Embedded newform 1024.2.g.a.385.3

• Label: 1024.2.g.a.385.3
• Level: $1024$
• Weight: $2$
• Character: 1024.385
• Analytic conductor: $8.177$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1024 = 2^{10} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1024.g (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.17668116698\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(4\) over \(\Q(\zeta_{8})\)
Coefficient field:	\(\Q(\zeta_{48})\)
Defining polynomial:	\( x^{16} - x^{8} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 2^{8} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			385.3
Root			\(0.608761 - 0.793353i\) of defining polynomial
Character	\(\chi\)	\(=\)	1024.385
Dual form			1024.2.g.a.641.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.860919 - 0.356604i) q^{3} +(0.366025 - 0.883663i) q^{5} +(2.35207 + 2.35207i) q^{7} +(-1.50731 + 1.50731i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{5} - 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1024\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(1023\)
\(\chi(n)\)	\(e\left(\frac{5}{8}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		0			0
							\(3\)	0.860919	−	0.356604i	0.497052	−	0.205886i	−0.120052	−	0.992768i	\(-0.538306\pi\)
0.617103	+	0.786882i	\(0.288306\pi\)
\(5\)	0.366025	−	0.883663i	0.163692	−	0.395186i	−0.820656	−	0.571422i	\(-0.806392\pi\)
							0.984348	+	0.176236i	\(0.0563920\pi\)
\(7\)	2.35207	+	2.35207i	0.889000	+	0.889000i	0.994427	−	0.105427i	\(-0.0336209\pi\)
							−0.105427	+	0.994427i	\(0.533621\pi\)
\(11\)	−1.81739	−	0.752787i	−0.547963	−	0.226974i	0.0914869	−	0.995806i	\(-0.470838\pi\)
							−0.639450	+	0.768832i	\(0.720838\pi\)
\(13\)	0.848387	+	2.04819i	0.235300	+	0.568065i	0.996785	−	0.0801172i	\(-0.0255294\pi\)
							−0.761485	+	0.648182i	\(0.775529\pi\)
\(17\)			6.00997i			1.45763i	0.684710	+	0.728816i	\(0.259929\pi\)
							−0.684710	+	0.728816i	\(0.740071\pi\)
\(19\)	1.54713	+	3.73510i	0.354935	+	0.856890i	0.995996	+	0.0893996i	\(0.0284948\pi\)
							−0.641060	+	0.767490i	\(0.721505\pi\)
\(23\)	2.91236	−	2.91236i	0.607269	−	0.607269i	−0.334962	−	0.942232i	\(-0.608724\pi\)
							0.942232	+	0.334962i	\(0.108724\pi\)
\(29\)	9.69383	−	4.01532i	1.80010	−	0.745625i	0.813694	−	0.581293i	\(-0.197453\pi\)
							0.986405	−	0.164332i	\(-0.0525469\pi\)
\(31\)	−7.52140			−1.35088			−0.675442	−	0.737413i	\(-0.736047\pi\)
							−0.675442	+	0.737413i	\(0.736047\pi\)
\(37\)	0.994652	−	2.40130i	0.163520	−	0.394772i	−0.820788	−	0.571233i	\(-0.806465\pi\)
							0.984308	+	0.176462i	\(0.0564652\pi\)
\(41\)	−1.37894	+	1.37894i	−0.215354	+	0.215354i	−0.806537	−	0.591183i	\(-0.798661\pi\)
							0.591183	+	0.806537i	\(0.298661\pi\)
\(43\)	−10.8651	−	4.50046i	−1.65691	−	0.686314i	−0.659074	−	0.752078i	\(-0.729052\pi\)
							−0.997835	+	0.0657637i	\(0.979052\pi\)
\(47\)		−	3.33173i		−	0.485983i	−0.970028	−	0.242991i	\(-0.921871\pi\)
							0.970028	−	0.242991i	\(-0.0781286\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1024.2.g.a.385.3	yes	16
4.3	odd	2	inner	1024.2.g.a.385.2	✓	16
8.3	odd	2		1024.2.g.f.385.3	yes	16
8.5	even	2		1024.2.g.f.385.2	yes	16
16.3	odd	4		1024.2.g.g.897.2	yes	16
16.5	even	4		1024.2.g.d.897.2	yes	16
16.11	odd	4		1024.2.g.d.897.3	yes	16
16.13	even	4		1024.2.g.g.897.3	yes	16
32.3	odd	8	inner	1024.2.g.a.641.2	yes	16
32.5	even	8		1024.2.g.d.129.2	yes	16
32.11	odd	8		1024.2.g.g.129.2	yes	16
32.13	even	8		1024.2.g.f.641.2	yes	16
32.19	odd	8		1024.2.g.f.641.3	yes	16
32.21	even	8		1024.2.g.g.129.3	yes	16
32.27	odd	8		1024.2.g.d.129.3	yes	16
32.29	even	8	inner	1024.2.g.a.641.3	yes	16
64.3	odd	16		4096.2.a.i.1.8		8
64.29	even	16		4096.2.a.i.1.7		8
64.35	odd	16		4096.2.a.s.1.1		8
64.61	even	16		4096.2.a.s.1.2		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1024.2.g.a.385.2	✓	16	4.3	odd	2	inner
1024.2.g.a.385.3	yes	16	1.1	even	1	trivial
1024.2.g.a.641.2	yes	16	32.3	odd	8	inner
1024.2.g.a.641.3	yes	16	32.29	even	8	inner
1024.2.g.d.129.2	yes	16	32.5	even	8
1024.2.g.d.129.3	yes	16	32.27	odd	8
1024.2.g.d.897.2	yes	16	16.5	even	4
1024.2.g.d.897.3	yes	16	16.11	odd	4
1024.2.g.f.385.2	yes	16	8.5	even	2
1024.2.g.f.385.3	yes	16	8.3	odd	2
1024.2.g.f.641.2	yes	16	32.13	even	8
1024.2.g.f.641.3	yes	16	32.19	odd	8
1024.2.g.g.129.2	yes	16	32.11	odd	8
1024.2.g.g.129.3	yes	16	32.21	even	8
1024.2.g.g.897.2	yes	16	16.3	odd	4
1024.2.g.g.897.3	yes	16	16.13	even	4
4096.2.a.i.1.7		8	64.29	even	16
4096.2.a.i.1.8		8	64.3	odd	16
4096.2.a.s.1.1		8	64.35	odd	16
4096.2.a.s.1.2		8	64.61	even	16