Embedded newform 400.2.bl.a.109.47

• Label: 400.2.bl.a.109.47
• 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			109.47
Character	\(\chi\)	\(=\)	400.109
Dual form			400.2.bl.a.389.47

$q$-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 \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.bl.a.109.47	✓	464	1.1	even	1	trivial
400.2.bl.a.189.37	yes	464	25.14	even	10	inner
400.2.bl.a.309.37	yes	464	16.5	even	4	inner
400.2.bl.a.389.47	yes	464	400.389	even	20	inner