Embedded newform 475.2.bc.a.6.8

• Label: 475.2.bc.a.6.8
• Level: $475$
• Weight: $2$
• Character: 475.6
• Analytic conductor: $3.793$
• Analytic rank: $0$
• Dimension: $1152$
• CM: no
• 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			6.8
Character	\(\chi\)	\(=\)	475.6
Dual form			475.2.bc.a.396.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.79524 - 1.12179i) q^{2} +(-0.560140 + 0.160617i) q^{3} +(1.08772 + 2.23017i) q^{4} +(-1.94606 + 1.10130i) q^{5} +(1.18576 + 0.340012i) q^{6} +(-1.52823 + 2.64697i) q^{7} +(0.106498 - 1.01326i) q^{8} +(-2.25619 + 1.40982i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1152 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 24 q^{5} - 18 q^{6} - 30 q^{7} - 9 q^{8} - 18 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{7}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.79524	−	1.12179i	−1.26942	−	0.793224i	−0.283268	−	0.959041i	\(-0.591419\pi\)
							−0.986157	+	0.165816i	\(0.946974\pi\)
\(3\)	−0.560140	+	0.160617i	−0.323397	+	0.0927325i	−0.433408	−	0.901198i	\(-0.642689\pi\)
							0.110011	+	0.993930i	\(0.464911\pi\)
\(5\)	−1.94606	+	1.10130i	−0.870303	+	0.492517i
							\(7\)	−1.52823	+	2.64697i	−0.577616	+	1.00046i	0.418136	+	0.908384i	\(0.362683\pi\)
−0.995752	+	0.0920757i	\(0.970650\pi\)
\(11\)	0.275390	−	0.305852i	0.0830333	−	0.0922178i	−0.700196	−	0.713950i	\(-0.746904\pi\)
							0.783230	+	0.621733i	\(0.213571\pi\)
\(13\)	−0.127584	−	3.65354i	−0.0353855	−	1.01331i	−0.879135	−	0.476572i	\(-0.841879\pi\)
							0.843750	−	0.536737i	\(-0.180343\pi\)
\(17\)	−4.10028	+	0.286720i	−0.994465	+	0.0695397i	−0.557706	−	0.830039i	\(-0.688318\pi\)
							−0.436759	+	0.899579i	\(0.643874\pi\)
\(19\)	3.72868	−	2.25764i	0.855419	−	0.517937i
							\(23\)	0.408444	−	0.0574031i	0.0851666	−	0.0119694i	−0.0964611	−	0.995337i	\(-0.530752\pi\)
0.181628	+	0.983367i	\(0.441863\pi\)
\(29\)	3.00947	+	0.210443i	0.558845	+	0.0390783i	0.346387	−	0.938092i	\(-0.387409\pi\)
							0.212458	+	0.977170i	\(0.431853\pi\)
\(31\)	4.28759	−	1.90896i	0.770073	−	0.342859i	0.0161917	−	0.999869i	\(-0.494846\pi\)
							0.753882	+	0.657010i	\(0.228179\pi\)
\(37\)	−0.480495	+	1.47881i	−0.0789929	+	0.243115i	−0.982753	−	0.184925i	\(-0.940796\pi\)
							0.903760	+	0.428040i	\(0.140796\pi\)
\(41\)	0.418228	−	0.535307i	0.0653162	−	0.0836010i	−0.754243	−	0.656595i	\(-0.771996\pi\)
							0.819559	+	0.572994i	\(0.194218\pi\)
\(43\)	−1.37612	−	7.80435i	−0.209856	−	1.19015i	−0.889613	−	0.456715i	\(-0.849026\pi\)
							0.679757	−	0.733438i	\(-0.262085\pi\)
\(47\)	−4.65049	−	0.325194i	−0.678344	−	0.0474344i	−0.273571	−	0.961852i	\(-0.588205\pi\)
							−0.404773	+	0.914417i	\(0.632649\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.bc.a.6.8	✓	1152
19.16	even	9	inner	475.2.bc.a.206.41	yes	1152
25.21	even	5	inner	475.2.bc.a.196.41	yes	1152
475.396	even	45	inner	475.2.bc.a.396.8	yes	1152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.bc.a.6.8	✓	1152	1.1	even	1	trivial
475.2.bc.a.196.41	yes	1152	25.21	even	5	inner
475.2.bc.a.206.41	yes	1152	19.16	even	9	inner
475.2.bc.a.396.8	yes	1152	475.396	even	45	inner