Embedded newform 475.2.bc.a.206.38

• Label: 475.2.bc.a.206.38
• Level: $475$
• Weight: $2$
• Character: 475.206
• Analytic conductor: $3.793$
• Analytic rank: $0$
• Dimension: $1152$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	475.bc (of order \(45\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.79289409601\)
Analytic rank:	\(0\)
Dimension:	\(1152\)
Relative dimension:	\(48\) over \(\Q(\zeta_{45})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{45}]$

Embedding invariants

Embedding label			206.38
Character	\(\chi\)	\(=\)	475.206
Dual form			475.2.bc.a.196.38

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.655951 - 1.62353i) q^{2} +(-0.221181 + 0.327915i) q^{3} +(-0.766914 - 0.740600i) q^{4} +(0.336766 - 2.21056i) q^{5} +(0.387297 + 0.574191i) q^{6} +(1.66674 + 2.88687i) q^{7} +(1.49386 - 0.665109i) q^{8} +(1.06521 + 2.63649i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1152 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 24 q^{5} - 18 q^{6} - 30 q^{7} - 9 q^{8} - 18 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{2}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.655951	−	1.62353i	0.463827	−	1.14801i	−0.496226	−	0.868194i	\(-0.665281\pi\)
							0.960053	−	0.279819i	\(-0.0902744\pi\)
\(3\)	−0.221181	+	0.327915i	−0.127699	+	0.189322i	−0.887465	−	0.460875i	\(-0.847536\pi\)
							0.759766	+	0.650197i	\(0.225313\pi\)
\(5\)	0.336766	−	2.21056i	0.150606	−	0.988594i
							\(7\)	1.66674	+	2.88687i	0.629967	+	1.09113i	0.987558	+	0.157256i	\(0.0502648\pi\)
−0.357591	+	0.933878i	\(0.616402\pi\)
\(11\)	2.91076	+	0.618700i	0.877626	+	0.186545i	0.624629	−	0.780922i	\(-0.285250\pi\)
							0.252997	+	0.967467i	\(0.418584\pi\)
\(13\)	0.606734	−	0.776584i	0.168278	−	0.215386i	−0.696478	−	0.717578i	\(-0.745251\pi\)
							0.864756	+	0.502192i	\(0.167473\pi\)
\(17\)	−0.624786	−	2.50588i	−0.151533	−	0.607765i	−0.997248	−	0.0741313i	\(-0.976382\pi\)
							0.845716	−	0.533634i	\(-0.179174\pi\)
\(19\)	−4.19110	−	1.19778i	−0.961505	−	0.274789i
							\(23\)	−0.315421	+	0.167713i	−0.0657699	+	0.0349705i	−0.502031	−	0.864850i	\(-0.667414\pi\)
0.436261	+	0.899820i	\(0.356302\pi\)
\(29\)	−1.92364	+	7.71529i	−0.357211	+	1.43269i	0.475857	+	0.879523i	\(0.342138\pi\)
							−0.833068	+	0.553171i	\(0.813418\pi\)
\(31\)	0.799934	−	7.61086i	0.143672	−	1.36695i	−0.650615	−	0.759408i	\(-0.725489\pi\)
							0.794287	−	0.607543i	\(-0.207845\pi\)
\(37\)	−1.80821	+	5.56510i	−0.297268	+	0.914897i	0.685182	+	0.728372i	\(0.259723\pi\)
							−0.982450	+	0.186525i	\(0.940277\pi\)
\(41\)	0.168009	+	4.81116i	0.0262387	+	0.751377i	0.939208	+	0.343349i	\(0.111561\pi\)
							−0.912969	+	0.408028i	\(0.866217\pi\)
\(43\)	1.80177	−	10.2183i	0.274767	−	1.55828i	−0.464933	−	0.885346i	\(-0.653922\pi\)
							0.739700	−	0.672936i	\(-0.234967\pi\)
\(47\)	2.41568	−	9.68878i	0.352364	−	1.41325i	−0.488764	−	0.872416i	\(-0.662552\pi\)
							0.841127	−	0.540837i	\(-0.181893\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.bc.a.206.38	yes	1152
19.6	even	9	inner	475.2.bc.a.6.11	✓	1152
25.21	even	5	inner	475.2.bc.a.396.11	yes	1152
475.196	even	45	inner	475.2.bc.a.196.38	yes	1152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.bc.a.6.11	✓	1152	19.6	even	9	inner
475.2.bc.a.196.38	yes	1152	475.196	even	45	inner
475.2.bc.a.206.38	yes	1152	1.1	even	1	trivial
475.2.bc.a.396.11	yes	1152	25.21	even	5	inner