Embedded newform 21.33.h.b.11.4

• Label: 21.33.h.b.11.4
• Level: $21$
• Weight: $33$
• Character: 21.11
• 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			11.4
Character	\(\chi\)	\(=\)	21.11
Dual form			21.33.h.b.2.4

$q$-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 \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).

\(n\)	\(8\)	\(10\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)

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	−0.589714	+	0.807612i	\(0.700759\pi\)
							−0.994270	+	0.106901i	\(0.965907\pi\)
\(3\)	−9.97571e6	+	4.18749e7i	−0.231741	+	0.972777i
							\(5\)	−1.39609e11	+	8.06034e10i	−0.914943	+	0.528242i	−0.882018	−	0.471215i	\(-0.843815\pi\)
−0.0329246	+	0.999458i	\(0.510482\pi\)
\(7\)	2.67513e13	−	1.97179e13i	0.804963	−	0.593325i
							\(11\)	2.54609e15	+	1.46999e15i	0.0554104	+	0.0319912i	0.527449	−	0.849587i	\(-0.323148\pi\)
−0.472039	+	0.881578i	\(0.656482\pi\)
\(13\)	−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	0.659128	−	0.752031i	\(-0.270925\pi\)
							−0.321714	+	0.946837i	\(0.604259\pi\)
\(19\)	−2.00418e20	−	3.47134e20i	−0.694830	−	1.20348i	−0.970238	−	0.242153i	\(-0.922146\pi\)
							0.275408	−	0.961327i	\(-0.411187\pi\)
\(23\)	1.36744e21	−	7.89490e20i	0.222978	−	0.128736i	−0.384350	−	0.923187i	\(-0.625575\pi\)
							0.607328	+	0.794451i	\(0.292241\pi\)
\(29\)		−	2.17970e23i		−	0.871021i	−0.900184	−	0.435511i	\(-0.856568\pi\)
							0.900184	−	0.435511i	\(-0.143432\pi\)
\(31\)	2.63413e23	−	4.56245e23i	0.362119	−	0.627208i	−0.626191	−	0.779670i	\(-0.715387\pi\)
							0.988309	+	0.152462i	\(0.0487202\pi\)
\(37\)	−9.66007e24	−	1.67317e25i	−0.782984	−	1.35617i	−0.930196	−	0.367063i	\(-0.880364\pi\)
							0.147213	−	0.989105i	\(-0.452970\pi\)
\(41\)		−	1.22429e26i		−	1.92018i	−0.279690	−	0.960090i	\(-0.590232\pi\)
							0.279690	−	0.960090i	\(-0.409768\pi\)
\(43\)	−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	−0.967270	−	0.253749i	\(-0.918336\pi\)
							−0.263882	+	0.964555i	\(0.585003\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.33.h.b.2.4	✓	164	21.2	odd	6	inner
21.33.h.b.2.79	yes	164	7.2	even	3	inner
21.33.h.b.11.4	yes	164	1.1	even	1	trivial
21.33.h.b.11.79	yes	164	3.2	odd	2	inner