Embedded newform 400.2.be.a.141.5

• Label: 400.2.be.a.141.5
• Level: $400$
• Weight: $2$
• Character: 400.141
• 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.be (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			141.5
Character	\(\chi\)	\(=\)	400.141
Dual form			400.2.be.a.261.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.38726 - 0.274806i) q^{2} +(0.652342 - 1.28029i) q^{3} +(1.84896 + 0.762452i) q^{4} +(2.03559 + 0.925395i) q^{5} +(-1.25680 + 1.59683i) q^{6} -1.67999i q^{7} +(-2.35546 - 1.56582i) q^{8} +(0.549753 + 0.756670i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 464 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 8 q^{5} - 6 q^{6} + 12 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(177\)	\(351\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.38726	−	0.274806i	−0.980939	−	0.194317i
							\(3\)	0.652342	−	1.28029i	0.376630	−	0.739178i	−0.622423	−	0.782681i	\(-0.713852\pi\)
0.999053	+	0.0435027i	\(0.0138517\pi\)
\(5\)	2.03559	+	0.925395i	0.910345	+	0.413849i
							\(7\)		−	1.67999i		−	0.634977i	−0.948262	−	0.317488i	\(-0.897161\pi\)
0.948262	−	0.317488i	\(-0.102839\pi\)
\(11\)	−0.376292	+	0.0595987i	−0.113456	+	0.0179697i	−0.212904	−	0.977073i	\(-0.568292\pi\)
							0.0994481	+	0.995043i	\(0.468292\pi\)
\(13\)	5.88037	+	0.931359i	1.63092	+	0.258313i	0.903724	−	0.428116i	\(-0.140822\pi\)
							0.727198	+	0.686428i	\(0.240822\pi\)
\(17\)	1.49793	−	4.61015i	0.363301	−	1.11813i	−0.587737	−	0.809052i	\(-0.699981\pi\)
							0.951038	−	0.309073i	\(-0.100019\pi\)
\(19\)	−5.97303	+	3.04341i	−1.37031	+	0.698206i	−0.975385	−	0.220509i	\(-0.929228\pi\)
							−0.394921	+	0.918715i	\(0.629228\pi\)
\(23\)	−4.03408	+	5.55244i	−0.841164	+	1.15776i	0.144577	+	0.989494i	\(0.453818\pi\)
							−0.985741	+	0.168270i	\(0.946182\pi\)
\(29\)	0.964763	−	1.89345i	0.179152	−	0.351606i	−0.783915	−	0.620869i	\(-0.786780\pi\)
							0.963067	+	0.269263i	\(0.0867801\pi\)
\(31\)	1.74337	−	5.36555i	0.313119	−	0.963681i	−0.663403	−	0.748262i	\(-0.730888\pi\)
							0.976522	−	0.215418i	\(-0.0691115\pi\)
\(37\)	0.233235	−	1.47258i	0.0383435	−	0.242091i	−0.961071	−	0.276301i	\(-0.910891\pi\)
							0.999415	+	0.0342095i	\(0.0108913\pi\)
\(41\)	−0.668698	−	0.920384i	−0.104433	−	0.143740i	0.753602	−	0.657331i	\(-0.228315\pi\)
							−0.858035	+	0.513591i	\(0.828315\pi\)
\(43\)	4.99167	−	4.99167i	0.761222	−	0.761222i	−0.215321	−	0.976543i	\(-0.569080\pi\)
							0.976543	+	0.215321i	\(0.0690799\pi\)
\(47\)	−1.93439	−	5.95344i	−0.282160	−	0.868399i	−0.987236	−	0.159267i	\(-0.949087\pi\)
							0.705076	−	0.709132i	\(-0.250913\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.2.be.a.141.5	yes	464
16.5	even	4	inner	400.2.be.a.341.19	yes	464
25.11	even	5	inner	400.2.be.a.61.19	✓	464
400.261	even	20	inner	400.2.be.a.261.5	yes	464

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.be.a.61.19	✓	464	25.11	even	5	inner
400.2.be.a.141.5	yes	464	1.1	even	1	trivial
400.2.be.a.261.5	yes	464	400.261	even	20	inner
400.2.be.a.341.19	yes	464	16.5	even	4	inner