Embedded newform 100.4.l.b.3.9

• Label: 100.4.l.b.3.9
• Level: $100$
• Weight: $4$
• Character: 100.3
• Analytic conductor: $5.900$
• Analytic rank: $0$
• Dimension: $336$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 100 = 2^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	100.l (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.90019100057\)
Analytic rank:	\(0\)
Dimension:	\(336\)
Relative dimension:	\(42\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			3.9
Character	\(\chi\)	\(=\)	100.3
Dual form			100.4.l.b.67.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.33331 - 1.59865i) q^{2} +(0.894258 - 0.141637i) q^{3} +(2.88864 + 7.46028i) q^{4} +(2.87150 + 10.8053i) q^{5} +(-2.31301 - 1.09912i) q^{6} +(-10.5284 - 10.5284i) q^{7} +(5.18627 - 22.0250i) q^{8} +(-24.8989 + 8.09014i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 4 q^{2} - 10 q^{4} - 20 q^{5} - 6 q^{6} + 2 q^{8} - 20 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(51\)	\(77\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{7}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.33331	−	1.59865i	−0.824949	−	0.565208i
							\(3\)	0.894258	−	0.141637i	0.172100	−	0.0272580i	−0.0697896	−	0.997562i	\(-0.522233\pi\)
0.241890	+	0.970304i	\(0.422233\pi\)
\(5\)	2.87150	+	10.8053i	0.256835	+	0.966455i
							\(7\)	−10.5284	−	10.5284i	−0.568482	−	0.568482i	0.363221	−	0.931703i	\(-0.381677\pi\)
−0.931703	+	0.363221i	\(0.881677\pi\)
\(11\)	−43.8959	−	14.2626i	−1.20319	−	0.390941i	−0.362257	−	0.932078i	\(-0.617994\pi\)
							−0.840934	+	0.541138i	\(0.817994\pi\)
\(13\)	−0.627000	+	1.23056i	−0.0133768	+	0.0262534i	−0.897598	−	0.440815i	\(-0.854690\pi\)
							0.884221	+	0.467068i	\(0.154690\pi\)
\(17\)	−8.44838	+	53.3409i	−0.120531	+	0.761005i	0.851187	+	0.524862i	\(0.175883\pi\)
							−0.971718	+	0.236142i	\(0.924117\pi\)
\(19\)	−34.4402	+	25.0223i	−0.415849	+	0.302132i	−0.775965	−	0.630775i	\(-0.782737\pi\)
							0.360117	+	0.932907i	\(0.382737\pi\)
\(23\)	6.23129	+	12.2296i	0.0564919	+	0.110872i	0.917526	−	0.397675i	\(-0.130183\pi\)
							−0.861034	+	0.508547i	\(0.830183\pi\)
\(29\)	−65.0560	+	89.5419i	−0.416572	+	0.573363i	−0.964806	−	0.262962i	\(-0.915300\pi\)
							0.548234	+	0.836325i	\(0.315300\pi\)
\(31\)	−110.271	−	151.775i	−0.638880	−	0.879343i	0.359675	−	0.933078i	\(-0.382888\pi\)
							−0.998555	+	0.0537345i	\(0.982888\pi\)
\(37\)	−55.3194	−	28.1866i	−0.245796	−	0.125239i	0.326753	−	0.945110i	\(-0.394045\pi\)
							−0.572549	+	0.819870i	\(0.694045\pi\)
\(41\)	102.835	+	316.495i	0.391712	+	1.20557i	0.931493	+	0.363760i	\(0.118507\pi\)
							−0.539780	+	0.841806i	\(0.681493\pi\)
\(43\)	265.734	−	265.734i	0.942421	−	0.942421i	−0.0560091	−	0.998430i	\(-0.517838\pi\)
							0.998430	+	0.0560091i	\(0.0178376\pi\)
\(47\)	77.2703	+	487.865i	0.239809	+	1.51410i	0.754260	+	0.656576i	\(0.227996\pi\)
							−0.514451	+	0.857520i	\(0.672004\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	100.4.l.b.3.9	✓	336
4.3	odd	2	inner	100.4.l.b.3.21	yes	336
25.17	odd	20	inner	100.4.l.b.67.21	yes	336
100.67	even	20	inner	100.4.l.b.67.9	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
100.4.l.b.3.9	✓	336	1.1	even	1	trivial
100.4.l.b.3.21	yes	336	4.3	odd	2	inner
100.4.l.b.67.9	yes	336	100.67	even	20	inner
100.4.l.b.67.21	yes	336	25.17	odd	20	inner