Embedded newform 4.21.b.b.3.6

• Label: 4.21.b.b.3.6
• 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			3.6
Root			\(77.2239 + 239.948i\) of defining polynomial
Character	\(\chi\)	\(=\)	4.3
Dual form			4.21.b.b.3.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(356.896 + 959.792i) q^{2} +46975.6i q^{3} +(-793827. + 685091. i) q^{4} -4.32154e6 q^{5} +(-4.50868e7 + 1.67654e7i) q^{6} +1.10972e8i q^{7} +(-9.40859e8 - 5.17403e8i) q^{8} +1.28008e9 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 396 q^{2} - 1316080 q^{4} + 18568080 q^{5} + 99460608 q^{6} + 2177274816 q^{8} - 10411362168 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	356.896	+	959.792i	0.348531	+	0.937297i
							\(3\)			46975.6i			0.795536i	0.917486	+	0.397768i	\(0.130215\pi\)
−0.917486	+	0.397768i	\(0.869785\pi\)
\(5\)	−4.32154e6			−0.442526			−0.221263	−	0.975214i	\(-0.571018\pi\)
							−0.221263	+	0.975214i	\(0.571018\pi\)
\(7\)			1.10972e8i			0.392854i	0.980518	+	0.196427i	\(0.0629339\pi\)
							−0.980518	+	0.196427i	\(0.937066\pi\)
\(11\)		−	2.26825e10i		−	0.874510i	−0.899337	−	0.437255i	\(-0.855951\pi\)
							0.899337	−	0.437255i	\(-0.144049\pi\)
\(13\)	−2.51114e11			−1.82154			−0.910769	−	0.412916i	\(-0.864510\pi\)
							−0.910769	+	0.412916i	\(0.864510\pi\)
\(17\)	1.40803e12			0.698430			0.349215	−	0.937043i	\(-0.386448\pi\)
							0.349215	+	0.937043i	\(0.386448\pi\)
\(19\)			1.01893e13i			1.66191i	0.556339	+	0.830956i	\(0.312206\pi\)
							−0.556339	+	0.830956i	\(0.687794\pi\)
\(23\)			4.06234e13i			0.980613i	0.871550	+	0.490307i	\(0.163115\pi\)
							−0.871550	+	0.490307i	\(0.836885\pi\)
\(29\)	−2.78623e14			−0.662273			−0.331137	−	0.943583i	\(-0.607432\pi\)
							−0.331137	+	0.943583i	\(0.607432\pi\)
\(31\)			5.16079e14i			0.629650i	0.949150	+	0.314825i	\(0.101946\pi\)
							−0.949150	+	0.314825i	\(0.898054\pi\)
\(37\)	8.25764e14			0.171727			0.0858636	−	0.996307i	\(-0.472635\pi\)
							0.0858636	+	0.996307i	\(0.472635\pi\)
\(41\)	−2.23881e14			−0.0166793			−0.00833967	−	0.999965i	\(-0.502655\pi\)
							−0.00833967	+	0.999965i	\(0.502655\pi\)
\(43\)		−	2.43150e16i		−	1.12509i	−0.826765	−	0.562547i	\(-0.809821\pi\)
							0.826765	−	0.562547i	\(-0.190179\pi\)
\(47\)			7.49929e16i			1.42574i	0.701295	+	0.712872i	\(0.252606\pi\)
							−0.701295	+	0.712872i	\(0.747394\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4.21.b.b.3.6	yes	8
3.2	odd	2		36.21.d.b.19.3		8
4.3	odd	2	inner	4.21.b.b.3.5	✓	8
8.3	odd	2		64.21.c.d.63.6		8
8.5	even	2		64.21.c.d.63.3		8
12.11	even	2		36.21.d.b.19.4		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4.21.b.b.3.5	✓	8	4.3	odd	2	inner
4.21.b.b.3.6	yes	8	1.1	even	1	trivial
36.21.d.b.19.3		8	3.2	odd	2
36.21.d.b.19.4		8	12.11	even	2
64.21.c.d.63.3		8	8.5	even	2
64.21.c.d.63.6		8	8.3	odd	2