Embedded newform 4.21.b.b.3.8

• Label: 4.21.b.b.3.8
• 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			3.8
Root			\(228.158 + 88.6584i\) of defining polynomial
Character	\(\chi\)	\(=\)	4.3
Dual form			4.21.b.b.3.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(960.631 + 354.634i) q^{2} -55423.5i q^{3} +(797046. + 681344. i) q^{4} +2.53495e6 q^{5} +(1.96550e7 - 5.32415e7i) q^{6} -4.45714e8i q^{7} +(5.24039e8 + 9.37179e8i) q^{8} +4.15025e8 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 396 q^{2} - 1316080 q^{4} + 18568080 q^{5} + 99460608 q^{6} + 2177274816 q^{8} - 10411362168 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	960.631	+	354.634i	0.938116	+	0.346322i
							\(3\)		−	55423.5i		−	0.938601i	−0.883039	−	0.469301i	\(-0.844506\pi\)
0.883039	−	0.469301i	\(-0.155494\pi\)
\(5\)	2.53495e6			0.259579			0.129789	−	0.991542i	\(-0.458570\pi\)
							0.129789	+	0.991542i	\(0.458570\pi\)
\(7\)		−	4.45714e8i		−	1.57789i	−0.614466	−	0.788944i	\(-0.710628\pi\)
							0.614466	−	0.788944i	\(-0.289372\pi\)
\(11\)		−	1.75386e10i		−	0.676189i	−0.941112	−	0.338094i	\(-0.890218\pi\)
							0.941112	−	0.338094i	\(-0.109782\pi\)
\(13\)	1.80795e11			1.31145			0.655727	−	0.754998i	\(-0.272362\pi\)
							0.655727	+	0.754998i	\(0.272362\pi\)
\(17\)	−2.03482e12			−1.00934			−0.504669	−	0.863313i	\(-0.668386\pi\)
							−0.504669	+	0.863313i	\(0.668386\pi\)
\(19\)			3.24237e12i			0.528843i	0.964407	+	0.264421i	\(0.0851810\pi\)
							−0.964407	+	0.264421i	\(0.914819\pi\)
\(23\)			4.86053e13i			1.17329i	0.809844	+	0.586645i	\(0.199552\pi\)
							−0.809844	+	0.586645i	\(0.800448\pi\)
\(29\)	−3.34913e14			−0.796070			−0.398035	−	0.917370i	\(-0.630308\pi\)
							−0.398035	+	0.917370i	\(0.630308\pi\)
\(31\)			1.26906e15i			1.54833i	0.632983	+	0.774166i	\(0.281830\pi\)
							−0.632983	+	0.774166i	\(0.718170\pi\)
\(37\)	5.85360e15			1.21732			0.608661	−	0.793430i	\(-0.291707\pi\)
							0.608661	+	0.793430i	\(0.291707\pi\)
\(41\)	1.81810e16			1.35450			0.677250	−	0.735753i	\(-0.263172\pi\)
							0.677250	+	0.735753i	\(0.263172\pi\)
\(43\)			1.47461e16i			0.682327i	0.940004	+	0.341164i	\(0.110821\pi\)
							−0.940004	+	0.341164i	\(0.889179\pi\)
\(47\)		−	2.40368e16i		−	0.456982i	−0.973546	−	0.228491i	\(-0.926621\pi\)
							0.973546	−	0.228491i	\(-0.0733791\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4.21.b.b.3.8	yes	8
3.2	odd	2		36.21.d.b.19.1		8
4.3	odd	2	inner	4.21.b.b.3.7	✓	8
8.3	odd	2		64.21.c.d.63.2		8
8.5	even	2		64.21.c.d.63.7		8
12.11	even	2		36.21.d.b.19.2		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4.21.b.b.3.7	✓	8	4.3	odd	2	inner
4.21.b.b.3.8	yes	8	1.1	even	1	trivial
36.21.d.b.19.1		8	3.2	odd	2
36.21.d.b.19.2		8	12.11	even	2
64.21.c.d.63.2		8	8.3	odd	2
64.21.c.d.63.7		8	8.5	even	2