Embedded newform 21.5.h.b.2.3

• Label: 21.5.h.b.2.3
• 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			2.3
Root			\(-4.14633 - 2.39389i\) of defining polynomial
Character	\(\chi\)	\(=\)	21.2
Dual form			21.5.h.b.11.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.14633 - 2.39389i) q^{2} +(2.75140 + 8.56912i) q^{3} +(3.46138 + 5.99529i) q^{4} +(30.3128 + 17.5011i) q^{5} +(9.10526 - 42.1170i) q^{6} +(48.8998 - 3.13209i) q^{7} +43.4597i q^{8} +(-65.8596 + 47.1542i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{3} + 78 q^{4} + 28 q^{6} - 168 q^{7} + 62 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.14633	−	2.39389i	−1.03658	−	0.598472i	−0.117720	−	0.993047i	\(-0.537558\pi\)
							−0.918864	+	0.394575i	\(0.870892\pi\)
\(3\)	2.75140	+	8.56912i	0.305711	+	0.952124i
							\(5\)	30.3128	+	17.5011i	1.21251	+	0.700044i	0.963306	−	0.268407i	\(-0.0864972\pi\)
0.249206	+	0.968451i	\(0.419831\pi\)
\(7\)	48.8998	−	3.13209i	0.997955	−	0.0639201i
							\(11\)	−21.3991	+	12.3548i	−0.176852	+	0.102105i	−0.585813	−	0.810446i	\(-0.699225\pi\)
0.408961	+	0.912552i	\(0.365891\pi\)
\(13\)	−105.789			−0.625968			−0.312984	−	0.949758i	\(-0.601329\pi\)
							−0.312984	+	0.949758i	\(0.601329\pi\)
\(17\)	365.923	−	211.266i	1.26617	−	0.731024i	0.291909	−	0.956446i	\(-0.405709\pi\)
							0.974261	+	0.225422i	\(0.0723761\pi\)
\(19\)	−13.4836	+	23.3543i	−0.0373508	+	0.0646935i	−0.884096	−	0.467304i	\(-0.845225\pi\)
							0.846746	+	0.531998i	\(0.178559\pi\)
\(23\)	−580.127	−	334.936i	−1.09665	−	0.633150i	−0.161309	−	0.986904i	\(-0.551572\pi\)
							−0.935338	+	0.353754i	\(0.884905\pi\)
\(29\)		−	332.267i		−	0.395086i	−0.980294	−	0.197543i	\(-0.936704\pi\)
							0.980294	−	0.197543i	\(-0.0632962\pi\)
\(31\)	35.0201	+	60.6566i	0.0364413	+	0.0631182i	0.883671	−	0.468109i	\(-0.155064\pi\)
							−0.847230	+	0.531227i	\(0.821731\pi\)
\(37\)	601.101	−	1041.14i	0.439080	−	0.760509i	−0.558539	−	0.829479i	\(-0.688638\pi\)
							0.997619	+	0.0689693i	\(0.0219711\pi\)
\(41\)			1615.22i			0.960867i	0.877031	+	0.480433i	\(0.159521\pi\)
							−0.877031	+	0.480433i	\(0.840479\pi\)
\(43\)	−309.151			−0.167199			−0.0835996	−	0.996499i	\(-0.526642\pi\)
							−0.0835996	+	0.996499i	\(0.526642\pi\)
\(47\)	−2986.36	−	1724.17i	−1.35190	−	0.780523i	−0.363389	−	0.931638i	\(-0.618380\pi\)
							−0.988516	+	0.151115i	\(0.951714\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.5.h.b.2.3	✓	16
3.2	odd	2	inner	21.5.h.b.2.6	yes	16
7.2	even	3		147.5.b.c.50.6		8
7.3	odd	6		147.5.h.b.116.6		16
7.4	even	3	inner	21.5.h.b.11.6	yes	16
7.5	odd	6		147.5.b.f.50.6		8
7.6	odd	2		147.5.h.b.128.3		16
21.2	odd	6		147.5.b.c.50.3		8
21.5	even	6		147.5.b.f.50.3		8
21.11	odd	6	inner	21.5.h.b.11.3	yes	16
21.17	even	6		147.5.h.b.116.3		16
21.20	even	2		147.5.h.b.128.6		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.5.h.b.2.3	✓	16	1.1	even	1	trivial
21.5.h.b.2.6	yes	16	3.2	odd	2	inner
21.5.h.b.11.3	yes	16	21.11	odd	6	inner
21.5.h.b.11.6	yes	16	7.4	even	3	inner
147.5.b.c.50.3		8	21.2	odd	6
147.5.b.c.50.6		8	7.2	even	3
147.5.b.f.50.3		8	21.5	even	6
147.5.b.f.50.6		8	7.5	odd	6
147.5.h.b.116.3		16	21.17	even	6
147.5.h.b.116.6		16	7.3	odd	6
147.5.h.b.128.3		16	7.6	odd	2
147.5.h.b.128.6		16	21.20	even	2