Embedded newform 400.2.bm.a.387.19

• Label: 400.2.bm.a.387.19
• Level: $400$
• Weight: $2$
• Character: 400.387
• 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			387.19
Character	\(\chi\)	\(=\)	400.387
Dual form			400.2.bm.a.123.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.731336 - 1.21043i) q^{2} +(2.47039 - 0.802680i) q^{3} +(-0.930297 + 1.77047i) q^{4} +(2.21235 - 0.324824i) q^{5} +(-2.77828 - 2.40322i) q^{6} +(-0.345636 - 0.345636i) q^{7} +(2.82339 - 0.168743i) q^{8} +(3.03150 - 2.20252i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 464 q - 8 q^{2} - 10 q^{3} - 10 q^{4} - 8 q^{5} - 6 q^{6} - 16 q^{7} - 2 q^{8} + 108 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(177\)	\(351\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{9}{20}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.731336	−	1.21043i	−0.517132	−	0.855905i
							\(3\)	2.47039	−	0.802680i	1.42628	−	0.463427i	0.508690	−	0.860949i	\(-0.330130\pi\)
0.917593	+	0.397522i	\(0.130130\pi\)
\(5\)	2.21235	−	0.324824i	0.989393	−	0.145266i
							\(7\)	−0.345636	−	0.345636i	−0.130638	−	0.130638i	0.638764	−	0.769402i	\(-0.279446\pi\)
−0.769402	+	0.638764i	\(0.779446\pi\)
\(11\)	0.0590135	+	0.372596i	0.0177932	+	0.112342i	0.994986	−	0.100010i	\(-0.0318876\pi\)
							−0.977193	+	0.212352i	\(0.931888\pi\)
\(13\)	−1.02336	+	0.743514i	−0.283829	+	0.206214i	−0.720586	−	0.693366i	\(-0.756127\pi\)
							0.436757	+	0.899580i	\(0.356127\pi\)
\(17\)	−3.12352	+	1.59151i	−0.757565	+	0.385999i	−0.789707	−	0.613485i	\(-0.789767\pi\)
							0.0321414	+	0.999483i	\(0.489767\pi\)
\(19\)	5.22229	−	2.66089i	1.19807	−	0.610450i	0.262964	−	0.964806i	\(-0.415300\pi\)
							0.935111	+	0.354356i	\(0.115300\pi\)
\(23\)	−1.29389	+	0.204932i	−0.269794	+	0.0427312i	−0.289866	−	0.957067i	\(-0.593611\pi\)
							0.0200715	+	0.999799i	\(0.493611\pi\)
\(29\)	3.65318	+	1.86139i	0.678378	+	0.345651i	0.758997	−	0.651094i	\(-0.225690\pi\)
							−0.0806190	+	0.996745i	\(0.525690\pi\)
\(31\)	−5.97506	−	1.94142i	−1.07315	−	0.348688i	−0.281438	−	0.959579i	\(-0.590811\pi\)
							−0.791715	+	0.610891i	\(0.790811\pi\)
\(37\)	−5.49014	+	3.98882i	−0.902574	+	0.655758i	−0.939126	−	0.343573i	\(-0.888363\pi\)
							0.0365518	+	0.999332i	\(0.488363\pi\)
\(41\)	−0.583190	−	0.802693i	−0.0910790	−	0.125360i	0.761046	−	0.648698i	\(-0.224686\pi\)
							−0.852125	+	0.523339i	\(0.824686\pi\)
\(43\)	2.14145			0.326568			0.163284	−	0.986579i	\(-0.447791\pi\)
							0.163284	+	0.986579i	\(0.447791\pi\)
\(47\)	−6.02758	−	3.07121i	−0.879213	−	0.447981i	−0.0447199	−	0.999000i	\(-0.514240\pi\)
							−0.834493	+	0.551018i	\(0.814240\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.2.bm.a.387.19	yes	464
16.11	odd	4		400.2.bd.a.187.34	✓	464
25.23	odd	20		400.2.bd.a.323.34	yes	464
400.123	even	20	inner	400.2.bm.a.123.19	yes	464

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.bd.a.187.34	✓	464	16.11	odd	4
400.2.bd.a.323.34	yes	464	25.23	odd	20
400.2.bm.a.123.19	yes	464	400.123	even	20	inner
400.2.bm.a.387.19	yes	464	1.1	even	1	trivial