Embedded newform 21.5.h.b.2.1

• Label: 21.5.h.b.2.1
• Level: $21$
• Weight: $5$
• Character: 21.2
• Analytic conductor: $2.171$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	21.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.17076922476\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	x^16 - 103*x^14 + 7227*x^12 - 270898*x^10 + 7374256*x^8 - 115494792*x^6 + 1245573504*x^4 - 2908017504*x^2 + 5639409216
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 3^{7} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			2.1
Root			\(-6.18666 - 3.57187i\) of defining polynomial
Character	\(\chi\)	\(=\)	21.2
Dual form			21.5.h.b.11.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.18666 - 3.57187i) q^{2} +(-7.57871 - 4.85420i) q^{3} +(17.5165 + 30.3395i) q^{4} +(13.6392 + 7.87461i) q^{5} +(29.5483 + 57.1014i) q^{6} +(-44.8988 + 19.6239i) q^{7} -135.967i q^{8} +(33.8736 + 73.5771i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{3} + 78 q^{4} + 28 q^{6} - 168 q^{7} + 62 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.18666	−	3.57187i	−1.54667	−	0.892968i	−0.998393	−	0.0566676i	\(-0.981952\pi\)
							−0.548272	−	0.836300i	\(-0.684714\pi\)
\(3\)	−7.57871	−	4.85420i	−0.842078	−	0.539355i
							\(5\)	13.6392	+	7.87461i	0.545569	+	0.314984i	0.747333	−	0.664450i	\(-0.231334\pi\)
−0.201764	+	0.979434i	\(0.564667\pi\)
\(7\)	−44.8988	+	19.6239i	−0.916302	+	0.400488i
							\(11\)	−135.910	+	78.4677i	−1.12322	+	0.648494i	−0.942222	−	0.334989i	\(-0.891267\pi\)
−0.181002	+	0.983483i	\(0.557934\pi\)
\(13\)	44.9724			0.266109			0.133054	−	0.991109i	\(-0.457521\pi\)
							0.133054	+	0.991109i	\(0.457521\pi\)
\(17\)	−160.681	+	92.7690i	−0.555988	+	0.321000i	−0.751534	−	0.659695i	\(-0.770686\pi\)
							0.195546	+	0.980695i	\(0.437352\pi\)
\(19\)	−223.565	+	387.226i	−0.619293	+	1.07265i	0.370321	+	0.928904i	\(0.379248\pi\)
							−0.989615	+	0.143744i	\(0.954086\pi\)
\(23\)	−194.505	−	112.298i	−0.367684	−	0.212283i	0.304762	−	0.952429i	\(-0.401423\pi\)
							−0.672446	+	0.740146i	\(0.734756\pi\)
\(29\)		−	535.091i		−	0.636255i	−0.948048	−	0.318128i	\(-0.896946\pi\)
							0.948048	−	0.318128i	\(-0.103054\pi\)
\(31\)	131.805	+	228.294i	0.137154	+	0.237558i	0.926418	−	0.376496i	\(-0.122871\pi\)
							−0.789264	+	0.614054i	\(0.789538\pi\)
\(37\)	26.0980	−	45.2031i	0.0190636	−	0.0330191i	−0.856336	−	0.516419i	\(-0.827265\pi\)
							0.875400	+	0.483400i	\(0.160598\pi\)
\(41\)			2065.09i			1.22849i	0.789115	+	0.614245i	\(0.210539\pi\)
							−0.789115	+	0.614245i	\(0.789461\pi\)
\(43\)	−1948.75			−1.05395			−0.526974	−	0.849881i	\(-0.676674\pi\)
							−0.526974	+	0.849881i	\(0.676674\pi\)
\(47\)	1629.29	+	940.671i	0.737569	+	0.425836i	0.821185	−	0.570662i	\(-0.193314\pi\)
							−0.0836158	+	0.996498i	\(0.526647\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.5.h.b.2.1	✓	16
3.2	odd	2	inner	21.5.h.b.2.8	yes	16
7.2	even	3		147.5.b.c.50.8		8
7.3	odd	6		147.5.h.b.116.8		16
7.4	even	3	inner	21.5.h.b.11.8	yes	16
7.5	odd	6		147.5.b.f.50.8		8
7.6	odd	2		147.5.h.b.128.1		16
21.2	odd	6		147.5.b.c.50.1		8
21.5	even	6		147.5.b.f.50.1		8
21.11	odd	6	inner	21.5.h.b.11.1	yes	16
21.17	even	6		147.5.h.b.116.1		16
21.20	even	2		147.5.h.b.128.8		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.5.h.b.2.1	✓	16	1.1	even	1	trivial
21.5.h.b.2.8	yes	16	3.2	odd	2	inner
21.5.h.b.11.1	yes	16	21.11	odd	6	inner
21.5.h.b.11.8	yes	16	7.4	even	3	inner
147.5.b.c.50.1		8	21.2	odd	6
147.5.b.c.50.8		8	7.2	even	3
147.5.b.f.50.1		8	21.5	even	6
147.5.b.f.50.8		8	7.5	odd	6
147.5.h.b.116.1		16	21.17	even	6
147.5.h.b.116.8		16	7.3	odd	6
147.5.h.b.128.1		16	7.6	odd	2
147.5.h.b.128.8		16	21.20	even	2