Embedded newform 475.2.bc.a.396.8

• Label: 475.2.bc.a.396.8
• Level: $475$
• Weight: $2$
• Character: 475.396
• 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			396.8
Character	\(\chi\)	\(=\)	475.396
Dual form			475.2.bc.a.6.8

$q$-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 \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{2}{9}\right)\)

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.bc.a.6.8	✓	1152	475.6	even	45	inner
475.2.bc.a.196.41	yes	1152	19.6	even	9	inner
475.2.bc.a.206.41	yes	1152	25.6	even	5	inner
475.2.bc.a.396.8	yes	1152	1.1	even	1	trivial