Embedded newform 400.2.be.a.21.9

• Label: 400.2.be.a.21.9
• 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			21.9
Character	\(\chi\)	\(=\)	400.21
Dual form			400.2.be.a.381.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.32191 - 0.502558i) q^{2} +(-2.84258 + 0.450220i) q^{3} +(1.49487 + 1.32867i) q^{4} +(2.19785 + 0.411629i) q^{5} +(3.98388 + 0.833413i) q^{6} -0.416200i q^{7} +(-1.30834 - 2.50764i) q^{8} +(5.02439 - 1.63252i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 464 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 8 q^{5} - 6 q^{6} + 12 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.32191	−	0.502558i	−0.934729	−	0.355362i
							\(3\)	−2.84258	+	0.450220i	−1.64116	+	0.259935i	−0.907646	−	0.419736i	\(-0.862123\pi\)
−0.733518	+	0.679670i	\(0.762123\pi\)
\(5\)	2.19785	+	0.411629i	0.982910	+	0.184086i
							\(7\)		−	0.416200i		−	0.157309i	−0.996902	−	0.0786544i	\(-0.974938\pi\)
0.996902	−	0.0786544i	\(-0.0250624\pi\)
\(11\)	−1.69974	+	0.866062i	−0.512492	+	0.261128i	−0.691057	−	0.722801i	\(-0.742855\pi\)
							0.178565	+	0.983928i	\(0.442855\pi\)
\(13\)	−2.54723	−	1.29788i	−0.706476	−	0.359967i	0.0635513	−	0.997979i	\(-0.479757\pi\)
							−0.770027	+	0.638011i	\(0.779757\pi\)
\(17\)	1.14178	−	0.829551i	0.276922	−	0.201196i	−0.440652	−	0.897678i	\(-0.645253\pi\)
							0.717574	+	0.696483i	\(0.245253\pi\)
\(19\)	−0.250744	+	1.58314i	−0.0575247	+	0.363196i	0.942088	+	0.335366i	\(0.108860\pi\)
							−0.999613	+	0.0278307i	\(0.991140\pi\)
\(23\)	6.83445	+	2.22065i	1.42508	+	0.463037i	0.917213	−	0.398398i	\(-0.130434\pi\)
							0.507869	+	0.861435i	\(0.330434\pi\)
\(29\)	8.14228	−	1.28961i	1.51198	−	0.239475i	0.655320	−	0.755351i	\(-0.272534\pi\)
							0.856663	+	0.515876i	\(0.172534\pi\)
\(31\)	−1.72092	+	1.25032i	−0.309087	+	0.224565i	−0.731505	−	0.681836i	\(-0.761182\pi\)
							0.422418	+	0.906401i	\(0.361182\pi\)
\(37\)	−4.26615	+	8.37279i	−0.701351	+	1.37648i	0.215204	+	0.976569i	\(0.430958\pi\)
							−0.916555	+	0.399909i	\(0.869042\pi\)
\(41\)	−4.61740	+	1.50029i	−0.721117	+	0.234305i	−0.646508	−	0.762908i	\(-0.723771\pi\)
							−0.0746098	+	0.997213i	\(0.523771\pi\)
\(43\)	0.171232	+	0.171232i	0.0261127	+	0.0261127i	0.720043	−	0.693930i	\(-0.244122\pi\)
							−0.693930	+	0.720043i	\(0.744122\pi\)
\(47\)	8.36006	+	6.07394i	1.21944	+	0.885975i	0.996053	−	0.0887552i	\(-0.0282889\pi\)
							0.223386	+	0.974730i	\(0.428289\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.2.be.a.21.9	✓	464
16.13	even	4	inner	400.2.be.a.221.55	yes	464
25.6	even	5	inner	400.2.be.a.181.55	yes	464
400.381	even	20	inner	400.2.be.a.381.9	yes	464

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.be.a.21.9	✓	464	1.1	even	1	trivial
400.2.be.a.181.55	yes	464	25.6	even	5	inner
400.2.be.a.221.55	yes	464	16.13	even	4	inner
400.2.be.a.381.9	yes	464	400.381	even	20	inner