Embedded newform 21.33.h.b.2.15

• Label: 21.33.h.b.2.15
• 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			2.15
Character	\(\chi\)	\(=\)	21.2
Dual form			21.33.h.b.11.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-80445.4 - 46445.2i) q^{2} +(4.22184e7 + 8.40380e6i) q^{3} +(2.16683e9 + 3.75305e9i) q^{4} +(1.68281e11 + 9.71571e10i) q^{5} +(-3.00596e12 - 2.63689e12i) q^{6} +(-2.65601e13 - 1.99748e13i) q^{7} -3.59344e12i q^{8} +(1.71177e15 + 7.09591e14i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−80445.4	−	46445.2i	−1.22750	−	0.708697i	−0.260993	−	0.965341i	\(-0.584050\pi\)
							−0.966506	+	0.256643i	\(0.917383\pi\)
\(3\)	4.22184e7	+	8.40380e6i	0.980758	+	0.195225i
							\(5\)	1.68281e11	+	9.71571e10i	1.10285	+	0.636729i	0.936967	−	0.349418i	\(-0.113621\pi\)
0.165879	+	0.986146i	\(0.446954\pi\)
\(7\)	−2.65601e13	−	1.99748e13i	−0.799209	−	0.601053i
							\(11\)	−5.49722e16	+	3.17382e16i	−1.19636	+	0.690716i	−0.959741	−	0.280887i	\(-0.909371\pi\)
−0.236615	+	0.971604i	\(0.576038\pi\)
\(13\)	−4.92731e17			−0.740485			−0.370243	−	0.928935i	\(-0.620725\pi\)
							−0.370243	+	0.928935i	\(0.620725\pi\)
\(17\)	6.46679e19	−	3.73360e19i	1.32894	−	0.767265i	0.343806	−	0.939041i	\(-0.388284\pi\)
							0.985136	+	0.171776i	\(0.0549504\pi\)
\(19\)	−3.80322e19	+	6.58737e19i	−0.131854	+	0.228378i	−0.924391	−	0.381445i	\(-0.875426\pi\)
							0.792537	+	0.609824i	\(0.208760\pi\)
\(23\)	−1.17642e21	−	6.79204e20i	−0.191830	−	0.110753i	0.401009	−	0.916074i	\(-0.368660\pi\)
							−0.592839	+	0.805321i	\(0.701993\pi\)
\(29\)			1.16492e23i			0.465508i	0.972536	+	0.232754i	\(0.0747737\pi\)
							−0.972536	+	0.232754i	\(0.925226\pi\)
\(31\)	−8.42920e22	−	1.45998e23i	−0.115878	−	0.200706i	0.802253	−	0.596985i	\(-0.203635\pi\)
							−0.918130	+	0.396279i	\(0.870301\pi\)
\(37\)	−2.55862e24	+	4.43166e24i	−0.207386	+	0.359202i	−0.950890	−	0.309528i	\(-0.899829\pi\)
							0.743505	+	0.668731i	\(0.233162\pi\)
\(41\)		−	8.98422e25i		−	1.40909i	−0.709659	−	0.704545i	\(-0.751151\pi\)
							0.709659	−	0.704545i	\(-0.248849\pi\)
\(43\)	−1.62344e26			−1.18834			−0.594170	−	0.804339i	\(-0.702519\pi\)
							−0.594170	+	0.804339i	\(0.702519\pi\)
\(47\)	−2.28099e26	−	1.31693e26i	−0.402307	−	0.232272i	0.285172	−	0.958476i	\(-0.407949\pi\)
							−0.687479	+	0.726204i	\(0.741283\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.33.h.b.2.15	✓	164
3.2	odd	2	inner	21.33.h.b.2.68	yes	164
7.4	even	3	inner	21.33.h.b.11.68	yes	164
21.11	odd	6	inner	21.33.h.b.11.15	yes	164

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.33.h.b.2.15	✓	164	1.1	even	1	trivial
21.33.h.b.2.68	yes	164	3.2	odd	2	inner
21.33.h.b.11.15	yes	164	21.11	odd	6	inner
21.33.h.b.11.68	yes	164	7.4	even	3	inner