Embedded newform 400.2.bd.a.163.21

• Label: 400.2.bd.a.163.21
• Level: $400$
• Weight: $2$
• Character: 400.163
• Analytic conductor: $3.194$
• Analytic rank: $0$
• Dimension: $464$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	400.bd (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			163.21
Character	\(\chi\)	\(=\)	400.163
Dual form			400.2.bd.a.27.21

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.784982 + 1.17635i) q^{2} +(0.494319 + 1.52136i) q^{3} +(-0.767608 - 1.84683i) q^{4} +(-0.776173 - 2.09703i) q^{5} +(-2.17768 - 0.612744i) q^{6} +(0.325173 - 0.325173i) q^{7} +(2.77508 + 0.546750i) q^{8} +(0.356877 - 0.259287i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 464 q - 8 q^{2} - 6 q^{3} - 10 q^{4} - 8 q^{5} - 6 q^{6} - 16 q^{7} - 2 q^{8} - 108 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(177\)	\(351\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{19}{20}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.784982	+	1.17635i	−0.555066	+	0.831806i
							\(3\)	0.494319	+	1.52136i	0.285395	+	0.878355i	0.986280	+	0.165081i	\(0.0527886\pi\)
−0.700885	+	0.713274i	\(0.747211\pi\)
\(5\)	−0.776173	−	2.09703i	−0.347115	−	0.937823i
							\(7\)	0.325173	−	0.325173i	0.122904	−	0.122904i	−0.642980	−	0.765883i	\(-0.722302\pi\)
0.765883	+	0.642980i	\(0.222302\pi\)
\(11\)	−0.445299	−	2.81151i	−0.134263	−	0.847702i	−0.959252	−	0.282553i	\(-0.908819\pi\)
							0.824989	−	0.565149i	\(-0.191181\pi\)
\(13\)	2.52007	+	3.46857i	0.698941	+	0.962009i	0.999965	+	0.00839265i	\(0.00267150\pi\)
							−0.301024	+	0.953616i	\(0.597329\pi\)
\(17\)	2.50508	+	4.91649i	0.607570	+	1.19242i	0.965921	+	0.258837i	\(0.0833392\pi\)
							−0.358351	+	0.933587i	\(0.616661\pi\)
\(19\)	7.23952	−	3.68872i	1.66086	−	0.846251i	0.665894	−	0.746047i	\(-0.268050\pi\)
							0.994967	−	0.100204i	\(-0.0319495\pi\)
\(23\)	−1.02294	−	6.45861i	−0.213298	−	1.34671i	−0.829228	−	0.558911i	\(-0.811219\pi\)
							0.615929	−	0.787801i	\(-0.288781\pi\)
\(29\)	5.16039	+	2.62935i	0.958260	+	0.488258i	0.861894	−	0.507088i	\(-0.169278\pi\)
							0.0963658	+	0.995346i	\(0.469278\pi\)
\(31\)	−4.34364	−	1.41134i	−0.780142	−	0.253483i	−0.108241	−	0.994125i	\(-0.534522\pi\)
							−0.671901	+	0.740641i	\(0.734522\pi\)
\(37\)	−1.33116	−	1.83219i	−0.218842	−	0.301210i	0.685454	−	0.728116i	\(-0.259604\pi\)
							−0.904296	+	0.426906i	\(0.859604\pi\)
\(41\)	−2.90660	−	4.00060i	−0.453935	−	0.624788i	0.519302	−	0.854591i	\(-0.326192\pi\)
							−0.973237	+	0.229803i	\(0.926192\pi\)
\(43\)			4.98111i			0.759613i	0.925066	+	0.379806i	\(0.124009\pi\)
							−0.925066	+	0.379806i	\(0.875991\pi\)
\(47\)	2.95207	−	5.79376i	0.430604	−	0.845107i	−0.569135	−	0.822244i	\(-0.692722\pi\)
							0.999739	−	0.0228632i	\(-0.00727822\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.2.bd.a.163.21	yes	464
16.11	odd	4		400.2.bm.a.363.14	yes	464
25.2	odd	20		400.2.bm.a.227.14	yes	464
400.27	even	20	inner	400.2.bd.a.27.21	✓	464

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.bd.a.27.21	✓	464	400.27	even	20	inner
400.2.bd.a.163.21	yes	464	1.1	even	1	trivial
400.2.bm.a.227.14	yes	464	25.2	odd	20
400.2.bm.a.363.14	yes	464	16.11	odd	4