Embedded newform 400.2.bm.a.147.54

• Label: 400.2.bm.a.147.54
• Level: $400$
• Weight: $2$
• Character: 400.147
• Analytic conductor: $3.194$
• Analytic rank: $0$
• Dimension: $464$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	400.bm (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			147.54
Character	\(\chi\)	\(=\)	400.147
Dual form			400.2.bm.a.283.54

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.36130 - 0.383224i) q^{2} +(1.52148 - 2.09413i) q^{3} +(1.70628 - 1.04337i) q^{4} +(-2.19349 + 0.434303i) q^{5} +(1.26867 - 3.43381i) q^{6} +(-0.912906 - 0.912906i) q^{7} +(1.92291 - 2.07422i) q^{8} +(-1.14345 - 3.51919i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 464 q - 8 q^{2} - 10 q^{3} - 10 q^{4} - 8 q^{5} - 6 q^{6} - 16 q^{7} - 2 q^{8} + 108 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(177\)	\(351\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{17}{20}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.36130	−	0.383224i	0.962585	−	0.270980i
							\(3\)	1.52148	−	2.09413i	0.878425	−	1.20905i	−0.0984291	−	0.995144i	\(-0.531382\pi\)
0.976855	−	0.213905i	\(-0.0686182\pi\)
\(5\)	−2.19349	+	0.434303i	−0.980957	+	0.194226i
							\(7\)	−0.912906	−	0.912906i	−0.345046	−	0.345046i	0.513214	−	0.858260i	\(-0.328455\pi\)
−0.858260	+	0.513214i	\(0.828455\pi\)
\(11\)	−1.23891	+	0.631254i	−0.373544	+	0.190330i	−0.630676	−	0.776046i	\(-0.717222\pi\)
							0.257132	+	0.966376i	\(0.417222\pi\)
\(13\)	0.633547	+	1.94986i	0.175714	+	0.540793i	0.999665	−	0.0258678i	\(-0.00823490\pi\)
							−0.823951	+	0.566661i	\(0.808235\pi\)
\(17\)	−1.61335	−	0.255529i	−0.391295	−	0.0619750i	−0.0423127	−	0.999104i	\(-0.513473\pi\)
							−0.348982	+	0.937129i	\(0.613473\pi\)
\(19\)	7.18562	+	1.13809i	1.64849	+	0.261096i	0.910438	−	0.413645i	\(-0.135744\pi\)
							0.738055	+	0.674740i	\(0.235744\pi\)
\(23\)	2.93279	+	5.75592i	0.611529	+	1.20019i	0.964382	+	0.264514i	\(0.0852114\pi\)
							−0.352853	+	0.935679i	\(0.614789\pi\)
\(29\)	−1.61627	+	0.255991i	−0.300133	+	0.0475364i	−0.304685	−	0.952453i	\(-0.598551\pi\)
							0.00455181	+	0.999990i	\(0.498551\pi\)
\(31\)	0.943047	+	1.29799i	0.169376	+	0.233126i	0.885264	−	0.465089i	\(-0.153978\pi\)
							−0.715888	+	0.698215i	\(0.753978\pi\)
\(37\)	−0.187698	−	0.577674i	−0.0308573	−	0.0949691i	0.934442	−	0.356116i	\(-0.115899\pi\)
							−0.965299	+	0.261147i	\(0.915899\pi\)
\(41\)	−1.27266	+	0.413511i	−0.198755	+	0.0645795i	−0.406703	−	0.913560i	\(-0.633322\pi\)
							0.207948	+	0.978140i	\(0.433322\pi\)
\(43\)	−10.1224			−1.54365			−0.771823	−	0.635837i	\(-0.780655\pi\)
							−0.771823	+	0.635837i	\(0.780655\pi\)
\(47\)	−4.36134	+	0.690768i	−0.636167	+	0.100759i	−0.466184	−	0.884688i	\(-0.654371\pi\)
							−0.169983	+	0.985447i	\(0.554371\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.2.bm.a.147.54	yes	464
16.11	odd	4		400.2.bd.a.347.46	yes	464
25.8	odd	20		400.2.bd.a.83.46	✓	464
400.283	even	20	inner	400.2.bm.a.283.54	yes	464

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.bd.a.83.46	✓	464	25.8	odd	20
400.2.bd.a.347.46	yes	464	16.11	odd	4
400.2.bm.a.147.54	yes	464	1.1	even	1	trivial
400.2.bm.a.283.54	yes	464	400.283	even	20	inner