Embedded newform 4.17.b.b.3.3

• Label: 4.17.b.b.3.3
• Level: $4$
• Weight: $17$
• Character: 4.3
• Analytic conductor: $6.493$
• Analytic rank: $0$
• Dimension: $6$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 4 = 2^{2} \)
Weight:	\( k \)	\(=\)	\( 17 \)
Character orbit:	\([\chi]\)	\(=\)	4.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.49298175427\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial:	\( x^{6} - x^{5} + 5152x^{4} + 242526x^{3} + 17329473x^{2} + 402444531x + 64957563630 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{30}\cdot 3^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3.3
Root			\(11.8147 + 63.8186i\) of defining polynomial
Character	\(\chi\)	\(=\)	4.3
Dual form			4.17.b.b.3.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(19.2587 - 255.275i) q^{2} -11183.9i q^{3} +(-64794.2 - 9832.53i) q^{4} +389860. q^{5} +(-2.85496e6 - 215387. i) q^{6} +1.06777e6i q^{7} +(-3.75785e6 + 1.63509e7i) q^{8} -8.20322e7 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 164 q^{2} - 160368 q^{4} - 506740 q^{5} - 1187136 q^{6} - 33829184 q^{8} - 137574522 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	19.2587	−	255.275i	0.0752294	−	0.997166i
							\(3\)		−	11183.9i		−	1.70460i	−0.523055	−	0.852299i	\(-0.675208\pi\)
0.523055	−	0.852299i	\(-0.324792\pi\)
\(5\)	389860.			0.998042			0.499021	−	0.866590i	\(-0.333693\pi\)
							0.499021	+	0.866590i	\(0.333693\pi\)
\(7\)			1.06777e6i			0.185223i	0.995702	+	0.0926113i	\(0.0295214\pi\)
							−0.995702	+	0.0926113i	\(0.970479\pi\)
\(11\)		−	1.26927e8i		−	0.592126i	−0.955168	−	0.296063i	\(-0.904326\pi\)
							0.955168	−	0.296063i	\(-0.0956739\pi\)
\(13\)	1.12108e9			1.37433			0.687165	−	0.726501i	\(-0.258855\pi\)
							0.687165	+	0.726501i	\(0.258855\pi\)
\(17\)	−7.43906e9			−1.06642			−0.533208	−	0.845984i	\(-0.679014\pi\)
							−0.533208	+	0.845984i	\(0.679014\pi\)
\(19\)		−	1.76792e10i		−	1.04096i	−0.853874	−	0.520479i	\(-0.825753\pi\)
							0.853874	−	0.520479i	\(-0.174247\pi\)
\(23\)		−	8.72327e9i		−	0.111393i	−0.998448	−	0.0556963i	\(-0.982262\pi\)
							0.998448	−	0.0556963i	\(-0.0177379\pi\)
\(29\)	2.42638e11			0.485038			0.242519	−	0.970147i	\(-0.422026\pi\)
							0.242519	+	0.970147i	\(0.422026\pi\)
\(31\)			6.45199e10i			0.0756485i	0.999284	+	0.0378243i	\(0.0120427\pi\)
							−0.999284	+	0.0378243i	\(0.987957\pi\)
\(37\)	2.41678e12			0.688055			0.344028	−	0.938960i	\(-0.388209\pi\)
							0.344028	+	0.938960i	\(0.388209\pi\)
\(41\)	5.45949e12			0.683725			0.341862	−	0.939750i	\(-0.388942\pi\)
							0.341862	+	0.939750i	\(0.388942\pi\)
\(43\)			1.67228e13i			1.43075i	0.698743	+	0.715373i	\(0.253743\pi\)
							−0.698743	+	0.715373i	\(0.746257\pi\)
\(47\)		−	1.33646e13i		−	0.561272i	−0.959814	−	0.280636i	\(-0.909455\pi\)
							0.959814	−	0.280636i	\(-0.0905454\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4.17.b.b.3.3	✓	6
3.2	odd	2		36.17.d.b.19.4		6
4.3	odd	2	inner	4.17.b.b.3.4	yes	6
8.3	odd	2		64.17.c.d.63.1		6
8.5	even	2		64.17.c.d.63.6		6
12.11	even	2		36.17.d.b.19.3		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4.17.b.b.3.3	✓	6	1.1	even	1	trivial
4.17.b.b.3.4	yes	6	4.3	odd	2	inner
36.17.d.b.19.3		6	12.11	even	2
36.17.d.b.19.4		6	3.2	odd	2
64.17.c.d.63.1		6	8.3	odd	2
64.17.c.d.63.6		6	8.5	even	2