Embedded newform 475.2.v.b.113.38

• Label: 475.2.v.b.113.38
• 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			113.38
Character	\(\chi\)	\(=\)	475.113
Dual form			475.2.v.b.227.38

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.307848 + 1.94368i) q^{2} +(-1.17623 - 0.599319i) q^{3} +(-1.78099 + 0.578679i) q^{4} +(1.54688 + 1.61467i) q^{5} +(0.802782 - 2.47071i) q^{6} +(1.03104 - 1.03104i) q^{7} +(0.113781 + 0.223308i) q^{8} +(-0.739023 - 1.01718i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 20 q^{4} - 16 q^{5} - 12 q^{6} - 8 q^{7} - 20 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.307848	+	1.94368i	0.217681	+	1.37439i	0.818273	+	0.574830i	\(0.194932\pi\)
							−0.600592	+	0.799556i	\(0.705068\pi\)
\(3\)	−1.17623	−	0.599319i	−0.679097	−	0.346017i	0.0801843	−	0.996780i	\(-0.474449\pi\)
							−0.759281	+	0.650763i	\(0.774449\pi\)
\(5\)	1.54688	+	1.61467i	0.691787	+	0.722101i
							\(7\)	1.03104	−	1.03104i	0.389697	−	0.389697i	−0.484883	−	0.874579i	\(-0.661138\pi\)
0.874579	+	0.484883i	\(0.161138\pi\)
\(11\)	4.78165	+	3.47408i	1.44172	+	1.04747i	0.987679	+	0.156494i	\(0.0500191\pi\)
							0.454044	+	0.890979i	\(0.349981\pi\)
\(13\)	0.338705	−	2.13850i	0.0939400	−	0.593114i	−0.895146	−	0.445773i	\(-0.852929\pi\)
							0.989086	−	0.147340i	\(-0.0470713\pi\)
\(17\)	0.948191	+	1.86093i	0.229970	+	0.451342i	0.976939	−	0.213517i	\(-0.0684918\pi\)
							−0.746969	+	0.664858i	\(0.768492\pi\)
\(19\)	−0.704317	+	4.30162i	−0.161581	+	0.986859i
							\(23\)	−0.733866	−	4.63345i	−0.153022	−	0.966140i	−0.938005	−	0.346623i	\(-0.887328\pi\)
0.784983	−	0.619517i	\(-0.212672\pi\)
\(29\)	0.471434	+	1.45093i	0.0875431	+	0.269430i	0.985239	−	0.171186i	\(-0.0547600\pi\)
							−0.897696	+	0.440616i	\(0.854760\pi\)
\(31\)	−3.97517	−	1.29161i	−0.713962	−	0.231980i	−0.0705588	−	0.997508i	\(-0.522478\pi\)
							−0.643404	+	0.765527i	\(0.722478\pi\)
\(37\)	1.85614	+	0.293984i	0.305148	+	0.0483306i	0.307131	−	0.951667i	\(-0.400631\pi\)
							−0.00198293	+	0.999998i	\(0.500631\pi\)
\(41\)	2.82273	+	3.88515i	0.440836	+	0.606758i	0.970398	−	0.241513i	\(-0.0776436\pi\)
							−0.529562	+	0.848271i	\(0.677644\pi\)
\(43\)	−4.99176	−	4.99176i	−0.761236	−	0.761236i	0.215310	−	0.976546i	\(-0.430924\pi\)
							−0.976546	+	0.215310i	\(0.930924\pi\)
\(47\)	−6.10591	−	3.11112i	−0.890639	−	0.453803i	−0.0520981	−	0.998642i	\(-0.516591\pi\)
							−0.838541	+	0.544839i	\(0.816591\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.v.b.113.38	yes	368
19.18	odd	2	inner	475.2.v.b.113.9	✓	368
25.2	odd	20	inner	475.2.v.b.227.9	yes	368
475.227	even	20	inner	475.2.v.b.227.38	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.v.b.113.9	✓	368	19.18	odd	2	inner
475.2.v.b.113.38	yes	368	1.1	even	1	trivial
475.2.v.b.227.9	yes	368	25.2	odd	20	inner
475.2.v.b.227.38	yes	368	475.227	even	20	inner