Embedded newform 320.2.bj.a.267.35

• Label: 320.2.bj.a.267.35
• Level: $320$
• Weight: $2$
• Character: 320.267
• 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.bj (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			267.35
Character	\(\chi\)	\(=\)	320.267
Dual form			320.2.bj.a.163.35

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.937743 - 1.05860i) q^{2} +(2.62954 + 1.75700i) q^{3} +(-0.241278 - 1.98539i) q^{4} +(-2.11102 - 0.737284i) q^{5} +(4.32580 - 1.13602i) q^{6} +(1.26588 - 0.524344i) q^{7} +(-2.32800 - 1.60637i) q^{8} +(2.67937 + 6.46857i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 8 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(191\)	\(257\)	\(261\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{5}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.937743	−	1.05860i	0.663084	−	0.748545i
							\(3\)	2.62954	+	1.75700i	1.51817	+	1.01441i	0.985806	+	0.167887i	\(0.0536943\pi\)
0.532359	+	0.846519i	\(0.321306\pi\)
\(5\)	−2.11102	−	0.737284i	−0.944078	−	0.329723i
							\(7\)	1.26588	−	0.524344i	0.478457	−	0.198183i	−0.130403	−	0.991461i	\(-0.541627\pi\)
0.608860	+	0.793278i	\(0.291627\pi\)
\(11\)	4.55356	+	0.905758i	1.37295	+	0.273096i	0.825811	−	0.563946i	\(-0.190718\pi\)
							0.547137	+	0.837043i	\(0.315718\pi\)
\(13\)	0.00342843	+	0.0172359i	0.000950876	+	0.00478037i	0.981258	−	0.192700i	\(-0.0617243\pi\)
							−0.980307	+	0.197480i	\(0.936724\pi\)
\(17\)	−5.02857			−1.21961			−0.609804	−	0.792552i	\(-0.708752\pi\)
							−0.609804	+	0.792552i	\(0.708752\pi\)
\(19\)	−4.58118	−	3.06104i	−1.05099	−	0.702252i	−0.0949521	−	0.995482i	\(-0.530270\pi\)
							−0.956042	+	0.293230i	\(0.905270\pi\)
\(23\)	0.774747	−	1.87041i	0.161546	−	0.390007i	−0.822292	−	0.569065i	\(-0.807305\pi\)
							0.983838	+	0.179059i	\(0.0573052\pi\)
\(29\)	−7.18704	+	1.42959i	−1.33460	+	0.265468i	−0.810275	−	0.586050i	\(-0.800682\pi\)
							−0.524324	+	0.851519i	\(0.675682\pi\)
\(31\)	−1.51262			−0.271675			−0.135837	−	0.990731i	\(-0.543372\pi\)
							−0.135837	+	0.990731i	\(0.543372\pi\)
\(37\)	2.71465	+	0.539977i	0.446285	+	0.0887717i	0.413117	−	0.910678i	\(-0.364440\pi\)
							0.0331687	+	0.999450i	\(0.489440\pi\)
\(41\)	−0.0258104	−	0.0623119i	−0.00403091	−	0.00973148i	0.921851	−	0.387544i	\(-0.126676\pi\)
							−0.925882	+	0.377812i	\(0.876676\pi\)
\(43\)	3.55252	+	5.31672i	0.541754	+	0.810792i	0.996822	−	0.0796645i	\(-0.0253849\pi\)
							−0.455068	+	0.890457i	\(0.650385\pi\)
\(47\)			5.66309i			0.826046i	0.910720	+	0.413023i	\(0.135527\pi\)
							−0.910720	+	0.413023i	\(0.864473\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.2.bj.a.267.35	yes	368
5.3	odd	4		320.2.bd.a.203.12	✓	368
64.35	odd	16		320.2.bd.a.227.12	yes	368
320.163	even	16	inner	320.2.bj.a.163.35	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.bd.a.203.12	✓	368	5.3	odd	4
320.2.bd.a.227.12	yes	368	64.35	odd	16
320.2.bj.a.163.35	yes	368	320.163	even	16	inner
320.2.bj.a.267.35	yes	368	1.1	even	1	trivial