Embedded newform 320.2.bf.a.229.7

• Label: 320.2.bf.a.229.7
• Level: $320$
• Weight: $2$
• Character: 320.229
• Analytic conductor: $2.555$
• Analytic rank: $0$
• Dimension: $368$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	320.bf (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			229.7
Character	\(\chi\)	\(=\)	320.229
Dual form			320.2.bf.a.109.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.29070 + 0.578016i) q^{2} +(-1.32779 - 0.887198i) q^{3} +(1.33179 - 1.49209i) q^{4} +(-2.01398 + 0.971546i) q^{5} +(2.22658 + 0.377622i) q^{6} +(2.49852 + 1.03492i) q^{7} +(-0.856491 + 2.69563i) q^{8} +(-0.172155 - 0.415618i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 16 q^{4} - 8 q^{5} - 16 q^{6} - 16 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.29070	+	0.578016i	−0.912660	+	0.408719i
							\(3\)	−1.32779	−	0.887198i	−0.766598	−	0.512224i	0.109717	−	0.993963i	\(-0.465005\pi\)
−0.876315	+	0.481739i	\(0.840005\pi\)
\(5\)	−2.01398	+	0.971546i	−0.900677	+	0.434489i
							\(7\)	2.49852	+	1.03492i	0.944351	+	0.391163i	0.801105	−	0.598524i	\(-0.204246\pi\)
0.143246	+	0.989687i	\(0.454246\pi\)
\(11\)	0.425142	−	0.284071i	0.128185	−	0.0856505i	−0.489825	−	0.871821i	\(-0.662939\pi\)
							0.618010	+	0.786170i	\(0.287939\pi\)
\(13\)	2.48384	−	0.494067i	0.688893	−	0.137029i	0.161783	−	0.986826i	\(-0.448275\pi\)
							0.527110	+	0.849797i	\(0.323275\pi\)
\(17\)	−4.97326	−	4.97326i	−1.20619	−	1.20619i	−0.972251	−	0.233941i	\(-0.924838\pi\)
							−0.233941	−	0.972251i	\(-0.575162\pi\)
\(19\)	−6.54697	+	1.30227i	−1.50198	+	0.298762i	−0.876470	−	0.481456i	\(-0.840108\pi\)
							−0.625508	+	0.780218i	\(0.715108\pi\)
\(23\)	−1.66726	−	4.02513i	−0.347648	−	0.839297i	−0.996897	−	0.0787207i	\(-0.974916\pi\)
							0.649248	−	0.760576i	\(-0.275084\pi\)
\(29\)	−7.69679	−	5.14283i	−1.42926	−	0.955000i	−0.998622	−	0.0524742i	\(-0.983289\pi\)
							−0.430636	−	0.902526i	\(-0.641711\pi\)
\(31\)		−	3.69276i		−	0.663239i	−0.943413	−	0.331619i	\(-0.892405\pi\)
							0.943413	−	0.331619i	\(-0.107595\pi\)
\(37\)	1.07816	−	5.42029i	0.177249	−	0.891090i	−0.785120	−	0.619344i	\(-0.787399\pi\)
							0.962369	−	0.271747i	\(-0.0876013\pi\)
\(41\)	7.15059	−	2.96187i	1.11673	−	0.462566i	0.253482	−	0.967340i	\(-0.418424\pi\)
							0.863252	+	0.504774i	\(0.168424\pi\)
\(43\)	−2.99129	+	1.99872i	−0.456168	+	0.304802i	−0.762343	−	0.647174i	\(-0.775951\pi\)
							0.306175	+	0.951975i	\(0.400951\pi\)
\(47\)	6.48347	+	6.48347i	0.945711	+	0.945711i	0.998600	−	0.0528890i	\(-0.0168430\pi\)
							−0.0528890	+	0.998600i	\(0.516843\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.2.bf.a.229.7	yes	368
5.4	even	2	inner	320.2.bf.a.229.40	yes	368
64.45	even	16	inner	320.2.bf.a.109.40	yes	368
320.109	even	16	inner	320.2.bf.a.109.7	✓	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.bf.a.109.7	✓	368	320.109	even	16	inner
320.2.bf.a.109.40	yes	368	64.45	even	16	inner
320.2.bf.a.229.7	yes	368	1.1	even	1	trivial
320.2.bf.a.229.40	yes	368	5.4	even	2	inner