Embedded newform 400.2.bd.a.83.43

• Label: 400.2.bd.a.83.43
• Level: $400$
• Weight: $2$
• Character: 400.83
• 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			83.43
Character	\(\chi\)	\(=\)	400.83
Dual form			400.2.bd.a.347.43

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.952117 - 1.04569i) q^{2} +(-0.761606 + 0.553339i) q^{3} +(-0.186945 - 1.99124i) q^{4} +(-1.68069 - 1.47488i) q^{5} +(-0.146516 + 1.32325i) q^{6} +(-2.97017 + 2.97017i) q^{7} +(-2.26022 - 1.70041i) q^{8} +(-0.653192 + 2.01032i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 464 q - 8 q^{2} - 6 q^{3} - 10 q^{4} - 8 q^{5} - 6 q^{6} - 16 q^{7} - 2 q^{8} - 108 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.952117	−	1.04569i	0.673249	−	0.739416i
							\(3\)	−0.761606	+	0.553339i	−0.439713	+	0.319470i	−0.785521	−	0.618835i	\(-0.787605\pi\)
0.345808	+	0.938305i	\(0.387605\pi\)
\(5\)	−1.68069	−	1.47488i	−0.751628	−	0.659587i
							\(7\)	−2.97017	+	2.97017i	−1.12262	+	1.12262i	−0.131273	+	0.991346i	\(0.541906\pi\)
−0.991346	+	0.131273i	\(0.958094\pi\)
\(11\)	2.26116	−	4.43778i	0.681766	−	1.33804i	−0.247593	−	0.968864i	\(-0.579640\pi\)
							0.929359	−	0.369177i	\(-0.120360\pi\)
\(13\)	−5.55144	−	1.80377i	−1.53969	−	0.500276i	−0.588400	−	0.808570i	\(-0.700242\pi\)
							−0.951291	+	0.308294i	\(0.900242\pi\)
\(17\)	−3.45405	+	0.547068i	−0.837730	+	0.132683i	−0.560536	−	0.828130i	\(-0.689405\pi\)
							−0.277195	+	0.960814i	\(0.589405\pi\)
\(19\)	0.984436	+	6.21548i	0.225845	+	1.42593i	0.796450	+	0.604705i	\(0.206709\pi\)
							−0.570605	+	0.821225i	\(0.693291\pi\)
\(23\)	3.63461	−	7.13332i	0.757868	−	1.48740i	−0.111783	−	0.993733i	\(-0.535656\pi\)
							0.869651	−	0.493667i	\(-0.164344\pi\)
\(29\)	0.385001	−	2.43080i	0.0714928	−	0.451388i	−0.925810	−	0.377990i	\(-0.876616\pi\)
							0.997303	−	0.0733984i	\(-0.0233845\pi\)
\(31\)	0.486804	−	0.670028i	0.0874326	−	0.120341i	−0.763060	−	0.646328i	\(-0.776304\pi\)
							0.850493	+	0.525987i	\(0.176304\pi\)
\(37\)	−2.80279	−	0.910681i	−0.460775	−	0.149715i	0.0694254	−	0.997587i	\(-0.477883\pi\)
							−0.530201	+	0.847872i	\(0.677883\pi\)
\(41\)	−3.03456	−	0.985987i	−0.473918	−	0.153985i	0.0623128	−	0.998057i	\(-0.480152\pi\)
							−0.536231	+	0.844071i	\(0.680152\pi\)
\(43\)		−	2.12253i		−	0.323682i	−0.986817	−	0.161841i	\(-0.948257\pi\)
							0.986817	−	0.161841i	\(-0.0517432\pi\)
\(47\)	−2.49278	−	0.394818i	−0.363610	−	0.0575901i	−0.0280434	−	0.999607i	\(-0.508928\pi\)
							−0.335566	+	0.942017i	\(0.608928\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.bd.a.83.43	✓	464	1.1	even	1	trivial
400.2.bd.a.347.43	yes	464	400.347	even	20	inner
400.2.bm.a.147.25	yes	464	25.22	odd	20
400.2.bm.a.283.25	yes	464	16.11	odd	4