Embedded newform 21.33.h.b.2.16

• Label: 21.33.h.b.2.16
• 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			2.16
Character	\(\chi\)	\(=\)	21.2
Dual form			21.33.h.b.11.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-77640.0 - 44825.5i) q^{2} +(-2.98329e7 - 3.10325e7i) q^{3} +(1.87116e9 + 3.24095e9i) q^{4} +(-1.87248e11 - 1.08108e11i) q^{5} +(9.25177e11 + 3.74664e12i) q^{6} +(-3.25948e13 - 6.48148e12i) q^{7} +4.95447e13i q^{8} +(-7.30174e13 + 1.85158e15i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−77640.0	−	44825.5i	−1.18469	−	0.683983i	−0.227597	−	0.973755i	\(-0.573087\pi\)
							−0.957095	+	0.289773i	\(0.906420\pi\)
\(3\)	−2.98329e7	−	3.10325e7i	−0.693035	−	0.720904i
							\(5\)	−1.87248e11	−	1.08108e11i	−1.22715	−	0.708494i	−0.260716	−	0.965416i	\(-0.583959\pi\)
−0.966432	+	0.256921i	\(0.917292\pi\)
\(7\)	−3.25948e13	−	6.48148e12i	−0.980797	−	0.195032i
							\(11\)	6.79138e16	−	3.92101e16i	1.47800	−	0.853325i	0.478312	−	0.878190i	\(-0.341249\pi\)
0.999691	+	0.0248649i	\(0.00791556\pi\)
\(13\)	−9.87051e16			−0.148336			−0.0741679	−	0.997246i	\(-0.523630\pi\)
							−0.0741679	+	0.997246i	\(0.523630\pi\)
\(17\)	−1.30043e18	+	7.50803e17i	−0.0267242	+	0.0154292i	−0.513303	−	0.858208i	\(-0.671578\pi\)
							0.486578	+	0.873637i	\(0.338245\pi\)
\(19\)	1.90435e20	−	3.29844e20i	0.660222	−	1.14354i	−0.320335	−	0.947304i	\(-0.603796\pi\)
							0.980557	−	0.196234i	\(-0.0628711\pi\)
\(23\)	−7.02456e21	−	4.05563e21i	−1.14544	−	0.661322i	−0.197671	−	0.980268i	\(-0.563338\pi\)
							−0.947773	+	0.318946i	\(0.896671\pi\)
\(29\)		−	9.97964e22i		−	0.398792i	−0.979919	−	0.199396i	\(-0.936102\pi\)
							0.979919	−	0.199396i	\(-0.0638981\pi\)
\(31\)	4.09457e23	+	7.09200e23i	0.562887	+	0.974948i	0.997243	+	0.0742072i	\(0.0236426\pi\)
							−0.434356	+	0.900741i	\(0.643024\pi\)
\(37\)	1.02293e25	−	1.77176e25i	0.829121	−	1.43608i	−0.0696081	−	0.997574i	\(-0.522175\pi\)
							0.898729	−	0.438505i	\(-0.144492\pi\)
\(41\)			2.78554e24i			0.0436886i	0.999761	+	0.0218443i	\(0.00695381\pi\)
							−0.999761	+	0.0218443i	\(0.993046\pi\)
\(43\)	1.29107e25			0.0945052			0.0472526	−	0.998883i	\(-0.484953\pi\)
							0.0472526	+	0.998883i	\(0.484953\pi\)
\(47\)	−3.01411e26	−	1.74020e26i	−0.531610	−	0.306925i	0.210062	−	0.977688i	\(-0.432633\pi\)
							−0.741672	+	0.670763i	\(0.765967\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.33.h.b.2.16	✓	164
3.2	odd	2	inner	21.33.h.b.2.67	yes	164
7.4	even	3	inner	21.33.h.b.11.67	yes	164
21.11	odd	6	inner	21.33.h.b.11.16	yes	164

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.33.h.b.2.16	✓	164	1.1	even	1	trivial
21.33.h.b.2.67	yes	164	3.2	odd	2	inner
21.33.h.b.11.16	yes	164	21.11	odd	6	inner
21.33.h.b.11.67	yes	164	7.4	even	3	inner