Embedded newform 21.33.h.b.2.11

• Label: 21.33.h.b.2.11
• 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			2.11
Character	\(\chi\)	\(=\)	21.2
Dual form			21.33.h.b.11.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-89987.0 - 51954.0i) q^{2} +(-1.34905e6 - 4.30256e7i) q^{3} +(3.25096e9 + 5.63082e9i) q^{4} +(2.01263e11 + 1.16199e11i) q^{5} +(-2.11396e12 + 3.94183e12i) q^{6} +(2.96316e13 - 1.50465e13i) q^{7} -2.29320e14i q^{8} +(-1.84938e15 + 1.16087e14i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−89987.0	−	51954.0i	−1.37309	−	0.792756i	−0.381777	−	0.924255i	\(-0.624688\pi\)
							−0.991316	+	0.131499i	\(0.958021\pi\)
\(3\)	−1.34905e6	−	4.30256e7i	−0.0313392	−	0.999509i
							\(5\)	2.01263e11	+	1.16199e11i	1.31900	+	0.761523i	0.983567	−	0.180543i	\(-0.0577855\pi\)
0.335429	+	0.942066i	\(0.391119\pi\)
\(7\)	2.96316e13	−	1.50465e13i	0.891633	−	0.452759i
							\(11\)	2.24326e16	−	1.29515e16i	0.488199	−	0.281862i	−0.235628	−	0.971843i	\(-0.575715\pi\)
0.723827	+	0.689982i	\(0.242381\pi\)
\(13\)	7.52921e17			1.13150			0.565751	−	0.824576i	\(-0.308586\pi\)
							0.565751	+	0.824576i	\(0.308586\pi\)
\(17\)	6.31507e19	−	3.64601e19i	1.29776	−	0.749264i	0.317746	−	0.948176i	\(-0.397074\pi\)
							0.980017	+	0.198912i	\(0.0637407\pi\)
\(19\)	1.80765e20	−	3.13095e20i	0.626697	−	1.08547i	−0.361513	−	0.932367i	\(-0.617740\pi\)
							0.988210	−	0.153104i	\(-0.0489268\pi\)
\(23\)	−9.36222e21	−	5.40528e21i	−1.52663	−	0.881400i	−0.999500	−	0.0316145i	\(-0.989935\pi\)
							−0.527129	−	0.849785i	\(-0.676732\pi\)
\(29\)			3.72386e22i			0.148808i	0.997228	+	0.0744038i	\(0.0237054\pi\)
							−0.997228	+	0.0744038i	\(0.976295\pi\)
\(31\)	1.08265e23	+	1.87520e23i	0.148833	+	0.257786i	0.930796	−	0.365538i	\(-0.119115\pi\)
							−0.781963	+	0.623324i	\(0.785782\pi\)
\(37\)	−1.87123e24	+	3.24107e24i	−0.151670	+	0.262700i	−0.931842	−	0.362865i	\(-0.881798\pi\)
							0.780171	+	0.625566i	\(0.215132\pi\)
\(41\)		−	4.32983e25i		−	0.679093i	−0.940589	−	0.339547i	\(-0.889726\pi\)
							0.940589	−	0.339547i	\(-0.110274\pi\)
\(43\)	−1.56074e26			−1.14245			−0.571223	−	0.820795i	\(-0.693531\pi\)
							−0.571223	+	0.820795i	\(0.693531\pi\)
\(47\)	7.84344e26	+	4.52841e26i	1.38338	+	0.798693i	0.992558	−	0.121774i	\(-0.0388583\pi\)
							0.390820	+	0.920467i	\(0.372192\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.33.h.b.2.11	✓	164
3.2	odd	2	inner	21.33.h.b.2.72	yes	164
7.4	even	3	inner	21.33.h.b.11.72	yes	164
21.11	odd	6	inner	21.33.h.b.11.11	yes	164

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.33.h.b.2.11	✓	164	1.1	even	1	trivial
21.33.h.b.2.72	yes	164	3.2	odd	2	inner
21.33.h.b.11.11	yes	164	21.11	odd	6	inner
21.33.h.b.11.72	yes	164	7.4	even	3	inner