Embedded newform 21.33.h.b.2.4

• Label: 21.33.h.b.2.4
• 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			2.4
Character	\(\chi\)	\(=\)	21.2
Dual form			21.33.h.b.11.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-103808. - 59933.5i) q^{2} +(-9.97571e6 - 4.18749e7i) q^{3} +(5.03657e9 + 8.72360e9i) q^{4} +(-1.39609e11 - 8.06034e10i) q^{5} +(-1.47415e12 + 4.94482e12i) q^{6} +(2.67513e13 + 1.97179e13i) q^{7} -6.92614e14i q^{8} +(-1.65399e15 + 8.35463e14i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−103808.	−	59933.5i	−1.58398	−	0.914513i	−0.994270	−	0.106901i	\(-0.965907\pi\)
							−0.589714	−	0.807612i	\(-0.700759\pi\)
\(3\)	−9.97571e6	−	4.18749e7i	−0.231741	−	0.972777i
							\(5\)	−1.39609e11	−	8.06034e10i	−0.914943	−	0.528242i	−0.0329246	−	0.999458i	\(-0.510482\pi\)
−0.882018	+	0.471215i	\(0.843815\pi\)
\(7\)	2.67513e13	+	1.97179e13i	0.804963	+	0.593325i
							\(11\)	2.54609e15	−	1.46999e15i	0.0554104	−	0.0319912i	−0.472039	−	0.881578i	\(-0.656482\pi\)
0.527449	+	0.849587i	\(0.323148\pi\)
\(13\)	−1.02086e18			−1.53417			−0.767087	−	0.641543i	\(-0.778294\pi\)
							−0.767087	+	0.641543i	\(0.778294\pi\)
\(17\)	1.64190e19	−	9.47951e18i	0.337415	−	0.194806i	−0.321714	−	0.946837i	\(-0.604259\pi\)
							0.659128	+	0.752031i	\(0.270925\pi\)
\(19\)	−2.00418e20	+	3.47134e20i	−0.694830	+	1.20348i	0.275408	+	0.961327i	\(0.411187\pi\)
							−0.970238	+	0.242153i	\(0.922146\pi\)
\(23\)	1.36744e21	+	7.89490e20i	0.222978	+	0.128736i	0.607328	−	0.794451i	\(-0.292241\pi\)
							−0.384350	+	0.923187i	\(0.625575\pi\)
\(29\)			2.17970e23i			0.871021i	0.900184	+	0.435511i	\(0.143432\pi\)
							−0.900184	+	0.435511i	\(0.856568\pi\)
\(31\)	2.63413e23	+	4.56245e23i	0.362119	+	0.627208i	0.988309	−	0.152462i	\(-0.0487202\pi\)
							−0.626191	+	0.779670i	\(0.715387\pi\)
\(37\)	−9.66007e24	+	1.67317e25i	−0.782984	+	1.35617i	0.147213	+	0.989105i	\(0.452970\pi\)
							−0.930196	+	0.367063i	\(0.880364\pi\)
\(41\)			1.22429e26i			1.92018i	0.279690	+	0.960090i	\(0.409768\pi\)
							−0.279690	+	0.960090i	\(0.590232\pi\)
\(43\)	−1.75166e25			−0.128220			−0.0641100	−	0.997943i	\(-0.520421\pi\)
							−0.0641100	+	0.997943i	\(0.520421\pi\)
\(47\)	−6.98035e26	−	4.03011e26i	−1.23115	−	0.710806i	−0.263882	−	0.964555i	\(-0.585003\pi\)
							−0.967270	+	0.253749i	\(0.918336\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.33.h.b.2.4	✓	164
3.2	odd	2	inner	21.33.h.b.2.79	yes	164
7.4	even	3	inner	21.33.h.b.11.79	yes	164
21.11	odd	6	inner	21.33.h.b.11.4	yes	164

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.33.h.b.2.4	✓	164	1.1	even	1	trivial
21.33.h.b.2.79	yes	164	3.2	odd	2	inner
21.33.h.b.11.4	yes	164	21.11	odd	6	inner
21.33.h.b.11.79	yes	164	7.4	even	3	inner