Embedded newform 400.2.be.a.21.1

• Label: 400.2.be.a.21.1
• 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			21.1
Character	\(\chi\)	\(=\)	400.21
Dual form			400.2.be.a.381.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41077 - 0.0986613i) q^{2} +(2.12918 - 0.337229i) q^{3} +(1.98053 + 0.278376i) q^{4} +(0.968273 + 2.01555i) q^{5} +(-3.03705 + 0.265684i) q^{6} -1.96518i q^{7} +(-2.76661 - 0.588127i) q^{8} +(1.56651 - 0.508991i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 464 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 8 q^{5} - 6 q^{6} + 12 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41077	−	0.0986613i	−0.997564	−	0.0697641i
							\(3\)	2.12918	−	0.337229i	1.22928	−	0.194699i	0.492188	−	0.870489i	\(-0.336197\pi\)
0.737094	+	0.675790i	\(0.236197\pi\)
\(5\)	0.968273	+	2.01555i	0.433025	+	0.901382i
							\(7\)		−	1.96518i		−	0.742767i	−0.928480	−	0.371384i	\(-0.878883\pi\)
0.928480	−	0.371384i	\(-0.121117\pi\)
\(11\)	5.19238	−	2.64565i	1.56556	−	0.797693i	0.565919	−	0.824461i	\(-0.308521\pi\)
							0.999642	+	0.0267674i	\(0.00852135\pi\)
\(13\)	−2.74457	−	1.39843i	−0.761206	−	0.387854i	0.0298841	−	0.999553i	\(-0.490486\pi\)
							−0.791090	+	0.611699i	\(0.790486\pi\)
\(17\)	1.68537	−	1.22449i	0.408762	−	0.296983i	−0.364338	−	0.931267i	\(-0.618705\pi\)
							0.773100	+	0.634284i	\(0.218705\pi\)
\(19\)	−0.306047	+	1.93230i	−0.0702119	+	0.443300i	0.927391	+	0.374095i	\(0.122046\pi\)
							−0.997602	+	0.0692057i	\(0.977954\pi\)
\(23\)	5.58309	+	1.81406i	1.16415	+	0.378257i	0.826458	−	0.562998i	\(-0.190352\pi\)
							0.337697	+	0.941255i	\(0.390352\pi\)
\(29\)	−3.33223	+	0.527774i	−0.618780	+	0.0980052i	−0.457952	−	0.888977i	\(-0.651417\pi\)
							−0.160829	+	0.986982i	\(0.551417\pi\)
\(31\)	−2.31504	+	1.68198i	−0.415794	+	0.302092i	−0.775943	−	0.630803i	\(-0.782726\pi\)
							0.360149	+	0.932895i	\(0.382726\pi\)
\(37\)	−3.74243	+	7.34493i	−0.615251	+	1.20750i	0.347645	+	0.937626i	\(0.386982\pi\)
							−0.962896	+	0.269873i	\(0.913018\pi\)
\(41\)	10.6714	−	3.46735i	1.66659	−	0.541509i	0.684356	−	0.729148i	\(-0.260084\pi\)
							0.982238	+	0.187639i	\(0.0600836\pi\)
\(43\)	−0.295251	−	0.295251i	−0.0450254	−	0.0450254i	0.684236	−	0.729261i	\(-0.260136\pi\)
							−0.729261	+	0.684236i	\(0.760136\pi\)
\(47\)	−4.33120	−	3.14680i	−0.631770	−	0.459008i	0.225243	−	0.974303i	\(-0.427682\pi\)
							−0.857013	+	0.515295i	\(0.827682\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.2.be.a.21.1	✓	464
16.13	even	4	inner	400.2.be.a.221.48	yes	464
25.6	even	5	inner	400.2.be.a.181.48	yes	464
400.381	even	20	inner	400.2.be.a.381.1	yes	464

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.be.a.21.1	✓	464	1.1	even	1	trivial
400.2.be.a.181.48	yes	464	25.6	even	5	inner
400.2.be.a.221.48	yes	464	16.13	even	4	inner
400.2.be.a.381.1	yes	464	400.381	even	20	inner