Embedded newform 21.33.h.b.2.13

• Label: 21.33.h.b.2.13
• 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			2.13
Character	\(\chi\)	\(=\)	21.2
Dual form			21.33.h.b.11.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-85030.6 - 49092.5i) q^{2} +(-1.46438e6 + 4.30218e7i) q^{3} +(2.67265e9 + 4.62917e9i) q^{4} +(9.94676e10 + 5.74276e10i) q^{5} +(2.23656e12 - 3.58628e12i) q^{6} +(-2.61520e13 + 2.05061e13i) q^{7} -1.03128e14i q^{8} +(-1.84873e15 - 1.26000e14i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−85030.6	−	49092.5i	−1.29746	−	0.749091i	−0.317499	−	0.948259i	\(-0.602843\pi\)
							−0.979965	+	0.199167i	\(0.936176\pi\)
\(3\)	−1.46438e6	+	4.30218e7i	−0.0340183	+	0.999421i
							\(5\)	9.94676e10	+	5.74276e10i	0.651871	+	0.376358i	0.789173	−	0.614172i	\(-0.210510\pi\)
−0.137302	+	0.990529i	\(0.543843\pi\)
\(7\)	−2.61520e13	+	2.05061e13i	−0.786931	+	0.617041i
							\(11\)	3.10898e16	−	1.79497e16i	0.676605	−	0.390638i	−0.121969	−	0.992534i	\(-0.538921\pi\)
0.798575	+	0.601896i	\(0.205588\pi\)
\(13\)	3.29625e17			0.495366			0.247683	−	0.968841i	\(-0.420331\pi\)
							0.247683	+	0.968841i	\(0.420331\pi\)
\(17\)	2.87284e19	−	1.65864e19i	0.590376	−	0.340854i	−0.174870	−	0.984592i	\(-0.555951\pi\)
							0.765246	+	0.643738i	\(0.222617\pi\)
\(19\)	1.00160e20	−	1.73482e20i	0.347245	−	0.601446i	−0.638514	−	0.769610i	\(-0.720451\pi\)
							0.985759	+	0.168164i	\(0.0537838\pi\)
\(23\)	6.94874e21	+	4.01186e21i	1.13308	+	0.654185i	0.944708	−	0.327913i	\(-0.106345\pi\)
							0.188373	+	0.982098i	\(0.439679\pi\)
\(29\)		−	7.09309e22i		−	0.283444i	−0.989906	−	0.141722i	\(-0.954736\pi\)
							0.989906	−	0.141722i	\(-0.0452639\pi\)
\(31\)	−1.65521e23	−	2.86691e23i	−0.227544	−	0.394118i	0.729535	−	0.683943i	\(-0.239736\pi\)
							−0.957080	+	0.289825i	\(0.906403\pi\)
\(37\)	−6.45315e24	+	1.11772e25i	−0.523051	+	0.905951i	0.476589	+	0.879126i	\(0.341873\pi\)
							−0.999640	+	0.0268249i	\(0.991460\pi\)
\(41\)			9.66749e25i			1.51625i	0.652107	+	0.758127i	\(0.273885\pi\)
							−0.652107	+	0.758127i	\(0.726115\pi\)
\(43\)	−1.55836e26			−1.14070			−0.570352	−	0.821400i	\(-0.693193\pi\)
							−0.570352	+	0.821400i	\(0.693193\pi\)
\(47\)	−2.94689e26	−	1.70139e26i	−0.519754	−	0.300080i	0.217080	−	0.976154i	\(-0.430347\pi\)
							−0.736834	+	0.676074i	\(0.763680\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.33.h.b.2.13	✓	164
3.2	odd	2	inner	21.33.h.b.2.70	yes	164
7.4	even	3	inner	21.33.h.b.11.70	yes	164
21.11	odd	6	inner	21.33.h.b.11.13	yes	164

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.33.h.b.2.13	✓	164	1.1	even	1	trivial
21.33.h.b.2.70	yes	164	3.2	odd	2	inner
21.33.h.b.11.13	yes	164	21.11	odd	6	inner
21.33.h.b.11.70	yes	164	7.4	even	3	inner