	\(-0.110680\pi\)
\(29\)	−5.79160	+	3.77599i	−1.07547	+	0.701185i	−0.956506	−	0.291714i	\(-0.905775\pi\)
							−0.118968	+	0.992898i	\(0.537958\pi\)
\(31\)	4.81281	+	7.09836i	0.864406	+	1.27490i	0.960146	+	0.279497i	\(0.0901678\pi\)
							−0.0957403	+	0.995406i	\(0.530522\pi\)
\(37\)	−1.35084	+	0.812774i	−0.222077	+	0.133619i	−0.622245	−	0.782822i	\(-0.713779\pi\)
							0.400168	+	0.916442i	\(0.368952\pi\)
\(41\)	−1.95425	−	11.9204i	−0.305202	−	1.86165i	−0.480305	−	0.877102i	\(-0.659474\pi\)
							0.175103	−	0.984550i	\(-0.443974\pi\)
\(43\)	−0.417018	−	4.60683i	−0.0635947	−	0.702535i	−0.963722	−	0.266908i	\(-0.913998\pi\)
							0.900127	−	0.435627i	\(-0.143473\pi\)
\(47\)	1.24399	+	0.945660i	0.181455	+	0.137939i	0.691959	−	0.721937i	\(-0.256748\pi\)
							−0.510504	+	0.859876i	\(0.670541\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	349.2.i.a.9.6	✓	1568
349.194	even	87	inner	349.2.i.a.194.6	yes	1568

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
349.2.i.a.9.6	✓	1568	1.1	even	1	trivial
349.2.i.a.194.6	yes	1568	349.194	even	87	inner