Embedded newform 21.33.h.b.11.3

• Label: 21.33.h.b.11.3
• 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			11.3
Character	\(\chi\)	\(=\)	21.11
Dual form			21.33.h.b.2.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-107705. + 62183.5i) q^{2} +(-3.45696e7 - 2.56508e7i) q^{3} +(5.58609e9 - 9.67539e9i) q^{4} +(4.70485e10 - 2.71635e10i) q^{5} +(5.31837e12 + 6.13057e11i) q^{6} +(-5.64089e12 + 3.27507e13i) q^{7} +8.55298e14i q^{8} +(5.37095e14 + 1.77347e15i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−107705.	+	62183.5i	−1.64345	+	0.948845i	−0.663852	+	0.747864i	\(0.731080\pi\)
							−0.979595	+	0.200981i	\(0.935587\pi\)
\(3\)	−3.45696e7	−	2.56508e7i	−0.803072	−	0.595882i
							\(5\)	4.70485e10	−	2.71635e10i	0.308337	−	0.178019i	−0.337845	−	0.941202i	\(-0.609698\pi\)
0.646182	+	0.763183i	\(0.276365\pi\)
\(7\)	−5.64089e12	+	3.27507e13i	−0.169738	+	0.985489i
							\(11\)	7.71847e16	+	4.45626e16i	1.67976	+	0.969812i	0.961809	+	0.273723i	\(0.0882551\pi\)
0.717955	+	0.696089i	\(0.245078\pi\)
\(13\)	−7.94366e17			−1.19379			−0.596894	−	0.802320i	\(-0.703599\pi\)
							−0.596894	+	0.802320i	\(0.703599\pi\)
\(17\)	2.93864e19	+	1.69662e19i	0.603898	+	0.348661i	0.770574	−	0.637351i	\(-0.219970\pi\)
							−0.166675	+	0.986012i	\(0.553303\pi\)
\(19\)	−8.82807e19	−	1.52907e20i	−0.306061	−	0.530113i	0.671436	−	0.741063i	\(-0.265678\pi\)
							−0.977497	+	0.210949i	\(0.932344\pi\)
\(23\)	−3.84855e21	+	2.22196e21i	−0.627555	+	0.362319i	−0.779805	−	0.626023i	\(-0.784682\pi\)
							0.152250	+	0.988342i	\(0.451348\pi\)
\(29\)		−	1.60786e23i		−	0.642510i	−0.946993	−	0.321255i	\(-0.895895\pi\)
							0.946993	−	0.321255i	\(-0.104105\pi\)
\(31\)	−1.34324e23	+	2.32656e23i	−0.184658	+	0.319836i	−0.943461	−	0.331483i	\(-0.892451\pi\)
							0.758803	+	0.651320i	\(0.225784\pi\)
\(37\)	−7.18410e24	−	1.24432e25i	−0.582297	−	1.00857i	−0.995206	−	0.0977961i	\(-0.968821\pi\)
							0.412909	−	0.910772i	\(-0.364513\pi\)
\(41\)			9.83994e25i			1.54330i	0.636047	+	0.771651i	\(0.280569\pi\)
							−0.636047	+	0.771651i	\(0.719431\pi\)
\(43\)	−9.19202e25			−0.672846			−0.336423	−	0.941711i	\(-0.609217\pi\)
							−0.336423	+	0.941711i	\(0.609217\pi\)
\(47\)	1.47399e26	−	8.51009e25i	0.259974	−	0.150096i	−0.364349	−	0.931263i	\(-0.618708\pi\)
							0.624323	+	0.781167i	\(0.285375\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.33.h.b.11.3	yes	164
3.2	odd	2	inner	21.33.h.b.11.80	yes	164
7.2	even	3	inner	21.33.h.b.2.80	yes	164
21.2	odd	6	inner	21.33.h.b.2.3	✓	164

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.33.h.b.2.3	✓	164	21.2	odd	6	inner
21.33.h.b.2.80	yes	164	7.2	even	3	inner
21.33.h.b.11.3	yes	164	1.1	even	1	trivial
21.33.h.b.11.80	yes	164	3.2	odd	2	inner