Embedded newform 21.33.h.b.2.6

• Label: 21.33.h.b.2.6
• 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			2.6
Character	\(\chi\)	\(=\)	21.2
Dual form			21.33.h.b.11.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-100224. - 57864.6i) q^{2} +(-4.15312e7 + 1.13217e7i) q^{3} +(4.54913e9 + 7.87933e9i) q^{4} +(-9.99516e10 - 5.77071e10i) q^{5} +(4.81756e12 + 1.26847e12i) q^{6} +(1.10609e13 + 3.13382e13i) q^{7} -5.55881e14i q^{8} +(1.59666e15 - 9.40407e14i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−100224.	−	57864.6i	−1.52930	−	0.882943i	−0.999391	−	0.0348901i	\(-0.988892\pi\)
							−0.529911	−	0.848053i	\(-0.677775\pi\)
\(3\)	−4.15312e7	+	1.13217e7i	−0.964793	+	0.263010i
							\(5\)	−9.99516e10	−	5.77071e10i	−0.655043	−	0.378189i	0.135343	−	0.990799i	\(-0.456786\pi\)
−0.790385	+	0.612610i	\(0.790120\pi\)
\(7\)	1.10609e13	+	3.13382e13i	0.332830	+	0.942987i
							\(11\)	−9.36632e15	+	5.40765e15i	−0.203838	+	0.117686i	−0.598445	−	0.801164i	\(-0.704214\pi\)
0.394606	+	0.918850i	\(0.370881\pi\)
\(13\)	8.57824e17			1.28915			0.644576	−	0.764540i	\(-0.277034\pi\)
							0.644576	+	0.764540i	\(0.277034\pi\)
\(17\)	−1.89042e19	+	1.09144e19i	−0.388486	+	0.224293i	−0.681504	−	0.731814i	\(-0.738674\pi\)
							0.293018	+	0.956107i	\(0.405341\pi\)
\(19\)	2.54174e20	−	4.40242e20i	0.881197	−	1.52628i	0.0311856	−	0.999514i	\(-0.490072\pi\)
							0.850011	−	0.526764i	\(-0.176595\pi\)
\(23\)	−2.78178e20	−	1.60606e20i	−0.0453605	−	0.0261889i	0.477148	−	0.878823i	\(-0.341670\pi\)
							−0.522509	+	0.852634i	\(0.675004\pi\)
\(29\)		−	2.13996e23i		−	0.855140i	−0.903982	−	0.427570i	\(-0.859370\pi\)
							0.903982	−	0.427570i	\(-0.140630\pi\)
\(31\)	3.07970e23	+	5.33420e23i	0.423372	+	0.733301i	0.996267	−	0.0863273i	\(-0.0275131\pi\)
							−0.572895	+	0.819629i	\(0.694180\pi\)
\(37\)	7.13482e23	−	1.23579e24i	0.0578303	−	0.100165i	−0.835661	−	0.549246i	\(-0.814915\pi\)
							0.893491	+	0.449081i	\(0.148248\pi\)
\(41\)		−	5.55387e25i		−	0.871072i	−0.900171	−	0.435536i	\(-0.856559\pi\)
							0.900171	−	0.435536i	\(-0.143441\pi\)
\(43\)	−1.63017e26			−1.19327			−0.596633	−	0.802514i	\(-0.703495\pi\)
							−0.596633	+	0.802514i	\(0.703495\pi\)
\(47\)	−1.40056e26	−	8.08613e25i	−0.247022	−	0.142618i	0.371378	−	0.928482i	\(-0.378885\pi\)
							−0.618400	+	0.785864i	\(0.712219\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.33.h.b.2.6	✓	164
3.2	odd	2	inner	21.33.h.b.2.77	yes	164
7.4	even	3	inner	21.33.h.b.11.77	yes	164
21.11	odd	6	inner	21.33.h.b.11.6	yes	164

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.33.h.b.2.6	✓	164	1.1	even	1	trivial
21.33.h.b.2.77	yes	164	3.2	odd	2	inner
21.33.h.b.11.6	yes	164	21.11	odd	6	inner
21.33.h.b.11.77	yes	164	7.4	even	3	inner