Embedded newform 349.2.e.a.123.16

• Label: 349.2.e.a.123.16
• Level: $349$
• Weight: $2$
• Character: 349.123
• Analytic conductor: $2.787$
• Analytic rank: $0$
• Dimension: $58$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 349 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	349.e (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.78677903054\)
Analytic rank:	\(0\)
Dimension:	\(58\)
Relative dimension:	\(29\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			123.16
Character	\(\chi\)	\(=\)	349.123
Dual form			349.2.e.a.227.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.228167 - 0.131732i) q^{2} +(0.150552 - 0.260763i) q^{3} +(-0.965293 + 1.67194i) q^{4} +(0.807178 - 1.39807i) q^{5} -0.0793300i q^{6} +(1.23534 - 0.713223i) q^{7} +1.03557i q^{8} +(1.45467 + 2.51956i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 58 q - 3 q^{2} + 27 q^{4} + 2 q^{5} - 29 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.228167	−	0.131732i	0.161338	−	0.0931487i	−0.417157	−	0.908834i	\(-0.636973\pi\)
							0.578495	+	0.815686i	\(0.303640\pi\)
\(3\)	0.150552	−	0.260763i	0.0869211	−	0.150552i	−0.819287	−	0.573383i	\(-0.805631\pi\)
							0.906208	+	0.422832i	\(0.138964\pi\)
\(5\)	0.807178	−	1.39807i	0.360981	−	0.625237i	−0.627142	−	0.778905i	\(-0.715775\pi\)
							0.988123	+	0.153668i	\(0.0491087\pi\)
\(7\)	1.23534	−	0.713223i	0.466914	−	0.269573i	−0.248033	−	0.968752i	\(-0.579784\pi\)
							0.714947	+	0.699179i	\(0.246451\pi\)
\(11\)			2.29478i			0.691903i	0.938252	+	0.345951i	\(0.112444\pi\)
							−0.938252	+	0.345951i	\(0.887556\pi\)
\(13\)	0.932614	−	0.538445i	0.258661	−	0.149338i	−0.365063	−	0.930983i	\(-0.618953\pi\)
							0.623723	+	0.781645i	\(0.285619\pi\)
\(17\)	6.95728			1.68739			0.843694	−	0.536824i	\(-0.180376\pi\)
							0.843694	+	0.536824i	\(0.180376\pi\)
\(19\)	0.461520	−	0.799377i	0.105880	−	0.183390i	−0.808217	−	0.588884i	\(-0.799567\pi\)
							0.914097	+	0.405495i	\(0.132901\pi\)
\(23\)	−2.46875	−	4.27600i	−0.514769	−	0.891607i	−0.999853	−	0.0171390i	\(-0.994544\pi\)
							0.485084	−	0.874468i	\(-0.338789\pi\)
\(29\)	−4.87947	+	8.45149i	−0.906095	+	1.56940i	−0.0866541	+	0.996238i	\(0.527618\pi\)
							−0.819441	+	0.573164i	\(0.805716\pi\)
\(31\)	10.1669			1.82603			0.913016	−	0.407924i	\(-0.133747\pi\)
							0.913016	+	0.407924i	\(0.133747\pi\)
\(37\)	−0.988427			−0.162496			−0.0812482	−	0.996694i	\(-0.525891\pi\)
							−0.0812482	+	0.996694i	\(0.525891\pi\)
\(41\)	−9.20028			−1.43684			−0.718421	−	0.695608i	\(-0.755135\pi\)
							−0.718421	+	0.695608i	\(0.755135\pi\)
\(43\)	−3.45983	−	1.99754i	−0.527620	−	0.304621i	0.212427	−	0.977177i	\(-0.431863\pi\)
							−0.740047	+	0.672556i	\(0.765197\pi\)
\(47\)		−	7.91493i		−	1.15451i	−0.816563	−	0.577256i	\(-0.804124\pi\)
							0.816563	−	0.577256i	\(-0.195876\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	349.2.e.a.123.16	✓	58
349.227	even	6	inner	349.2.e.a.227.16	yes	58

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
349.2.e.a.123.16	✓	58	1.1	even	1	trivial
349.2.e.a.227.16	yes	58	349.227	even	6	inner