Embedded newform 21.33.f.b.10.2

• Label: 21.33.f.b.10.2
• Level: $21$
• Weight: $33$
• Character: 21.10
• Analytic conductor: $136.220$
• Analytic rank: $0$
• Dimension: $44$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 33 \)
Character orbit:	\([\chi]\)	\(=\)	21.f (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(136.219975799\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Relative dimension:	\(22\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			10.2
Character	\(\chi\)	\(=\)	21.10
Dual form			21.33.f.b.19.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-61847.9 - 107124. i) q^{2} +(2.15234e7 + 1.24265e7i) q^{3} +(-5.50285e9 + 9.53122e9i) q^{4} +(-2.11428e11 + 1.22068e11i) q^{5} -3.07422e12i q^{6} +(3.05188e13 + 1.31541e13i) q^{7} +8.30090e14 q^{8} +(3.08837e14 + 5.34921e14i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	44 * q - 41757 * q^2 + 947027862 * q^3 - 54484687005 * q^4 + 153816227589 * q^5 - 14837835233530 * q^7 + 90590859203718 * q^8 + 13588814718246834 * q^9

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).

\(n\)	\(8\)	\(10\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−61847.9	−	107124.i	−0.943725	−	1.63458i	−0.758285	−	0.651923i	\(-0.773962\pi\)
							−0.185440	−	0.982656i	\(-0.559371\pi\)
\(3\)	2.15234e7	+	1.24265e7i	0.500000	+	0.288675i
							\(5\)	−2.11428e11	+	1.22068e11i	−1.38562	+	0.799986i	−0.992817	−	0.119639i	\(-0.961826\pi\)
−0.392798	+	0.919625i	\(0.628493\pi\)
\(7\)	3.05188e13	+	1.31541e13i	0.918330	+	0.395816i
							\(11\)	2.74097e16	−	4.74750e16i	0.596515	−	1.03319i	−0.396816	−	0.917898i	\(-0.629885\pi\)
0.993331	−	0.115297i	\(-0.0367819\pi\)
\(13\)			9.35749e16i			0.140626i	0.997525	+	0.0703130i	\(0.0223998\pi\)
							−0.997525	+	0.0703130i	\(0.977600\pi\)
\(17\)	−4.80012e19	−	2.77135e19i	−0.986437	−	0.569520i	−0.0822298	−	0.996613i	\(-0.526204\pi\)
							−0.904207	+	0.427094i	\(0.859537\pi\)
\(19\)	−6.23378e19	+	3.59908e19i	−0.216120	+	0.124777i	−0.604152	−	0.796869i	\(-0.706488\pi\)
							0.388033	+	0.921646i	\(0.373155\pi\)
\(23\)	3.59060e21	+	6.21910e21i	0.585493	+	1.01410i	0.994814	+	0.101713i	\(0.0324324\pi\)
							−0.409321	+	0.912391i	\(0.634234\pi\)
\(29\)	1.32856e23			0.530899			0.265449	−	0.964125i	\(-0.414480\pi\)
							0.265449	+	0.964125i	\(0.414480\pi\)
\(31\)	2.02862e23	+	1.17122e23i	0.278877	+	0.161010i	0.632915	−	0.774221i	\(-0.281858\pi\)
							−0.354038	+	0.935231i	\(0.615192\pi\)
\(37\)	−1.02494e25	−	1.77524e25i	−0.830749	−	1.43890i	−0.897445	−	0.441127i	\(-0.854579\pi\)
							0.0666954	−	0.997773i	\(-0.478754\pi\)
\(41\)			1.03865e26i			1.62903i	0.580144	+	0.814514i	\(0.302996\pi\)
							−0.580144	+	0.814514i	\(0.697004\pi\)
\(43\)	−2.20448e26			−1.61366			−0.806829	−	0.590785i	\(-0.798818\pi\)
							−0.806829	+	0.590785i	\(0.798818\pi\)
\(47\)	4.51326e26	−	2.60573e26i	0.796021	−	0.459583i	−0.0460569	−	0.998939i	\(-0.514666\pi\)
							0.842078	+	0.539356i	\(0.181332\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.33.f.b.10.2	✓	44
7.5	odd	6	inner	21.33.f.b.19.2	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.33.f.b.10.2	✓	44	1.1	even	1	trivial
21.33.f.b.19.2	yes	44	7.5	odd	6	inner