Embedded newform 400.2.be.a.21.20

• Label: 400.2.be.a.21.20
• 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			21.20
Character	\(\chi\)	\(=\)	400.21
Dual form			400.2.be.a.381.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.737939 - 1.20642i) q^{2} +(0.372869 - 0.0590566i) q^{3} +(-0.910891 + 1.78053i) q^{4} +(-0.553147 - 2.16657i) q^{5} +(-0.346402 - 0.406256i) q^{6} +3.39917i q^{7} +(2.82024 - 0.215005i) q^{8} +(-2.71763 + 0.883010i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 464 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 8 q^{5} - 6 q^{6} + 12 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.737939	−	1.20642i	−0.521802	−	0.853067i
							\(3\)	0.372869	−	0.0590566i	0.215276	−	0.0340964i	−0.0478647	−	0.998854i	\(-0.515242\pi\)
0.263141	+	0.964757i	\(0.415242\pi\)
\(5\)	−0.553147	−	2.16657i	−0.247375	−	0.968920i
							\(7\)			3.39917i			1.28477i	0.766384	+	0.642383i	\(0.222054\pi\)
−0.766384	+	0.642383i	\(0.777946\pi\)
\(11\)	−3.18763	+	1.62418i	−0.961106	+	0.489708i	−0.862854	−	0.505454i	\(-0.831325\pi\)
							−0.0982520	+	0.995162i	\(0.531325\pi\)
\(13\)	−4.76881	−	2.42983i	−1.32263	−	0.673914i	−0.357066	−	0.934079i	\(-0.616223\pi\)
							−0.965564	+	0.260165i	\(0.916223\pi\)
\(17\)	−4.02515	+	2.92444i	−0.976242	+	0.709282i	−0.956866	−	0.290531i	\(-0.906168\pi\)
							−0.0193767	+	0.999812i	\(0.506168\pi\)
\(19\)	0.681050	−	4.29998i	0.156244	−	0.986483i	−0.777589	−	0.628773i	\(-0.783558\pi\)
							0.933833	−	0.357710i	\(-0.116442\pi\)
\(23\)	4.87533	+	1.58409i	1.01658	+	0.330305i	0.769470	−	0.638683i	\(-0.220521\pi\)
							0.247106	+	0.968989i	\(0.420521\pi\)
\(29\)	3.94970	−	0.625571i	0.733441	−	0.116166i	0.221469	−	0.975168i	\(-0.428915\pi\)
							0.511972	+	0.859002i	\(0.328915\pi\)
\(31\)	−5.14258	+	3.73631i	−0.923635	+	0.671060i	−0.944426	−	0.328724i	\(-0.893382\pi\)
							0.0207909	+	0.999784i	\(0.493382\pi\)
\(37\)	−4.44306	+	8.72000i	−0.730435	+	1.43356i	0.164046	+	0.986453i	\(0.447545\pi\)
							−0.894481	+	0.447106i	\(0.852455\pi\)
\(41\)	7.99705	−	2.59840i	1.24893	−	0.405802i	0.391392	−	0.920224i	\(-0.371993\pi\)
							0.857537	+	0.514422i	\(0.171993\pi\)
\(43\)	−7.44483	−	7.44483i	−1.13533	−	1.13533i	−0.989277	−	0.146049i	\(-0.953344\pi\)
							−0.146049	−	0.989277i	\(-0.546656\pi\)
\(47\)	−0.0635165	−	0.0461475i	−0.00926484	−	0.00673130i	0.583143	−	0.812369i	\(-0.301823\pi\)
							−0.592408	+	0.805638i	\(0.701823\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.2.be.a.21.20	✓	464
16.13	even	4	inner	400.2.be.a.221.51	yes	464
25.6	even	5	inner	400.2.be.a.181.51	yes	464
400.381	even	20	inner	400.2.be.a.381.20	yes	464

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.be.a.21.20	✓	464	1.1	even	1	trivial
400.2.be.a.181.51	yes	464	25.6	even	5	inner
400.2.be.a.221.51	yes	464	16.13	even	4	inner
400.2.be.a.381.20	yes	464	400.381	even	20	inner