Embedded newform 349.2.h.a.17.10

• Label: 349.2.h.a.17.10
• Level: $349$
• Weight: $2$
• Character: 349.17
• Analytic conductor: $2.787$
• Analytic rank: $0$
• Dimension: $784$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 349 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	349.h (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.78677903054\)
Analytic rank:	\(0\)
Dimension:	\(784\)
Relative dimension:	\(28\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			17.10
Character	\(\chi\)	\(=\)	349.17
Dual form			349.2.h.a.308.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.622305 + 0.656959i) q^{2} +(-0.150428 - 0.177098i) q^{3} +(0.0639457 + 1.17941i) q^{4} +(0.905172 - 0.688094i) q^{5} +(0.209959 + 0.0113836i) q^{6} +(1.57394 - 2.61591i) q^{7} +(-2.19399 - 1.86359i) q^{8} +(0.476611 - 2.90720i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 784 q - 29 q^{2} - 25 q^{3} + 3 q^{4} - 21 q^{5} - 29 q^{6} - 29 q^{7} + 58 q^{8} - 57 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{43}{58}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.622305	+	0.656959i	−0.440036	+	0.464540i	−0.907855	−	0.419284i	\(-0.862281\pi\)
							0.467819	+	0.883824i	\(0.345040\pi\)
\(3\)	−0.150428	−	0.177098i	−0.0868499	−	0.102248i	0.717021	−	0.697052i	\(-0.245505\pi\)
							−0.803870	+	0.594804i	\(0.797229\pi\)
\(5\)	0.905172	−	0.688094i	0.404805	−	0.307725i	−0.382886	−	0.923796i	\(-0.625070\pi\)
							0.787691	+	0.616071i	\(0.211277\pi\)
\(7\)	1.57394	−	2.61591i	0.594894	−	0.988722i	−0.402418	−	0.915456i	\(-0.631830\pi\)
							0.997312	−	0.0732657i	\(-0.0233421\pi\)
\(11\)	−0.431421	+	1.95997i	−0.130078	+	0.590952i	0.865811	+	0.500372i	\(0.166803\pi\)
							−0.995889	+	0.0905806i	\(0.971128\pi\)
\(13\)	5.71471	−	1.58668i	1.58498	−	0.440066i	0.640314	−	0.768113i	\(-0.278804\pi\)
							0.944662	+	0.328047i	\(0.106390\pi\)
\(17\)	0.426906	+	0.502593i	0.103540	+	0.121897i	0.811491	−	0.584365i	\(-0.198656\pi\)
							−0.707951	+	0.706261i	\(0.750380\pi\)
\(19\)	−2.26345	−	5.68083i	−0.519271	−	1.30327i	−0.921729	−	0.387834i	\(-0.873223\pi\)
							0.402458	−	0.915438i	\(-0.368156\pi\)
\(23\)	1.55659	+	5.60633i	0.324571	+	1.16900i	0.928307	+	0.371816i	\(0.121265\pi\)
							−0.603735	+	0.797185i	\(0.706322\pi\)
\(29\)	4.55082	+	4.31076i	0.845066	+	0.800489i	0.981760	−	0.190124i	\(-0.0608891\pi\)
							−0.136694	+	0.990613i	\(0.543648\pi\)
\(31\)	0.885880	−	1.67095i	0.159109	−	0.300111i	−0.791122	−	0.611659i	\(-0.790503\pi\)
							0.950230	+	0.311548i	\(0.100847\pi\)
\(37\)	0.0716167	+	0.257940i	0.0117737	+	0.0424051i	0.969204	−	0.246261i	\(-0.0792020\pi\)
							−0.957430	+	0.288666i	\(0.906788\pi\)
\(41\)	4.65722	−	5.48290i	0.727335	−	0.856285i	−0.266876	−	0.963731i	\(-0.585991\pi\)
							0.994211	+	0.107446i	\(0.0342672\pi\)
\(43\)	1.04439	−	4.74469i	0.159267	−	0.723558i	−0.827507	−	0.561456i	\(-0.810241\pi\)
							0.986774	−	0.162103i	\(-0.0518275\pi\)
\(47\)	−0.262960	+	0.780438i	−0.0383567	+	0.113839i	−0.965121	−	0.261806i	\(-0.915682\pi\)
							0.926764	+	0.375644i	\(0.122579\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	349.2.h.a.17.10	✓	784
349.308	even	58	inner	349.2.h.a.308.10	yes	784

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
349.2.h.a.17.10	✓	784	1.1	even	1	trivial
349.2.h.a.308.10	yes	784	349.308	even	58	inner