Embedded newform 21.33.h.b.2.20

• Label: 21.33.h.b.2.20
• 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			2.20
Character	\(\chi\)	\(=\)	21.2
Dual form			21.33.h.b.11.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-65272.0 - 37684.8i) q^{2} +(-3.95790e7 - 1.69270e7i) q^{3} +(6.92806e8 + 1.19998e9i) q^{4} +(3.97688e10 + 2.29605e10i) q^{5} +(1.94551e12 + 2.59639e12i) q^{6} +(3.31510e13 + 2.33200e12i) q^{7} +2.19277e14i q^{8} +(1.27997e15 + 1.33991e15i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−65272.0	−	37684.8i	−0.995972	−	0.575025i	−0.0889178	−	0.996039i	\(-0.528341\pi\)
							−0.907054	+	0.421014i	\(0.861674\pi\)
\(3\)	−3.95790e7	−	1.69270e7i	−0.919443	−	0.393224i
							\(5\)	3.97688e10	+	2.29605e10i	0.260629	+	0.150474i	0.624621	−	0.780928i	\(-0.285253\pi\)
−0.363993	+	0.931402i	\(0.618587\pi\)
\(7\)	3.31510e13	+	2.33200e12i	0.997535	+	0.0701715i
							\(11\)	5.57292e16	−	3.21753e16i	1.21283	−	0.700227i	0.249455	−	0.968386i	\(-0.419749\pi\)
0.963375	+	0.268159i	\(0.0864153\pi\)
\(13\)	2.90734e17			0.436921			0.218460	−	0.975846i	\(-0.429897\pi\)
							0.218460	+	0.975846i	\(0.429897\pi\)
\(17\)	−6.44360e19	+	3.72021e19i	−1.32418	+	0.764513i	−0.984392	−	0.175990i	\(-0.943687\pi\)
							−0.339784	+	0.940503i	\(0.610354\pi\)
\(19\)	−1.45412e20	+	2.51860e20i	−0.504129	+	0.873176i	0.495860	+	0.868402i	\(0.334853\pi\)
							−0.999989	+	0.00477384i	\(0.998480\pi\)
\(23\)	2.02673e21	+	1.17014e21i	0.330485	+	0.190806i	0.656056	−	0.754712i	\(-0.272223\pi\)
							−0.325572	+	0.945517i	\(0.605557\pi\)
\(29\)			3.17119e23i			1.26723i	0.773650	+	0.633614i	\(0.218429\pi\)
							−0.773650	+	0.633614i	\(0.781571\pi\)
\(31\)	−5.67143e23	−	9.82321e23i	−0.779661	−	1.35041i	−0.932137	−	0.362105i	\(-0.882058\pi\)
							0.152476	−	0.988307i	\(-0.451275\pi\)
\(37\)	−3.41278e24	+	5.91110e24i	−0.276618	+	0.479116i	−0.970542	−	0.240932i	\(-0.922547\pi\)
							0.693924	+	0.720048i	\(0.255880\pi\)
\(41\)		−	6.04876e25i		−	0.948691i	−0.880339	−	0.474346i	\(-0.842685\pi\)
							0.880339	−	0.474346i	\(-0.157315\pi\)
\(43\)	−9.54238e25			−0.698492			−0.349246	−	0.937031i	\(-0.613562\pi\)
							−0.349246	+	0.937031i	\(0.613562\pi\)
\(47\)	−1.73785e25	−	1.00335e25i	−0.0306512	−	0.0176965i	0.484596	−	0.874738i	\(-0.338967\pi\)
							−0.515247	+	0.857042i	\(0.672300\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.33.h.b.2.20	✓	164
3.2	odd	2	inner	21.33.h.b.2.63	yes	164
7.4	even	3	inner	21.33.h.b.11.63	yes	164
21.11	odd	6	inner	21.33.h.b.11.20	yes	164

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.33.h.b.2.20	✓	164	1.1	even	1	trivial
21.33.h.b.2.63	yes	164	3.2	odd	2	inner
21.33.h.b.11.20	yes	164	21.11	odd	6	inner
21.33.h.b.11.63	yes	164	7.4	even	3	inner