Embedded newform 349.2.h.a.17.12

• Label: 349.2.h.a.17.12
• 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			17.12
Character	\(\chi\)	\(=\)	349.17
Dual form			349.2.h.a.308.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.357823 + 0.377749i) q^{2} +(2.21687 + 2.60991i) q^{3} +(0.0936207 + 1.72673i) q^{4} +(-0.811402 + 0.616812i) q^{5} +(-1.77914 - 0.0964621i) q^{6} +(0.990354 - 1.64598i) q^{7} +(-1.47891 - 1.25619i) q^{8} +(-1.41173 + 8.61120i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 784 q - 29 q^{2} - 25 q^{3} + 3 q^{4} - 21 q^{5} - 29 q^{6} - 29 q^{7} + 58 q^{8} - 57 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.357823	+	0.377749i	−0.253019	+	0.267109i	−0.840115	−	0.542408i	\(-0.817513\pi\)
							0.587096	+	0.809517i	\(0.300271\pi\)
\(3\)	2.21687	+	2.60991i	1.27991	+	1.50683i	0.755093	+	0.655617i	\(0.227591\pi\)
							0.524820	+	0.851213i	\(0.324133\pi\)
\(5\)	−0.811402	+	0.616812i	−0.362870	+	0.275847i	−0.770683	−	0.637219i	\(-0.780085\pi\)
							0.407813	+	0.913066i	\(0.366292\pi\)
\(7\)	0.990354	−	1.64598i	0.374319	−	0.622123i	−0.611242	−	0.791444i	\(-0.709330\pi\)
							0.985561	+	0.169321i	\(0.0541575\pi\)
\(11\)	0.887229	−	4.03072i	0.267510	−	1.21531i	−0.630800	−	0.775945i	\(-0.717273\pi\)
							0.898310	−	0.439363i	\(-0.144796\pi\)
\(13\)	3.66538	−	1.01769i	1.01659	−	0.282256i	0.281021	−	0.959702i	\(-0.409327\pi\)
							0.735573	+	0.677446i	\(0.236913\pi\)
\(17\)	3.18560	+	3.75038i	0.772621	+	0.909600i	0.997905	−	0.0646933i	\(-0.0206069\pi\)
							−0.225284	+	0.974293i	\(0.572331\pi\)
\(19\)	−0.661581	−	1.66044i	−0.151777	−	0.380932i	0.833353	−	0.552741i	\(-0.186418\pi\)
							−0.985130	+	0.171809i	\(0.945039\pi\)
\(23\)	−2.48880	−	8.96385i	−0.518951	−	1.86909i	−0.494108	−	0.869401i	\(-0.664505\pi\)
							−0.0248429	−	0.999691i	\(-0.507909\pi\)
\(29\)	−4.89716	−	4.63884i	−0.909380	−	0.861411i	0.0814287	−	0.996679i	\(-0.474052\pi\)
							−0.990809	+	0.135268i	\(0.956810\pi\)
\(31\)	3.03041	−	5.71597i	0.544278	−	1.02662i	−0.446963	−	0.894553i	\(-0.647494\pi\)
							0.991241	−	0.132065i	\(-0.0421609\pi\)
\(37\)	0.852445	+	3.07023i	0.140141	+	0.504742i	0.999983	−	0.00581775i	\(-0.00185186\pi\)
							−0.859842	+	0.510560i	\(0.829438\pi\)
\(41\)	0.913051	−	1.07493i	0.142595	−	0.167875i	−0.686229	−	0.727386i	\(-0.740735\pi\)
							0.828823	+	0.559511i	\(0.189011\pi\)
\(43\)	0.256467	−	1.16514i	0.0391108	−	0.177682i	−0.953016	−	0.302920i	\(-0.902038\pi\)
							0.992127	+	0.125238i	\(0.0399695\pi\)
\(47\)	−2.30396	+	6.83791i	−0.336067	+	0.997412i	0.637782	+	0.770217i	\(0.279852\pi\)
							−0.973849	+	0.227195i	\(0.927044\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	349.2.h.a.17.12	✓	784
349.308	even	58	inner	349.2.h.a.308.12	yes	784

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
349.2.h.a.17.12	✓	784	1.1	even	1	trivial
349.2.h.a.308.12	yes	784	349.308	even	58	inner