Embedded newform 400.2.bd.a.323.29

• Label: 400.2.bd.a.323.29
• Level: $400$
• Weight: $2$
• Character: 400.323
• 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			323.29
Character	\(\chi\)	\(=\)	400.323
Dual form			400.2.bd.a.187.29

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.150704 + 1.40616i) q^{2} +(0.269711 - 0.830084i) q^{3} +(-1.95458 - 0.423828i) q^{4} +(-2.21657 - 0.294681i) q^{5} +(1.12658 + 0.504353i) q^{6} +(0.0768366 - 0.0768366i) q^{7} +(0.890533 - 2.68458i) q^{8} +(1.81076 + 1.31559i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 464 q - 8 q^{2} - 6 q^{3} - 10 q^{4} - 8 q^{5} - 6 q^{6} - 16 q^{7} - 2 q^{8} - 108 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(177\)	\(351\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{11}{20}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.150704	+	1.40616i	−0.106564	+	0.994306i
							\(3\)	0.269711	−	0.830084i	0.155717	−	0.479249i	−0.842515	−	0.538672i	\(-0.818926\pi\)
0.998233	+	0.0594231i	\(0.0189261\pi\)
\(5\)	−2.21657	−	0.294681i	−0.991278	−	0.131785i
							\(7\)	0.0768366	−	0.0768366i	0.0290415	−	0.0290415i	−0.692437	−	0.721478i	\(-0.743463\pi\)
0.721478	+	0.692437i	\(0.243463\pi\)
\(11\)	5.02356	+	0.795654i	1.51466	+	0.239899i	0.857749	−	0.514068i	\(-0.171862\pi\)
							0.656912	+	0.753967i	\(0.271862\pi\)
\(13\)	1.22492	−	1.68595i	0.339730	−	0.467599i	−0.604632	−	0.796505i	\(-0.706680\pi\)
							0.944363	+	0.328906i	\(0.106680\pi\)
\(17\)	1.70624	+	0.869374i	0.413825	+	0.210854i	0.648492	−	0.761222i	\(-0.275400\pi\)
							−0.234667	+	0.972076i	\(0.575400\pi\)
\(19\)	−1.02078	+	2.00339i	−0.234183	+	0.459610i	−0.977953	−	0.208826i	\(-0.933036\pi\)
							0.743770	+	0.668436i	\(0.233036\pi\)
\(23\)	4.91850	+	0.779014i	1.02558	+	0.162436i	0.646488	−	0.762924i	\(-0.276237\pi\)
							0.379090	+	0.925360i	\(0.376237\pi\)
\(29\)	−0.862725	−	1.69319i	−0.160204	−	0.314418i	0.796926	−	0.604077i	\(-0.206458\pi\)
							−0.957130	+	0.289659i	\(0.906458\pi\)
\(31\)	−5.10949	+	1.66017i	−0.917691	+	0.298176i	−0.729519	−	0.683960i	\(-0.760256\pi\)
							−0.188172	+	0.982136i	\(0.560256\pi\)
\(37\)	4.25009	−	5.84974i	0.698710	−	0.961692i	−0.301257	−	0.953543i	\(-0.597406\pi\)
							0.999967	−	0.00814872i	\(-0.00259385\pi\)
\(41\)	2.41144	−	3.31906i	0.376604	−	0.518351i	−0.578077	−	0.815982i	\(-0.696197\pi\)
							0.954681	+	0.297632i	\(0.0961967\pi\)
\(43\)		−	0.756843i		−	0.115418i	−0.998333	−	0.0577088i	\(-0.981621\pi\)
							0.998333	−	0.0577088i	\(-0.0183795\pi\)
\(47\)	−0.133866	+	0.0682083i	−0.0195264	+	0.00994920i	−0.463726	−	0.885978i	\(-0.653488\pi\)
							0.444200	+	0.895928i	\(0.353488\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.2.bd.a.323.29	yes	464
16.11	odd	4		400.2.bm.a.123.26	yes	464
25.12	odd	20		400.2.bm.a.387.26	yes	464
400.187	even	20	inner	400.2.bd.a.187.29	✓	464

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.bd.a.187.29	✓	464	400.187	even	20	inner
400.2.bd.a.323.29	yes	464	1.1	even	1	trivial
400.2.bm.a.123.26	yes	464	16.11	odd	4
400.2.bm.a.387.26	yes	464	25.12	odd	20