Embedded newform 21.18.g.b.17.17

• Label: 21.18.g.b.17.17
• Level: $21$
• Weight: $18$
• Character: 21.17
• Analytic conductor: $38.477$
• Analytic rank: $0$
• Dimension: $84$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	21.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(38.4766383424\)
Analytic rank:	\(0\)
Dimension:	\(84\)
Relative dimension:	\(42\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			17.17
Character	\(\chi\)	\(=\)	21.17
Dual form			21.18.g.b.5.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-173.612 + 100.235i) q^{2} +(7824.85 + 8240.86i) q^{3} +(-45441.9 + 78707.6i) q^{4} +(669619. + 1.15981e6i) q^{5} +(-2.18451e6 - 646390. i) q^{6} +(-1.44396e7 + 4.91194e6i) q^{7} -4.44955e7i q^{8} +(-6.68348e6 + 1.28967e8i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 19686 q^{3} + 2752510 q^{4} - 4340518 q^{7} - 98670996 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).

\(n\)	\(8\)	\(10\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−173.612	+	100.235i	−0.479541	+	0.276863i	−0.720225	−	0.693741i	\(-0.755961\pi\)
							0.240684	+	0.970603i	\(0.422628\pi\)
\(3\)	7824.85	+	8240.86i	0.688566	+	0.725174i
							\(5\)	669619.	+	1.15981e6i	0.766625	+	1.32783i	0.939383	+	0.342869i	\(0.111399\pi\)
−0.172758	+	0.984964i	\(0.555268\pi\)
\(7\)	−1.44396e7	+	4.91194e6i	−0.946724	+	0.322047i
							\(11\)	−8.95936e8	−	5.17269e8i	−1.26020	−	0.727576i	−0.287086	−	0.957905i	\(-0.592687\pi\)
−0.973113	+	0.230328i	\(0.926020\pi\)
\(13\)		−	4.13964e9i		−	1.40749i	−0.710454	−	0.703743i	\(-0.751510\pi\)
							0.710454	−	0.703743i	\(-0.248490\pi\)
\(17\)	−8.45977e9	+	1.46528e10i	−0.294132	+	0.509452i	−0.974783	−	0.223156i	\(-0.928364\pi\)
							0.680650	+	0.732608i	\(0.261697\pi\)
\(19\)	8.35028e10	−	4.82104e10i	1.12796	−	0.651231i	0.184543	−	0.982825i	\(-0.440920\pi\)
							0.943422	+	0.331594i	\(0.107586\pi\)
\(23\)	−2.19788e11	+	1.26895e11i	−0.585217	+	0.337875i	−0.763204	−	0.646158i	\(-0.776375\pi\)
							0.177987	+	0.984033i	\(0.443042\pi\)
\(29\)			3.83058e12i			1.42194i	0.703222	+	0.710970i	\(0.251744\pi\)
							−0.703222	+	0.710970i	\(0.748256\pi\)
\(31\)	4.87661e12	+	2.81551e12i	1.02694	+	0.592902i	0.916106	−	0.400937i	\(-0.131315\pi\)
							0.110831	+	0.993839i	\(0.464649\pi\)
\(37\)	−1.12894e13	−	1.95539e13i	−0.528394	−	0.915206i	−0.999452	−	0.0331033i	\(-0.989461\pi\)
							0.471058	−	0.882102i	\(-0.343872\pi\)
\(41\)	1.05968e13			0.207258			0.103629	−	0.994616i	\(-0.466955\pi\)
							0.103629	+	0.994616i	\(0.466955\pi\)
\(43\)	−3.14571e12			−0.0410428			−0.0205214	−	0.999789i	\(-0.506533\pi\)
							−0.0205214	+	0.999789i	\(0.506533\pi\)
\(47\)	−9.16424e13	−	1.58729e14i	−0.561390	−	0.972356i	−0.997376	−	0.0724023i	\(-0.976933\pi\)
							0.435986	−	0.899954i	\(-0.356400\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.18.g.b.17.17	yes	84
3.2	odd	2	inner	21.18.g.b.17.26	yes	84
7.5	odd	6	inner	21.18.g.b.5.26	yes	84
21.5	even	6	inner	21.18.g.b.5.17	✓	84

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.18.g.b.5.17	✓	84	21.5	even	6	inner
21.18.g.b.5.26	yes	84	7.5	odd	6	inner
21.18.g.b.17.17	yes	84	1.1	even	1	trivial
21.18.g.b.17.26	yes	84	3.2	odd	2	inner