Embedded newform 349.2.h.a.17.13

• Label: 349.2.h.a.17.13
• 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			17.13
Character	\(\chi\)	\(=\)	349.17
Dual form			349.2.h.a.308.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.262216 + 0.276818i) q^{2} +(-0.127909 - 0.150586i) q^{3} +(0.100407 + 1.85189i) q^{4} +(3.35848 - 2.55305i) q^{5} +(0.0752247 + 0.00407857i) q^{6} +(0.247523 - 0.411386i) q^{7} +(-1.12018 - 0.951491i) q^{8} +(0.479031 - 2.92196i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 784 q - 29 q^{2} - 25 q^{3} + 3 q^{4} - 21 q^{5} - 29 q^{6} - 29 q^{7} + 58 q^{8} - 57 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.262216	+	0.276818i	−0.185415	+	0.195740i	−0.812076	−	0.583552i	\(-0.801662\pi\)
							0.626661	+	0.779292i	\(0.284421\pi\)
\(3\)	−0.127909	−	0.150586i	−0.0738483	−	0.0869409i	0.723997	−	0.689803i	\(-0.242303\pi\)
							−0.797846	+	0.602862i	\(0.794027\pi\)
\(5\)	3.35848	−	2.55305i	1.50196	−	1.14176i	0.550149	−	0.835067i	\(-0.314571\pi\)
							0.951809	−	0.306692i	\(-0.0992222\pi\)
\(7\)	0.247523	−	0.411386i	0.0935549	−	0.155489i	−0.806545	−	0.591172i	\(-0.798665\pi\)
							0.900100	+	0.435683i	\(0.143493\pi\)
\(11\)	1.11393	−	5.06062i	0.335861	−	1.52583i	−0.441779	−	0.897124i	\(-0.645652\pi\)
							0.777640	−	0.628709i	\(-0.216417\pi\)
\(13\)	−4.56188	+	1.26660i	−1.26524	+	0.351292i	−0.834411	−	0.551143i	\(-0.814192\pi\)
							−0.430828	+	0.902434i	\(0.641778\pi\)
\(17\)	4.43503	+	5.22132i	1.07565	+	1.26636i	0.962063	+	0.272828i	\(0.0879590\pi\)
							0.113590	+	0.993528i	\(0.463765\pi\)
\(19\)	1.25082	+	3.13933i	0.286959	+	0.720212i	0.999821	+	0.0189082i	\(0.00601901\pi\)
							−0.712863	+	0.701304i	\(0.752602\pi\)
\(23\)	−0.730707	−	2.63177i	−0.152363	−	0.548761i	−0.999839	−	0.0179284i	\(-0.994293\pi\)
							0.847476	−	0.530833i	\(-0.178121\pi\)
\(29\)	2.02236	+	1.91568i	0.375542	+	0.355732i	0.852071	−	0.523427i	\(-0.175347\pi\)
							−0.476529	+	0.879159i	\(0.658105\pi\)
\(31\)	0.115667	−	0.218172i	0.0207745	−	0.0391848i	−0.872908	−	0.487886i	\(-0.837768\pi\)
							0.893682	+	0.448701i	\(0.148113\pi\)
\(37\)	1.03105	+	3.71351i	0.169504	+	0.610497i	0.998673	+	0.0514995i	\(0.0164001\pi\)
							−0.829169	+	0.558997i	\(0.811186\pi\)
\(41\)	−0.0838845	+	0.0987565i	−0.0131006	+	0.0154232i	−0.768673	−	0.639642i	\(-0.779083\pi\)
							0.755573	+	0.655065i	\(0.227359\pi\)
\(43\)	−2.15198	+	9.77655i	−0.328174	+	1.49091i	0.466852	+	0.884335i	\(0.345388\pi\)
							−0.795026	+	0.606575i	\(0.792543\pi\)
\(47\)	−2.98456	+	8.85785i	−0.435342	+	1.29205i	0.475653	+	0.879633i	\(0.342212\pi\)
							−0.910995	+	0.412417i	\(0.864685\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	349.2.h.a.17.13	✓	784
349.308	even	58	inner	349.2.h.a.308.13	yes	784

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
349.2.h.a.17.13	✓	784	1.1	even	1	trivial
349.2.h.a.308.13	yes	784	349.308	even	58	inner