Embedded newform 475.2.v.b.113.27

• Label: 475.2.v.b.113.27
• Level: $475$
• Weight: $2$
• Character: 475.113
• Analytic conductor: $3.793$
• Analytic rank: $0$
• Dimension: $368$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	475.v (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.79289409601\)
Analytic rank:	\(0\)
Dimension:	\(368\)
Relative dimension:	\(46\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			113.27
Character	\(\chi\)	\(=\)	475.113
Dual form			475.2.v.b.227.27

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.119250 + 0.752912i) q^{2} +(2.74225 + 1.39725i) q^{3} +(1.34946 - 0.438465i) q^{4} +(-2.19248 - 0.439353i) q^{5} +(-0.724991 + 2.23129i) q^{6} +(2.38864 - 2.38864i) q^{7} +(1.18320 + 2.32216i) q^{8} +(3.80429 + 5.23615i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 20 q^{4} - 16 q^{5} - 12 q^{6} - 8 q^{7} - 20 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{19}{20}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.119250	+	0.752912i	0.0843221	+	0.532389i	0.993302	+	0.115549i	\(0.0368626\pi\)
							−0.908980	+	0.416840i	\(0.863137\pi\)
\(3\)	2.74225	+	1.39725i	1.58324	+	0.806701i	0.999984	−	0.00558807i	\(-0.00177875\pi\)
							0.583255	+	0.812289i	\(0.301779\pi\)
\(5\)	−2.19248	−	0.439353i	−0.980507	−	0.196485i
							\(7\)	2.38864	−	2.38864i	0.902819	−	0.902819i	−0.0928599	−	0.995679i	\(-0.529601\pi\)
0.995679	+	0.0928599i	\(0.0296009\pi\)
\(11\)	−3.69396	−	2.68382i	−1.11377	−	0.809203i	−0.130519	−	0.991446i	\(-0.541664\pi\)
							−0.983254	+	0.182243i	\(0.941664\pi\)
\(13\)	−0.649772	+	4.10250i	−0.180214	+	1.13783i	0.717274	+	0.696791i	\(0.245390\pi\)
							−0.897489	+	0.441038i	\(0.854610\pi\)
\(17\)	−0.350920	−	0.688719i	−0.0851106	−	0.167039i	0.844528	−	0.535511i	\(-0.179881\pi\)
							−0.929639	+	0.368472i	\(0.879881\pi\)
\(19\)	−2.47938	−	3.58506i	−0.568809	−	0.822469i
							\(23\)	1.15058	+	7.26447i	0.239912	+	1.51475i	0.753922	+	0.656963i	\(0.228159\pi\)
−0.514010	+	0.857784i	\(0.671841\pi\)
\(29\)	−2.66427	−	8.19977i	−0.494742	−	1.52266i	−0.817358	−	0.576130i	\(-0.804562\pi\)
							0.322616	−	0.946530i	\(-0.395438\pi\)
\(31\)	1.19271	+	0.387536i	0.214217	+	0.0696034i	0.414160	−	0.910204i	\(-0.364075\pi\)
							−0.199942	+	0.979808i	\(0.564075\pi\)
\(37\)	−5.77466	−	0.914616i	−0.949348	−	0.150362i	−0.337489	−	0.941329i	\(-0.609578\pi\)
							−0.611858	+	0.790967i	\(0.709578\pi\)
\(41\)	−0.0652943	−	0.0898699i	−0.0101973	−	0.0140353i	0.803888	−	0.594781i	\(-0.202761\pi\)
							−0.814085	+	0.580745i	\(0.802761\pi\)
\(43\)	−5.11966	−	5.11966i	−0.780740	−	0.780740i	0.199216	−	0.979956i	\(-0.436161\pi\)
							−0.979956	+	0.199216i	\(0.936161\pi\)
\(47\)	−8.23188	−	4.19435i	−1.20074	−	0.611809i	−0.264917	−	0.964271i	\(-0.585344\pi\)
							−0.935826	+	0.352462i	\(0.885344\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.v.b.113.27	yes	368
19.18	odd	2	inner	475.2.v.b.113.20	✓	368
25.2	odd	20	inner	475.2.v.b.227.20	yes	368
475.227	even	20	inner	475.2.v.b.227.27	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.v.b.113.20	✓	368	19.18	odd	2	inner
475.2.v.b.113.27	yes	368	1.1	even	1	trivial
475.2.v.b.227.20	yes	368	25.2	odd	20	inner
475.2.v.b.227.27	yes	368	475.227	even	20	inner