Embedded newform 400.2.bl.a.189.45

• Label: 400.2.bl.a.189.45
• Level: $400$
• Weight: $2$
• Character: 400.189
• 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.bl (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			189.45
Character	\(\chi\)	\(=\)	400.189
Dual form			400.2.bl.a.309.45

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.970204 + 1.02893i) q^{2} +(-0.133943 - 0.262877i) q^{3} +(-0.117407 + 1.99655i) q^{4} +(-0.662410 + 2.13570i) q^{5} +(0.140531 - 0.392863i) q^{6} -0.849930 q^{7} +(-2.16823 + 1.81626i) q^{8} +(1.71219 - 2.35663i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 464 q - 10 q^{2} - 10 q^{3} - 6 q^{4} - 8 q^{5} - 6 q^{6} - 40 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(177\)	\(351\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{3}{10}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.970204	+	1.02893i	0.686038	+	0.727566i
							\(3\)	−0.133943	−	0.262877i	−0.0773318	−	0.151772i	0.849103	−	0.528227i	\(-0.177143\pi\)
−0.926435	+	0.376455i	\(0.877143\pi\)
\(5\)	−0.662410	+	2.13570i	−0.296239	+	0.955114i
							\(7\)	−0.849930			−0.321243			−0.160622	−	0.987016i	\(-0.551350\pi\)
−0.160622	+	0.987016i	\(0.551350\pi\)
\(11\)	−0.801667	+	5.06153i	−0.241712	+	1.52611i	0.506262	+	0.862380i	\(0.331027\pi\)
							−0.747974	+	0.663728i	\(0.768973\pi\)
\(13\)	0.623022	−	0.0986770i	0.172795	−	0.0273681i	−0.0694372	−	0.997586i	\(-0.522120\pi\)
							0.242232	+	0.970218i	\(0.422120\pi\)
\(17\)	−2.00851	+	0.652606i	−0.487136	+	0.158280i	−0.542279	−	0.840198i	\(-0.682439\pi\)
							0.0551430	+	0.998478i	\(0.482439\pi\)
\(19\)	−2.34605	+	4.60438i	−0.538221	+	1.05632i	0.448483	+	0.893791i	\(0.351964\pi\)
							−0.986704	+	0.162527i	\(0.948036\pi\)
\(23\)	5.46310	−	3.96917i	1.13913	−	0.827630i	0.152136	−	0.988360i	\(-0.451385\pi\)
							0.986998	+	0.160730i	\(0.0513847\pi\)
\(29\)	7.06277	−	3.59866i	1.31152	−	0.668255i	0.348408	−	0.937343i	\(-0.386722\pi\)
							0.963116	+	0.269088i	\(0.0867222\pi\)
\(31\)	1.85644	+	5.71352i	0.333426	+	1.02618i	0.967492	+	0.252900i	\(0.0813845\pi\)
							−0.634067	+	0.773278i	\(0.718615\pi\)
\(37\)	−0.174498	−	1.10174i	−0.0286873	−	0.181125i	0.969184	−	0.246338i	\(-0.0792273\pi\)
							−0.997871	+	0.0652132i	\(0.979227\pi\)
\(41\)	0.885943	−	1.21940i	0.138361	−	0.190438i	−0.734213	−	0.678919i	\(-0.762449\pi\)
							0.872575	+	0.488481i	\(0.162449\pi\)
\(43\)	2.53336	+	2.53336i	0.386334	+	0.386334i	0.873378	−	0.487043i	\(-0.161925\pi\)
							−0.487043	+	0.873378i	\(0.661925\pi\)
\(47\)	9.68456	+	3.14671i	1.41264	+	0.458994i	0.913256	−	0.407387i	\(-0.133560\pi\)
							0.499383	+	0.866381i	\(0.333560\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.2.bl.a.189.45	yes	464
16.5	even	4	inner	400.2.bl.a.389.38	yes	464
25.9	even	10	inner	400.2.bl.a.109.38	✓	464
400.309	even	20	inner	400.2.bl.a.309.45	yes	464

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.bl.a.109.38	✓	464	25.9	even	10	inner
400.2.bl.a.189.45	yes	464	1.1	even	1	trivial
400.2.bl.a.309.45	yes	464	400.309	even	20	inner
400.2.bl.a.389.38	yes	464	16.5	even	4	inner