Embedded newform 21.33.h.b.2.14

• Label: 21.33.h.b.2.14
• 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			2.14
Character	\(\chi\)	\(=\)	21.2
Dual form			21.33.h.b.11.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-81147.0 - 46850.3i) q^{2} +(2.92478e7 + 3.15846e7i) q^{3} +(2.24241e9 + 3.88397e9i) q^{4} +(1.46995e10 + 8.48674e9i) q^{5} +(-8.93631e11 - 3.93326e12i) q^{6} +(5.53322e12 - 3.27691e13i) q^{7} -1.77891e13i q^{8} +(-1.42148e14 + 1.84756e15i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−81147.0	−	46850.3i	−1.23821	−	0.714878i	−0.269478	−	0.963007i	\(-0.586851\pi\)
							−0.968727	+	0.248128i	\(0.920185\pi\)
\(3\)	2.92478e7	+	3.15846e7i	0.679444	+	0.733727i
							\(5\)	1.46995e10	+	8.48674e9i	0.0963345	+	0.0556187i	0.547393	−	0.836876i	\(-0.315620\pi\)
−0.451059	+	0.892494i	\(0.648954\pi\)
\(7\)	5.53322e12	−	3.27691e13i	0.166498	−	0.986042i
							\(11\)	3.46655e16	−	2.00141e16i	0.754421	−	0.435565i	−0.0728679	−	0.997342i	\(-0.523215\pi\)
0.827289	+	0.561776i	\(0.189882\pi\)
\(13\)	7.68983e17			1.15564			0.577820	−	0.816164i	\(-0.303903\pi\)
							0.577820	+	0.816164i	\(0.303903\pi\)
\(17\)	−5.61030e19	+	3.23911e19i	−1.15293	+	0.665645i	−0.949600	−	0.313464i	\(-0.898510\pi\)
							−0.203332	+	0.979110i	\(0.565177\pi\)
\(19\)	−9.39650e19	+	1.62752e20i	−0.325768	+	0.564247i	−0.981668	−	0.190601i	\(-0.938956\pi\)
							0.655899	+	0.754848i	\(0.272290\pi\)
\(23\)	−5.77728e21	−	3.33551e21i	−0.942058	−	0.543898i	−0.0514534	−	0.998675i	\(-0.516385\pi\)
							−0.890605	+	0.454778i	\(0.849719\pi\)
\(29\)		−	2.09025e23i		−	0.835277i	−0.908613	−	0.417638i	\(-0.862858\pi\)
							0.908613	−	0.417638i	\(-0.137142\pi\)
\(31\)	4.76006e23	+	8.24466e23i	0.654373	+	1.13341i	0.982051	+	0.188617i	\(0.0604005\pi\)
							−0.327678	+	0.944789i	\(0.606266\pi\)
\(37\)	−1.05619e25	+	1.82938e25i	−0.856084	+	1.48278i	0.0195512	+	0.999809i	\(0.493776\pi\)
							−0.875635	+	0.482973i	\(0.839557\pi\)
\(41\)			5.73927e25i			0.900151i	0.892991	+	0.450075i	\(0.148603\pi\)
							−0.892991	+	0.450075i	\(0.851397\pi\)
\(43\)	2.04424e26			1.49636			0.748181	−	0.663494i	\(-0.230927\pi\)
							0.748181	+	0.663494i	\(0.230927\pi\)
\(47\)	−9.38645e26	−	5.41927e26i	−1.65552	−	0.955817i	−0.974743	−	0.223330i	\(-0.928307\pi\)
							−0.680781	−	0.732487i	\(-0.738359\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.33.h.b.2.14	✓	164
3.2	odd	2	inner	21.33.h.b.2.69	yes	164
7.4	even	3	inner	21.33.h.b.11.69	yes	164
21.11	odd	6	inner	21.33.h.b.11.14	yes	164

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.33.h.b.2.14	✓	164	1.1	even	1	trivial
21.33.h.b.2.69	yes	164	3.2	odd	2	inner
21.33.h.b.11.14	yes	164	21.11	odd	6	inner
21.33.h.b.11.69	yes	164	7.4	even	3	inner