Embedded newform 400.2.be.a.21.2

• Label: 400.2.be.a.21.2
• 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			21.2
Character	\(\chi\)	\(=\)	400.21
Dual form			400.2.be.a.381.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.39702 - 0.219860i) q^{2} +(-1.62017 + 0.256610i) q^{3} +(1.90332 + 0.614297i) q^{4} +(-1.31621 + 1.80765i) q^{5} +(2.31983 - 0.00227828i) q^{6} +2.84840i q^{7} +(-2.52392 - 1.27665i) q^{8} +(-0.294054 + 0.0955441i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 464 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 8 q^{5} - 6 q^{6} + 12 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.39702	−	0.219860i	−0.987841	−	0.155464i
							\(3\)	−1.62017	+	0.256610i	−0.935408	+	0.148154i	−0.605488	−	0.795855i	\(-0.707022\pi\)
−0.329920	+	0.944009i	\(0.607022\pi\)
\(5\)	−1.31621	+	1.80765i	−0.588628	+	0.808404i
							\(7\)			2.84840i			1.07659i	0.842755	+	0.538297i	\(0.180932\pi\)
−0.842755	+	0.538297i	\(0.819068\pi\)
\(11\)	1.91450	−	0.975487i	0.577244	−	0.294120i	−0.140890	−	0.990025i	\(-0.544997\pi\)
							0.718134	+	0.695905i	\(0.244997\pi\)
\(13\)	0.644828	+	0.328556i	0.178843	+	0.0911252i	0.541120	−	0.840945i	\(-0.318000\pi\)
							−0.362277	+	0.932071i	\(0.618000\pi\)
\(17\)	−4.49655	+	3.26693i	−1.09057	+	0.792348i	−0.979496	−	0.201464i	\(-0.935430\pi\)
							−0.111078	+	0.993812i	\(0.535430\pi\)
\(19\)	1.01181	−	6.38830i	0.232124	−	1.46558i	−0.546169	−	0.837675i	\(-0.683914\pi\)
							0.778294	−	0.627901i	\(-0.216086\pi\)
\(23\)	−0.148518	−	0.0482565i	−0.0309682	−	0.0100622i	0.293492	−	0.955962i	\(-0.405183\pi\)
							−0.324460	+	0.945899i	\(0.605183\pi\)
\(29\)	−6.21331	+	0.984092i	−1.15378	+	0.182741i	−0.703869	−	0.710330i	\(-0.748546\pi\)
							−0.449915	+	0.893072i	\(0.648546\pi\)
\(31\)	−2.85427	+	2.07375i	−0.512641	+	0.372456i	−0.813825	−	0.581111i	\(-0.802618\pi\)
							0.301183	+	0.953566i	\(0.402618\pi\)
\(37\)	−0.0522876	+	0.102620i	−0.00859602	+	0.0168706i	−0.895265	−	0.445535i	\(-0.853014\pi\)
							0.886669	+	0.462405i	\(0.153014\pi\)
\(41\)	−8.54990	+	2.77803i	−1.33527	+	0.433855i	−0.887712	−	0.460400i	\(-0.847706\pi\)
							−0.447558	+	0.894255i	\(0.647706\pi\)
\(43\)	−7.99717	−	7.99717i	−1.21956	−	1.21956i	−0.967787	−	0.251770i	\(-0.918987\pi\)
							−0.251770	−	0.967787i	\(-0.581013\pi\)
\(47\)	−2.40635	−	1.74832i	−0.351003	−	0.255018i	0.398287	−	0.917261i	\(-0.369605\pi\)
							−0.749290	+	0.662242i	\(0.769605\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.2.be.a.21.2	✓	464
16.13	even	4	inner	400.2.be.a.221.49	yes	464
25.6	even	5	inner	400.2.be.a.181.49	yes	464
400.381	even	20	inner	400.2.be.a.381.2	yes	464

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.be.a.21.2	✓	464	1.1	even	1	trivial
400.2.be.a.181.49	yes	464	25.6	even	5	inner
400.2.be.a.221.49	yes	464	16.13	even	4	inner
400.2.be.a.381.2	yes	464	400.381	even	20	inner