Embedded newform 4.21.b.b.3.3

• Label: 4.21.b.b.3.3
• 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			3.3
Root			\(-98.5328 - 240.932i\) of defining polynomial
Character	\(\chi\)	\(=\)	4.3
Dual form			4.21.b.b.3.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-346.131 - 963.727i) q^{2} +109031. i q^{3} +(-808962. + 667152. i) q^{4} -3.29975e6 q^{5} +(1.05076e8 - 3.77390e7i) q^{6} -3.27338e8i q^{7} +(9.22959e8 + 5.48696e8i) q^{8} -8.40096e9 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 396 q^{2} - 1316080 q^{4} + 18568080 q^{5} + 99460608 q^{6} + 2177274816 q^{8} - 10411362168 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−346.131	−	963.727i	−0.338019	−	0.941139i
							\(3\)			109031.i			1.84645i	0.384262	+	0.923224i	\(0.374456\pi\)
−0.384262	+	0.923224i	\(0.625544\pi\)
\(5\)	−3.29975e6			−0.337894			−0.168947	−	0.985625i	\(-0.554037\pi\)
							−0.168947	+	0.985625i	\(0.554037\pi\)
\(7\)		−	3.27338e8i		−	1.15882i	−0.815036	−	0.579410i	\(-0.803283\pi\)
							0.815036	−	0.579410i	\(-0.196717\pi\)
\(11\)		−	1.94822e10i		−	0.751124i	−0.926797	−	0.375562i	\(-0.877450\pi\)
							0.926797	−	0.375562i	\(-0.122550\pi\)
\(13\)	4.61706e10			0.334913			0.167456	−	0.985879i	\(-0.446445\pi\)
							0.167456	+	0.985879i	\(0.446445\pi\)
\(17\)	−1.68568e12			−0.836156			−0.418078	−	0.908411i	\(-0.637296\pi\)
							−0.418078	+	0.908411i	\(0.637296\pi\)
\(19\)		−	2.49285e12i		−	0.406593i	−0.979117	−	0.203296i	\(-0.934835\pi\)
							0.979117	−	0.203296i	\(-0.0651655\pi\)
\(23\)		−	3.59575e13i		−	0.867983i	−0.900917	−	0.433992i	\(-0.857105\pi\)
							0.900917	−	0.433992i	\(-0.142895\pi\)
\(29\)	1.22416e14			0.290976			0.145488	−	0.989360i	\(-0.453525\pi\)
							0.145488	+	0.989360i	\(0.453525\pi\)
\(31\)			5.44497e14i			0.664321i	0.943223	+	0.332161i	\(0.107778\pi\)
							−0.943223	+	0.332161i	\(0.892222\pi\)
\(37\)	−7.92560e15			−1.64822			−0.824110	−	0.566430i	\(-0.808324\pi\)
							−0.824110	+	0.566430i	\(0.808324\pi\)
\(41\)	−4.28370e15			−0.319140			−0.159570	−	0.987187i	\(-0.551011\pi\)
							−0.159570	+	0.987187i	\(0.551011\pi\)
\(43\)			1.54669e16i			0.715680i	0.933783	+	0.357840i	\(0.116487\pi\)
							−0.933783	+	0.357840i	\(0.883513\pi\)
\(47\)			5.92849e15i			0.112711i	0.998411	+	0.0563554i	\(0.0179480\pi\)
							−0.998411	+	0.0563554i	\(0.982052\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4.21.b.b.3.3	✓	8
3.2	odd	2		36.21.d.b.19.6		8
4.3	odd	2	inner	4.21.b.b.3.4	yes	8
8.3	odd	2		64.21.c.d.63.8		8
8.5	even	2		64.21.c.d.63.1		8
12.11	even	2		36.21.d.b.19.5		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4.21.b.b.3.3	✓	8	1.1	even	1	trivial
4.21.b.b.3.4	yes	8	4.3	odd	2	inner
36.21.d.b.19.5		8	12.11	even	2
36.21.d.b.19.6		8	3.2	odd	2
64.21.c.d.63.1		8	8.5	even	2
64.21.c.d.63.8		8	8.3	odd	2