Embedded newform 400.2.bm.a.123.19

• Label: 400.2.bm.a.123.19
• Level: $400$
• Weight: $2$
• Character: 400.123
• 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			123.19
Character	\(\chi\)	\(=\)	400.123
Dual form			400.2.bm.a.387.19

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

\(n\)	\(101\)	\(177\)	\(351\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{11}{20}\right)\)	\(-1\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.bd.a.187.34	✓	464	25.12	odd	20
400.2.bd.a.323.34	yes	464	16.3	odd	4
400.2.bm.a.123.19	yes	464	1.1	even	1	trivial
400.2.bm.a.387.19	yes	464	400.387	even	20	inner