Embedded newform 400.2.be.a.21.14

• Label: 400.2.be.a.21.14
• 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			21.14
Character	\(\chi\)	\(=\)	400.21
Dual form			400.2.be.a.381.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.03320 + 0.965656i) q^{2} +(-3.03259 + 0.480315i) q^{3} +(0.135016 - 1.99544i) q^{4} +(0.812902 + 2.08307i) q^{5} +(2.66946 - 3.42470i) q^{6} +2.37186i q^{7} +(1.78741 + 2.19207i) q^{8} +(6.11271 - 1.98614i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 464 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 8 q^{5} - 6 q^{6} + 12 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.03320	+	0.965656i	−0.730585	+	0.682822i
							\(3\)	−3.03259	+	0.480315i	−1.75086	+	0.277310i	−0.947866	−	0.318670i	\(-0.896764\pi\)
−0.802999	+	0.595980i	\(0.796764\pi\)
\(5\)	0.812902	+	2.08307i	0.363541	+	0.931578i
							\(7\)			2.37186i			0.896478i	0.893914	+	0.448239i	\(0.147949\pi\)
−0.893914	+	0.448239i	\(0.852051\pi\)
\(11\)	2.06286	−	1.05108i	0.621976	−	0.316913i	−0.114452	−	0.993429i	\(-0.536511\pi\)
							0.736428	+	0.676516i	\(0.236511\pi\)
\(13\)	3.82714	+	1.95003i	1.06146	+	0.540840i	0.895394	−	0.445275i	\(-0.146894\pi\)
							0.166064	+	0.986115i	\(0.446894\pi\)
\(17\)	1.32777	−	0.964683i	0.322032	−	0.233970i	−0.415010	−	0.909817i	\(-0.636222\pi\)
							0.737042	+	0.675847i	\(0.236222\pi\)
\(19\)	−1.29970	+	8.20600i	−0.298172	+	1.88259i	0.150073	+	0.988675i	\(0.452049\pi\)
							−0.448246	+	0.893910i	\(0.647951\pi\)
\(23\)	−4.98377	−	1.61932i	−1.03919	−	0.337653i	−0.260771	−	0.965401i	\(-0.583977\pi\)
							−0.778416	+	0.627748i	\(0.783977\pi\)
\(29\)	−6.84890	+	1.08476i	−1.27181	+	0.201435i	−0.755603	−	0.655030i	\(-0.772656\pi\)
							−0.516206	+	0.856465i	\(0.672656\pi\)
\(31\)	6.27551	−	4.55942i	1.12711	−	0.818897i	0.141842	−	0.989889i	\(-0.454697\pi\)
							0.985272	+	0.170993i	\(0.0546975\pi\)
\(37\)	−0.901742	+	1.76977i	−0.148246	+	0.290948i	−0.953173	−	0.302425i	\(-0.902204\pi\)
							0.804928	+	0.593373i	\(0.202204\pi\)
\(41\)	−0.208601	+	0.0677786i	−0.0325780	+	0.0105852i	−0.325261	−	0.945624i	\(-0.605452\pi\)
							0.292683	+	0.956210i	\(0.405452\pi\)
\(43\)	0.794765	+	0.794765i	0.121201	+	0.121201i	0.765105	−	0.643905i	\(-0.222687\pi\)
							−0.643905	+	0.765105i	\(0.722687\pi\)
\(47\)	−4.97842	−	3.61703i	−0.726177	−	0.527598i	0.162175	−	0.986762i	\(-0.448149\pi\)
							−0.888352	+	0.459164i	\(0.848149\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.2.be.a.21.14	✓	464
16.13	even	4	inner	400.2.be.a.221.33	yes	464
25.6	even	5	inner	400.2.be.a.181.33	yes	464
400.381	even	20	inner	400.2.be.a.381.14	yes	464

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.be.a.21.14	✓	464	1.1	even	1	trivial
400.2.be.a.181.33	yes	464	25.6	even	5	inner
400.2.be.a.221.33	yes	464	16.13	even	4	inner
400.2.be.a.381.14	yes	464	400.381	even	20	inner