Embedded newform 400.2.be.a.181.44

• Label: 400.2.be.a.181.44
• Level: $400$
• Weight: $2$
• Character: 400.181
• 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			181.44
Character	\(\chi\)	\(=\)	400.181
Dual form			400.2.be.a.221.44

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.971795 + 1.02743i) q^{2} +(0.129189 - 0.815667i) q^{3} +(-0.111228 + 1.99690i) q^{4} +(-1.25843 + 1.84834i) q^{5} +(0.963587 - 0.659929i) q^{6} -1.98090i q^{7} +(-2.15977 + 1.82630i) q^{8} +(2.20455 + 0.716301i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 464 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 8 q^{5} - 6 q^{6} + 12 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(177\)	\(351\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.971795	+	1.02743i	0.687163	+	0.726503i
							\(3\)	0.129189	−	0.815667i	0.0745873	−	0.470926i	−0.921917	−	0.387387i	\(-0.873378\pi\)
0.996505	−	0.0835388i	\(-0.0266222\pi\)
\(5\)	−1.25843	+	1.84834i	−0.562789	+	0.826601i
							\(7\)		−	1.98090i		−	0.748711i	−0.927285	−	0.374355i	\(-0.877864\pi\)
0.927285	−	0.374355i	\(-0.122136\pi\)
\(11\)	−2.03455	+	3.99302i	−0.613439	+	1.20394i	0.350186	+	0.936680i	\(0.386118\pi\)
							−0.963624	+	0.267261i	\(0.913882\pi\)
\(13\)	1.98889	+	3.90341i	0.551618	+	1.08261i	0.983538	+	0.180700i	\(0.0578363\pi\)
							−0.431920	+	0.901912i	\(0.642164\pi\)
\(17\)	4.10311	+	2.98108i	0.995150	+	0.723019i	0.961043	−	0.276399i	\(-0.0891412\pi\)
							0.0341073	+	0.999418i	\(0.489141\pi\)
\(19\)	−2.61050	+	0.413463i	−0.598890	+	0.0948549i	−0.448517	−	0.893774i	\(-0.648048\pi\)
							−0.150373	+	0.988629i	\(0.548048\pi\)
\(23\)	3.61334	−	1.17405i	0.753434	−	0.244806i	0.0929760	−	0.995668i	\(-0.470362\pi\)
							0.660458	+	0.750863i	\(0.270362\pi\)
\(29\)	1.33058	−	8.40093i	0.247082	−	1.56001i	−0.482362	−	0.875972i	\(-0.660221\pi\)
							0.729444	−	0.684041i	\(-0.239779\pi\)
\(31\)	−7.30533	−	5.30763i	−1.31208	−	0.953278i	−0.999995	−	0.00322280i	\(-0.998974\pi\)
							−0.312080	−	0.950056i	\(-0.601026\pi\)
\(37\)	−4.78434	+	2.43774i	−0.786540	+	0.400762i	−0.800646	−	0.599138i	\(-0.795510\pi\)
							0.0141055	+	0.999901i	\(0.495510\pi\)
\(41\)	5.54147	+	1.80053i	0.865432	+	0.281196i	0.707896	−	0.706317i	\(-0.249645\pi\)
							0.157537	+	0.987513i	\(0.449645\pi\)
\(43\)	3.41136	+	3.41136i	0.520228	+	0.520228i	0.917640	−	0.397412i	\(-0.130092\pi\)
							−0.397412	+	0.917640i	\(0.630092\pi\)
\(47\)	2.50317	−	1.81866i	0.365124	−	0.265278i	−0.390062	−	0.920789i	\(-0.627546\pi\)
							0.755186	+	0.655510i	\(0.227546\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.2.be.a.181.44	yes	464
16.13	even	4	inner	400.2.be.a.381.28	yes	464
25.21	even	5	inner	400.2.be.a.21.28	✓	464
400.221	even	20	inner	400.2.be.a.221.44	yes	464

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.be.a.21.28	✓	464	25.21	even	5	inner
400.2.be.a.181.44	yes	464	1.1	even	1	trivial
400.2.be.a.221.44	yes	464	400.221	even	20	inner
400.2.be.a.381.28	yes	464	16.13	even	4	inner