Embedded newform 400.2.be.a.141.20

• Label: 400.2.be.a.141.20
• Level: $400$
• Weight: $2$
• Character: 400.141
• 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			141.20
Character	\(\chi\)	\(=\)	400.141
Dual form			400.2.be.a.261.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.690341 - 1.23427i) q^{2} +(1.40727 - 2.76191i) q^{3} +(-1.04686 + 1.70414i) q^{4} +(2.16931 - 0.542315i) q^{5} +(-4.38045 + 0.169714i) q^{6} +0.132643i q^{7} +(2.82606 + 0.115671i) q^{8} +(-3.88442 - 5.34645i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 464 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 8 q^{5} - 6 q^{6} + 12 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.690341	−	1.23427i	−0.488145	−	0.872763i
							\(3\)	1.40727	−	2.76191i	0.812485	−	1.59459i	0.00848455	−	0.999964i	\(-0.497299\pi\)
0.804001	−	0.594628i	\(-0.202701\pi\)
\(5\)	2.16931	−	0.542315i	0.970144	−	0.242530i
							\(7\)			0.132643i			0.0501345i	0.999686	+	0.0250673i	\(0.00797999\pi\)
−0.999686	+	0.0250673i	\(0.992020\pi\)
\(11\)	−4.62690	+	0.732828i	−1.39506	+	0.220956i	−0.808284	−	0.588792i	\(-0.799604\pi\)
							−0.586777	+	0.809748i	\(0.699604\pi\)
\(13\)	0.185179	+	0.0293295i	0.0513595	+	0.00813454i	0.182061	−	0.983287i	\(-0.441723\pi\)
							−0.130702	+	0.991422i	\(0.541723\pi\)
\(17\)	0.766309	−	2.35846i	0.185857	−	0.572010i	−0.814105	−	0.580718i	\(-0.802772\pi\)
							0.999962	+	0.00870805i	\(0.00277189\pi\)
\(19\)	0.0229751	−	0.0117064i	0.00527084	−	0.00268563i	−0.451353	−	0.892345i	\(-0.649059\pi\)
							0.456624	+	0.889660i	\(0.349059\pi\)
\(23\)	2.58295	−	3.55513i	0.538583	−	0.741296i	−0.449825	−	0.893117i	\(-0.648514\pi\)
							0.988408	+	0.151820i	\(0.0485136\pi\)
\(29\)	3.93919	−	7.73109i	0.731488	−	1.43563i	−0.162116	−	0.986772i	\(-0.551832\pi\)
							0.893605	−	0.448855i	\(-0.148168\pi\)
\(31\)	−2.77289	+	8.53407i	−0.498025	+	1.53276i	0.314163	+	0.949369i	\(0.398276\pi\)
							−0.812188	+	0.583395i	\(0.801724\pi\)
\(37\)	−0.157987	+	0.997488i	−0.0259728	+	0.163986i	−0.997266	−	0.0739015i	\(-0.976455\pi\)
							0.971293	+	0.237888i	\(0.0764550\pi\)
\(41\)	6.93741	+	9.54853i	1.08344	+	1.49123i	0.855674	+	0.517515i	\(0.173143\pi\)
							0.227768	+	0.973716i	\(0.426857\pi\)
\(43\)	−4.84884	+	4.84884i	−0.739441	+	0.739441i	−0.972470	−	0.233029i	\(-0.925136\pi\)
							0.233029	+	0.972470i	\(0.425136\pi\)
\(47\)	1.41906	+	4.36741i	0.206991	+	0.637052i	0.999626	+	0.0273555i	\(0.00870862\pi\)
							−0.792635	+	0.609696i	\(0.791291\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.2.be.a.141.20	yes	464
16.5	even	4	inner	400.2.be.a.341.5	yes	464
25.11	even	5	inner	400.2.be.a.61.5	✓	464
400.261	even	20	inner	400.2.be.a.261.20	yes	464

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.be.a.61.5	✓	464	25.11	even	5	inner
400.2.be.a.141.20	yes	464	1.1	even	1	trivial
400.2.be.a.261.20	yes	464	400.261	even	20	inner
400.2.be.a.341.5	yes	464	16.5	even	4	inner