Embedded newform 400.2.bd.a.27.21

• Label: 400.2.bd.a.27.21
• Level: $400$
• Weight: $2$
• Character: 400.27
• 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			27.21
Character	\(\chi\)	\(=\)	400.27
Dual form			400.2.bd.a.163.21

$q$-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 \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

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

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.bd.a.27.21	✓	464	1.1	even	1	trivial
400.2.bd.a.163.21	yes	464	400.163	even	20	inner
400.2.bm.a.227.14	yes	464	16.3	odd	4
400.2.bm.a.363.14	yes	464	25.13	odd	20