Embedded newform 475.2.v.b.227.31

• Label: 475.2.v.b.227.31
• Level: $475$
• Weight: $2$
• Character: 475.227
• 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			227.31
Character	\(\chi\)	\(=\)	475.227
Dual form			475.2.v.b.113.31

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.187582 - 1.18434i) q^{2} +(1.65503 - 0.843279i) q^{3} +(0.534629 + 0.173712i) q^{4} +(-2.13438 + 0.666643i) q^{5} +(-0.688280 - 2.11831i) q^{6} +(-3.01019 - 3.01019i) q^{7} +(1.39479 - 2.73742i) q^{8} +(0.264646 - 0.364253i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 20 q^{4} - 16 q^{5} - 12 q^{6} - 8 q^{7} - 20 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.187582	−	1.18434i	0.132640	−	0.837458i	−0.828216	−	0.560409i	\(-0.810644\pi\)
							0.960856	−	0.277048i	\(-0.0893562\pi\)
\(3\)	1.65503	−	0.843279i	0.955532	−	0.486868i	0.0945591	−	0.995519i	\(-0.469856\pi\)
							0.860972	+	0.508652i	\(0.169856\pi\)
\(5\)	−2.13438	+	0.666643i	−0.954525	+	0.298132i
							\(7\)	−3.01019	−	3.01019i	−1.13774	−	1.13774i	−0.988854	−	0.148891i	\(-0.952430\pi\)
−0.148891	−	0.988854i	\(-0.547570\pi\)
\(11\)	3.77293	−	2.74119i	1.13758	−	0.826501i	0.150801	−	0.988564i	\(-0.451815\pi\)
							0.986780	+	0.162063i	\(0.0518147\pi\)
\(13\)	0.156437	+	0.987704i	0.0433878	+	0.273940i	0.999839	−	0.0179531i	\(-0.00571496\pi\)
							−0.956451	+	0.291893i	\(0.905715\pi\)
\(17\)	2.40957	−	4.72904i	0.584406	−	1.14696i	−0.389716	−	0.920935i	\(-0.627427\pi\)
							0.974121	−	0.226025i	\(-0.0725733\pi\)
\(19\)	−3.96595	−	1.80865i	−0.909852	−	0.414933i
							\(23\)	−0.503584	+	3.17950i	−0.105005	+	0.662973i	0.877898	+	0.478847i	\(0.158945\pi\)
−0.982903	+	0.184125i	\(0.941055\pi\)
\(29\)	−2.74099	+	8.43591i	−0.508990	+	1.56651i	0.284969	+	0.958537i	\(0.408017\pi\)
							−0.793959	+	0.607972i	\(0.791983\pi\)
\(31\)	6.28092	−	2.04079i	1.12809	−	0.366538i	0.315238	−	0.949013i	\(-0.397916\pi\)
							0.812849	+	0.582475i	\(0.197916\pi\)
\(37\)	9.94479	−	1.57510i	1.63491	−	0.258945i	0.729658	−	0.683812i	\(-0.239679\pi\)
							0.905256	+	0.424867i	\(0.139679\pi\)
\(41\)	−4.13217	+	5.68745i	−0.645337	+	0.888231i	−0.998886	−	0.0471855i	\(-0.984975\pi\)
							0.353549	+	0.935416i	\(0.384975\pi\)
\(43\)	5.07203	−	5.07203i	0.773477	−	0.773477i	−0.205236	−	0.978713i	\(-0.565796\pi\)
							0.978713	+	0.205236i	\(0.0657961\pi\)
\(47\)	−2.49935	+	1.27348i	−0.364568	+	0.185757i	−0.626672	−	0.779283i	\(-0.715584\pi\)
							0.262104	+	0.965040i	\(0.415584\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.v.b.227.31	yes	368
19.18	odd	2	inner	475.2.v.b.227.16	yes	368
25.13	odd	20	inner	475.2.v.b.113.16	✓	368
475.113	even	20	inner	475.2.v.b.113.31	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.v.b.113.16	✓	368	25.13	odd	20	inner
475.2.v.b.113.31	yes	368	475.113	even	20	inner
475.2.v.b.227.16	yes	368	19.18	odd	2	inner
475.2.v.b.227.31	yes	368	1.1	even	1	trivial