Embedded newform 100.2.l.b.87.1

• Label: 100.2.l.b.87.1
• Level: $100$
• Weight: $2$
• Character: 100.87
• Analytic conductor: $0.799$
• Analytic rank: $0$
• Dimension: $96$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 100 = 2^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	100.l (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.798504020213\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(12\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			87.1
Character	\(\chi\)	\(=\)	100.87
Dual form			100.2.l.b.23.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41083 + 0.0976968i) q^{2} +(0.0867473 - 0.170251i) q^{3} +(1.98091 - 0.275668i) q^{4} +(-1.18066 - 1.89896i) q^{5} +(-0.105753 + 0.248671i) q^{6} +(1.89427 - 1.89427i) q^{7} +(-2.76781 + 0.582451i) q^{8} +(1.74190 + 2.39751i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 10 q^{2} - 10 q^{4} - 20 q^{5} - 6 q^{6} - 10 q^{8} - 20 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/100\mathbb{Z}\right)^\times\).

\(n\)	\(51\)	\(77\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{9}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41083	+	0.0976968i	−0.997611	+	0.0690820i
							\(3\)	0.0867473	−	0.170251i	0.0500836	−	0.0982946i	−0.864608	−	0.502447i	\(-0.832433\pi\)
0.914692	+	0.404152i	\(0.132433\pi\)
\(5\)	−1.18066	−	1.89896i	−0.528006	−	0.849240i
							\(7\)	1.89427	−	1.89427i	0.715965	−	0.715965i	−0.251811	−	0.967776i	\(-0.581026\pi\)
0.967776	+	0.251811i	\(0.0810262\pi\)
\(11\)	2.99934	−	4.12824i	0.904336	−	1.24471i	−0.0647283	−	0.997903i	\(-0.520618\pi\)
							0.969064	−	0.246809i	\(-0.0793819\pi\)
\(13\)	0.433317	−	2.73586i	0.120181	−	0.758790i	−0.851824	−	0.523828i	\(-0.824503\pi\)
							0.972005	−	0.234962i	\(-0.0754966\pi\)
\(17\)	−4.16697	+	2.12318i	−1.01064	+	0.514947i	−0.879238	−	0.476383i	\(-0.841948\pi\)
							−0.131402	+	0.991329i	\(0.541948\pi\)
\(19\)	−1.55045	+	4.77181i	−0.355699	+	1.09473i	0.599905	+	0.800072i	\(0.295205\pi\)
							−0.955603	+	0.294657i	\(0.904795\pi\)
\(23\)	0.699923	+	4.41914i	0.145944	+	0.921455i	0.946619	+	0.322355i	\(0.104475\pi\)
							−0.800675	+	0.599100i	\(0.795525\pi\)
\(29\)	−0.211843	+	0.0688321i	−0.0393383	+	0.0127818i	−0.328620	−	0.944462i	\(-0.606584\pi\)
							0.289282	+	0.957244i	\(0.406584\pi\)
\(31\)	−7.01920	−	2.28068i	−1.26069	−	0.409622i	−0.398947	−	0.916974i	\(-0.630624\pi\)
							−0.861739	+	0.507352i	\(0.830624\pi\)
\(37\)	4.32186	+	0.684516i	0.710510	+	0.112534i	0.501220	−	0.865320i	\(-0.332885\pi\)
							0.209290	+	0.977854i	\(0.432885\pi\)
\(41\)	2.54404	−	1.84835i	0.397312	−	0.288664i	−0.371133	−	0.928580i	\(-0.621031\pi\)
							0.768445	+	0.639915i	\(0.221031\pi\)
\(43\)	2.68807	+	2.68807i	0.409927	+	0.409927i	0.881713	−	0.471786i	\(-0.156391\pi\)
							−0.471786	+	0.881713i	\(0.656391\pi\)
\(47\)	7.84709	+	3.99829i	1.14462	+	0.583211i	0.920264	−	0.391298i	\(-0.127974\pi\)
							0.224351	+	0.974508i	\(0.427974\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	100.2.l.b.87.1	yes	96
3.2	odd	2		900.2.bj.d.487.12		96
4.3	odd	2	inner	100.2.l.b.87.12	yes	96
5.2	odd	4		500.2.l.d.43.5		96
5.3	odd	4		500.2.l.e.43.8		96
5.4	even	2		500.2.l.f.207.12		96
12.11	even	2		900.2.bj.d.487.1		96
20.3	even	4		500.2.l.e.43.7		96
20.7	even	4		500.2.l.d.43.6		96
20.19	odd	2		500.2.l.f.207.1		96
25.2	odd	20		500.2.l.f.343.1		96
25.11	even	5		500.2.l.e.407.7		96
25.14	even	10		500.2.l.d.407.6		96
25.23	odd	20	inner	100.2.l.b.23.12	yes	96
75.23	even	20		900.2.bj.d.523.1		96
100.11	odd	10		500.2.l.e.407.8		96
100.23	even	20	inner	100.2.l.b.23.1	✓	96
100.27	even	20		500.2.l.f.343.12		96
100.39	odd	10		500.2.l.d.407.5		96
300.23	odd	20		900.2.bj.d.523.12		96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
100.2.l.b.23.1	✓	96	100.23	even	20	inner
100.2.l.b.23.12	yes	96	25.23	odd	20	inner
100.2.l.b.87.1	yes	96	1.1	even	1	trivial
100.2.l.b.87.12	yes	96	4.3	odd	2	inner
500.2.l.d.43.5		96	5.2	odd	4
500.2.l.d.43.6		96	20.7	even	4
500.2.l.d.407.5		96	100.39	odd	10
500.2.l.d.407.6		96	25.14	even	10
500.2.l.e.43.7		96	20.3	even	4
500.2.l.e.43.8		96	5.3	odd	4
500.2.l.e.407.7		96	25.11	even	5
500.2.l.e.407.8		96	100.11	odd	10
500.2.l.f.207.1		96	20.19	odd	2
500.2.l.f.207.12		96	5.4	even	2
500.2.l.f.343.1		96	25.2	odd	20
500.2.l.f.343.12		96	100.27	even	20
900.2.bj.d.487.1		96	12.11	even	2
900.2.bj.d.487.12		96	3.2	odd	2
900.2.bj.d.523.1		96	75.23	even	20
900.2.bj.d.523.12		96	300.23	odd	20