Embedded newform 400.2.bl.a.189.14

• Label: 400.2.bl.a.189.14
• 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			189.14
Character	\(\chi\)	\(=\)	400.189
Dual form			400.2.bl.a.309.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.09102 + 0.899821i) q^{2} +(1.16138 + 2.27933i) q^{3} +(0.380645 - 1.96344i) q^{4} +(0.211114 + 2.22608i) q^{5} +(-3.31808 - 1.44176i) q^{6} +0.405092 q^{7} +(1.35146 + 2.48467i) q^{8} +(-2.08320 + 2.86728i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 464 q - 10 q^{2} - 10 q^{3} - 6 q^{4} - 8 q^{5} - 6 q^{6} - 40 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.09102	+	0.899821i	−0.771467	+	0.636269i
							\(3\)	1.16138	+	2.27933i	0.670522	+	1.31597i	0.936049	+	0.351870i	\(0.114454\pi\)
−0.265527	+	0.964103i	\(0.585546\pi\)
\(5\)	0.211114	+	2.22608i	0.0944129	+	0.995533i
							\(7\)	0.405092			0.153110			0.0765552	−	0.997065i	\(-0.475608\pi\)
0.0765552	+	0.997065i	\(0.475608\pi\)
\(11\)	−0.148300	+	0.936329i	−0.0447141	+	0.282314i	−0.999907	−	0.0136717i	\(-0.995648\pi\)
							0.955192	+	0.295986i	\(0.0956480\pi\)
\(13\)	−1.95740	+	0.310022i	−0.542885	+	0.0859846i	−0.421854	−	0.906664i	\(-0.638620\pi\)
							−0.121032	+	0.992649i	\(0.538620\pi\)
\(17\)	3.96184	−	1.28728i	0.960886	−	0.312211i	0.213755	−	0.976887i	\(-0.431431\pi\)
							0.747131	+	0.664676i	\(0.231431\pi\)
\(19\)	−2.51737	+	4.94061i	−0.577523	+	1.13345i	0.398780	+	0.917047i	\(0.369434\pi\)
							−0.976303	+	0.216407i	\(0.930566\pi\)
\(23\)	4.30446	−	3.12737i	0.897542	−	0.652102i	−0.0402917	−	0.999188i	\(-0.512829\pi\)
							0.937833	+	0.347086i	\(0.112829\pi\)
\(29\)	−4.08194	+	2.07985i	−0.757997	+	0.386219i	−0.789871	−	0.613273i	\(-0.789852\pi\)
							0.0318740	+	0.999492i	\(0.489852\pi\)
\(31\)	−0.642556	−	1.97758i	−0.115406	−	0.355184i	0.876625	−	0.481174i	\(-0.159789\pi\)
							−0.992032	+	0.125990i	\(0.959789\pi\)
\(37\)	−1.08298	−	6.83767i	−0.178041	−	1.12411i	−0.901193	−	0.433418i	\(-0.857308\pi\)
							0.723152	−	0.690689i	\(-0.242692\pi\)
\(41\)	6.12796	−	8.43441i	0.957026	−	1.31723i	0.00869110	−	0.999962i	\(-0.497234\pi\)
							0.948335	−	0.317271i	\(-0.102766\pi\)
\(43\)	4.87314	+	4.87314i	0.743148	+	0.743148i	0.973182	−	0.230035i	\(-0.0738840\pi\)
							−0.230035	+	0.973182i	\(0.573884\pi\)
\(47\)	0.222136	+	0.0721764i	0.0324019	+	0.0105280i	0.325173	−	0.945655i	\(-0.394577\pi\)
							−0.292771	+	0.956183i	\(0.594577\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.2.bl.a.189.14	yes	464
16.5	even	4	inner	400.2.bl.a.389.49	yes	464
25.9	even	10	inner	400.2.bl.a.109.49	✓	464
400.309	even	20	inner	400.2.bl.a.309.14	yes	464

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.bl.a.109.49	✓	464	25.9	even	10	inner
400.2.bl.a.189.14	yes	464	1.1	even	1	trivial
400.2.bl.a.309.14	yes	464	400.309	even	20	inner
400.2.bl.a.389.49	yes	464	16.5	even	4	inner