Embedded newform 349.2.g.a.31.2

• Label: 349.2.g.a.31.2
• Level: $349$
• Weight: $2$
• Character: 349.31
• Analytic conductor: $2.787$
• Analytic rank: $0$
• Dimension: $756$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 349 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	349.g (of order \(29\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.78677903054\)
Analytic rank:	\(0\)
Dimension:	\(756\)
Relative dimension:	\(27\) over \(\Q(\zeta_{29})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{29}]$

Embedding invariants

Embedding label			31.2
Character	\(\chi\)	\(=\)	349.31
Dual form			349.2.g.a.304.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.409444 - 2.49750i) q^{2} +(-0.175952 + 0.331880i) q^{3} +(-4.17455 + 1.40657i) q^{4} +(-1.14986 + 1.08920i) q^{5} +(0.900913 + 0.303553i) q^{6} +(-0.0267563 - 0.00290993i) q^{7} +(2.85122 + 5.37796i) q^{8} +(1.60438 + 2.36628i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 756 q - 26 q^{2} - 25 q^{3} - 46 q^{4} - 19 q^{5} - 9 q^{6} - 19 q^{7} - 95 q^{8} - 44 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{13}{29}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.409444	−	2.49750i	−0.289521	−	1.76600i	−0.585373	−	0.810764i	\(-0.699052\pi\)
							0.295852	−	0.955234i	\(-0.404396\pi\)
\(3\)	−0.175952	+	0.331880i	−0.101586	+	0.191611i	−0.929095	−	0.369841i	\(-0.879412\pi\)
							0.827509	+	0.561452i	\(0.189757\pi\)
\(5\)	−1.14986	+	1.08920i	−0.514231	+	0.487106i	−0.900243	−	0.435388i	\(-0.856611\pi\)
							0.386012	+	0.922494i	\(0.373852\pi\)
\(7\)	−0.0267563	−	0.00290993i	−0.0101129	−	0.00109985i	0.103061	−	0.994675i	\(-0.467136\pi\)
							−0.113174	+	0.993575i	\(0.536102\pi\)
\(11\)	0.779399	+	2.80714i	0.234998	+	0.846385i	0.983141	+	0.182848i	\(0.0585317\pi\)
							−0.748143	+	0.663537i	\(0.769055\pi\)
\(13\)	0.107822	+	1.98867i	0.0299046	+	0.551557i	0.974648	+	0.223745i	\(0.0718284\pi\)
							−0.944743	+	0.327812i	\(0.893689\pi\)
\(17\)	0.545014	−	1.02801i	0.132185	−	0.249328i	−0.808580	−	0.588386i	\(-0.799763\pi\)
							0.940765	+	0.339058i	\(0.110108\pi\)
\(19\)	−1.75909	−	2.07096i	−0.403563	−	0.475111i	0.522413	−	0.852692i	\(-0.325032\pi\)
							−0.925977	+	0.377581i	\(0.876756\pi\)
\(23\)	−0.0101997	−	0.188122i	−0.00212678	−	0.0392262i	0.997277	−	0.0737415i	\(-0.0234940\pi\)
							−0.999404	+	0.0345154i	\(0.989011\pi\)
\(29\)	1.45233	+	8.85882i	0.269691	+	1.64504i	0.680085	+	0.733133i	\(0.261943\pi\)
							−0.410395	+	0.911908i	\(0.634609\pi\)
\(31\)	5.11259	−	1.12537i	0.918249	−	0.202122i	0.269390	−	0.963031i	\(-0.413178\pi\)
							0.648858	+	0.760909i	\(0.275247\pi\)
\(37\)	−0.384747	−	7.09625i	−0.0632521	−	1.16662i	−0.842071	−	0.539367i	\(-0.818664\pi\)
							0.778819	−	0.627249i	\(-0.215819\pi\)
\(41\)	1.80885	+	3.41185i	0.282495	+	0.532842i	0.983633	−	0.180184i	\(-0.0576692\pi\)
							−0.701138	+	0.713026i	\(0.747324\pi\)
\(43\)	0.944601	+	3.40215i	0.144050	+	0.518823i	0.999998	+	0.00176078i	\(0.000560473\pi\)
							−0.855948	+	0.517062i	\(0.827026\pi\)
\(47\)	−1.13639	+	2.85213i	−0.165760	+	0.416026i	−0.988268	−	0.152727i	\(-0.951195\pi\)
							0.822509	+	0.568753i	\(0.192574\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	349.2.g.a.31.2	✓	756
349.304	even	29	inner	349.2.g.a.304.2	yes	756

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
349.2.g.a.31.2	✓	756	1.1	even	1	trivial
349.2.g.a.304.2	yes	756	349.304	even	29	inner