Embedded newform 100.4.l.b.67.17

• Label: 100.4.l.b.67.17
• Level: $100$
• Weight: $4$
• Character: 100.67
• 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			67.17
Character	\(\chi\)	\(=\)	100.67
Dual form			100.4.l.b.3.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.18699 - 2.56730i) q^{2} +(7.22567 + 1.14443i) q^{3} +(-5.18210 + 6.09474i) q^{4} +(-7.02886 + 8.69454i) q^{5} +(-5.63871 - 19.9089i) q^{6} +(11.6198 - 11.6198i) q^{7} +(21.7982 + 6.06960i) q^{8} +(25.2220 + 8.19513i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 4 q^{2} - 10 q^{4} - 20 q^{5} - 6 q^{6} + 2 q^{8} - 20 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(51\)	\(77\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{13}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.18699	−	2.56730i	−0.419665	−	0.907679i
							\(3\)	7.22567	+	1.14443i	1.39058	+	0.220246i	0.806397	−	0.591375i	\(-0.201415\pi\)
0.584184	+	0.811621i	\(0.301415\pi\)
\(5\)	−7.02886	+	8.69454i	−0.628681	+	0.777663i
							\(7\)	11.6198	−	11.6198i	0.627412	−	0.627412i	−0.320004	−	0.947416i	\(-0.603684\pi\)
0.947416	+	0.320004i	\(0.103684\pi\)
\(11\)	65.9267	−	21.4209i	1.80706	−	0.587149i	0.807064	−	0.590464i	\(-0.201055\pi\)
							0.999995	+	0.00331463i	\(0.00105508\pi\)
\(13\)	18.1806	+	35.6814i	0.387876	+	0.761249i	0.999554	−	0.0298572i	\(-0.00950526\pi\)
							−0.611678	+	0.791107i	\(0.709505\pi\)
\(17\)	7.87771	+	49.7379i	0.112390	+	0.709601i	0.977957	+	0.208808i	\(0.0669583\pi\)
							−0.865567	+	0.500793i	\(0.833042\pi\)
\(19\)	46.2790	+	33.6237i	0.558797	+	0.405990i	0.831018	−	0.556245i	\(-0.187758\pi\)
							−0.272222	+	0.962235i	\(0.587758\pi\)
\(23\)	6.85594	−	13.4555i	0.0621549	−	0.121986i	−0.857839	−	0.513919i	\(-0.828193\pi\)
							0.919994	+	0.391933i	\(0.128193\pi\)
\(29\)	−134.554	−	185.198i	−0.861591	−	1.18588i	−0.981188	−	0.193056i	\(-0.938160\pi\)
							0.119597	−	0.992823i	\(-0.461840\pi\)
\(31\)	−131.944	+	181.605i	−0.764445	+	1.05217i	0.232387	+	0.972623i	\(0.425346\pi\)
							−0.996831	+	0.0795442i	\(0.974654\pi\)
\(37\)	−215.758	+	109.934i	−0.958660	+	0.488462i	−0.862029	−	0.506859i	\(-0.830807\pi\)
							−0.0966307	+	0.995320i	\(0.530807\pi\)
\(41\)	−32.8623	+	101.140i	−0.125176	+	0.385253i	−0.993933	−	0.109986i	\(-0.964919\pi\)
							0.868757	+	0.495239i	\(0.164919\pi\)
\(43\)	−230.751	−	230.751i	−0.818353	−	0.818353i	0.167516	−	0.985869i	\(-0.446425\pi\)
							−0.985869	+	0.167516i	\(0.946425\pi\)
\(47\)	54.4120	−	343.544i	0.168868	−	1.06619i	−0.747033	−	0.664787i	\(-0.768522\pi\)
							0.915901	−	0.401405i	\(-0.131478\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	100.4.l.b.67.17	yes	336
4.3	odd	2	inner	100.4.l.b.67.39	yes	336
25.3	odd	20	inner	100.4.l.b.3.39	yes	336
100.3	even	20	inner	100.4.l.b.3.17	✓	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
100.4.l.b.3.17	✓	336	100.3	even	20	inner
100.4.l.b.3.39	yes	336	25.3	odd	20	inner
100.4.l.b.67.17	yes	336	1.1	even	1	trivial
100.4.l.b.67.39	yes	336	4.3	odd	2	inner