Embedded newform 100.4.l.b.3.8

• Label: 100.4.l.b.3.8
• 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			3.8
Character	\(\chi\)	\(=\)	100.3
Dual form			100.4.l.b.67.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.45543 - 1.40388i) q^{2} +(-3.77500 + 0.597901i) q^{3} +(4.05822 + 6.89426i) q^{4} +(-11.1789 - 0.177504i) q^{5} +(10.1086 + 3.83155i) q^{6} +(9.86563 + 9.86563i) q^{7} +(-0.285929 - 22.6256i) q^{8} +(-11.7854 + 3.82931i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 4 q^{2} - 10 q^{4} - 20 q^{5} - 6 q^{6} + 2 q^{8} - 20 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.45543	−	1.40388i	−0.868124	−	0.496348i
							\(3\)	−3.77500	+	0.597901i	−0.726499	+	0.115066i	−0.508720	−	0.860932i	\(-0.669881\pi\)
−0.217779	+	0.975998i	\(0.569881\pi\)
\(5\)	−11.1789	−	0.177504i	−0.999874	−	0.0158764i
							\(7\)	9.86563	+	9.86563i	0.532694	+	0.532694i	0.921373	−	0.388679i	\(-0.127069\pi\)
−0.388679	+	0.921373i	\(0.627069\pi\)
\(11\)	26.9221	+	8.74752i	0.737938	+	0.239771i	0.653783	−	0.756682i	\(-0.273181\pi\)
							0.0841552	+	0.996453i	\(0.473181\pi\)
\(13\)	27.3373	−	53.6525i	0.583231	−	1.14466i	−0.391270	−	0.920276i	\(-0.627964\pi\)
							0.974502	−	0.224380i	\(-0.0720357\pi\)
\(17\)	12.0184	−	75.8815i	0.171465	−	1.08259i	−0.740421	−	0.672143i	\(-0.765374\pi\)
							0.911886	−	0.410443i	\(-0.134626\pi\)
\(19\)	30.8285	−	22.3982i	0.372239	−	0.270447i	−0.385900	−	0.922541i	\(-0.626109\pi\)
							0.758139	+	0.652093i	\(0.226109\pi\)
\(23\)	−63.0536	−	123.750i	−0.571634	−	1.12190i	−0.978082	−	0.208221i	\(-0.933233\pi\)
							0.406448	−	0.913674i	\(-0.366767\pi\)
\(29\)	66.8440	−	92.0029i	0.428022	−	0.589121i	−0.539476	−	0.842001i	\(-0.681378\pi\)
							0.967498	+	0.252880i	\(0.0813777\pi\)
\(31\)	167.272	+	230.230i	0.969126	+	1.33389i	0.942487	+	0.334242i	\(0.108480\pi\)
							0.0266390	+	0.999645i	\(0.491520\pi\)
\(37\)	29.0995	+	14.8270i	0.129296	+	0.0658794i	0.517442	−	0.855718i	\(-0.326884\pi\)
							−0.388146	+	0.921598i	\(0.626884\pi\)
\(41\)	64.0136	+	197.014i	0.243835	+	0.750448i	0.995826	+	0.0912741i	\(0.0290939\pi\)
							−0.751990	+	0.659174i	\(0.770906\pi\)
\(43\)	−69.9527	+	69.9527i	−0.248086	+	0.248086i	−0.820185	−	0.572099i	\(-0.806129\pi\)
							0.572099	+	0.820185i	\(0.306129\pi\)
\(47\)	44.0453	+	278.091i	0.136695	+	0.863059i	0.956779	+	0.290816i	\(0.0939267\pi\)
							−0.820084	+	0.572243i	\(0.806073\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	100.4.l.b.3.8	✓	336
4.3	odd	2	inner	100.4.l.b.3.22	yes	336
25.17	odd	20	inner	100.4.l.b.67.22	yes	336
100.67	even	20	inner	100.4.l.b.67.8	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
100.4.l.b.3.8	✓	336	1.1	even	1	trivial
100.4.l.b.3.22	yes	336	4.3	odd	2	inner
100.4.l.b.67.8	yes	336	100.67	even	20	inner
100.4.l.b.67.22	yes	336	25.17	odd	20	inner