Embedded newform 400.2.be.a.21.28

• Label: 400.2.be.a.21.28
• Level: $400$
• Weight: $2$
• Character: 400.21
• 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			21.28
Character	\(\chi\)	\(=\)	400.21
Dual form			400.2.be.a.381.28

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.182290 - 1.40242i) q^{2} +(0.815667 - 0.129189i) q^{3} +(-1.93354 + 0.511293i) q^{4} +(1.84834 - 1.25843i) q^{5} +(-0.329865 - 1.12035i) q^{6} -1.98090i q^{7} +(1.06951 + 2.61842i) q^{8} +(-2.20455 + 0.716301i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 464 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 8 q^{5} - 6 q^{6} + 12 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.182290	−	1.40242i	−0.128899	−	0.991658i
							\(3\)	0.815667	−	0.129189i	0.470926	−	0.0745873i	0.0835388	−	0.996505i	\(-0.473378\pi\)
0.387387	+	0.921917i	\(0.373378\pi\)
\(5\)	1.84834	−	1.25843i	0.826601	−	0.562789i
							\(7\)		−	1.98090i		−	0.748711i	−0.927285	−	0.374355i	\(-0.877864\pi\)
0.927285	−	0.374355i	\(-0.122136\pi\)
\(11\)	3.99302	−	2.03455i	1.20394	−	0.613439i	0.267261	−	0.963624i	\(-0.413882\pi\)
							0.936680	+	0.350186i	\(0.113882\pi\)
\(13\)	−3.90341	−	1.98889i	−1.08261	−	0.551618i	−0.180700	−	0.983538i	\(-0.557836\pi\)
							−0.901912	+	0.431920i	\(0.857836\pi\)
\(17\)	4.10311	−	2.98108i	0.995150	−	0.723019i	0.0341073	−	0.999418i	\(-0.489141\pi\)
							0.961043	+	0.276399i	\(0.0891412\pi\)
\(19\)	−0.413463	+	2.61050i	−0.0948549	+	0.598890i	0.893774	+	0.448517i	\(0.148048\pi\)
							−0.988629	+	0.150373i	\(0.951952\pi\)
\(23\)	−3.61334	−	1.17405i	−0.753434	−	0.244806i	−0.0929760	−	0.995668i	\(-0.529638\pi\)
							−0.660458	+	0.750863i	\(0.729638\pi\)
\(29\)	8.40093	−	1.33058i	1.56001	−	0.247082i	0.684041	−	0.729444i	\(-0.260221\pi\)
							0.875972	+	0.482362i	\(0.160221\pi\)
\(31\)	−7.30533	+	5.30763i	−1.31208	+	0.953278i	−0.312080	+	0.950056i	\(0.601026\pi\)
							−0.999995	+	0.00322280i	\(0.998974\pi\)
\(37\)	2.43774	−	4.78434i	0.400762	−	0.786540i	−0.599138	−	0.800646i	\(-0.704490\pi\)
							0.999901	+	0.0141055i	\(0.00449006\pi\)
\(41\)	−5.54147	+	1.80053i	−0.865432	+	0.281196i	−0.707896	−	0.706317i	\(-0.750355\pi\)
							−0.157537	+	0.987513i	\(0.550355\pi\)
\(43\)	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	0.755186	−	0.655510i	\(-0.227546\pi\)
							−0.390062	+	0.920789i	\(0.627546\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.be.a.21.28	✓	464	1.1	even	1	trivial
400.2.be.a.181.44	yes	464	25.6	even	5	inner
400.2.be.a.221.44	yes	464	16.13	even	4	inner
400.2.be.a.381.28	yes	464	400.381	even	20	inner