Embedded newform 21.33.h.b.11.5

• Label: 21.33.h.b.11.5
• Level: $21$
• Weight: $33$
• Character: 21.11
• 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			11.5
Character	\(\chi\)	\(=\)	21.11
Dual form			21.33.h.b.2.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-102259. + 59039.4i) q^{2} +(1.92280e7 + 3.85137e7i) q^{3} +(4.82383e9 - 8.35512e9i) q^{4} +(7.05959e10 - 4.07586e10i) q^{5} +(-4.24007e12 - 2.80317e12i) q^{6} +(-3.32321e13 - 2.35838e11i) q^{7} +6.32040e14i q^{8} +(-1.11359e15 + 1.48108e15i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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 \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).

\(n\)	\(8\)	\(10\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−102259.	+	59039.4i	−1.56035	+	0.900870i	−0.563133	+	0.826366i	\(0.690404\pi\)
							−0.997221	+	0.0745040i	\(0.976263\pi\)
\(3\)	1.92280e7	+	3.85137e7i	0.446677	+	0.894695i
							\(5\)	7.05959e10	−	4.07586e10i	0.462657	−	0.267115i	−0.250504	−	0.968116i	\(-0.580596\pi\)
0.713161	+	0.701000i	\(0.247263\pi\)
\(7\)	−3.32321e13	−	2.35838e11i	−0.999975	−	0.00709652i
							\(11\)	−1.31558e15	−	7.59552e14i	−0.0286309	−	0.0165301i	0.485616	−	0.874172i	\(-0.338595\pi\)
−0.514247	+	0.857642i	\(0.671929\pi\)
\(13\)	−2.46605e17			−0.370603			−0.185302	−	0.982682i	\(-0.559326\pi\)
							−0.185302	+	0.982682i	\(0.559326\pi\)
\(17\)	−7.67063e19	−	4.42864e19i	−1.57633	−	0.910097i	−0.995365	−	0.0961719i	\(-0.969340\pi\)
							−0.580970	−	0.813925i	\(-0.697327\pi\)
\(19\)	7.94407e19	+	1.37595e20i	0.275414	+	0.477031i	0.970239	−	0.242148i	\(-0.0778518\pi\)
							−0.694826	+	0.719178i	\(0.744518\pi\)
\(23\)	6.90671e20	−	3.98759e20i	0.112623	−	0.0650228i	−0.442631	−	0.896704i	\(-0.645955\pi\)
							0.555253	+	0.831681i	\(0.312621\pi\)
\(29\)			1.45142e23i			0.579996i	0.957027	+	0.289998i	\(0.0936546\pi\)
							−0.957027	+	0.289998i	\(0.906345\pi\)
\(31\)	−2.06605e23	+	3.57851e23i	−0.284023	+	0.491943i	−0.972372	−	0.233437i	\(-0.925003\pi\)
							0.688348	+	0.725380i	\(0.258336\pi\)
\(37\)	4.46922e23	+	7.74091e23i	0.0362246	+	0.0627429i	0.883569	−	0.468300i	\(-0.155133\pi\)
							−0.847345	+	0.531043i	\(0.821800\pi\)
\(41\)			9.55257e25i			1.49823i	0.662440	+	0.749115i	\(0.269521\pi\)
							−0.662440	+	0.749115i	\(0.730479\pi\)
\(43\)	−1.25738e26			−0.920387			−0.460193	−	0.887819i	\(-0.652220\pi\)
							−0.460193	+	0.887819i	\(0.652220\pi\)
\(47\)	−9.02607e26	+	5.21120e26i	−1.59196	+	0.919120i	−0.598993	+	0.800754i	\(0.704432\pi\)
							−0.992970	+	0.118366i	\(0.962235\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.33.h.b.11.5	yes	164
3.2	odd	2	inner	21.33.h.b.11.78	yes	164
7.2	even	3	inner	21.33.h.b.2.78	yes	164
21.2	odd	6	inner	21.33.h.b.2.5	✓	164

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.33.h.b.2.5	✓	164	21.2	odd	6	inner
21.33.h.b.2.78	yes	164	7.2	even	3	inner
21.33.h.b.11.5	yes	164	1.1	even	1	trivial
21.33.h.b.11.78	yes	164	3.2	odd	2	inner