Embedded newform 349.2.g.a.31.13

• Label: 349.2.g.a.31.13
• 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			31.13
Character	\(\chi\)	\(=\)	349.31
Dual form			349.2.g.a.304.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0384208 - 0.234357i) q^{2} +(-0.0824196 + 0.155460i) q^{3} +(1.84186 - 0.620595i) q^{4} +(-0.451145 + 0.427347i) q^{5} +(0.0395997 + 0.0133427i) q^{6} +(1.45469 + 0.158207i) q^{7} +(-0.438686 - 0.827450i) q^{8} +(1.66619 + 2.45744i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 756 q - 26 q^{2} - 25 q^{3} - 46 q^{4} - 19 q^{5} - 9 q^{6} - 19 q^{7} - 95 q^{8} - 44 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0384208	−	0.234357i	−0.0271676	−	0.165715i	0.969758	−	0.244068i	\(-0.0784821\pi\)
							−0.996926	+	0.0783531i	\(0.975034\pi\)
\(3\)	−0.0824196	+	0.155460i	−0.0475850	+	0.0897548i	−0.906164	−	0.422926i	\(-0.861003\pi\)
							0.858579	+	0.512681i	\(0.171348\pi\)
\(5\)	−0.451145	+	0.427347i	−0.201758	+	0.191116i	−0.781907	−	0.623395i	\(-0.785753\pi\)
							0.580149	+	0.814510i	\(0.302994\pi\)
\(7\)	1.45469	+	0.158207i	0.549822	+	0.0597967i	0.378814	−	0.925473i	\(-0.376332\pi\)
							0.171008	+	0.985270i	\(0.445298\pi\)
\(11\)	−0.416923	−	1.50162i	−0.125707	−	0.452756i	0.873726	−	0.486418i	\(-0.161697\pi\)
							−0.999433	+	0.0336623i	\(0.989283\pi\)
\(13\)	0.190506	+	3.51367i	0.0528367	+	0.974516i	0.897997	+	0.440003i	\(0.145022\pi\)
							−0.845160	+	0.534514i	\(0.820495\pi\)
\(17\)	2.19248	−	4.13546i	0.531756	−	1.00300i	−0.461510	−	0.887135i	\(-0.652692\pi\)
							0.993265	−	0.115862i	\(-0.0369632\pi\)
\(19\)	−2.08334	−	2.45270i	−0.477951	−	0.562688i	0.469358	−	0.883008i	\(-0.344485\pi\)
							−0.947309	+	0.320320i	\(0.896209\pi\)
\(23\)	0.0420665	+	0.775871i	0.00877148	+	0.161780i	0.999637	+	0.0269292i	\(0.00857287\pi\)
							−0.990866	+	0.134851i	\(0.956944\pi\)
\(29\)	−0.338866	−	2.06699i	−0.0629258	−	0.383830i	−0.999448	−	0.0332220i	\(-0.989423\pi\)
							0.936522	−	0.350608i	\(-0.114025\pi\)
\(31\)	6.54012	−	1.43959i	1.17464	−	0.258558i	0.415566	−	0.909563i	\(-0.363583\pi\)
							0.759073	+	0.651005i	\(0.225652\pi\)
\(37\)	0.623451	+	11.4989i	0.102495	+	1.89041i	0.373212	+	0.927746i	\(0.378257\pi\)
							−0.270717	+	0.962659i	\(0.587261\pi\)
\(41\)	−5.64430	−	10.6463i	−0.881492	−	1.66267i	−0.739870	−	0.672750i	\(-0.765113\pi\)
							−0.141623	−	0.989921i	\(-0.545232\pi\)
\(43\)	−2.23829	−	8.06161i	−0.341337	−	1.22938i	−0.912681	−	0.408673i	\(-0.865992\pi\)
							0.571344	−	0.820711i	\(-0.306422\pi\)
\(47\)	−0.277768	+	0.697145i	−0.0405166	+	0.101689i	−0.947843	−	0.318739i	\(-0.896741\pi\)
							0.907326	+	0.420428i	\(0.138120\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	349.2.g.a.31.13	✓	756
349.304	even	29	inner	349.2.g.a.304.13	yes	756

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
349.2.g.a.31.13	✓	756	1.1	even	1	trivial
349.2.g.a.304.13	yes	756	349.304	even	29	inner