Embedded newform 475.2.v.b.113.35

• Label: 475.2.v.b.113.35
• 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			113.35
Character	\(\chi\)	\(=\)	475.113
Dual form			475.2.v.b.227.35

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.260669 + 1.64580i) q^{2} +(-2.22247 - 1.13241i) q^{3} +(-0.738599 + 0.239985i) q^{4} +(-1.39458 - 1.74789i) q^{5} +(1.28439 - 3.95293i) q^{6} +(0.297379 - 0.297379i) q^{7} +(0.925485 + 1.81637i) q^{8} +(1.89369 + 2.60644i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 20 q^{4} - 16 q^{5} - 12 q^{6} - 8 q^{7} - 20 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.260669	+	1.64580i	0.184321	+	1.16376i	0.890249	+	0.455474i	\(0.150530\pi\)
							−0.705928	+	0.708283i	\(0.749470\pi\)
\(3\)	−2.22247	−	1.13241i	−1.28315	−	0.653795i	−0.326541	−	0.945183i	\(-0.605883\pi\)
							−0.956605	+	0.291388i	\(0.905883\pi\)
\(5\)	−1.39458	−	1.74789i	−0.623677	−	0.781682i
							\(7\)	0.297379	−	0.297379i	0.112399	−	0.112399i	−0.648671	−	0.761069i	\(-0.724675\pi\)
0.761069	+	0.648671i	\(0.224675\pi\)
\(11\)	−2.82021	−	2.04900i	−0.850325	−	0.617798i	0.0749103	−	0.997190i	\(-0.476133\pi\)
							−0.925236	+	0.379393i	\(0.876133\pi\)
\(13\)	−0.837987	+	5.29084i	−0.232416	+	1.46742i	0.545004	+	0.838433i	\(0.316528\pi\)
							−0.777420	+	0.628982i	\(0.783472\pi\)
\(17\)	2.60187	+	5.10646i	0.631046	+	1.23850i	0.956166	+	0.292825i	\(0.0945952\pi\)
							−0.325120	+	0.945673i	\(0.605405\pi\)
\(19\)	1.86147	+	3.94144i	0.427050	+	0.904228i
							\(23\)	−0.127289	−	0.803673i	−0.0265417	−	0.167577i	0.970857	−	0.239659i	\(-0.0770357\pi\)
−0.997399	+	0.0720818i	\(0.977036\pi\)
\(29\)	−1.14557	−	3.52569i	−0.212727	−	0.654705i	−0.999307	−	0.0372175i	\(-0.988151\pi\)
							0.786581	−	0.617488i	\(-0.211849\pi\)
\(31\)	−7.78219	−	2.52859i	−1.39772	−	0.454148i	−0.489269	−	0.872133i	\(-0.662736\pi\)
							−0.908454	+	0.417985i	\(0.862736\pi\)
\(37\)	−2.45451	−	0.388756i	−0.403518	−	0.0639110i	−0.0486233	−	0.998817i	\(-0.515483\pi\)
							−0.354895	+	0.934906i	\(0.615483\pi\)
\(41\)	4.06730	+	5.59816i	0.635206	+	0.874286i	0.998349	−	0.0574476i	\(-0.0182962\pi\)
							−0.363143	+	0.931734i	\(0.618296\pi\)
\(43\)	6.56136	+	6.56136i	1.00060	+	1.00060i	1.00000		0.000597994i	\(0.000190347\pi\)
							0.000597994		1.00000i	\(0.499810\pi\)
\(47\)	−3.21660	−	1.63894i	−0.469190	−	0.239064i	0.203377	−	0.979100i	\(-0.434808\pi\)
							−0.672567	+	0.740036i	\(0.734808\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.v.b.113.35	yes	368
19.18	odd	2	inner	475.2.v.b.113.12	✓	368
25.2	odd	20	inner	475.2.v.b.227.12	yes	368
475.227	even	20	inner	475.2.v.b.227.35	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.v.b.113.12	✓	368	19.18	odd	2	inner
475.2.v.b.113.35	yes	368	1.1	even	1	trivial
475.2.v.b.227.12	yes	368	25.2	odd	20	inner
475.2.v.b.227.35	yes	368	475.227	even	20	inner