Embedded newform 21.5.h.b.2.2

• Label: 21.5.h.b.2.2
• 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			2.2
Root			\(-4.46939 - 2.58040i\) of defining polynomial
Character	\(\chi\)	\(=\)	21.2
Dual form			21.5.h.b.11.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.46939 - 2.58040i) q^{2} +(8.54954 - 2.81164i) q^{3} +(5.31698 + 9.20927i) q^{4} +(-31.7353 - 18.3224i) q^{5} +(-45.4664 - 9.49493i) q^{6} +(-18.9163 - 45.2015i) q^{7} +27.6932i q^{8} +(65.1893 - 48.0765i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{3} + 78 q^{4} + 28 q^{6} - 168 q^{7} + 62 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.46939	−	2.58040i	−1.11735	−	0.645101i	−0.176625	−	0.984278i	\(-0.556518\pi\)
							−0.940723	+	0.339177i	\(0.889851\pi\)
\(3\)	8.54954	−	2.81164i	0.949949	−	0.312405i
							\(5\)	−31.7353	−	18.3224i	−1.26941	−	0.732896i	−0.294536	−	0.955640i	\(-0.595165\pi\)
−0.974877	+	0.222744i	\(0.928498\pi\)
\(7\)	−18.9163	−	45.2015i	−0.386047	−	0.922479i
							\(11\)	32.7370	−	18.9007i	0.270554	−	0.156204i	−0.358585	−	0.933497i	\(-0.616741\pi\)
0.629139	+	0.777292i	\(0.283407\pi\)
\(13\)	127.139			0.752300			0.376150	−	0.926559i	\(-0.377248\pi\)
							0.376150	+	0.926559i	\(0.377248\pi\)
\(17\)	248.343	−	143.381i	0.859318	−	0.496127i	−0.00446598	−	0.999990i	\(-0.501422\pi\)
							0.863784	+	0.503863i	\(0.168088\pi\)
\(19\)	−161.952	+	280.509i	−0.448620	+	0.777032i	−0.998296	−	0.0583452i	\(-0.981418\pi\)
							0.549677	+	0.835378i	\(0.314751\pi\)
\(23\)	303.425	+	175.183i	0.573583	+	0.331158i	0.758579	−	0.651581i	\(-0.225894\pi\)
							−0.184996	+	0.982739i	\(0.559227\pi\)
\(29\)		−	210.131i		−	0.249858i	−0.992166	−	0.124929i	\(-0.960130\pi\)
							0.992166	−	0.124929i	\(-0.0398703\pi\)
\(31\)	−658.024	−	1139.73i	−0.684728	−	1.18598i	−0.973522	−	0.228593i	\(-0.926588\pi\)
							0.288794	−	0.957391i	\(-0.406746\pi\)
\(37\)	−497.558	+	861.795i	−0.363446	+	0.629507i	−0.988525	−	0.151054i	\(-0.951733\pi\)
							0.625079	+	0.780561i	\(0.285067\pi\)
\(41\)		−	288.879i		−	0.171850i	−0.996302	−	0.0859249i	\(-0.972615\pi\)
							0.996302	−	0.0859249i	\(-0.0273845\pi\)
\(43\)	1811.84			0.979902			0.489951	−	0.871750i	\(-0.337015\pi\)
							0.489951	+	0.871750i	\(0.337015\pi\)
\(47\)	−91.1152	−	52.6054i	−0.0412473	−	0.0238141i	0.479235	−	0.877687i	\(-0.340914\pi\)
							−0.520482	+	0.853873i	\(0.674248\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.5.h.b.2.2	✓	16
3.2	odd	2	inner	21.5.h.b.2.7	yes	16
7.2	even	3		147.5.b.c.50.7		8
7.3	odd	6		147.5.h.b.116.7		16
7.4	even	3	inner	21.5.h.b.11.7	yes	16
7.5	odd	6		147.5.b.f.50.7		8
7.6	odd	2		147.5.h.b.128.2		16
21.2	odd	6		147.5.b.c.50.2		8
21.5	even	6		147.5.b.f.50.2		8
21.11	odd	6	inner	21.5.h.b.11.2	yes	16
21.17	even	6		147.5.h.b.116.2		16
21.20	even	2		147.5.h.b.128.7		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.5.h.b.2.2	✓	16	1.1	even	1	trivial
21.5.h.b.2.7	yes	16	3.2	odd	2	inner
21.5.h.b.11.2	yes	16	21.11	odd	6	inner
21.5.h.b.11.7	yes	16	7.4	even	3	inner
147.5.b.c.50.2		8	21.2	odd	6
147.5.b.c.50.7		8	7.2	even	3
147.5.b.f.50.2		8	21.5	even	6
147.5.b.f.50.7		8	7.5	odd	6
147.5.h.b.116.2		16	21.17	even	6
147.5.h.b.116.7		16	7.3	odd	6
147.5.h.b.128.2		16	7.6	odd	2
147.5.h.b.128.7		16	21.20	even	2