Embedded newform 349.2.g.a.31.8

• Label: 349.2.g.a.31.8
• 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			31.8
Character	\(\chi\)	\(=\)	349.31
Dual form			349.2.g.a.304.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.222646 - 1.35808i) q^{2} +(0.0623339 - 0.117574i) q^{3} +(0.100492 - 0.0338599i) q^{4} +(2.59545 - 2.45854i) q^{5} +(-0.173554 - 0.0584770i) q^{6} +(-4.60163 - 0.500458i) q^{7} +(-1.35762 - 2.56074i) q^{8} +(1.67362 + 2.46841i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 756 q - 26 q^{2} - 25 q^{3} - 46 q^{4} - 19 q^{5} - 9 q^{6} - 19 q^{7} - 95 q^{8} - 44 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.222646	−	1.35808i	−0.157435	−	0.960309i	−0.940853	−	0.338814i	\(-0.889974\pi\)
							0.783419	−	0.621494i	\(-0.213474\pi\)
\(3\)	0.0623339	−	0.117574i	0.0359885	−	0.0678815i	−0.864866	−	0.502003i	\(-0.832596\pi\)
							0.900854	+	0.434122i	\(0.142941\pi\)
\(5\)	2.59545	−	2.45854i	1.16072	−	1.09949i	0.167189	−	0.985925i	\(-0.446531\pi\)
							0.993530	−	0.113567i	\(-0.0362275\pi\)
\(7\)	−4.60163	−	0.500458i	−1.73925	−	0.189155i	−0.817236	−	0.576303i	\(-0.804495\pi\)
							−0.922018	+	0.387148i	\(0.873460\pi\)
\(11\)	−1.06502	−	3.83585i	−0.321115	−	1.15655i	−0.931288	−	0.364284i	\(-0.881314\pi\)
							0.610173	−	0.792268i	\(-0.291100\pi\)
\(13\)	0.233516	+	4.30695i	0.0647657	+	1.19453i	0.832596	+	0.553881i	\(0.186854\pi\)
							−0.767830	+	0.640653i	\(0.778664\pi\)
\(17\)	2.82875	−	5.33558i	0.686072	−	1.29407i	−0.258579	−	0.965990i	\(-0.583254\pi\)
							0.944650	−	0.328079i	\(-0.106401\pi\)
\(19\)	1.94623	+	2.29128i	0.446495	+	0.525655i	0.938659	−	0.344848i	\(-0.112070\pi\)
							−0.492163	+	0.870503i	\(0.663794\pi\)
\(23\)	−0.0346818	−	0.639667i	−0.00723165	−	0.133380i	−0.999920	−	0.0126719i	\(-0.995966\pi\)
							0.992688	−	0.120708i	\(-0.0385165\pi\)
\(29\)	1.25796	+	7.67320i	0.233597	+	1.42488i	0.799629	+	0.600494i	\(0.205029\pi\)
							−0.566032	+	0.824383i	\(0.691522\pi\)
\(31\)	−0.487621	+	0.107334i	−0.0875794	+	0.0192777i	−0.258544	−	0.966000i	\(-0.583243\pi\)
							0.170965	+	0.985277i	\(0.445312\pi\)
\(37\)	−0.399914	−	7.37598i	−0.0657455	−	1.21260i	−0.826262	−	0.563285i	\(-0.809537\pi\)
							0.760517	−	0.649318i	\(-0.224946\pi\)
\(41\)	−2.80349	−	5.28794i	−0.437831	−	0.825838i	0.562168	−	0.827023i	\(-0.309967\pi\)
							−0.999999	+	0.00118556i	\(0.999623\pi\)
\(43\)	3.40334	+	12.2577i	0.519004	+	1.86928i	0.493752	+	0.869603i	\(0.335625\pi\)
							0.0252520	+	0.999681i	\(0.491961\pi\)
\(47\)	1.40270	−	3.52051i	0.204605	−	0.513519i	−0.790487	−	0.612479i	\(-0.790172\pi\)
							0.995091	+	0.0989601i	\(0.0315516\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	349.2.g.a.31.8	✓	756
349.304	even	29	inner	349.2.g.a.304.8	yes	756

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
349.2.g.a.31.8	✓	756	1.1	even	1	trivial
349.2.g.a.304.8	yes	756	349.304	even	29	inner