Embedded newform 400.2.bl.a.389.47

• Label: 400.2.bl.a.389.47
• Level: $400$
• Weight: $2$
• Character: 400.389
• 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			389.47
Character	\(\chi\)	\(=\)	400.389
Dual form			400.2.bl.a.109.47

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.08235 + 0.910234i) q^{2} +(2.23494 - 1.13876i) q^{3} +(0.342948 + 1.97038i) q^{4} +(-1.70670 + 1.44471i) q^{5} +(3.45552 + 0.801789i) q^{6} +1.03629 q^{7} +(-1.42232 + 2.44479i) q^{8} +(1.93484 - 2.66309i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 464 q - 10 q^{2} - 10 q^{3} - 6 q^{4} - 8 q^{5} - 6 q^{6} - 40 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.08235	+	0.910234i	0.765335	+	0.643633i
							\(3\)	2.23494	−	1.13876i	1.29035	−	0.657464i	0.332055	−	0.943260i	\(-0.392258\pi\)
0.958291	+	0.285796i	\(0.0922579\pi\)
\(5\)	−1.70670	+	1.44471i	−0.763259	+	0.646093i
							\(7\)	1.03629			0.391680			0.195840	−	0.980636i	\(-0.437257\pi\)
0.195840	+	0.980636i	\(0.437257\pi\)
\(11\)	3.45339	+	0.546964i	1.04124	+	0.164916i	0.653556	−	0.756878i	\(-0.273276\pi\)
							0.387681	+	0.921794i	\(0.373276\pi\)
\(13\)	−0.657502	−	4.15131i	−0.182358	−	1.15137i	−0.893749	−	0.448568i	\(-0.851934\pi\)
							0.711390	−	0.702797i	\(-0.248066\pi\)
\(17\)	−1.05165	+	0.341703i	−0.255064	+	0.0828752i	−0.433758	−	0.901029i	\(-0.642813\pi\)
							0.178694	+	0.983905i	\(0.442813\pi\)
\(19\)	−0.953910	−	0.486041i	−0.218842	−	0.111506i	0.341133	−	0.940015i	\(-0.389189\pi\)
							−0.559975	+	0.828509i	\(0.689189\pi\)
\(23\)	−3.71583	+	2.69971i	−0.774804	+	0.562928i	−0.903415	−	0.428767i	\(-0.858948\pi\)
							0.128611	+	0.991695i	\(0.458948\pi\)
\(29\)	0.525186	+	1.03074i	0.0975246	+	0.191403i	0.934614	−	0.355663i	\(-0.115745\pi\)
							−0.837090	+	0.547066i	\(0.815745\pi\)
\(31\)	−1.38830	−	4.27275i	−0.249346	−	0.767408i	−0.994891	−	0.100953i	\(-0.967811\pi\)
							0.745545	−	0.666455i	\(-0.232189\pi\)
\(37\)	−5.85001	+	0.926551i	−0.961736	+	0.152324i	−0.617510	−	0.786563i	\(-0.711859\pi\)
							−0.344226	+	0.938887i	\(0.611859\pi\)
\(41\)	6.02492	−	8.29259i	0.940935	−	1.29509i	−0.0145035	−	0.999895i	\(-0.504617\pi\)
							0.955438	−	0.295191i	\(-0.0953832\pi\)
\(43\)	8.20962	−	8.20962i	1.25195	−	1.25195i	0.297112	−	0.954842i	\(-0.403976\pi\)
							0.954842	−	0.297112i	\(-0.0960236\pi\)
\(47\)	2.50436	+	0.813716i	0.365299	+	0.118693i	0.485914	−	0.874007i	\(-0.338487\pi\)
							−0.120615	+	0.992699i	\(0.538487\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.2.bl.a.389.47	yes	464
16.13	even	4	inner	400.2.bl.a.189.37	yes	464
25.9	even	10	inner	400.2.bl.a.309.37	yes	464
400.109	even	20	inner	400.2.bl.a.109.47	✓	464

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.bl.a.109.47	✓	464	400.109	even	20	inner
400.2.bl.a.189.37	yes	464	16.13	even	4	inner
400.2.bl.a.309.37	yes	464	25.9	even	10	inner
400.2.bl.a.389.47	yes	464	1.1	even	1	trivial