Embedded newform 475.2.v.b.227.40

• Label: 475.2.v.b.227.40
• Level: $475$
• Weight: $2$
• Character: 475.227
• Analytic conductor: $3.793$
• Analytic rank: $0$
• Dimension: $368$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	475.v (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.79289409601\)
Analytic rank:	\(0\)
Dimension:	\(368\)
Relative dimension:	\(46\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			227.40
Character	\(\chi\)	\(=\)	475.227
Dual form			475.2.v.b.113.40

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.342643 - 2.16336i) q^{2} +(1.83797 - 0.936494i) q^{3} +(-2.66063 - 0.864490i) q^{4} +(2.23140 + 0.144412i) q^{5} +(-1.39621 - 4.29709i) q^{6} +(2.03794 + 2.03794i) q^{7} +(-0.793074 + 1.55649i) q^{8} +(0.737769 - 1.01545i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 20 q^{4} - 16 q^{5} - 12 q^{6} - 8 q^{7} - 20 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{1}{20}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.342643	−	2.16336i	0.242285	−	1.52973i	−0.503766	−	0.863840i	\(-0.668052\pi\)
							0.746051	−	0.665889i	\(-0.231948\pi\)
\(3\)	1.83797	−	0.936494i	1.06115	−	0.540685i	0.165855	−	0.986150i	\(-0.446962\pi\)
							0.895299	+	0.445465i	\(0.146962\pi\)
\(5\)	2.23140	+	0.144412i	0.997912	+	0.0645831i
							\(7\)	2.03794	+	2.03794i	0.770270	+	0.770270i	0.978154	−	0.207883i	\(-0.0666575\pi\)
−0.207883	+	0.978154i	\(0.566657\pi\)
\(11\)	−3.85460	+	2.80053i	−1.16221	+	0.844392i	−0.990055	−	0.140678i	\(-0.955072\pi\)
							−0.172151	+	0.985071i	\(0.555072\pi\)
\(13\)	−0.439617	−	2.77563i	−0.121928	−	0.769822i	−0.970564	−	0.240845i	\(-0.922576\pi\)
							0.848636	−	0.528978i	\(-0.177424\pi\)
\(17\)	3.20131	−	6.28292i	0.776431	−	1.52383i	−0.0737117	−	0.997280i	\(-0.523484\pi\)
							0.850143	−	0.526552i	\(-0.176516\pi\)
\(19\)	−4.19944	+	1.16820i	−0.963418	+	0.268003i
							\(23\)	−0.767791	+	4.84764i	−0.160096	+	1.01080i	0.768538	+	0.639805i	\(0.220985\pi\)
−0.928633	+	0.370999i	\(0.879015\pi\)
\(29\)	1.29261	−	3.97824i	0.240031	−	0.738740i	−0.756383	−	0.654129i	\(-0.773035\pi\)
							0.996414	−	0.0846108i	\(-0.0269647\pi\)
\(31\)	−5.55484	+	1.80488i	−0.997678	+	0.324165i	−0.761937	−	0.647651i	\(-0.775752\pi\)
							−0.235741	+	0.971816i	\(0.575752\pi\)
\(37\)	8.85636	−	1.40271i	1.45598	−	0.230604i	0.622266	−	0.782806i	\(-0.286212\pi\)
							0.833711	+	0.552202i	\(0.186212\pi\)
\(41\)	−5.47309	+	7.53307i	−0.854754	+	1.17647i	0.128041	+	0.991769i	\(0.459131\pi\)
							−0.982795	+	0.184699i	\(0.940869\pi\)
\(43\)	−4.90816	+	4.90816i	−0.748488	+	0.748488i	−0.974195	−	0.225707i	\(-0.927531\pi\)
							0.225707	+	0.974195i	\(0.427531\pi\)
\(47\)	5.61240	−	2.85966i	0.818653	−	0.417124i	0.00607810	−	0.999982i	\(-0.498065\pi\)
							0.812575	+	0.582857i	\(0.198065\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.v.b.227.40	yes	368
19.18	odd	2	inner	475.2.v.b.227.7	yes	368
25.13	odd	20	inner	475.2.v.b.113.7	✓	368
475.113	even	20	inner	475.2.v.b.113.40	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.v.b.113.7	✓	368	25.13	odd	20	inner
475.2.v.b.113.40	yes	368	475.113	even	20	inner
475.2.v.b.227.7	yes	368	19.18	odd	2	inner
475.2.v.b.227.40	yes	368	1.1	even	1	trivial