Embedded newform 400.2.bm.a.283.25

• Label: 400.2.bm.a.283.25
• Level: $400$
• Weight: $2$
• Character: 400.283
• 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.bm (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			283.25
Character	\(\chi\)	\(=\)	400.283
Dual form			400.2.bm.a.147.25

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.286342 + 1.38492i) q^{2} +(-0.553339 - 0.761606i) q^{3} +(-1.83602 - 0.793123i) q^{4} +(1.47488 - 1.68069i) q^{5} +(1.21321 - 0.548251i) q^{6} +(-2.97017 + 2.97017i) q^{7} +(1.62414 - 2.31563i) q^{8} +(0.653192 - 2.01032i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 464 q - 8 q^{2} - 10 q^{3} - 10 q^{4} - 8 q^{5} - 6 q^{6} - 16 q^{7} - 2 q^{8} + 108 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(177\)	\(351\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{3}{20}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.286342	+	1.38492i	−0.202475	+	0.979288i
							\(3\)	−0.553339	−	0.761606i	−0.319470	−	0.439713i	0.618835	−	0.785521i	\(-0.287605\pi\)
−0.938305	+	0.345808i	\(0.887605\pi\)
\(5\)	1.47488	−	1.68069i	0.659587	−	0.751628i
							\(7\)	−2.97017	+	2.97017i	−1.12262	+	1.12262i	−0.131273	+	0.991346i	\(0.541906\pi\)
−0.991346	+	0.131273i	\(0.958094\pi\)
\(11\)	−4.43778	−	2.26116i	−1.33804	−	0.681766i	−0.369177	−	0.929359i	\(-0.620360\pi\)
							−0.968864	+	0.247593i	\(0.920360\pi\)
\(13\)	−1.80377	+	5.55144i	−0.500276	+	1.53969i	0.308294	+	0.951291i	\(0.400242\pi\)
							−0.808570	+	0.588400i	\(0.799758\pi\)
\(17\)	−3.45405	+	0.547068i	−0.837730	+	0.132683i	−0.560536	−	0.828130i	\(-0.689405\pi\)
							−0.277195	+	0.960814i	\(0.589405\pi\)
\(19\)	−6.21548	+	0.984436i	−1.42593	+	0.225845i	−0.821225	−	0.570605i	\(-0.806709\pi\)
							−0.604705	+	0.796450i	\(0.706709\pi\)
\(23\)	3.63461	−	7.13332i	0.757868	−	1.48740i	−0.111783	−	0.993733i	\(-0.535656\pi\)
							0.869651	−	0.493667i	\(-0.164344\pi\)
\(29\)	−2.43080	−	0.385001i	−0.451388	−	0.0714928i	−0.0733984	−	0.997303i	\(-0.523384\pi\)
							−0.377990	+	0.925810i	\(0.623384\pi\)
\(31\)	−0.486804	+	0.670028i	−0.0874326	+	0.120341i	−0.850493	−	0.525987i	\(-0.823696\pi\)
							0.763060	+	0.646328i	\(0.223696\pi\)
\(37\)	0.910681	−	2.80279i	0.149715	−	0.460775i	−0.847872	−	0.530201i	\(-0.822117\pi\)
							0.997587	+	0.0694254i	\(0.0221166\pi\)
\(41\)	3.03456	+	0.985987i	0.473918	+	0.153985i	0.536231	−	0.844071i	\(-0.319848\pi\)
							−0.0623128	+	0.998057i	\(0.519848\pi\)
\(43\)	−2.12253			−0.323682			−0.161841	−	0.986817i	\(-0.551743\pi\)
							−0.161841	+	0.986817i	\(0.551743\pi\)
\(47\)	2.49278	+	0.394818i	0.363610	+	0.0575901i	0.335566	−	0.942017i	\(-0.391072\pi\)
							0.0280434	+	0.999607i	\(0.491072\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.2.bm.a.283.25	yes	464
16.3	odd	4		400.2.bd.a.83.43	✓	464
25.22	odd	20		400.2.bd.a.347.43	yes	464
400.147	even	20	inner	400.2.bm.a.147.25	yes	464

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.bd.a.83.43	✓	464	16.3	odd	4
400.2.bd.a.347.43	yes	464	25.22	odd	20
400.2.bm.a.147.25	yes	464	400.147	even	20	inner
400.2.bm.a.283.25	yes	464	1.1	even	1	trivial