Embedded newform 400.2.be.a.21.12

• Label: 400.2.be.a.21.12
• 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			21.12
Character	\(\chi\)	\(=\)	400.21
Dual form			400.2.be.a.381.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.10053 + 0.888164i) q^{2} +(0.211293 - 0.0334655i) q^{3} +(0.422329 - 1.95490i) q^{4} +(-1.64970 + 1.50947i) q^{5} +(-0.202811 + 0.224492i) q^{6} -3.25198i q^{7} +(1.27149 + 2.52652i) q^{8} +(-2.80964 + 0.912909i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 464 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 8 q^{5} - 6 q^{6} + 12 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.10053	+	0.888164i	−0.778192	+	0.628027i
							\(3\)	0.211293	−	0.0334655i	0.121990	−	0.0193213i	−0.0951409	−	0.995464i	\(-0.530330\pi\)
0.217131	+	0.976143i	\(0.430330\pi\)
\(5\)	−1.64970	+	1.50947i	−0.737767	+	0.675055i
							\(7\)		−	3.25198i		−	1.22913i	−0.788865	−	0.614566i	\(-0.789331\pi\)
0.788865	−	0.614566i	\(-0.210669\pi\)
\(11\)	3.16126	−	1.61074i	0.953157	−	0.485658i	0.0929881	−	0.995667i	\(-0.470358\pi\)
							0.860169	+	0.510010i	\(0.170358\pi\)
\(13\)	5.19028	+	2.64458i	1.43952	+	0.733474i	0.987365	−	0.158463i	\(-0.0506539\pi\)
							0.452159	+	0.891937i	\(0.350654\pi\)
\(17\)	2.01877	−	1.46672i	0.489623	−	0.355732i	−0.315417	−	0.948953i	\(-0.602144\pi\)
							0.805039	+	0.593222i	\(0.202144\pi\)
\(19\)	0.365121	−	2.30529i	0.0837646	−	0.528869i	−0.909749	−	0.415158i	\(-0.863726\pi\)
							0.993514	−	0.113711i	\(-0.0362737\pi\)
\(23\)	6.76707	+	2.19875i	1.41103	+	0.458472i	0.912741	−	0.408539i	\(-0.133962\pi\)
							0.498290	+	0.867011i	\(0.333962\pi\)
\(29\)	3.53718	−	0.560235i	0.656838	−	0.104033i	0.180886	−	0.983504i	\(-0.442104\pi\)
							0.475952	+	0.879471i	\(0.342104\pi\)
\(31\)	0.377024	−	0.273924i	0.0677156	−	0.0491983i	−0.553412	−	0.832907i	\(-0.686675\pi\)
							0.621128	+	0.783709i	\(0.286675\pi\)
\(37\)	3.54273	−	6.95300i	0.582421	−	1.14307i	−0.392341	−	0.919820i	\(-0.628335\pi\)
							0.974762	−	0.223247i	\(-0.0716655\pi\)
\(41\)	−8.02224	+	2.60659i	−1.25286	+	0.407080i	−0.858946	−	0.512066i	\(-0.828880\pi\)
							−0.393918	+	0.919146i	\(0.628880\pi\)
\(43\)	1.97888	+	1.97888i	0.301776	+	0.301776i	0.841709	−	0.539932i	\(-0.181550\pi\)
							−0.539932	+	0.841709i	\(0.681550\pi\)
\(47\)	7.34488	+	5.33636i	1.07136	+	0.778389i	0.976156	−	0.217069i	\(-0.0696497\pi\)
							0.0952040	+	0.995458i	\(0.469650\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.2.be.a.21.12	✓	464
16.13	even	4	inner	400.2.be.a.221.35	yes	464
25.6	even	5	inner	400.2.be.a.181.35	yes	464
400.381	even	20	inner	400.2.be.a.381.12	yes	464

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.be.a.21.12	✓	464	1.1	even	1	trivial
400.2.be.a.181.35	yes	464	25.6	even	5	inner
400.2.be.a.221.35	yes	464	16.13	even	4	inner
400.2.be.a.381.12	yes	464	400.381	even	20	inner