Embedded newform 775.2.bz.a.58.19

• Label: 775.2.bz.a.58.19
• Level: $775$
• Weight: $2$
• Character: 775.58
• Analytic conductor: $6.188$
• Analytic rank: $0$
• Dimension: $624$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 775 = 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	775.bz (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.18840615665\)
Analytic rank:	\(0\)
Dimension:	\(624\)
Relative dimension:	\(78\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			58.19
Character	\(\chi\)	\(=\)	775.58
Dual form			775.2.bz.a.147.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.270232 + 1.70618i) q^{2} +(0.232154 + 0.118288i) q^{3} +(-0.935906 - 0.304094i) q^{4} +(-1.45606 - 1.69703i) q^{5} +(-0.264556 + 0.364130i) q^{6} +(-0.629780 + 3.97627i) q^{7} +(-0.796737 + 1.56368i) q^{8} +(-1.72345 - 2.37213i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 624 q - 6 q^{2} - 10 q^{4} - 16 q^{5} + 2 q^{7} - 20 q^{8} - 20 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/775\mathbb{Z}\right)^\times\).

\(n\)	\(251\)	\(652\)
\(\chi(n)\)	\(e\left(\frac{1}{10}\right)\)	\(e\left(\frac{3}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.270232	+	1.70618i	−0.191083	+	1.20645i	0.686538	+	0.727094i	\(0.259130\pi\)
							−0.877621	+	0.479356i	\(0.840870\pi\)
\(3\)	0.232154	+	0.118288i	0.134034	+	0.0682937i	0.519721	−	0.854336i	\(-0.326036\pi\)
							−0.385688	+	0.922629i	\(0.626036\pi\)
\(5\)	−1.45606	−	1.69703i	−0.651168	−	0.758934i
							\(7\)	−0.629780	+	3.97627i	−0.238034	+	1.50289i	0.521966	+	0.852966i	\(0.325199\pi\)
−0.760000	+	0.649923i	\(0.774801\pi\)
\(11\)	1.76057	−	2.42322i	0.530832	−	0.730628i	−0.456425	−	0.889762i	\(-0.650870\pi\)
							0.987257	+	0.159134i	\(0.0508702\pi\)
\(13\)	−2.45238	+	0.388419i	−0.680168	+	0.107728i	−0.486953	−	0.873428i	\(-0.661892\pi\)
							−0.193215	+	0.981156i	\(0.561892\pi\)
\(17\)	−0.800499	+	1.57107i	−0.194149	+	0.381040i	−0.967474	−	0.252971i	\(-0.918592\pi\)
							0.773325	+	0.634010i	\(0.218592\pi\)
\(19\)	−3.27834	−	1.06520i	−0.752102	−	0.244373i	−0.0922166	−	0.995739i	\(-0.529395\pi\)
							−0.659885	+	0.751366i	\(0.729395\pi\)
\(23\)	−0.278062	−	0.545728i	−0.0579800	−	0.113792i	0.860198	−	0.509960i	\(-0.170340\pi\)
							−0.918178	+	0.396168i	\(0.870340\pi\)
\(29\)	−2.56277	−	1.86196i	−0.475894	−	0.345757i	0.323840	−	0.946112i	\(-0.395026\pi\)
							−0.799734	+	0.600355i	\(0.795026\pi\)
\(31\)	−5.12788	−	2.16907i	−0.920994	−	0.389576i
							\(37\)	0.599269	−	0.305343i	0.0985192	−	0.0501981i	−0.404037	−	0.914743i	\(-0.632393\pi\)
0.502556	+	0.864545i	\(0.332393\pi\)
\(41\)	6.48439			1.01269			0.506346	−	0.862330i	\(-0.330996\pi\)
							0.506346	+	0.862330i	\(0.330996\pi\)
\(43\)	−8.74091	−	4.45372i	−1.33298	−	0.679185i	−0.365184	−	0.930935i	\(-0.618994\pi\)
							−0.967792	+	0.251750i	\(0.918994\pi\)
\(47\)	−3.11475	−	1.58704i	−0.454333	−	0.231494i	0.211821	−	0.977308i	\(-0.432061\pi\)
							−0.666154	+	0.745814i	\(0.732061\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	775.2.bz.a.58.19	yes	624
25.22	odd	20		775.2.br.a.647.60	yes	624
31.23	odd	10		775.2.br.a.333.60	✓	624
775.147	even	20	inner	775.2.bz.a.147.19	yes	624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
775.2.br.a.333.60	✓	624	31.23	odd	10
775.2.br.a.647.60	yes	624	25.22	odd	20
775.2.bz.a.58.19	yes	624	1.1	even	1	trivial
775.2.bz.a.147.19	yes	624	775.147	even	20	inner