Embedded newform 475.2.n.a.39.7

• Label: 475.2.n.a.39.7
• Level: $475$
• Weight: $2$
• Character: 475.39
• Analytic conductor: $3.793$
• Analytic rank: $0$
• Dimension: $80$
• 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:	\(80\)
Relative dimension:	\(20\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			39.7
Character	\(\chi\)	\(=\)	475.39
Dual form			475.2.n.a.134.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.708585 - 0.975283i) q^{2} +(-1.48229 - 0.481625i) q^{3} +(0.168949 - 0.519971i) q^{4} +(-1.59939 + 1.56267i) q^{5} +(0.580606 + 1.78692i) q^{6} -2.31635i q^{7} +(-2.91986 + 0.948721i) q^{8} +(-0.461836 - 0.335543i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 16 q^{4} - 3 q^{5} + 6 q^{6} + 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.708585	−	0.975283i	−0.501045	−	0.689629i	0.481332	−	0.876538i	\(-0.340153\pi\)
							−0.982377	+	0.186909i	\(0.940153\pi\)
\(3\)	−1.48229	−	0.481625i	−0.855799	−	0.278066i	−0.151926	−	0.988392i	\(-0.548548\pi\)
							−0.703873	+	0.710326i	\(0.748548\pi\)
\(5\)	−1.59939	+	1.56267i	−0.715271	+	0.698847i
							\(7\)		−	2.31635i		−	0.875498i	−0.899097	−	0.437749i	\(-0.855776\pi\)
0.899097	−	0.437749i	\(-0.144224\pi\)
\(11\)	2.91430	−	2.11737i	0.878696	−	0.638410i	−0.0542103	−	0.998530i	\(-0.517264\pi\)
							0.932906	+	0.360120i	\(0.117264\pi\)
\(13\)	−3.05967	+	4.21128i	−0.848601	+	1.16800i	0.135568	+	0.990768i	\(0.456714\pi\)
							−0.984169	+	0.177231i	\(0.943286\pi\)
\(17\)	−2.61335	+	0.849128i	−0.633830	+	0.205944i	−0.608271	−	0.793730i	\(-0.708137\pi\)
							−0.0255590	+	0.999673i	\(0.508137\pi\)
\(19\)	−0.309017	−	0.951057i	−0.0708934	−	0.218187i
							\(23\)	4.43918	+	6.11000i	0.925633	+	1.27402i	0.961539	+	0.274669i	\(0.0885682\pi\)
−0.0359063	+	0.999355i	\(0.511432\pi\)
\(29\)	−2.20964	+	6.80057i	−0.410320	+	1.26283i	0.506051	+	0.862503i	\(0.331105\pi\)
							−0.916371	+	0.400331i	\(0.868895\pi\)
\(31\)	−0.648666	−	1.99639i	−0.116504	−	0.358562i	0.875754	−	0.482758i	\(-0.160365\pi\)
							−0.992258	+	0.124196i	\(0.960365\pi\)
\(37\)	1.04745	−	1.44169i	0.172199	−	0.237012i	−0.714191	−	0.699951i	\(-0.753205\pi\)
							0.886390	+	0.462939i	\(0.153205\pi\)
\(41\)	3.05247	+	2.21775i	0.476716	+	0.346354i	0.800053	−	0.599929i	\(-0.204805\pi\)
							−0.323337	+	0.946284i	\(0.604805\pi\)
\(43\)			1.24098i			0.189247i	0.995513	+	0.0946236i	\(0.0301648\pi\)
							−0.995513	+	0.0946236i	\(0.969835\pi\)
\(47\)	−8.48796	−	2.75790i	−1.23810	−	0.402282i	−0.384455	−	0.923144i	\(-0.625611\pi\)
							−0.853641	+	0.520862i	\(0.825611\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.n.a.39.7	✓	80
25.9	even	10	inner	475.2.n.a.134.7	yes	80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.n.a.39.7	✓	80	1.1	even	1	trivial
475.2.n.a.134.7	yes	80	25.9	even	10	inner