Embedded newform 349.2.h.a.17.5

• Label: 349.2.h.a.17.5
• 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			17.5
Character	\(\chi\)	\(=\)	349.17
Dual form			349.2.h.a.308.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.53663 + 1.62220i) q^{2} +(0.865190 + 1.01858i) q^{3} +(-0.162026 - 2.98840i) q^{4} +(1.35343 - 1.02885i) q^{5} +(-2.98183 - 0.161670i) q^{6} +(0.922190 - 1.53269i) q^{7} +(1.69073 + 1.43612i) q^{8} +(0.196394 - 1.19795i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 784 q - 29 q^{2} - 25 q^{3} + 3 q^{4} - 21 q^{5} - 29 q^{6} - 29 q^{7} + 58 q^{8} - 57 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.53663	+	1.62220i	−1.08656	+	1.14707i	−0.0981683	+	0.995170i	\(0.531298\pi\)
							−0.988396	+	0.151902i	\(0.951460\pi\)
\(3\)	0.865190	+	1.01858i	0.499518	+	0.588078i	0.952918	−	0.303230i	\(-0.0980649\pi\)
							−0.453400	+	0.891307i	\(0.649789\pi\)
\(5\)	1.35343	−	1.02885i	0.605272	−	0.460116i	−0.257104	−	0.966384i	\(-0.582768\pi\)
							0.862376	+	0.506268i	\(0.168975\pi\)
\(7\)	0.922190	−	1.53269i	0.348555	−	0.579303i	−0.632138	−	0.774856i	\(-0.717822\pi\)
							0.980693	+	0.195553i	\(0.0626501\pi\)
\(11\)	0.698354	−	3.17265i	0.210562	−	0.956591i	−0.746201	−	0.665721i	\(-0.768124\pi\)
							0.956762	−	0.290870i	\(-0.0939449\pi\)
\(13\)	3.09216	−	0.858533i	0.857610	−	0.238114i	0.189229	−	0.981933i	\(-0.439401\pi\)
							0.668381	+	0.743819i	\(0.266988\pi\)
\(17\)	−2.25455	−	2.65426i	−0.546808	−	0.643752i	0.417456	−	0.908697i	\(-0.362922\pi\)
							−0.964263	+	0.264946i	\(0.914646\pi\)
\(19\)	−0.0612986	−	0.153848i	−0.0140629	−	0.0352951i	0.921778	−	0.387719i	\(-0.126737\pi\)
							−0.935841	+	0.352424i	\(0.885358\pi\)
\(23\)	−1.95425	−	7.03856i	−0.407488	−	1.46764i	−0.828193	−	0.560443i	\(-0.810631\pi\)
							0.420705	−	0.907198i	\(-0.361783\pi\)
\(29\)	1.22827	+	1.16348i	0.228085	+	0.216053i	0.793204	−	0.608956i	\(-0.208411\pi\)
							−0.565119	+	0.825009i	\(0.691170\pi\)
\(31\)	−4.32967	+	8.16662i	−0.777632	+	1.46677i	0.103872	+	0.994591i	\(0.466877\pi\)
							−0.881503	+	0.472178i	\(0.843468\pi\)
\(37\)	−0.0864706	−	0.311439i	−0.0142157	−	0.0512003i	0.956102	−	0.293034i	\(-0.0946649\pi\)
							−0.970318	+	0.241834i	\(0.922251\pi\)
\(41\)	−6.52859	+	7.68604i	−1.01959	+	1.20036i	−0.0400353	+	0.999198i	\(0.512747\pi\)
							−0.979558	+	0.201160i	\(0.935529\pi\)
\(43\)	−1.02625	+	4.66231i	−0.156502	+	0.710995i	0.831369	+	0.555720i	\(0.187557\pi\)
							−0.987871	+	0.155275i	\(0.950374\pi\)
\(47\)	−1.17045	+	3.47376i	−0.170727	+	0.506700i	−0.998523	−	0.0543295i	\(-0.982698\pi\)
							0.827796	+	0.561029i	\(0.189594\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	349.2.h.a.17.5	✓	784
349.308	even	58	inner	349.2.h.a.308.5	yes	784

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
349.2.h.a.17.5	✓	784	1.1	even	1	trivial
349.2.h.a.308.5	yes	784	349.308	even	58	inner