Embedded newform 21.33.h.b.2.17

• Label: 21.33.h.b.2.17
• 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			2.17
Character	\(\chi\)	\(=\)	21.2
Dual form			21.33.h.b.11.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-76577.7 - 44212.2i) q^{2} +(-1.26540e7 + 4.11448e7i) q^{3} +(1.76195e9 + 3.05178e9i) q^{4} +(-2.57749e11 - 1.48811e11i) q^{5} +(2.78812e12 - 2.59132e12i) q^{6} +(3.15580e13 - 1.04173e13i) q^{7} +6.81818e13i q^{8} +(-1.53277e15 - 1.04129e15i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−76577.7	−	44212.2i	−1.16848	−	0.674624i	−0.215161	−	0.976579i	\(-0.569028\pi\)
							−0.953322	+	0.301955i	\(0.902361\pi\)
\(3\)	−1.26540e7	+	4.11448e7i	−0.293960	+	0.955818i
							\(5\)	−2.57749e11	−	1.48811e11i	−1.68918	−	0.975251i	−0.955143	−	0.296146i	\(-0.904299\pi\)
−0.734041	−	0.679105i	\(-0.762368\pi\)
\(7\)	3.15580e13	−	1.04173e13i	0.949601	−	0.313463i
							\(11\)	3.72259e16	−	2.14924e16i	0.810144	−	0.467737i	−0.0368621	−	0.999320i	\(-0.511736\pi\)
0.847006	+	0.531584i	\(0.178403\pi\)
\(13\)	3.10657e17			0.466861			0.233430	−	0.972373i	\(-0.425005\pi\)
							0.233430	+	0.972373i	\(0.425005\pi\)
\(17\)	2.66416e19	−	1.53815e19i	0.547491	−	0.316094i	−0.200619	−	0.979669i	\(-0.564295\pi\)
							0.748109	+	0.663575i	\(0.230962\pi\)
\(19\)	−1.75293e20	+	3.03616e20i	−0.607725	+	1.05261i	0.383890	+	0.923379i	\(0.374584\pi\)
							−0.991615	+	0.129231i	\(0.958749\pi\)
\(23\)	9.62321e21	+	5.55596e21i	1.56919	+	0.905970i	0.996264	+	0.0863598i	\(0.0275235\pi\)
							0.572922	+	0.819610i	\(0.305810\pi\)
\(29\)		−	1.85980e23i		−	0.743186i	−0.928396	−	0.371593i	\(-0.878812\pi\)
							0.928396	−	0.371593i	\(-0.121188\pi\)
\(31\)	1.35532e23	+	2.34748e23i	0.186317	+	0.322711i	0.944020	−	0.329889i	\(-0.107011\pi\)
							−0.757702	+	0.652600i	\(0.773678\pi\)
\(37\)	4.77122e24	−	8.26399e24i	0.386724	−	0.669826i	−0.605282	−	0.796011i	\(-0.706940\pi\)
							0.992007	+	0.126185i	\(0.0402731\pi\)
\(41\)		−	6.27853e25i		−	0.984727i	−0.870390	−	0.492364i	\(-0.836133\pi\)
							0.870390	−	0.492364i	\(-0.163867\pi\)
\(43\)	−3.24566e25			−0.237579			−0.118789	−	0.992919i	\(-0.537901\pi\)
							−0.118789	+	0.992919i	\(0.537901\pi\)
\(47\)	2.70699e26	+	1.56288e26i	0.477442	+	0.275651i	0.719350	−	0.694648i	\(-0.244440\pi\)
							−0.241908	+	0.970299i	\(0.577773\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.33.h.b.2.17	✓	164
3.2	odd	2	inner	21.33.h.b.2.66	yes	164
7.4	even	3	inner	21.33.h.b.11.66	yes	164
21.11	odd	6	inner	21.33.h.b.11.17	yes	164

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.33.h.b.2.17	✓	164	1.1	even	1	trivial
21.33.h.b.2.66	yes	164	3.2	odd	2	inner
21.33.h.b.11.17	yes	164	21.11	odd	6	inner
21.33.h.b.11.66	yes	164	7.4	even	3	inner