Embedded newform 349.2.h.a.17.20

• Label: 349.2.h.a.17.20
• Level: $349$
• Weight: $2$
• Character: 349.17
• Analytic conductor: $2.787$
• Analytic rank: $0$
• Dimension: $784$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 349 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	349.h (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.78677903054\)
Analytic rank:	\(0\)
Dimension:	\(784\)
Relative dimension:	\(28\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			17.20
Character	\(\chi\)	\(=\)	349.17
Dual form			349.2.h.a.308.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.875876 - 0.924651i) q^{2} +(-0.774622 - 0.911956i) q^{3} +(0.0204571 + 0.377309i) q^{4} +(2.88562 - 2.19359i) q^{5} +(-1.52171 - 0.0825049i) q^{6} +(0.0984930 - 0.163697i) q^{7} +(2.30822 + 1.96062i) q^{8} +(0.253722 - 1.54764i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 784 q - 29 q^{2} - 25 q^{3} + 3 q^{4} - 21 q^{5} - 29 q^{6} - 29 q^{7} + 58 q^{8} - 57 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{43}{58}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.875876	−	0.924651i	0.619338	−	0.653827i	−0.338557	−	0.940946i	\(-0.609939\pi\)
							0.957895	+	0.287119i	\(0.0926975\pi\)
\(3\)	−0.774622	−	0.911956i	−0.447228	−	0.526518i	0.491638	−	0.870800i	\(-0.336398\pi\)
							−0.938866	+	0.344282i	\(0.888122\pi\)
\(5\)	2.88562	−	2.19359i	1.29049	−	0.981004i	0.290781	−	0.956790i	\(-0.406085\pi\)
							0.999707	−	0.0242141i	\(-0.00770835\pi\)
\(7\)	0.0984930	−	0.163697i	0.0372269	−	0.0618715i	−0.837685	−	0.546153i	\(-0.816092\pi\)
							0.874912	+	0.484282i	\(0.160919\pi\)
\(11\)	−1.01854	+	4.62728i	−0.307101	+	1.39518i	0.529939	+	0.848036i	\(0.322215\pi\)
							−0.837041	+	0.547141i	\(0.815716\pi\)
\(13\)	1.17804	−	0.327080i	0.326728	−	0.0907157i	−0.100291	−	0.994958i	\(-0.531977\pi\)
							0.427019	+	0.904242i	\(0.359564\pi\)
\(17\)	−4.76738	−	5.61259i	−1.15626	−	1.36125i	−0.921114	−	0.389294i	\(-0.872719\pi\)
							−0.235146	−	0.971960i	\(-0.575557\pi\)
\(19\)	0.933746	+	2.34352i	0.214216	+	0.537641i	0.996340	−	0.0854816i	\(-0.0272429\pi\)
							−0.782124	+	0.623123i	\(0.785864\pi\)
\(23\)	−1.53047	−	5.51226i	−0.319125	−	1.14939i	−0.932969	−	0.359958i	\(-0.882791\pi\)
							0.613843	−	0.789428i	\(-0.289623\pi\)
\(29\)	2.11870	+	2.00694i	0.393434	+	0.372680i	0.858690	−	0.512496i	\(-0.171279\pi\)
							−0.465256	+	0.885176i	\(0.654038\pi\)
\(31\)	−1.05288	+	1.98594i	−0.189103	+	0.356686i	−0.959876	−	0.280423i	\(-0.909525\pi\)
							0.770774	+	0.637109i	\(0.219870\pi\)
\(37\)	2.03620	+	7.33373i	0.334749	+	1.20566i	0.919057	+	0.394125i	\(0.128952\pi\)
							−0.584308	+	0.811532i	\(0.698634\pi\)
\(41\)	−1.73334	+	2.04065i	−0.270702	+	0.318696i	−0.880669	−	0.473731i	\(-0.842907\pi\)
							0.609967	+	0.792427i	\(0.291183\pi\)
\(43\)	−0.201437	+	0.915140i	−0.0307189	+	0.139558i	−0.989484	−	0.144642i	\(-0.953797\pi\)
							0.958765	+	0.284200i	\(0.0917279\pi\)
\(47\)	1.87958	−	5.57840i	0.274165	−	0.813693i	−0.718793	−	0.695224i	\(-0.755305\pi\)
							0.992958	−	0.118468i	\(-0.0377984\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	349.2.h.a.17.20	✓	784
349.308	even	58	inner	349.2.h.a.308.20	yes	784

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
349.2.h.a.17.20	✓	784	1.1	even	1	trivial
349.2.h.a.308.20	yes	784	349.308	even	58	inner