Embedded newform 4.21.b.b.3.2

• Label: 4.21.b.b.3.2
• Level: $4$
• Weight: $21$
• Character: 4.3
• Analytic conductor: $10.141$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 4 = 2^{2} \)
Weight:	\( k \)	\(=\)	\( 21 \)
Character orbit:	\([\chi]\)	\(=\)	4.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.1405506041\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial:	x^8 - 3*x^7 + 41744*x^6 - 12461904*x^5 + 2339673984*x^4 - 888904297728*x^3 + 163438296621056*x^2 - 23541970369228800*x + 18139117546354278400
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{56}\cdot 3^{5}\cdot 5^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3.2
Root			\(-205.349 + 167.786i\) of defining polynomial
Character	\(\chi\)	\(=\)	4.3
Dual form			4.21.b.b.3.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-773.395 + 671.146i) q^{2} +44571.4i q^{3} +(147703. - 1.03812e6i) q^{4} +1.43704e7 q^{5} +(-2.99139e7 - 3.44713e7i) q^{6} -2.43808e8i q^{7} +(5.82498e8 + 9.02008e8i) q^{8} +1.50018e9 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 396 q^{2} - 1316080 q^{4} + 18568080 q^{5} + 99460608 q^{6} + 2177274816 q^{8} - 10411362168 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−773.395	+	671.146i	−0.755268	+	0.655416i
							\(3\)			44571.4i			0.754820i	0.926046	+	0.377410i	\(0.123185\pi\)
−0.926046	+	0.377410i	\(0.876815\pi\)
\(5\)	1.43704e7			1.47153			0.735763	−	0.677239i	\(-0.236824\pi\)
							0.735763	+	0.677239i	\(0.236824\pi\)
\(7\)		−	2.43808e8i		−	0.863113i	−0.902086	−	0.431556i	\(-0.857965\pi\)
							0.902086	−	0.431556i	\(-0.142035\pi\)
\(11\)			3.70669e10i			1.42909i	0.699590	+	0.714544i	\(0.253366\pi\)
							−0.699590	+	0.714544i	\(0.746634\pi\)
\(13\)	2.55906e9			0.0185630			0.00928149	−	0.999957i	\(-0.497046\pi\)
							0.00928149	+	0.999957i	\(0.497046\pi\)
\(17\)	1.09503e12			0.543173			0.271587	−	0.962414i	\(-0.412452\pi\)
							0.271587	+	0.962414i	\(0.412452\pi\)
\(19\)		−	5.54993e12i		−	0.905214i	−0.891710	−	0.452607i	\(-0.850494\pi\)
							0.891710	−	0.452607i	\(-0.149506\pi\)
\(23\)			6.68756e13i			1.61432i	0.590333	+	0.807160i	\(0.298997\pi\)
							−0.590333	+	0.807160i	\(0.701003\pi\)
\(29\)	5.41501e14			1.28712			0.643560	−	0.765396i	\(-0.277457\pi\)
							0.643560	+	0.765396i	\(0.277457\pi\)
\(31\)			2.99596e14i			0.365527i	0.983157	+	0.182763i	\(0.0585043\pi\)
							−0.983157	+	0.182763i	\(0.941496\pi\)
\(37\)	2.28472e15			0.475134			0.237567	−	0.971371i	\(-0.423650\pi\)
							0.237567	+	0.971371i	\(0.423650\pi\)
\(41\)	−2.45924e16			−1.83216			−0.916078	−	0.400999i	\(-0.868663\pi\)
							−0.916078	+	0.400999i	\(0.868663\pi\)
\(43\)			3.95328e15i			0.182925i	0.995809	+	0.0914625i	\(0.0291542\pi\)
							−0.995809	+	0.0914625i	\(0.970846\pi\)
\(47\)		−	4.94701e16i		−	0.940512i	−0.882530	−	0.470256i	\(-0.844161\pi\)
							0.882530	−	0.470256i	\(-0.155839\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4.21.b.b.3.2	yes	8
3.2	odd	2		36.21.d.b.19.7		8
4.3	odd	2	inner	4.21.b.b.3.1	✓	8
8.3	odd	2		64.21.c.d.63.5		8
8.5	even	2		64.21.c.d.63.4		8
12.11	even	2		36.21.d.b.19.8		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4.21.b.b.3.1	✓	8	4.3	odd	2	inner
4.21.b.b.3.2	yes	8	1.1	even	1	trivial
36.21.d.b.19.7		8	3.2	odd	2
36.21.d.b.19.8		8	12.11	even	2
64.21.c.d.63.4		8	8.5	even	2
64.21.c.d.63.5		8	8.3	odd	2