Embedded newform 475.2.v.b.113.36

• Label: 475.2.v.b.113.36
• Level: $475$
• Weight: $2$
• Character: 475.113
• 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			113.36
Character	\(\chi\)	\(=\)	475.113
Dual form			475.2.v.b.227.36

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.288838 + 1.82365i) q^{2} +(2.78020 + 1.41658i) q^{3} +(-1.34017 + 0.435448i) q^{4} +(0.851139 - 2.06774i) q^{5} +(-1.78033 + 5.47929i) q^{6} +(-2.17838 + 2.17838i) q^{7} +(0.495285 + 0.972052i) q^{8} +(3.95946 + 5.44974i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 20 q^{4} - 16 q^{5} - 12 q^{6} - 8 q^{7} - 20 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{19}{20}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.288838	+	1.82365i	0.204239	+	1.28952i	0.850327	+	0.526254i	\(0.176404\pi\)
							−0.646088	+	0.763263i	\(0.723596\pi\)
\(3\)	2.78020	+	1.41658i	1.60515	+	0.817865i	0.999759	+	0.0219379i	\(0.00698361\pi\)
							0.605392	+	0.795928i	\(0.293016\pi\)
\(5\)	0.851139	−	2.06774i	0.380641	−	0.924723i
							\(7\)	−2.17838	+	2.17838i	−0.823350	+	0.823350i	−0.986587	−	0.163237i	\(-0.947806\pi\)
0.163237	+	0.986587i	\(0.447806\pi\)
\(11\)	−1.41180	−	1.02573i	−0.425673	−	0.309269i	0.354243	−	0.935153i	\(-0.384738\pi\)
							−0.779916	+	0.625884i	\(0.784738\pi\)
\(13\)	0.699994	−	4.41959i	0.194143	−	1.22577i	−0.677459	−	0.735560i	\(-0.736919\pi\)
							0.871603	−	0.490213i	\(-0.163081\pi\)
\(17\)	−2.07136	−	4.06527i	−0.502378	−	0.985972i	−0.993387	−	0.114817i	\(-0.963372\pi\)
							0.491009	−	0.871155i	\(-0.336628\pi\)
\(19\)	3.71591	−	2.27861i	0.852487	−	0.522748i
							\(23\)	0.261795	+	1.65291i	0.0545881	+	0.344656i	0.999833	+	0.0182929i	\(0.00582312\pi\)
−0.945245	+	0.326363i	\(0.894177\pi\)
\(29\)	1.56295	+	4.81028i	0.290233	+	0.893246i	0.984781	+	0.173799i	\(0.0556044\pi\)
							−0.694548	+	0.719446i	\(0.744396\pi\)
\(31\)	−3.39623	−	1.10350i	−0.609981	−	0.198195i	−0.0122941	−	0.999924i	\(-0.503913\pi\)
							−0.597687	+	0.801730i	\(0.703913\pi\)
\(37\)	−2.41730	−	0.382863i	−0.397402	−	0.0629424i	−0.0454651	−	0.998966i	\(-0.514477\pi\)
							−0.351937	+	0.936024i	\(0.614477\pi\)
\(41\)	2.97386	+	4.09316i	0.464439	+	0.639245i	0.975422	−	0.220346i	\(-0.0707186\pi\)
							−0.510983	+	0.859591i	\(0.670719\pi\)
\(43\)	−1.12827	−	1.12827i	−0.172060	−	0.172060i	0.615824	−	0.787884i	\(-0.288823\pi\)
							−0.787884	+	0.615824i	\(0.788823\pi\)
\(47\)	−9.07295	−	4.62290i	−1.32343	−	0.674319i	−0.357689	−	0.933841i	\(-0.616435\pi\)
							−0.965737	+	0.259522i	\(0.916435\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.v.b.113.36	yes	368
19.18	odd	2	inner	475.2.v.b.113.11	✓	368
25.2	odd	20	inner	475.2.v.b.227.11	yes	368
475.227	even	20	inner	475.2.v.b.227.36	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.v.b.113.11	✓	368	19.18	odd	2	inner
475.2.v.b.113.36	yes	368	1.1	even	1	trivial
475.2.v.b.227.11	yes	368	25.2	odd	20	inner
475.2.v.b.227.36	yes	368	475.227	even	20	inner