Embedded newform 320.2.bf.a.229.17

• Label: 320.2.bf.a.229.17
• 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			229.17
Character	\(\chi\)	\(=\)	320.229
Dual form			320.2.bf.a.109.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.656591 - 1.25255i) q^{2} +(1.44172 + 0.963326i) q^{3} +(-1.13778 + 1.64483i) q^{4} +(1.41181 - 1.73401i) q^{5} +(0.259996 - 2.43834i) q^{6} +(-2.06273 - 0.854410i) q^{7} +(2.80729 + 0.345141i) q^{8} +(0.00250772 + 0.00605416i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 16 q^{4} - 8 q^{5} - 16 q^{6} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.656591	−	1.25255i	−0.464280	−	0.885688i
							\(3\)	1.44172	+	0.963326i	0.832377	+	0.556177i	0.897151	−	0.441723i	\(-0.145633\pi\)
−0.0647741	+	0.997900i	\(0.520633\pi\)
\(5\)	1.41181	−	1.73401i	0.631380	−	0.775474i
							\(7\)	−2.06273	−	0.854410i	−0.779638	−	0.322937i	−0.0428686	−	0.999081i	\(-0.513650\pi\)
−0.736769	+	0.676144i	\(0.763650\pi\)
\(11\)	4.25638	−	2.84402i	1.28335	−	0.857504i	0.288360	−	0.957522i	\(-0.406890\pi\)
							0.994986	+	0.100018i	\(0.0318900\pi\)
\(13\)	−2.24095	+	0.445753i	−0.621528	+	0.123630i	−0.495802	−	0.868435i	\(-0.665126\pi\)
							−0.125726	+	0.992065i	\(0.540126\pi\)
\(17\)	1.49139	+	1.49139i	0.361715	+	0.361715i	0.864444	−	0.502729i	\(-0.167671\pi\)
							−0.502729	+	0.864444i	\(0.667671\pi\)
\(19\)	6.78349	−	1.34932i	1.55624	−	0.309555i	0.659359	−	0.751829i	\(-0.270828\pi\)
							0.896880	+	0.442273i	\(0.145828\pi\)
\(23\)	1.96401	+	4.74154i	0.409525	+	0.988680i	0.985263	+	0.171046i	\(0.0547146\pi\)
							−0.575739	+	0.817634i	\(0.695285\pi\)
\(29\)	−1.63270	−	1.09094i	−0.303185	−	0.202582i	0.394666	−	0.918825i	\(-0.370860\pi\)
							−0.697851	+	0.716243i	\(0.745860\pi\)
\(31\)			4.02324i			0.722596i	0.932450	+	0.361298i	\(0.117666\pi\)
							−0.932450	+	0.361298i	\(0.882334\pi\)
\(37\)	−0.862457	+	4.33586i	−0.141787	+	0.712812i	0.842843	+	0.538159i	\(0.180880\pi\)
							−0.984630	+	0.174652i	\(0.944120\pi\)
\(41\)	−8.18555	+	3.39057i	−1.27837	+	0.529518i	−0.915498	−	0.402322i	\(-0.868203\pi\)
							−0.362871	+	0.931840i	\(0.618203\pi\)
\(43\)	−3.13582	+	2.09529i	−0.478208	+	0.319528i	−0.771207	−	0.636584i	\(-0.780347\pi\)
							0.292999	+	0.956113i	\(0.405347\pi\)
\(47\)	−4.72576	−	4.72576i	−0.689323	−	0.689323i	0.272759	−	0.962082i	\(-0.412064\pi\)
							−0.962082	+	0.272759i	\(0.912064\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.2.bf.a.229.17	yes	368
5.4	even	2	inner	320.2.bf.a.229.30	yes	368
64.45	even	16	inner	320.2.bf.a.109.30	yes	368
320.109	even	16	inner	320.2.bf.a.109.17	✓	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.bf.a.109.17	✓	368	320.109	even	16	inner
320.2.bf.a.109.30	yes	368	64.45	even	16	inner
320.2.bf.a.229.17	yes	368	1.1	even	1	trivial
320.2.bf.a.229.30	yes	368	5.4	even	2	inner