Embedded newform 416.2.bd.a.83.16

• Label: 416.2.bd.a.83.16
• Level: $416$
• Weight: $2$
• Character: 416.83
• Analytic conductor: $3.322$
• Analytic rank: $0$
• Dimension: $216$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	416.bd (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.32177672409\)
Analytic rank:	\(0\)
Dimension:	\(216\)
Relative dimension:	\(54\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			83.16
Character	\(\chi\)	\(=\)	416.83
Dual form			416.2.bd.a.411.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.873613 + 1.11212i) q^{2} +(-0.260922 - 0.108078i) q^{3} +(-0.473600 - 1.94312i) q^{4} +(0.260195 - 0.628165i) q^{5} +(0.348140 - 0.195758i) q^{6} -0.282606i q^{7} +(2.57471 + 1.17084i) q^{8} +(-2.06492 - 2.06492i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q - 4 q^{2} - 8 q^{3} - 4 q^{5} + 8 q^{6} - 4 q^{8} - 8 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/416\mathbb{Z}\right)^\times\).

\(n\)	\(261\)	\(287\)	\(353\)
\(\chi(n)\)	\(e\left(\frac{7}{8}\right)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.873613	+	1.11212i	−0.617738	+	0.786384i
							\(3\)	−0.260922	−	0.108078i	−0.150644	−	0.0623986i	0.306088	−	0.952003i	\(-0.400980\pi\)
−0.456731	+	0.889605i	\(0.650980\pi\)
\(5\)	0.260195	−	0.628165i	0.116363	−	0.280924i	−0.854958	−	0.518697i	\(-0.826417\pi\)
							0.971321	+	0.237773i	\(0.0764173\pi\)
\(7\)		−	0.282606i		−	0.106815i	−0.998573	−	0.0534075i	\(-0.982992\pi\)
							0.998573	−	0.0534075i	\(-0.0170082\pi\)
\(11\)	−2.03693	−	0.843724i	−0.614157	−	0.254392i	0.0538476	−	0.998549i	\(-0.482851\pi\)
							−0.668005	+	0.744157i	\(0.732851\pi\)
\(13\)	−0.892268	+	3.49340i	−0.247471	+	0.968895i
							\(17\)	−6.04570			−1.46630			−0.733149	−	0.680068i	\(-0.761950\pi\)
−0.733149	+	0.680068i	\(0.761950\pi\)
\(19\)	−2.40819	−	5.81388i	−0.552476	−	1.33380i	−0.915613	−	0.402060i	\(-0.868294\pi\)
							0.363137	−	0.931736i	\(-0.381706\pi\)
\(23\)	−2.04676	−	2.04676i	−0.426779	−	0.426779i	0.460750	−	0.887530i	\(-0.347580\pi\)
							−0.887530	+	0.460750i	\(0.847580\pi\)
\(29\)	1.09830	−	2.65152i	0.203948	−	0.492375i	−0.788500	−	0.615034i	\(-0.789142\pi\)
							0.992449	+	0.122659i	\(0.0391422\pi\)
\(31\)	−5.51050	−	5.51050i	−0.989715	−	0.989715i	0.0102329	−	0.999948i	\(-0.496743\pi\)
							−0.999948	+	0.0102329i	\(0.996743\pi\)
\(37\)	2.61117	+	1.08158i	0.429274	+	0.177811i	0.586850	−	0.809696i	\(-0.300368\pi\)
							−0.157576	+	0.987507i	\(0.550368\pi\)
\(41\)	−3.13573			−0.489719			−0.244860	−	0.969559i	\(-0.578742\pi\)
							−0.244860	+	0.969559i	\(0.578742\pi\)
\(43\)	0.517219	+	1.24868i	0.0788751	+	0.190421i	0.958398	−	0.285435i	\(-0.0921381\pi\)
							−0.879523	+	0.475857i	\(0.842138\pi\)
\(47\)	−7.13337	−	7.13337i	−1.04051	−	1.04051i	−0.999144	−	0.0413650i	\(-0.986829\pi\)
							−0.0413650	−	0.999144i	\(-0.513171\pi\)