Embedded newform 349.2.g.a.31.15

• Label: 349.2.g.a.31.15
• 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			31.15
Character	\(\chi\)	\(=\)	349.31
Dual form			349.2.g.a.304.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0663976 + 0.405008i) q^{2} +(-1.58682 + 2.99305i) q^{3} +(1.73568 - 0.584820i) q^{4} +(-2.67584 + 2.53469i) q^{5} +(-1.31757 - 0.443941i) q^{6} +(1.71607 + 0.186634i) q^{7} +(0.736585 + 1.38935i) q^{8} +(-4.75683 - 7.01579i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 756 q - 26 q^{2} - 25 q^{3} - 46 q^{4} - 19 q^{5} - 9 q^{6} - 19 q^{7} - 95 q^{8} - 44 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0663976	+	0.405008i	0.0469502	+	0.286384i	0.999858	−	0.0168780i	\(-0.00537268\pi\)
							−0.952907	+	0.303262i	\(0.901924\pi\)
\(3\)	−1.58682	+	2.99305i	−0.916149	+	1.72804i	−0.273515	+	0.961868i	\(0.588186\pi\)
							−0.642634	+	0.766173i	\(0.722159\pi\)
\(5\)	−2.67584	+	2.53469i	−1.19667	+	1.13355i	−0.208852	+	0.977947i	\(0.566973\pi\)
							−0.987820	+	0.155601i	\(0.950269\pi\)
\(7\)	1.71607	+	0.186634i	0.648614	+	0.0705410i	0.426511	−	0.904482i	\(-0.359743\pi\)
							0.222103	+	0.975023i	\(0.428708\pi\)
\(11\)	1.10397	+	3.97614i	0.332859	+	1.19885i	0.920829	+	0.389967i	\(0.127514\pi\)
							−0.587969	+	0.808883i	\(0.700072\pi\)
\(13\)	0.126999	+	2.34236i	0.0352232	+	0.649654i	0.961834	+	0.273633i	\(0.0882253\pi\)
							−0.926611	+	0.376021i	\(0.877292\pi\)
\(17\)	−0.118528	+	0.223567i	−0.0287472	+	0.0542230i	−0.897470	−	0.441076i	\(-0.854597\pi\)
							0.868722	+	0.495299i	\(0.164942\pi\)
\(19\)	−4.09450	−	4.82041i	−0.939342	−	1.10588i	−0.994217	−	0.107386i	\(-0.965752\pi\)
							0.0548754	−	0.998493i	\(-0.482524\pi\)
\(23\)	−0.298695	−	5.50911i	−0.0622823	−	1.14873i	−0.847949	−	0.530078i	\(-0.822163\pi\)
							0.785666	−	0.618650i	\(-0.212320\pi\)
\(29\)	−0.00667225	−	0.0406989i	−0.00123901	−	0.00755760i	0.986200	−	0.165560i	\(-0.0529430\pi\)
							−0.987439	+	0.158002i	\(0.949495\pi\)
\(31\)	−0.0620000	+	0.0136472i	−0.0111355	+	0.00245111i	−0.220535	−	0.975379i	\(-0.570780\pi\)
							0.209399	+	0.977830i	\(0.432849\pi\)
\(37\)	0.457318	+	8.43474i	0.0751827	+	1.38666i	0.756174	+	0.654371i	\(0.227066\pi\)
							−0.680991	+	0.732292i	\(0.738451\pi\)
\(41\)	4.00376	+	7.55189i	0.625282	+	1.17941i	0.970549	+	0.240902i	\(0.0774433\pi\)
							−0.345268	+	0.938504i	\(0.612212\pi\)
\(43\)	1.30911	+	4.71499i	0.199638	+	0.719030i	0.993786	+	0.111306i	\(0.0355034\pi\)
							−0.794148	+	0.607724i	\(0.792083\pi\)
\(47\)	−0.662886	+	1.66372i	−0.0966919	+	0.242678i	−0.969364	−	0.245628i	\(-0.921006\pi\)
							0.872672	+	0.488306i	\(0.162385\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	349.2.g.a.31.15	✓	756
349.304	even	29	inner	349.2.g.a.304.15	yes	756

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
349.2.g.a.31.15	✓	756	1.1	even	1	trivial
349.2.g.a.304.15	yes	756	349.304	even	29	inner