Embedded newform 320.2.bd.a.227.38

• Label: 320.2.bd.a.227.38
• Level: $320$
• Weight: $2$
• Character: 320.227
• 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			227.38
Character	\(\chi\)	\(=\)	320.227
Dual form			320.2.bd.a.203.38

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.18136 - 0.777420i) q^{2} +(-0.0766820 - 0.114763i) q^{3} +(0.791238 - 1.83683i) q^{4} +(0.840785 + 2.07197i) q^{5} +(-0.179808 - 0.0759623i) q^{6} +(-0.433915 + 1.04756i) q^{7} +(-0.493248 - 2.78509i) q^{8} +(1.14076 - 2.75404i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 8 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(191\)	\(257\)	\(261\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{11}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.18136	−	0.777420i	0.835350	−	0.549719i
							\(3\)	−0.0766820	−	0.114763i	−0.0442724	−	0.0662583i	0.808673	−	0.588258i	\(-0.200186\pi\)
−0.852946	+	0.522000i	\(0.825186\pi\)
\(5\)	0.840785	+	2.07197i	0.376011	+	0.926615i
							\(7\)	−0.433915	+	1.04756i	−0.164005	+	0.395942i	−0.984422	−	0.175823i	\(-0.943741\pi\)
0.820417	+	0.571765i	\(0.193741\pi\)
\(11\)	6.05432	−	1.20428i	1.82545	−	0.363104i	0.841314	−	0.540546i	\(-0.181782\pi\)
							0.984131	+	0.177443i	\(0.0567825\pi\)
\(13\)	−4.32452	−	0.860200i	−1.19941	−	0.238577i	−0.445331	−	0.895366i	\(-0.646914\pi\)
							−0.754074	+	0.656789i	\(0.771914\pi\)
\(17\)			6.12813i			1.48629i	0.669131	+	0.743145i	\(0.266667\pi\)
							−0.669131	+	0.743145i	\(0.733333\pi\)
\(19\)	1.99917	−	1.33580i	0.458641	−	0.306454i	−0.304702	−	0.952448i	\(-0.598557\pi\)
							0.763343	+	0.645994i	\(0.223557\pi\)
\(23\)	2.05334	−	0.850521i	0.428151	−	0.177346i	−0.158193	−	0.987408i	\(-0.550567\pi\)
							0.586344	+	0.810062i	\(0.300567\pi\)
\(29\)	−8.76264	−	1.74300i	−1.62718	−	0.323667i	−0.704640	−	0.709565i	\(-0.748892\pi\)
							−0.922541	+	0.385898i	\(0.873892\pi\)
\(31\)	−7.25450			−1.30295			−0.651473	−	0.758672i	\(-0.725849\pi\)
							−0.651473	+	0.758672i	\(0.725849\pi\)
\(37\)	0.816694	+	4.10580i	0.134264	+	0.674989i	0.988021	+	0.154318i	\(0.0493181\pi\)
							−0.853758	+	0.520671i	\(0.825682\pi\)
\(41\)	−2.16737	+	5.23249i	−0.338486	+	0.817177i	0.659376	+	0.751814i	\(0.270821\pi\)
							−0.997862	+	0.0653634i	\(0.979179\pi\)
\(43\)	−0.215546	−	0.144023i	−0.0328704	−	0.0219633i	0.539026	−	0.842289i	\(-0.318792\pi\)
							−0.571897	+	0.820326i	\(0.693792\pi\)
\(47\)	5.40102			0.787820			0.393910	−	0.919149i	\(-0.371122\pi\)
							0.393910	+	0.919149i	\(0.371122\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.2.bd.a.227.38	yes	368
5.3	odd	4		320.2.bj.a.163.15	yes	368
64.11	odd	16		320.2.bj.a.267.15	yes	368
320.203	even	16	inner	320.2.bd.a.203.38	✓	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.bd.a.203.38	✓	368	320.203	even	16	inner
320.2.bd.a.227.38	yes	368	1.1	even	1	trivial
320.2.bj.a.163.15	yes	368	5.3	odd	4
320.2.bj.a.267.15	yes	368	64.11	odd	16