Embedded newform 400.2.be.a.21.4

• Label: 400.2.be.a.21.4
• 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			21.4
Character	\(\chi\)	\(=\)	400.21
Dual form			400.2.be.a.381.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.37734 + 0.320840i) q^{2} +(0.993702 - 0.157387i) q^{3} +(1.79412 - 0.883811i) q^{4} +(-2.20186 - 0.389615i) q^{5} +(-1.31817 + 0.535595i) q^{6} +0.222360i q^{7} +(-2.18755 + 1.79293i) q^{8} +(-1.89050 + 0.614260i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 464 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 8 q^{5} - 6 q^{6} + 12 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.37734	+	0.320840i	−0.973925	+	0.226868i
							\(3\)	0.993702	−	0.157387i	0.573714	−	0.0908674i	0.137166	−	0.990548i	\(-0.456201\pi\)
0.436548	+	0.899681i	\(0.356201\pi\)
\(5\)	−2.20186	−	0.389615i	−0.984703	−	0.174241i
							\(7\)			0.222360i			0.0840443i	0.999117	+	0.0420222i	\(0.0133800\pi\)
−0.999117	+	0.0420222i	\(0.986620\pi\)
\(11\)	−1.41561	+	0.721291i	−0.426823	+	0.217477i	−0.654187	−	0.756332i	\(-0.726989\pi\)
							0.227364	+	0.973810i	\(0.426989\pi\)
\(13\)	−3.51760	−	1.79231i	−0.975607	−	0.497097i	−0.107894	−	0.994162i	\(-0.534411\pi\)
							−0.867713	+	0.497066i	\(0.834411\pi\)
\(17\)	−1.68389	+	1.22342i	−0.408405	+	0.296723i	−0.772956	−	0.634460i	\(-0.781222\pi\)
							0.364551	+	0.931183i	\(0.381222\pi\)
\(19\)	−1.13753	+	7.18210i	−0.260968	+	1.64769i	0.414320	+	0.910131i	\(0.364019\pi\)
							−0.675288	+	0.737554i	\(0.735981\pi\)
\(23\)	−4.29425	−	1.39529i	−0.895414	−	0.290938i	−0.175071	−	0.984556i	\(-0.556015\pi\)
							−0.720343	+	0.693618i	\(0.756015\pi\)
\(29\)	1.88714	−	0.298893i	0.350433	−	0.0555031i	0.0212631	−	0.999774i	\(-0.493231\pi\)
							0.329170	+	0.944271i	\(0.393231\pi\)
\(31\)	7.10795	−	5.16423i	1.27663	−	0.927523i	0.277181	−	0.960818i	\(-0.410600\pi\)
							0.999446	+	0.0332944i	\(0.0105999\pi\)
\(37\)	−3.16130	+	6.20441i	−0.519715	+	1.02000i	0.470754	+	0.882265i	\(0.343982\pi\)
							−0.990469	+	0.137734i	\(0.956018\pi\)
\(41\)	−9.90557	+	3.21851i	−1.54699	+	0.502648i	−0.953295	−	0.302042i	\(-0.902332\pi\)
							−0.593696	+	0.804690i	\(0.702332\pi\)
\(43\)	−8.51645	−	8.51645i	−1.29875	−	1.29875i	−0.929220	−	0.369526i	\(-0.879520\pi\)
							−0.369526	−	0.929220i	\(-0.620480\pi\)
\(47\)	−3.95824	−	2.87583i	−0.577369	−	0.419483i	0.260406	−	0.965499i	\(-0.416144\pi\)
							−0.837774	+	0.546016i	\(0.816144\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.2.be.a.21.4	✓	464
16.13	even	4	inner	400.2.be.a.221.43	yes	464
25.6	even	5	inner	400.2.be.a.181.43	yes	464
400.381	even	20	inner	400.2.be.a.381.4	yes	464

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.be.a.21.4	✓	464	1.1	even	1	trivial
400.2.be.a.181.43	yes	464	25.6	even	5	inner
400.2.be.a.221.43	yes	464	16.13	even	4	inner
400.2.be.a.381.4	yes	464	400.381	even	20	inner