Embedded newform 1024.2.g.d.385.3

• Label: 1024.2.g.d.385.3
• Level: $1024$
• Weight: $2$
• Character: 1024.385
• Analytic conductor: $8.177$
• Analytic rank: $0$
• Dimension: $16$
• 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			\(0.130526 - 0.991445i\) of defining polynomial
Character	\(\chi\)	\(=\)	1024.385
Dual form			1024.2.g.d.641.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.445644 - 0.184592i) q^{3} +(0.565826 - 1.36603i) q^{5} +(0.135131 + 0.135131i) q^{7} +(-1.95680 + 1.95680i) q^{9} +(3.12395 + 1.29398i) q^{11} +(0.951812 + 2.29788i) q^{13} -0.713208i q^{15} -3.11099i q^{17} +(2.48451 + 5.99813i) q^{19} +(0.0851642 + 0.0352762i) q^{21} +(5.18330 - 5.18330i) q^{23} +(1.98967 + 1.98967i) q^{25} +(-1.06460 + 2.57018i) q^{27} +(-4.33315 + 1.79485i) q^{29} +7.44503 q^{31} +1.63103 q^{33} +(0.261052 - 0.108131i) q^{35} +(3.49768 - 8.44414i) q^{37} +(0.848339 + 0.848339i) q^{39} +(-4.27792 + 4.27792i) q^{41} +(4.33227 + 1.79448i) q^{43} +(1.56583 + 3.78024i) q^{45} -12.0952i q^{47} -6.96348i q^{49} +(-0.574263 - 1.38639i) q^{51} +(3.42713 + 1.41956i) q^{53} +(3.53523 - 3.53523i) q^{55} +(2.21441 + 2.21441i) q^{57} +(-1.16425 + 2.81074i) q^{59} +(-8.72911 + 3.61571i) q^{61} -0.528846 q^{63} +3.67752 q^{65} +(7.15047 - 2.96182i) q^{67} +(1.35311 - 3.26670i) q^{69} +(-2.86020 - 2.86020i) q^{71} +(-2.49697 + 2.49697i) q^{73} +(1.25396 + 0.519408i) q^{75} +(0.247285 + 0.596999i) q^{77} +8.39967i q^{79} -6.96008i q^{81} +(5.42005 + 13.0852i) q^{83} +(-4.24969 - 1.76028i) q^{85} +(-1.59973 + 1.59973i) q^{87} +(-4.96713 - 4.96713i) q^{89} +(-0.181895 + 0.439133i) q^{91} +(3.31784 - 1.37429i) q^{93} +9.59940 q^{95} -2.87492 q^{97} +(-8.64500 + 3.58087i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	16 * q - 8 * q^5 + 16 * q^9 + 24 * q^13 - 48 * q^21 + 32 * q^25 - 8 * q^29 - 80 * q^33 + 8 * q^37 + 16 * q^41 + 8 * q^45 + 40 * q^53 + 16 * q^57 + 8 * q^61 - 32 * q^65 - 32 * q^73 + 32 * q^77 - 32 * q^85 - 32 * q^89 + 48 * q^93 - 16 * q^97

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 \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.

Currently showing only \(a_p\); display all \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		0			0
							\(3\)	0.445644	−	0.184592i	0.257293	−	0.106574i	−0.250309	−	0.968166i	\(-0.580532\pi\)
0.507602	+	0.861592i	\(0.330532\pi\)
\(5\)	0.565826	−	1.36603i	0.253045	−	0.610905i	−0.745402	−	0.666615i	\(-0.767742\pi\)
							0.998447	+	0.0557103i	\(0.0177423\pi\)
\(7\)	0.135131	+	0.135131i	0.0510746	+	0.0510746i	0.732183	−	0.681108i	\(-0.238502\pi\)
							−0.681108	+	0.732183i	\(0.738502\pi\)
\(11\)	3.12395	+	1.29398i	0.941907	+	0.390151i	0.800183	−	0.599756i	\(-0.204736\pi\)
							0.141724	+	0.989906i	\(0.454736\pi\)
\(13\)	0.951812	+	2.29788i	0.263985	+	0.637316i	0.999178	−	0.0405417i	\(-0.0129084\pi\)
							−0.735193	+	0.677858i	\(0.762908\pi\)
\(17\)		−	3.11099i		−	0.754525i	−0.926106	−	0.377263i	\(-0.876865\pi\)
							0.926106	−	0.377263i	\(-0.123135\pi\)
\(19\)	2.48451	+	5.99813i	0.569985	+	1.37607i	0.901567	+	0.432639i	\(0.142418\pi\)
							−0.331582	+	0.943426i	\(0.607582\pi\)
\(23\)	5.18330	−	5.18330i	1.08079	−	1.08079i	0.0843577	−	0.996436i	\(-0.473116\pi\)
							0.996436	−	0.0843577i	\(-0.0268838\pi\)
\(29\)	−4.33315	+	1.79485i	−0.804646	+	0.333295i	−0.746816	−	0.665031i	\(-0.768418\pi\)
							−0.0578306	+	0.998326i	\(0.518418\pi\)
\(31\)	7.44503			1.33717			0.668584	−	0.743637i	\(-0.266901\pi\)
							0.668584	+	0.743637i	\(0.266901\pi\)
\(37\)	3.49768	−	8.44414i	0.575015	−	1.38821i	−0.322224	−	0.946663i	\(-0.604431\pi\)
							0.897239	−	0.441545i	\(-0.145569\pi\)
\(41\)	−4.27792	+	4.27792i	−0.668098	+	0.668098i	−0.957276	−	0.289177i	\(-0.906618\pi\)
							0.289177	+	0.957276i	\(0.406618\pi\)
\(43\)	4.33227	+	1.79448i	0.660664	+	0.273656i	0.687718	−	0.725978i	\(-0.258613\pi\)
							−0.0270537	+	0.999634i	\(0.508613\pi\)
\(47\)		−	12.0952i		−	1.76426i	−0.471002	−	0.882132i	\(-0.656108\pi\)
							0.471002	−	0.882132i	\(-0.343892\pi\)
\(53\)	3.42713	+	1.41956i	0.470752	+	0.194992i	0.605432	−	0.795897i	\(-0.293001\pi\)
							−0.134680	+	0.990889i	\(0.543001\pi\)
\(59\)	−1.16425	+	2.81074i	−0.151572	+	0.365927i	−0.981367	−	0.192140i	\(-0.938457\pi\)
							0.829795	+	0.558068i	\(0.188457\pi\)
\(61\)	−8.72911	+	3.61571i	−1.11765	+	0.462945i	−0.863565	−	0.504238i	\(-0.831774\pi\)
							−0.254083	+	0.967183i	\(0.581774\pi\)
\(67\)	7.15047	−	2.96182i	0.873569	−	0.361844i	0.0995698	−	0.995031i	\(-0.468253\pi\)
							0.773999	+	0.633186i	\(0.218253\pi\)
\(71\)	−2.86020	−	2.86020i	−0.339444	−	0.339444i	0.516714	−	0.856158i	\(-0.327155\pi\)
							−0.856158	+	0.516714i	\(0.827155\pi\)
\(73\)	−2.49697	+	2.49697i	−0.292249	+	0.292249i	−0.837968	−	0.545719i	\(-0.816256\pi\)
							0.545719	+	0.837968i	\(0.316256\pi\)
\(79\)			8.39967i			0.945036i	0.881321	+	0.472518i	\(0.156655\pi\)
							−0.881321	+	0.472518i	\(0.843345\pi\)
\(83\)	5.42005	+	13.0852i	0.594928	+	1.43628i	0.878692	+	0.477389i	\(0.158417\pi\)
							−0.283764	+	0.958894i	\(0.591583\pi\)
\(89\)	−4.96713	−	4.96713i	−0.526514	−	0.526514i	0.393017	−	0.919531i	\(-0.371431\pi\)
							−0.919531	+	0.393017i	\(0.871431\pi\)
\(97\)	−2.87492			−0.291903			−0.145952	−	0.989292i	\(-0.546624\pi\)
							−0.145952	+	0.989292i	\(0.546624\pi\)

Currently showing only \(a_p\); display all \(a_n\)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1024.2.g.d.385.3	yes	16
4.3	odd	2	inner	1024.2.g.d.385.2	yes	16
8.3	odd	2		1024.2.g.g.385.3	yes	16
8.5	even	2		1024.2.g.g.385.2	yes	16
16.3	odd	4		1024.2.g.a.897.2	yes	16
16.5	even	4		1024.2.g.f.897.2	yes	16
16.11	odd	4		1024.2.g.f.897.3	yes	16
16.13	even	4		1024.2.g.a.897.3	yes	16
32.3	odd	8	inner	1024.2.g.d.641.2	yes	16
32.5	even	8		1024.2.g.f.129.2	yes	16
32.11	odd	8		1024.2.g.a.129.2	✓	16
32.13	even	8		1024.2.g.g.641.2	yes	16
32.19	odd	8		1024.2.g.g.641.3	yes	16
32.21	even	8		1024.2.g.a.129.3	yes	16
32.27	odd	8		1024.2.g.f.129.3	yes	16
32.29	even	8	inner	1024.2.g.d.641.3	yes	16
64.3	odd	16		4096.2.a.s.1.4		8
64.29	even	16		4096.2.a.s.1.3		8
64.35	odd	16		4096.2.a.i.1.5		8
64.61	even	16		4096.2.a.i.1.6		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1024.2.g.a.129.2	✓	16	32.11	odd	8
1024.2.g.a.129.3	yes	16	32.21	even	8
1024.2.g.a.897.2	yes	16	16.3	odd	4
1024.2.g.a.897.3	yes	16	16.13	even	4
1024.2.g.d.385.2	yes	16	4.3	odd	2	inner
1024.2.g.d.385.3	yes	16	1.1	even	1	trivial
1024.2.g.d.641.2	yes	16	32.3	odd	8	inner
1024.2.g.d.641.3	yes	16	32.29	even	8	inner
1024.2.g.f.129.2	yes	16	32.5	even	8
1024.2.g.f.129.3	yes	16	32.27	odd	8
1024.2.g.f.897.2	yes	16	16.5	even	4
1024.2.g.f.897.3	yes	16	16.11	odd	4
1024.2.g.g.385.2	yes	16	8.5	even	2
1024.2.g.g.385.3	yes	16	8.3	odd	2
1024.2.g.g.641.2	yes	16	32.13	even	8
1024.2.g.g.641.3	yes	16	32.19	odd	8
4096.2.a.i.1.5		8	64.35	odd	16
4096.2.a.i.1.6		8	64.61	even	16
4096.2.a.s.1.3		8	64.29	even	16
4096.2.a.s.1.4		8	64.3	odd	16