Embedded newform 400.2.bl.a.109.38

• Label: 400.2.bl.a.109.38
• Level: $400$
• Weight: $2$
• Character: 400.109
• 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			109.38
Character	\(\chi\)	\(=\)	400.109
Dual form			400.2.bl.a.389.38

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.678764 - 1.24068i) q^{2} +(-0.262877 - 0.133943i) q^{3} +(-1.07856 - 1.68425i) q^{4} +(-2.13570 + 0.662410i) q^{5} +(-0.344611 + 0.235230i) q^{6} +0.849930 q^{7} +(-2.82170 + 0.194932i) q^{8} +(-1.71219 - 2.35663i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 464 q - 10 q^{2} - 10 q^{3} - 6 q^{4} - 8 q^{5} - 6 q^{6} - 40 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(177\)	\(351\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{7}{10}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.678764	−	1.24068i	0.479959	−	0.877291i
							\(3\)	−0.262877	−	0.133943i	−0.151772	−	0.0773318i	0.376455	−	0.926435i	\(-0.377143\pi\)
−0.528227	+	0.849103i	\(0.677143\pi\)
\(5\)	−2.13570	+	0.662410i	−0.955114	+	0.296239i
							\(7\)	0.849930			0.321243			0.160622	−	0.987016i	\(-0.448650\pi\)
0.160622	+	0.987016i	\(0.448650\pi\)
\(11\)	−5.06153	+	0.801667i	−1.52611	+	0.241712i	−0.862380	−	0.506262i	\(-0.831027\pi\)
							−0.663728	+	0.747974i	\(0.731027\pi\)
\(13\)	−0.0986770	+	0.623022i	−0.0273681	+	0.172795i	−0.997586	−	0.0694372i	\(-0.977880\pi\)
							0.970218	+	0.242232i	\(0.0778797\pi\)
\(17\)	−2.00851	−	0.652606i	−0.487136	−	0.158280i	0.0551430	−	0.998478i	\(-0.482439\pi\)
							−0.542279	+	0.840198i	\(0.682439\pi\)
\(19\)	4.60438	−	2.34605i	1.05632	−	0.538221i	0.162527	−	0.986704i	\(-0.448036\pi\)
							0.893791	+	0.448483i	\(0.148036\pi\)
\(23\)	−5.46310	−	3.96917i	−1.13913	−	0.827630i	−0.152136	−	0.988360i	\(-0.548615\pi\)
							−0.986998	+	0.160730i	\(0.948615\pi\)
\(29\)	−3.59866	+	7.06277i	−0.668255	+	1.31152i	0.269088	+	0.963116i	\(0.413278\pi\)
							−0.937343	+	0.348408i	\(0.886722\pi\)
\(31\)	1.85644	−	5.71352i	0.333426	−	1.02618i	−0.634067	−	0.773278i	\(-0.718615\pi\)
							0.967492	−	0.252900i	\(-0.0813845\pi\)
\(37\)	1.10174	+	0.174498i	0.181125	+	0.0286873i	0.246338	−	0.969184i	\(-0.420773\pi\)
							−0.0652132	+	0.997871i	\(0.520773\pi\)
\(41\)	−0.885943	−	1.21940i	−0.138361	−	0.190438i	0.734213	−	0.678919i	\(-0.237551\pi\)
							−0.872575	+	0.488481i	\(0.837551\pi\)
\(43\)	−2.53336	−	2.53336i	−0.386334	−	0.386334i	0.487043	−	0.873378i	\(-0.338075\pi\)
							−0.873378	+	0.487043i	\(0.838075\pi\)
\(47\)	9.68456	−	3.14671i	1.41264	−	0.458994i	0.499383	−	0.866381i	\(-0.333560\pi\)
							0.913256	+	0.407387i	\(0.133560\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.bl.a.109.38	✓	464	1.1	even	1	trivial
400.2.bl.a.189.45	yes	464	25.14	even	10	inner
400.2.bl.a.309.45	yes	464	16.5	even	4	inner
400.2.bl.a.389.38	yes	464	400.389	even	20	inner