Embedded newform 349.2.g.a.31.14

• Label: 349.2.g.a.31.14
• 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			31.14
Character	\(\chi\)	\(=\)	349.31
Dual form			349.2.g.a.304.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.000709636 - 0.00432859i) q^{2} +(1.55115 - 2.92579i) q^{3} +(1.89529 - 0.638597i) q^{4} +(-1.92168 + 1.82032i) q^{5} +(-0.0137653 - 0.00463806i) q^{6} +(3.38702 + 0.368360i) q^{7} +(-0.00821841 - 0.0155016i) q^{8} +(-4.47059 - 6.59363i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 756 q - 26 q^{2} - 25 q^{3} - 46 q^{4} - 19 q^{5} - 9 q^{6} - 19 q^{7} - 95 q^{8} - 44 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.000709636	−	0.00432859i	−0.000501788	−	0.00306077i	0.986574	−	0.163312i	\(-0.0522178\pi\)
							−0.987076	+	0.160251i	\(0.948770\pi\)
\(3\)	1.55115	−	2.92579i	0.895559	−	1.68920i	0.188471	−	0.982079i	\(-0.439647\pi\)
							0.707088	−	0.707125i	\(-0.250008\pi\)
\(5\)	−1.92168	+	1.82032i	−0.859403	+	0.814070i	−0.984006	−	0.178137i	\(-0.942993\pi\)
							0.124602	+	0.992207i	\(0.460234\pi\)
\(7\)	3.38702	+	0.368360i	1.28017	+	0.139227i	0.722813	−	0.691044i	\(-0.242849\pi\)
							0.557360	+	0.830271i	\(0.311814\pi\)
\(11\)	−0.818825	−	2.94914i	−0.246885	−	0.889199i	−0.978287	−	0.207253i	\(-0.933548\pi\)
							0.731402	−	0.681946i	\(-0.238866\pi\)
\(13\)	0.283777	+	5.23397i	0.0787057	+	1.45164i	0.725103	+	0.688640i	\(0.241792\pi\)
							−0.646398	+	0.763001i	\(0.723725\pi\)
\(17\)	−2.14047	+	4.03736i	−0.519141	+	0.979203i	0.475878	+	0.879511i	\(0.342130\pi\)
							−0.995018	+	0.0996916i	\(0.968214\pi\)
\(19\)	−1.13365	−	1.33464i	−0.260077	−	0.306187i	0.616589	−	0.787285i	\(-0.288514\pi\)
							−0.876667	+	0.481098i	\(0.840238\pi\)
\(23\)	0.128841	+	2.37634i	0.0268653	+	0.495501i	0.980743	+	0.195304i	\(0.0625694\pi\)
							−0.953877	+	0.300196i	\(0.902948\pi\)
\(29\)	−0.00243899	−	0.0148772i	−0.000452909	−	0.00276262i	0.986599	−	0.163163i	\(-0.0521697\pi\)
							−0.987052	+	0.160401i	\(0.948721\pi\)
\(31\)	−4.10435	+	0.903435i	−0.737162	+	0.162262i	−0.567654	−	0.823267i	\(-0.692149\pi\)
							−0.169508	+	0.985529i	\(0.554218\pi\)
\(37\)	−0.536291	−	9.89131i	−0.0881657	−	1.62612i	−0.623739	−	0.781633i	\(-0.714387\pi\)
							0.535573	−	0.844489i	\(-0.320096\pi\)
\(41\)	−1.98099	−	3.73654i	−0.309379	−	0.583550i	0.679254	−	0.733904i	\(-0.262304\pi\)
							−0.988632	+	0.150353i	\(0.951959\pi\)
\(43\)	2.63514	+	9.49093i	0.401856	+	1.44735i	0.837104	+	0.547043i	\(0.184247\pi\)
							−0.435249	+	0.900310i	\(0.643340\pi\)
\(47\)	−3.46105	+	8.68657i	−0.504846	+	1.26707i	0.427013	+	0.904245i	\(0.359566\pi\)
							−0.931859	+	0.362821i	\(0.881813\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	349.2.g.a.31.14	✓	756
349.304	even	29	inner	349.2.g.a.304.14	yes	756

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
349.2.g.a.31.14	✓	756	1.1	even	1	trivial
349.2.g.a.304.14	yes	756	349.304	even	29	inner