Embedded newform 320.2.bd.a.43.20

• Label: 320.2.bd.a.43.20
• Level: $320$
• Weight: $2$
• Character: 320.43
• Analytic conductor: $2.555$
• Analytic rank: $0$
• Dimension: $368$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	320.bd (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.55521286468\)
Analytic rank:	\(0\)
Dimension:	\(368\)
Relative dimension:	\(46\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			43.20
Character	\(\chi\)	\(=\)	320.43
Dual form			320.2.bd.a.67.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.317156 - 1.37819i) q^{2} +(-0.627981 - 0.419603i) q^{3} +(-1.79882 + 0.874203i) q^{4} +(-2.23256 - 0.125169i) q^{5} +(-0.379126 + 0.998558i) q^{6} +(-0.150193 - 0.362597i) q^{7} +(1.77533 + 2.20187i) q^{8} +(-0.929757 - 2.24463i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 8 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\)	\(191\)	\(257\)	\(261\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{13}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.317156	−	1.37819i	−0.224263	−	0.974529i
							\(3\)	−0.627981	−	0.419603i	−0.362565	−	0.242258i	0.360922	−	0.932596i	\(-0.382462\pi\)
−0.723487	+	0.690338i	\(0.757462\pi\)
\(5\)	−2.23256	−	0.125169i	−0.998432	−	0.0559773i
							\(7\)	−0.150193	−	0.362597i	−0.0567675	−	0.137049i	0.892951	−	0.450154i	\(-0.148631\pi\)
−0.949719	+	0.313105i	\(0.898631\pi\)
\(11\)	−0.363316	+	1.82651i	−0.109544	+	0.550714i	0.886567	+	0.462601i	\(0.153084\pi\)
							−0.996110	+	0.0881130i	\(0.971916\pi\)
\(13\)	0.738429	+	3.71233i	0.204803	+	1.02962i	0.937215	+	0.348753i	\(0.113395\pi\)
							−0.732411	+	0.680862i	\(0.761605\pi\)
\(17\)			5.24935i			1.27315i	0.771213	+	0.636577i	\(0.219650\pi\)
							−0.771213	+	0.636577i	\(0.780350\pi\)
\(19\)	−4.08446	+	6.11283i	−0.937040	+	1.40238i	−0.0216699	+	0.999765i	\(0.506898\pi\)
							−0.915370	+	0.402614i	\(0.868102\pi\)
\(23\)	0.861553	+	0.356867i	0.179646	+	0.0744119i	0.470694	−	0.882297i	\(-0.344004\pi\)
							−0.291047	+	0.956709i	\(0.594004\pi\)
\(29\)	−1.59381	−	8.01263i	−0.295963	−	1.48791i	−0.787100	−	0.616826i	\(-0.788418\pi\)
							0.491136	−	0.871083i	\(-0.336582\pi\)
\(31\)	−8.57939			−1.54090			−0.770452	−	0.637498i	\(-0.779969\pi\)
							−0.770452	+	0.637498i	\(0.779969\pi\)
\(37\)	−3.17524	−	0.631595i	−0.522007	−	0.103834i	−0.0729480	−	0.997336i	\(-0.523241\pi\)
							−0.449059	+	0.893502i	\(0.648241\pi\)
\(41\)	−3.35131	−	8.09078i	−0.523387	−	1.26357i	−0.935787	−	0.352565i	\(-0.885309\pi\)
							0.412400	−	0.911003i	\(-0.364691\pi\)
\(43\)	4.86620	+	7.28278i	0.742088	+	1.11061i	0.989895	+	0.141804i	\(0.0452902\pi\)
							−0.247807	+	0.968810i	\(0.579710\pi\)
\(47\)	−8.66100			−1.26334			−0.631668	−	0.775239i	\(-0.717629\pi\)
							−0.631668	+	0.775239i	\(0.717629\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.2.bd.a.43.20	✓	368
5.2	odd	4		320.2.bj.a.107.42	yes	368
64.3	odd	16		320.2.bj.a.3.42	yes	368
320.67	even	16	inner	320.2.bd.a.67.20	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.bd.a.43.20	✓	368	1.1	even	1	trivial
320.2.bd.a.67.20	yes	368	320.67	even	16	inner
320.2.bj.a.3.42	yes	368	64.3	odd	16
320.2.bj.a.107.42	yes	368	5.2	odd	4