Embedded newform 400.2.bm.a.147.10

• Label: 400.2.bm.a.147.10
• Level: $400$
• Weight: $2$
• Character: 400.147
• 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.bm (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			147.10
Character	\(\chi\)	\(=\)	400.147
Dual form			400.2.bm.a.283.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.23884 - 0.682116i) q^{2} +(0.504840 - 0.694852i) q^{3} +(1.06944 + 1.69006i) q^{4} +(-2.09876 - 0.771503i) q^{5} +(-1.09938 + 0.516450i) q^{6} +(2.12397 + 2.12397i) q^{7} +(-0.172042 - 2.82319i) q^{8} +(0.699094 + 2.15159i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 464 q - 8 q^{2} - 10 q^{3} - 10 q^{4} - 8 q^{5} - 6 q^{6} - 16 q^{7} - 2 q^{8} + 108 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(177\)	\(351\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{17}{20}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.23884	−	0.682116i	−0.875990	−	0.482329i
							\(3\)	0.504840	−	0.694852i	0.291469	−	0.401173i	−0.638021	−	0.770019i	\(-0.720247\pi\)
0.929491	+	0.368845i	\(0.120247\pi\)
\(5\)	−2.09876	−	0.771503i	−0.938593	−	0.345027i
							\(7\)	2.12397	+	2.12397i	0.802784	+	0.802784i	0.983530	−	0.180746i	\(-0.0578511\pi\)
−0.180746	+	0.983530i	\(0.557851\pi\)
\(11\)	−3.14038	+	1.60011i	−0.946861	+	0.482450i	−0.858032	−	0.513597i	\(-0.828313\pi\)
							−0.0888296	+	0.996047i	\(0.528313\pi\)
\(13\)	0.650228	+	2.00120i	0.180341	+	0.555032i	0.999837	−	0.0180527i	\(-0.00574665\pi\)
							−0.819496	+	0.573085i	\(0.805747\pi\)
\(17\)	2.76079	+	0.437265i	0.669589	+	0.106052i	0.481968	−	0.876189i	\(-0.339922\pi\)
							0.187621	+	0.982241i	\(0.439922\pi\)
\(19\)	0.887626	+	0.140586i	0.203635	+	0.0322527i	0.257419	−	0.966300i	\(-0.417128\pi\)
							−0.0537833	+	0.998553i	\(0.517128\pi\)
\(23\)	1.53495	+	3.01251i	0.320059	+	0.628151i	0.993846	−	0.110775i	\(-0.0353332\pi\)
							−0.673786	+	0.738926i	\(0.735333\pi\)
\(29\)	−5.07812	+	0.804296i	−0.942984	+	0.149354i	−0.608951	−	0.793208i	\(-0.708410\pi\)
							−0.334032	+	0.942562i	\(0.608410\pi\)
\(31\)	0.567859	+	0.781591i	0.101991	+	0.140378i	0.856962	−	0.515380i	\(-0.172349\pi\)
							−0.754971	+	0.655758i	\(0.772349\pi\)
\(37\)	−0.787780	−	2.42454i	−0.129510	−	0.398592i	0.865186	−	0.501452i	\(-0.167201\pi\)
							−0.994696	+	0.102860i	\(0.967201\pi\)
\(41\)	5.18299	−	1.68406i	0.809447	−	0.263005i	0.125083	−	0.992146i	\(-0.460080\pi\)
							0.684364	+	0.729141i	\(0.260080\pi\)
\(43\)	6.24375			0.952163			0.476082	−	0.879401i	\(-0.342057\pi\)
							0.476082	+	0.879401i	\(0.342057\pi\)
\(47\)	9.20201	−	1.45746i	1.34225	−	0.212592i	0.556352	−	0.830947i	\(-0.312201\pi\)
							0.785899	+	0.618355i	\(0.212201\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.2.bm.a.147.10	yes	464
16.11	odd	4		400.2.bd.a.347.29	yes	464
25.8	odd	20		400.2.bd.a.83.29	✓	464
400.283	even	20	inner	400.2.bm.a.283.10	yes	464

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.bd.a.83.29	✓	464	25.8	odd	20
400.2.bd.a.347.29	yes	464	16.11	odd	4
400.2.bm.a.147.10	yes	464	1.1	even	1	trivial
400.2.bm.a.283.10	yes	464	400.283	even	20	inner