Embedded newform 400.2.be.a.21.19

• Label: 400.2.be.a.21.19
• Level: $400$
• Weight: $2$
• Character: 400.21
• Analytic conductor: $3.194$
• Analytic rank: $0$
• Dimension: $464$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	400.be (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.19401608085\)
Analytic rank:	\(0\)
Dimension:	\(464\)
Relative dimension:	\(58\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			21.19
Character	\(\chi\)	\(=\)	400.21
Dual form			400.2.be.a.381.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.740418 - 1.20490i) q^{2} +(-1.22547 + 0.194095i) q^{3} +(-0.903562 + 1.78426i) q^{4} +(0.515282 + 2.17589i) q^{5} +(1.14122 + 1.33285i) q^{6} -3.96942i q^{7} +(2.81886 - 0.232396i) q^{8} +(-1.38907 + 0.451337i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 464 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 8 q^{5} - 6 q^{6} + 12 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(177\)	\(351\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.740418	−	1.20490i	−0.523555	−	0.851992i
							\(3\)	−1.22547	+	0.194095i	−0.707524	+	0.112061i	−0.499818	−	0.866130i	\(-0.666600\pi\)
−0.207706	+	0.978191i	\(0.566600\pi\)
\(5\)	0.515282	+	2.17589i	0.230441	+	0.973086i
							\(7\)		−	3.96942i		−	1.50030i	−0.661268	−	0.750149i	\(-0.729982\pi\)
0.661268	−	0.750149i	\(-0.270018\pi\)
\(11\)	0.199022	−	0.101407i	0.0600073	−	0.0305752i	−0.423729	−	0.905789i	\(-0.639279\pi\)
							0.483737	+	0.875214i	\(0.339279\pi\)
\(13\)	0.328612	+	0.167436i	0.0911406	+	0.0464384i	0.498967	−	0.866621i	\(-0.333713\pi\)
							−0.407826	+	0.913060i	\(0.633713\pi\)
\(17\)	−5.61109	+	4.07669i	−1.36089	+	0.988743i	−0.362501	+	0.931984i	\(0.618077\pi\)
							−0.998388	+	0.0567598i	\(0.981923\pi\)
\(19\)	−0.931262	+	5.87976i	−0.213646	+	1.34891i	0.614731	+	0.788737i	\(0.289265\pi\)
							−0.828377	+	0.560171i	\(0.810735\pi\)
\(23\)	−1.53627	−	0.499165i	−0.320335	−	0.104083i	0.144436	−	0.989514i	\(-0.453863\pi\)
							−0.464771	+	0.885431i	\(0.653863\pi\)
\(29\)	−4.95904	+	0.785435i	−0.920871	+	0.145852i	−0.598832	−	0.800875i	\(-0.704368\pi\)
							−0.322039	+	0.946726i	\(0.604368\pi\)
\(31\)	−6.40024	+	4.65005i	−1.14952	+	0.835173i	−0.988417	−	0.151765i	\(-0.951504\pi\)
							−0.161100	+	0.986938i	\(0.551504\pi\)
\(37\)	0.543675	−	1.06702i	0.0893796	−	0.175417i	−0.841989	−	0.539494i	\(-0.818615\pi\)
							0.931369	+	0.364077i	\(0.118615\pi\)
\(41\)	0.0358991	−	0.0116643i	0.00560650	−	0.00182166i	−0.306212	−	0.951963i	\(-0.599062\pi\)
							0.311819	+	0.950142i	\(0.399062\pi\)
\(43\)	−4.92314	−	4.92314i	−0.750772	−	0.750772i	0.223852	−	0.974623i	\(-0.428137\pi\)
							−0.974623	+	0.223852i	\(0.928137\pi\)
\(47\)	6.69201	+	4.86203i	0.976130	+	0.709200i	0.956840	−	0.290614i	\(-0.0938596\pi\)
							0.0192893	+	0.999814i	\(0.493860\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.2.be.a.21.19	✓	464
16.13	even	4	inner	400.2.be.a.221.52	yes	464
25.6	even	5	inner	400.2.be.a.181.52	yes	464
400.381	even	20	inner	400.2.be.a.381.19	yes	464

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.be.a.21.19	✓	464	1.1	even	1	trivial
400.2.be.a.181.52	yes	464	25.6	even	5	inner
400.2.be.a.221.52	yes	464	16.13	even	4	inner
400.2.be.a.381.19	yes	464	400.381	even	20	inner