Embedded newform 320.2.bd.a.43.3

• Label: 320.2.bd.a.43.3
• 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			43.3
Character	\(\chi\)	\(=\)	320.43
Dual form			320.2.bd.a.67.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.40925 - 0.118345i) q^{2} +(2.70757 + 1.80914i) q^{3} +(1.97199 + 0.333555i) q^{4} +(0.594278 - 2.15565i) q^{5} +(-3.60155 - 2.86996i) q^{6} +(1.00795 + 2.43340i) q^{7} +(-2.73956 - 0.703438i) q^{8} +(2.90990 + 7.02511i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 8 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.40925	−	0.118345i	−0.996492	−	0.0836823i
							\(3\)	2.70757	+	1.80914i	1.56322	+	1.04451i	0.971105	+	0.238652i	\(0.0767054\pi\)
0.592111	+	0.805856i	\(0.298295\pi\)
\(5\)	0.594278	−	2.15565i	0.265769	−	0.964037i
							\(7\)	1.00795	+	2.43340i	0.380968	+	0.919738i	0.991779	+	0.127962i	\(0.0408435\pi\)
−0.610811	+	0.791776i	\(0.709157\pi\)
\(11\)	−0.169020	+	0.849723i	−0.0509616	+	0.256201i	−0.997864	−	0.0653186i	\(-0.979194\pi\)
							0.946903	+	0.321520i	\(0.104194\pi\)
\(13\)	−0.733336	−	3.68673i	−0.203391	−	1.02251i	−0.938688	−	0.344769i	\(-0.887957\pi\)
							0.735297	−	0.677745i	\(-0.237043\pi\)
\(17\)		−	5.89871i		−	1.43065i	−0.698794	−	0.715323i	\(-0.746279\pi\)
							0.698794	−	0.715323i	\(-0.253721\pi\)
\(19\)	−3.70187	+	5.54023i	−0.849266	+	1.27102i	0.111530	+	0.993761i	\(0.464425\pi\)
							−0.960796	+	0.277256i	\(0.910575\pi\)
\(23\)	1.03310	+	0.427923i	0.215416	+	0.0892282i	0.487782	−	0.872966i	\(-0.337806\pi\)
							−0.272366	+	0.962194i	\(0.587806\pi\)
\(29\)	−1.30530	−	6.56219i	−0.242388	−	1.21857i	−0.889773	−	0.456404i	\(-0.849137\pi\)
							0.647385	−	0.762163i	\(-0.275863\pi\)
\(31\)	2.17557			0.390745			0.195372	−	0.980729i	\(-0.437408\pi\)
							0.195372	+	0.980729i	\(0.437408\pi\)
\(37\)	0.934319	+	0.185848i	0.153601	+	0.0305531i	0.271292	−	0.962497i	\(-0.412549\pi\)
							−0.117691	+	0.993050i	\(0.537549\pi\)
\(41\)	3.64719	+	8.80508i	0.569595	+	1.37512i	0.901897	+	0.431950i	\(0.142174\pi\)
							−0.332303	+	0.943173i	\(0.607826\pi\)
\(43\)	−2.45348	−	3.67190i	−0.374153	−	0.559959i	0.595837	−	0.803105i	\(-0.296820\pi\)
							−0.969990	+	0.243146i	\(0.921820\pi\)
\(47\)	−5.21266			−0.760344			−0.380172	−	0.924916i	\(-0.624135\pi\)
							−0.380172	+	0.924916i	\(0.624135\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.2.bd.a.43.3	✓	368
5.2	odd	4		320.2.bj.a.107.25	yes	368
64.3	odd	16		320.2.bj.a.3.25	yes	368
320.67	even	16	inner	320.2.bd.a.67.3	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.bd.a.43.3	✓	368	1.1	even	1	trivial
320.2.bd.a.67.3	yes	368	320.67	even	16	inner
320.2.bj.a.3.25	yes	368	64.3	odd	16
320.2.bj.a.107.25	yes	368	5.2	odd	4