Embedded newform 320.2.bd.a.43.19

• Label: 320.2.bd.a.43.19
• 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			43.19
Character	\(\chi\)	\(=\)	320.43
Dual form			320.2.bd.a.67.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.497102 - 1.32397i) q^{2} +(-0.705925 - 0.471684i) q^{3} +(-1.50578 + 1.31629i) q^{4} +(1.24689 + 1.85614i) q^{5} +(-0.273577 + 1.16910i) q^{6} +(-0.381149 - 0.920174i) q^{7} +(2.49126 + 1.33927i) q^{8} +(-0.872206 - 2.10569i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 8 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.497102	−	1.32397i	−0.351504	−	0.936186i
							\(3\)	−0.705925	−	0.471684i	−0.407566	−	0.272327i	0.334843	−	0.942274i	\(-0.391317\pi\)
−0.742409	+	0.669947i	\(0.766317\pi\)
\(5\)	1.24689	+	1.85614i	0.557628	+	0.830091i
							\(7\)	−0.381149	−	0.920174i	−0.144061	−	0.347793i	0.835336	−	0.549740i	\(-0.185273\pi\)
−0.979396	+	0.201947i	\(0.935273\pi\)
\(11\)	1.02334	−	5.14465i	0.308547	−	1.55117i	−0.446065	−	0.895001i	\(-0.647175\pi\)
							0.754612	−	0.656171i	\(-0.227825\pi\)
\(13\)	0.374370	+	1.88209i	0.103832	+	0.521997i	0.997336	+	0.0729425i	\(0.0232389\pi\)
							−0.893505	+	0.449054i	\(0.851761\pi\)
\(17\)		−	2.80391i		−	0.680048i	−0.940417	−	0.340024i	\(-0.889565\pi\)
							0.940417	−	0.340024i	\(-0.110435\pi\)
\(19\)	4.49470	−	6.72679i	1.03115	−	1.54323i	0.205819	−	0.978590i	\(-0.434014\pi\)
							0.825336	−	0.564642i	\(-0.190986\pi\)
\(23\)	1.65905	+	0.687201i	0.345936	+	0.143291i	0.548885	−	0.835898i	\(-0.315053\pi\)
							−0.202949	+	0.979189i	\(0.565053\pi\)
\(29\)	−0.852273	−	4.28466i	−0.158263	−	0.795642i	−0.975613	−	0.219500i	\(-0.929557\pi\)
							0.817349	−	0.576142i	\(-0.195443\pi\)
\(31\)	−3.71587			−0.667391			−0.333695	−	0.942681i	\(-0.608296\pi\)
							−0.333695	+	0.942681i	\(0.608296\pi\)
\(37\)	−1.91602	−	0.381120i	−0.314992	−	0.0626558i	0.0350625	−	0.999385i	\(-0.488837\pi\)
							−0.350055	+	0.936729i	\(0.613837\pi\)
\(41\)	3.45708	+	8.34614i	0.539906	+	1.30345i	0.924788	+	0.380482i	\(0.124242\pi\)
							−0.384882	+	0.922966i	\(0.625758\pi\)
\(43\)	−2.50061	−	3.74243i	−0.381340	−	0.570715i	0.590299	−	0.807184i	\(-0.299010\pi\)
							−0.971639	+	0.236469i	\(0.924010\pi\)
\(47\)	5.60597			0.817715			0.408857	−	0.912598i	\(-0.365927\pi\)
							0.408857	+	0.912598i	\(0.365927\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.2.bd.a.43.19	✓	368
5.2	odd	4		320.2.bj.a.107.41	yes	368
64.3	odd	16		320.2.bj.a.3.41	yes	368
320.67	even	16	inner	320.2.bd.a.67.19	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.bd.a.43.19	✓	368	1.1	even	1	trivial
320.2.bd.a.67.19	yes	368	320.67	even	16	inner
320.2.bj.a.3.41	yes	368	64.3	odd	16
320.2.bj.a.107.41	yes	368	5.2	odd	4