Embedded newform 400.2.be.a.61.19

• Label: 400.2.be.a.61.19
• Level: $400$
• Weight: $2$
• Character: 400.61
• 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			61.19
Character	\(\chi\)	\(=\)	400.61
Dual form			400.2.be.a.341.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.690042 + 1.23444i) q^{2} +(-1.28029 + 0.652342i) q^{3} +(-1.04769 - 1.70363i) q^{4} +(-0.925395 - 2.03559i) q^{5} +(0.0781784 - 2.03059i) q^{6} -1.67999i q^{7} +(2.82598 - 0.117729i) q^{8} +(-0.549753 + 0.756670i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 464 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 8 q^{5} - 6 q^{6} + 12 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.690042	+	1.23444i	−0.487933	+	0.872881i
							\(3\)	−1.28029	+	0.652342i	−0.739178	+	0.376630i	−0.782681	−	0.622423i	\(-0.786148\pi\)
0.0435027	+	0.999053i	\(0.486148\pi\)
\(5\)	−0.925395	−	2.03559i	−0.413849	−	0.910345i
							\(7\)		−	1.67999i		−	0.634977i	−0.948262	−	0.317488i	\(-0.897161\pi\)
0.948262	−	0.317488i	\(-0.102839\pi\)
\(11\)	−0.0595987	+	0.376292i	−0.0179697	+	0.113456i	−0.995043	−	0.0994481i	\(-0.968292\pi\)
							0.977073	+	0.212904i	\(0.0682923\pi\)
\(13\)	0.931359	+	5.88037i	0.258313	+	1.63092i	0.686428	+	0.727198i	\(0.259178\pi\)
							−0.428116	+	0.903724i	\(0.640822\pi\)
\(17\)	1.49793	+	4.61015i	0.363301	+	1.11813i	0.951038	+	0.309073i	\(0.100019\pi\)
							−0.587737	+	0.809052i	\(0.699981\pi\)
\(19\)	3.04341	−	5.97303i	0.698206	−	1.37031i	−0.220509	−	0.975385i	\(-0.570772\pi\)
							0.918715	−	0.394921i	\(-0.129228\pi\)
\(23\)	4.03408	+	5.55244i	0.841164	+	1.15776i	0.985741	+	0.168270i	\(0.0538179\pi\)
							−0.144577	+	0.989494i	\(0.546182\pi\)
\(29\)	−1.89345	+	0.964763i	−0.351606	+	0.179152i	−0.620869	−	0.783915i	\(-0.713220\pi\)
							0.269263	+	0.963067i	\(0.413220\pi\)
\(31\)	1.74337	+	5.36555i	0.313119	+	0.963681i	0.976522	+	0.215418i	\(0.0691115\pi\)
							−0.663403	+	0.748262i	\(0.730888\pi\)
\(37\)	1.47258	−	0.233235i	0.242091	−	0.0383435i	−0.0342095	−	0.999415i	\(-0.510891\pi\)
							0.276301	+	0.961071i	\(0.410891\pi\)
\(41\)	0.668698	−	0.920384i	0.104433	−	0.143740i	−0.753602	−	0.657331i	\(-0.771685\pi\)
							0.858035	+	0.513591i	\(0.171685\pi\)
\(43\)	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	0.705076	+	0.709132i	\(0.250913\pi\)
							−0.987236	+	0.159267i	\(0.949087\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.be.a.61.19	✓	464	1.1	even	1	trivial
400.2.be.a.141.5	yes	464	25.16	even	5	inner
400.2.be.a.261.5	yes	464	16.5	even	4	inner
400.2.be.a.341.19	yes	464	400.341	even	20	inner