Embedded newform 400.2.be.a.181.7

• Label: 400.2.be.a.181.7
• Level: $400$
• Weight: $2$
• Character: 400.181
• 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			181.7
Character	\(\chi\)	\(=\)	400.181
Dual form			400.2.be.a.221.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.32096 - 0.505044i) q^{2} +(-0.0284977 + 0.179927i) q^{3} +(1.48986 + 1.33428i) q^{4} +(1.98191 + 1.03540i) q^{5} +(0.128516 - 0.223284i) q^{6} +0.839485i q^{7} +(-1.29417 - 2.51498i) q^{8} +(2.82161 + 0.916796i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 464 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 8 q^{5} - 6 q^{6} + 12 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(177\)	\(351\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.32096	−	0.505044i	−0.934058	−	0.357120i
							\(3\)	−0.0284977	+	0.179927i	−0.0164531	+	0.103881i	−0.994548	−	0.104277i	\(-0.966747\pi\)
0.978095	+	0.208159i	\(0.0667470\pi\)
\(5\)	1.98191	+	1.03540i	0.886336	+	0.463043i
							\(7\)			0.839485i			0.317296i	0.987335	+	0.158648i	\(0.0507134\pi\)
−0.987335	+	0.158648i	\(0.949287\pi\)
\(11\)	−1.13833	+	2.23411i	−0.343221	+	0.673609i	−0.996508	−	0.0835017i	\(-0.973390\pi\)
							0.653287	+	0.757111i	\(0.273390\pi\)
\(13\)	−0.0565415	−	0.110969i	−0.0156818	−	0.0307772i	0.883033	−	0.469311i	\(-0.155498\pi\)
							−0.898715	+	0.438534i	\(0.855498\pi\)
\(17\)	−0.867590	−	0.630341i	−0.210421	−	0.152880i	0.477583	−	0.878587i	\(-0.341513\pi\)
							−0.688004	+	0.725707i	\(0.741513\pi\)
\(19\)	−1.56663	+	0.248130i	−0.359410	+	0.0569250i	−0.333528	−	0.942740i	\(-0.608239\pi\)
							−0.0258817	+	0.999665i	\(0.508239\pi\)
\(23\)	−3.63886	+	1.18234i	−0.758755	+	0.246535i	−0.662744	−	0.748846i	\(-0.730608\pi\)
							−0.0960110	+	0.995380i	\(0.530608\pi\)
\(29\)	−0.00352316	+	0.0222443i	−0.000654234	+	0.00413067i	−0.988013	−	0.154369i	\(-0.950666\pi\)
							0.987359	+	0.158499i	\(0.0506656\pi\)
\(31\)	4.71499	+	3.42564i	0.846837	+	0.615263i	0.924272	−	0.381734i	\(-0.124673\pi\)
							−0.0774350	+	0.996997i	\(0.524673\pi\)
\(37\)	3.85359	−	1.96350i	0.633526	−	0.322797i	−0.107569	−	0.994198i	\(-0.534307\pi\)
							0.741095	+	0.671400i	\(0.234307\pi\)
\(41\)	3.07859	+	1.00029i	0.480794	+	0.156220i	0.539379	−	0.842063i	\(-0.318659\pi\)
							−0.0585850	+	0.998282i	\(0.518659\pi\)
\(43\)	2.41770	+	2.41770i	0.368696	+	0.368696i	0.867002	−	0.498305i	\(-0.166044\pi\)
							−0.498305	+	0.867002i	\(0.666044\pi\)
\(47\)	6.85681	−	4.98176i	1.00017	−	0.726665i	0.0380441	−	0.999276i	\(-0.487887\pi\)
							0.962124	+	0.272611i	\(0.0878873\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.2.be.a.181.7	yes	464
16.13	even	4	inner	400.2.be.a.381.41	yes	464
25.21	even	5	inner	400.2.be.a.21.41	✓	464
400.221	even	20	inner	400.2.be.a.221.7	yes	464

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.be.a.21.41	✓	464	25.21	even	5	inner
400.2.be.a.181.7	yes	464	1.1	even	1	trivial
400.2.be.a.221.7	yes	464	400.221	even	20	inner
400.2.be.a.381.41	yes	464	16.13	even	4	inner