Embedded newform 21.33.h.b.2.18

• Label: 21.33.h.b.2.18
• Level: $21$
• Weight: $33$
• Character: 21.2
• Analytic conductor: $136.220$
• Analytic rank: $0$
• Dimension: $164$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 33 \)
Character orbit:	\([\chi]\)	\(=\)	21.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(136.219975799\)
Analytic rank:	\(0\)
Dimension:	\(164\)
Relative dimension:	\(82\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			2.18
Character	\(\chi\)	\(=\)	21.2
Dual form			21.33.h.b.11.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-74754.1 - 43159.3i) q^{2} +(3.41923e7 + 2.61516e7i) q^{3} +(1.57796e9 + 2.73311e9i) q^{4} +(-9.12997e10 - 5.27119e10i) q^{5} +(-1.42733e12 - 3.43066e12i) q^{6} +(3.71380e12 + 3.30248e13i) q^{7} +9.83202e13i q^{8} +(4.85204e14 + 1.78837e15i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	164 * q + 43046720 * q^3 + 176093659134 * q^4 + 3915256954876 * q^6 - 41868332499126 * q^7 + 1232720974221986 * q^9

Character values

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

\(n\)	\(8\)	\(10\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−74754.1	−	43159.3i	−1.14066	−	0.658558i	−0.194064	−	0.980989i	\(-0.562167\pi\)
							−0.946593	+	0.322431i	\(0.895500\pi\)
\(3\)	3.41923e7	+	2.61516e7i	0.794306	+	0.607517i
							\(5\)	−9.12997e10	−	5.27119e10i	−0.598342	−	0.345453i	0.170047	−	0.985436i	\(-0.445608\pi\)
−0.768389	+	0.639983i	\(0.778941\pi\)
\(7\)	3.71380e12	+	3.30248e13i	0.111751	+	0.993736i
							\(11\)	2.11514e16	−	1.22117e16i	0.460315	−	0.265763i	−0.251862	−	0.967763i	\(-0.581043\pi\)
0.712177	+	0.702000i	\(0.247709\pi\)
\(13\)	−1.14895e18			−1.72666			−0.863331	−	0.504637i	\(-0.831626\pi\)
							−0.863331	+	0.504637i	\(0.831626\pi\)
\(17\)	−1.82218e19	+	1.05204e19i	−0.374464	+	0.216197i	−0.675407	−	0.737445i	\(-0.736032\pi\)
							0.300943	+	0.953642i	\(0.402699\pi\)
\(19\)	−3.73927e18	+	6.47660e18i	−0.0129637	+	0.0224538i	−0.872434	−	0.488731i	\(-0.837460\pi\)
							0.859471	+	0.511185i	\(0.170793\pi\)
\(23\)	−1.71221e21	−	9.88544e20i	−0.279197	−	0.161195i	0.353863	−	0.935297i	\(-0.384868\pi\)
							−0.633060	+	0.774103i	\(0.718201\pi\)
\(29\)		−	8.94587e22i		−	0.357482i	−0.983896	−	0.178741i	\(-0.942798\pi\)
							0.983896	−	0.178741i	\(-0.0572025\pi\)
\(31\)	−1.54161e23	−	2.67014e23i	−0.211927	−	0.367069i	0.740390	−	0.672177i	\(-0.234641\pi\)
							−0.952318	+	0.305108i	\(0.901307\pi\)
\(37\)	7.35254e24	−	1.27350e25i	0.595950	−	1.03221i	−0.397463	−	0.917618i	\(-0.630109\pi\)
							0.993412	−	0.114597i	\(-0.0365575\pi\)
\(41\)		−	9.30479e25i		−	1.45937i	−0.683784	−	0.729684i	\(-0.739667\pi\)
							0.683784	−	0.729684i	\(-0.260333\pi\)
\(43\)	−8.55142e25			−0.625955			−0.312977	−	0.949761i	\(-0.601326\pi\)
							−0.312977	+	0.949761i	\(0.601326\pi\)
\(47\)	3.70669e26	+	2.14006e26i	0.653763	+	0.377450i	0.789896	−	0.613240i	\(-0.210134\pi\)
							−0.136134	+	0.990690i	\(0.543468\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.33.h.b.2.18	✓	164
3.2	odd	2	inner	21.33.h.b.2.65	yes	164
7.4	even	3	inner	21.33.h.b.11.65	yes	164
21.11	odd	6	inner	21.33.h.b.11.18	yes	164

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.33.h.b.2.18	✓	164	1.1	even	1	trivial
21.33.h.b.2.65	yes	164	3.2	odd	2	inner
21.33.h.b.11.18	yes	164	21.11	odd	6	inner
21.33.h.b.11.65	yes	164	7.4	even	3	inner