Embedded newform 320.2.bd.a.283.45

• Label: 320.2.bd.a.283.45
• Level: $320$
• Weight: $2$
• Character: 320.283
• 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			283.45
Character	\(\chi\)	\(=\)	320.283
Dual form			320.2.bd.a.147.45

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.40883 + 0.123259i) q^{2} +(-0.209831 - 1.05489i) q^{3} +(1.96961 + 0.347303i) q^{4} +(-0.408295 + 2.19848i) q^{5} +(-0.165591 - 1.51203i) q^{6} +(0.249628 - 0.103399i) q^{7} +(2.73205 + 0.732064i) q^{8} +(1.70288 - 0.705354i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 8 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(191\)	\(257\)	\(261\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{9}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.40883	+	0.123259i	0.996195	+	0.0871574i
							\(3\)	−0.209831	−	1.05489i	−0.121146	−	0.609041i	−0.992886	−	0.119072i	\(-0.962008\pi\)
0.871740	−	0.489969i	\(-0.162992\pi\)
\(5\)	−0.408295	+	2.19848i	−0.182595	+	0.983188i
							\(7\)	0.249628	−	0.103399i	0.0943505	−	0.0390813i	−0.335009	−	0.942215i	\(-0.608739\pi\)
0.429359	+	0.903134i	\(0.358739\pi\)
\(11\)	−0.477773	−	0.715038i	−0.144054	−	0.215592i	0.752425	−	0.658678i	\(-0.228884\pi\)
							−0.896479	+	0.443085i	\(0.853884\pi\)
\(13\)	1.75816	−	2.63127i	0.487626	−	0.729784i	−0.503312	−	0.864105i	\(-0.667885\pi\)
							0.990938	+	0.134321i	\(0.0428854\pi\)
\(17\)			5.79821i			1.40627i	0.711055	+	0.703136i	\(0.248218\pi\)
							−0.711055	+	0.703136i	\(0.751782\pi\)
\(19\)	−6.91106	+	1.37469i	−1.58551	+	0.315377i	−0.907621	−	0.419791i	\(-0.862103\pi\)
							−0.677885	+	0.735168i	\(0.737103\pi\)
\(23\)	2.31266	−	5.58326i	0.482223	−	1.16419i	−0.476327	−	0.879268i	\(-0.658032\pi\)
							0.958551	−	0.284922i	\(-0.0919677\pi\)
\(29\)	0.531148	−	0.794919i	0.0986317	−	0.147613i	−0.778872	−	0.627184i	\(-0.784208\pi\)
							0.877503	+	0.479571i	\(0.159208\pi\)
\(31\)	−3.54205			−0.636171			−0.318085	−	0.948062i	\(-0.603040\pi\)
							−0.318085	+	0.948062i	\(0.603040\pi\)
\(37\)	1.99593	−	1.33364i	0.328128	−	0.219248i	−0.380583	−	0.924747i	\(-0.624277\pi\)
							0.708712	+	0.705498i	\(0.249277\pi\)
\(41\)	−6.95767	+	2.88196i	−1.08661	+	0.450087i	−0.852822	−	0.522202i	\(-0.825111\pi\)
							−0.233783	+	0.972289i	\(0.575111\pi\)
\(43\)	−7.34484	−	1.46098i	−1.12008	−	0.222797i	−0.399881	−	0.916567i	\(-0.630948\pi\)
							−0.720197	+	0.693770i	\(0.755948\pi\)
\(47\)	9.88727			1.44221			0.721103	−	0.692828i	\(-0.243635\pi\)
							0.721103	+	0.692828i	\(0.243635\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.2.bd.a.283.45	yes	368
5.2	odd	4		320.2.bj.a.27.23	yes	368
64.19	odd	16		320.2.bj.a.83.23	yes	368
320.147	even	16	inner	320.2.bd.a.147.45	✓	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.bd.a.147.45	✓	368	320.147	even	16	inner
320.2.bd.a.283.45	yes	368	1.1	even	1	trivial
320.2.bj.a.27.23	yes	368	5.2	odd	4
320.2.bj.a.83.23	yes	368	64.19	odd	16