Embedded newform 400.2.bd.a.27.52

• Label: 400.2.bd.a.27.52
• Level: $400$
• Weight: $2$
• Character: 400.27
• Analytic conductor: $3.194$
• Analytic rank: $0$
• Dimension: $464$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	400.bd (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			27.52
Character	\(\chi\)	\(=\)	400.27
Dual form			400.2.bd.a.163.52

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.27353 - 0.614921i) q^{2} +(0.306781 - 0.944173i) q^{3} +(1.24374 - 1.56624i) q^{4} +(-0.152971 + 2.23083i) q^{5} +(-0.189899 - 1.39108i) q^{6} +(1.38601 + 1.38601i) q^{7} +(0.620829 - 2.75945i) q^{8} +(1.62970 + 1.18405i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 464 q - 8 q^{2} - 6 q^{3} - 10 q^{4} - 8 q^{5} - 6 q^{6} - 16 q^{7} - 2 q^{8} - 108 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(177\)	\(351\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{1}{20}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.27353	−	0.614921i	0.900520	−	0.434815i
							\(3\)	0.306781	−	0.944173i	0.177120	−	0.545119i	−0.822604	−	0.568615i	\(-0.807480\pi\)
0.999724	+	0.0234957i	\(0.00747962\pi\)
\(5\)	−0.152971	+	2.23083i	−0.0684108	+	0.997657i
							\(7\)	1.38601	+	1.38601i	0.523861	+	0.523861i	0.918735	−	0.394874i	\(-0.129212\pi\)
−0.394874	+	0.918735i	\(0.629212\pi\)
\(11\)	0.734974	−	4.64044i	0.221603	−	1.39915i	−0.586426	−	0.810003i	\(-0.699465\pi\)
							0.808029	−	0.589143i	\(-0.200535\pi\)
\(13\)	−0.701591	+	0.965657i	−0.194586	+	0.267825i	−0.895150	−	0.445765i	\(-0.852932\pi\)
							0.700564	+	0.713590i	\(0.252932\pi\)
\(17\)	−0.145513	+	0.285585i	−0.0352921	+	0.0692646i	−0.907979	−	0.419015i	\(-0.862375\pi\)
							0.872687	+	0.488280i	\(0.162375\pi\)
\(19\)	−3.33209	−	1.69779i	−0.764434	−	0.389499i	0.0278806	−	0.999611i	\(-0.491124\pi\)
							−0.792315	+	0.610113i	\(0.791124\pi\)
\(23\)	−0.610038	+	3.85163i	−0.127202	+	0.803120i	0.838773	+	0.544482i	\(0.183274\pi\)
							−0.965974	+	0.258638i	\(0.916726\pi\)
\(29\)	−4.20843	+	2.14430i	−0.781485	+	0.398187i	−0.798749	−	0.601664i	\(-0.794504\pi\)
							0.0172638	+	0.999851i	\(0.494504\pi\)
\(31\)	−2.48302	+	0.806783i	−0.445964	+	0.144902i	−0.523386	−	0.852096i	\(-0.675331\pi\)
							0.0774217	+	0.996998i	\(0.475331\pi\)
\(37\)	2.84768	−	3.91950i	0.468156	−	0.644361i	−0.508019	−	0.861346i	\(-0.669622\pi\)
							0.976175	+	0.216984i	\(0.0696221\pi\)
\(41\)	−5.63784	+	7.75983i	−0.880483	+	1.21188i	0.0958036	+	0.995400i	\(0.469458\pi\)
							−0.976287	+	0.216481i	\(0.930542\pi\)
\(43\)			1.81149i			0.276250i	0.990415	+	0.138125i	\(0.0441075\pi\)
							−0.990415	+	0.138125i	\(0.955892\pi\)
\(47\)	0.411115	+	0.806858i	0.0599673	+	0.117692i	0.919045	−	0.394153i	\(-0.128962\pi\)
							−0.859078	+	0.511845i	\(0.828962\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.2.bd.a.27.52	✓	464
16.3	odd	4		400.2.bm.a.227.47	yes	464
25.13	odd	20		400.2.bm.a.363.47	yes	464
400.163	even	20	inner	400.2.bd.a.163.52	yes	464

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.bd.a.27.52	✓	464	1.1	even	1	trivial
400.2.bd.a.163.52	yes	464	400.163	even	20	inner
400.2.bm.a.227.47	yes	464	16.3	odd	4
400.2.bm.a.363.47	yes	464	25.13	odd	20