Embedded newform 349.2.e.a.123.18

• Label: 349.2.e.a.123.18
• 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			123.18
Character	\(\chi\)	\(=\)	349.123
Dual form			349.2.e.a.227.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.531309 - 0.306751i) q^{2} +(0.869046 - 1.50523i) q^{3} +(-0.811807 + 1.40609i) q^{4} +(-1.66390 + 2.88196i) q^{5} -1.06632i q^{6} +(-1.62203 + 0.936479i) q^{7} +2.22310i q^{8} +(-0.0104823 - 0.0181558i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 58 q - 3 q^{2} + 27 q^{4} + 2 q^{5} - 29 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.531309	−	0.306751i	0.375692	−	0.216906i	−0.300250	−	0.953860i	\(-0.597070\pi\)
							0.675942	+	0.736955i	\(0.263737\pi\)
\(3\)	0.869046	−	1.50523i	0.501744	−	0.869046i	−0.498254	−	0.867031i	\(-0.666025\pi\)
							0.999998	−	0.00201497i	\(-0.000641386\pi\)
\(5\)	−1.66390	+	2.88196i	−0.744120	+	1.28885i	0.206485	+	0.978450i	\(0.433798\pi\)
							−0.950605	+	0.310404i	\(0.899536\pi\)
\(7\)	−1.62203	+	0.936479i	−0.613070	+	0.353956i	−0.774166	−	0.632983i	\(-0.781830\pi\)
							0.161096	+	0.986939i	\(0.448497\pi\)
\(11\)			1.41258i			0.425908i	0.977062	+	0.212954i	\(0.0683085\pi\)
							−0.977062	+	0.212954i	\(0.931692\pi\)
\(13\)	−0.199242	+	0.115033i	−0.0552599	+	0.0319043i	−0.527375	−	0.849632i	\(-0.676824\pi\)
							0.472116	+	0.881537i	\(0.343491\pi\)
\(17\)	2.97207			0.720833			0.360417	−	0.932791i	\(-0.382635\pi\)
							0.360417	+	0.932791i	\(0.382635\pi\)
\(19\)	−0.723681	+	1.25345i	−0.166024	+	0.287562i	−0.937018	−	0.349280i	\(-0.886426\pi\)
							0.770995	+	0.636842i	\(0.219760\pi\)
\(23\)	−0.563129	−	0.975369i	−0.117421	−	0.203378i	0.801324	−	0.598230i	\(-0.204129\pi\)
							−0.918745	+	0.394852i	\(0.870796\pi\)
\(29\)	0.522000	−	0.904130i	0.0969329	−	0.167893i	−0.813481	−	0.581592i	\(-0.802430\pi\)
							0.910414	+	0.413699i	\(0.135763\pi\)
\(31\)	−5.04008			−0.905225			−0.452612	−	0.891707i	\(-0.649508\pi\)
							−0.452612	+	0.891707i	\(0.649508\pi\)
\(37\)	8.62141			1.41735			0.708676	−	0.705534i	\(-0.249293\pi\)
							0.708676	+	0.705534i	\(0.249293\pi\)
\(41\)	9.91019			1.54771			0.773856	−	0.633362i	\(-0.218326\pi\)
							0.773856	+	0.633362i	\(0.218326\pi\)
\(43\)	−0.0676083	−	0.0390337i	−0.0103102	−	0.00595258i	0.494836	−	0.868986i	\(-0.335228\pi\)
							−0.505146	+	0.863034i	\(0.668561\pi\)
\(47\)		−	1.68060i		−	0.245140i	−0.992460	−	0.122570i	\(-0.960886\pi\)
							0.992460	−	0.122570i	\(-0.0391137\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	349.2.e.a.123.18	✓	58
349.227	even	6	inner	349.2.e.a.227.18	yes	58

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
349.2.e.a.123.18	✓	58	1.1	even	1	trivial
349.2.e.a.227.18	yes	58	349.227	even	6	inner