Embedded newform 475.2.n.b.39.18

• Label: 475.2.n.b.39.18
• Level: $475$
• Weight: $2$
• Character: 475.39
• Analytic conductor: $3.793$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	475.n (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.79289409601\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(24\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			39.18
Character	\(\chi\)	\(=\)	475.39
Dual form			475.2.n.b.134.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.923507 + 1.27110i) q^{2} +(3.21001 + 1.04299i) q^{3} +(-0.144793 + 0.445626i) q^{4} +(-1.48513 - 1.67165i) q^{5} +(1.63872 + 5.04345i) q^{6} -3.13596i q^{7} +(2.28838 - 0.743540i) q^{8} +(6.78926 + 4.93269i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 26 q^{4} - 4 q^{5} - 2 q^{6} + 28 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{3}{10}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.923507	+	1.27110i	0.653018	+	0.898803i	0.999225	−	0.0393555i	\(-0.0125305\pi\)
							−0.346207	+	0.938158i	\(0.612530\pi\)
\(3\)	3.21001	+	1.04299i	1.85330	+	0.602173i	0.996208	+	0.0870083i	\(0.0277307\pi\)
							0.857091	+	0.515165i	\(0.172269\pi\)
\(5\)	−1.48513	−	1.67165i	−0.664169	−	0.747583i
							\(7\)		−	3.13596i		−	1.18528i	−0.805466	−	0.592642i	\(-0.798085\pi\)
0.805466	−	0.592642i	\(-0.201915\pi\)
\(11\)	−4.29767	+	3.12244i	−1.29579	+	0.941450i	−0.999905	−	0.0137720i	\(-0.995616\pi\)
							−0.295890	+	0.955222i	\(0.595616\pi\)
\(13\)	−0.0967648	+	0.133185i	−0.0268377	+	0.0369390i	−0.822225	−	0.569163i	\(-0.807268\pi\)
							0.795387	+	0.606102i	\(0.207268\pi\)
\(17\)	−5.13676	+	1.66904i	−1.24585	+	0.404800i	−0.856431	−	0.516261i	\(-0.827323\pi\)
							−0.389417	+	0.921062i	\(0.627323\pi\)
\(19\)	0.309017	+	0.951057i	0.0708934	+	0.218187i
							\(23\)	−2.70062	−	3.71709i	−0.563119	−	0.775067i	0.428600	−	0.903494i	\(-0.359007\pi\)
−0.991719	+	0.128428i	\(0.959007\pi\)
\(29\)	−0.235928	+	0.726111i	−0.0438107	+	0.134835i	−0.970569	−	0.240822i	\(-0.922583\pi\)
							0.926759	+	0.375657i	\(0.122583\pi\)
\(31\)	1.33174	+	4.09868i	0.239188	+	0.736145i	0.996538	+	0.0831359i	\(0.0264936\pi\)
							−0.757350	+	0.653009i	\(0.773506\pi\)
\(37\)	3.19237	−	4.39392i	0.524822	−	0.722355i	−0.461508	−	0.887136i	\(-0.652692\pi\)
							0.986330	+	0.164781i	\(0.0526916\pi\)
\(41\)	−6.62063	−	4.81017i	−1.03397	−	0.751222i	−0.0648691	−	0.997894i	\(-0.520663\pi\)
							−0.969099	+	0.246672i	\(0.920663\pi\)
\(43\)		−	4.85523i		−	0.740416i	−0.928949	−	0.370208i	\(-0.879286\pi\)
							0.928949	−	0.370208i	\(-0.120714\pi\)
\(47\)	2.03123	+	0.659988i	0.296286	+	0.0962692i	0.453388	−	0.891313i	\(-0.350215\pi\)
							−0.157102	+	0.987582i	\(0.550215\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.n.b.39.18	✓	96
25.9	even	10	inner	475.2.n.b.134.18	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.n.b.39.18	✓	96	1.1	even	1	trivial
475.2.n.b.134.18	yes	96	25.9	even	10	inner