Embedded newform 400.2.be.a.21.8

• Label: 400.2.be.a.21.8
• 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			21.8
Character	\(\chi\)	\(=\)	400.21
Dual form			400.2.be.a.381.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.34997 + 0.421416i) q^{2} +(-1.51383 + 0.239767i) q^{3} +(1.64482 - 1.13779i) q^{4} +(0.647118 - 2.14038i) q^{5} +(1.94258 - 0.961631i) q^{6} +3.05783i q^{7} +(-1.74096 + 2.22914i) q^{8} +(-0.618970 + 0.201115i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 464 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 8 q^{5} - 6 q^{6} + 12 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.34997	+	0.421416i	−0.954570	+	0.297986i
							\(3\)	−1.51383	+	0.239767i	−0.874011	+	0.138430i	−0.577295	−	0.816536i	\(-0.695892\pi\)
−0.296717	+	0.954966i	\(0.595892\pi\)
\(5\)	0.647118	−	2.14038i	0.289400	−	0.957208i
							\(7\)			3.05783i			1.15575i	0.816124	+	0.577876i	\(0.196118\pi\)
−0.816124	+	0.577876i	\(0.803882\pi\)
\(11\)	3.07484	−	1.56671i	0.927100	−	0.472381i	0.0758367	−	0.997120i	\(-0.475837\pi\)
							0.851263	+	0.524739i	\(0.175837\pi\)
\(13\)	−1.79665	−	0.915441i	−0.498302	−	0.253898i	0.186729	−	0.982412i	\(-0.440211\pi\)
							−0.685031	+	0.728514i	\(0.740211\pi\)
\(17\)	2.59102	−	1.88249i	0.628415	−	0.456570i	−0.227436	−	0.973793i	\(-0.573034\pi\)
							0.855851	+	0.517223i	\(0.173034\pi\)
\(19\)	−0.0944698	+	0.596459i	−0.0216729	+	0.136837i	−0.996152	−	0.0876401i	\(-0.972067\pi\)
							0.974479	+	0.224477i	\(0.0720674\pi\)
\(23\)	2.61837	+	0.850761i	0.545968	+	0.177396i	0.568998	−	0.822339i	\(-0.307331\pi\)
							−0.0230296	+	0.999735i	\(0.507331\pi\)
\(29\)	7.25514	−	1.14910i	1.34724	−	0.213383i	0.559217	−	0.829022i	\(-0.311102\pi\)
							0.788028	+	0.615639i	\(0.211102\pi\)
\(31\)	1.94099	−	1.41021i	0.348612	−	0.253282i	−0.399674	−	0.916657i	\(-0.630877\pi\)
							0.748287	+	0.663376i	\(0.230877\pi\)
\(37\)	2.31467	−	4.54279i	0.380529	−	0.746831i	−0.618719	−	0.785613i	\(-0.712348\pi\)
							0.999248	+	0.0387822i	\(0.0123479\pi\)
\(41\)	4.59728	−	1.49375i	0.717974	−	0.233284i	0.0728293	−	0.997344i	\(-0.476797\pi\)
							0.645145	+	0.764061i	\(0.276797\pi\)
\(43\)	4.83386	+	4.83386i	0.737156	+	0.737156i	0.972027	−	0.234870i	\(-0.0754666\pi\)
							−0.234870	+	0.972027i	\(0.575467\pi\)
\(47\)	7.39916	+	5.37581i	1.07928	+	0.784142i	0.977557	−	0.210671i	\(-0.0675648\pi\)
							0.101722	+	0.994813i	\(0.467565\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.2.be.a.21.8	✓	464
16.13	even	4	inner	400.2.be.a.221.40	yes	464
25.6	even	5	inner	400.2.be.a.181.40	yes	464
400.381	even	20	inner	400.2.be.a.381.8	yes	464

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.be.a.21.8	✓	464	1.1	even	1	trivial
400.2.be.a.181.40	yes	464	25.6	even	5	inner
400.2.be.a.221.40	yes	464	16.13	even	4	inner
400.2.be.a.381.8	yes	464	400.381	even	20	inner