Embedded newform 1024.2.b.d.513.1

• Label: 1024.2.b.d.513.1
• Level: $1024$
• Weight: $2$
• Character: 1024.513
• Analytic conductor: $8.177$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -4
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1024 = 2^{10} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1024.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.17668116698\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-2}) \)
Defining polynomial:	\( x^{2} + 2 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 512)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			513.1
Root			\(1.41421i\) of defining polynomial
Character	\(\chi\)	\(=\)	1024.513
Dual form			1024.2.b.d.513.2

$q$-expansion

\(f(q)\)	\(=\)	\(q-4.24264i q^{5} +3.00000 q^{9} +1.41421i q^{13} +8.00000 q^{17} -13.0000 q^{25} -9.89949i q^{29} -7.07107i q^{37} -8.00000 q^{41} -12.7279i q^{45} -7.00000 q^{49} +7.07107i q^{53} -1.41421i q^{61} +6.00000 q^{65} -6.00000 q^{73} +9.00000 q^{81} -33.9411i q^{85} +10.0000 q^{89} +8.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{9} + 16 q^{17} - 26 q^{25} - 16 q^{41} - 14 q^{49} + 12 q^{65} - 12 q^{73} + 18 q^{81} + 20 q^{89} + 16 q^{97}+O(q^{100}) \)

Character values

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

\(n\)	\(5\)	\(1023\)
\(\chi(n)\)	\(-1\)	\(1\)

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	0	1.00000			\(0\)
−1.00000						\(\pi\)
\(5\)	− 4.24264i	− 1.89737i	−0.316228	−	0.948683i	\(-0.602416\pi\)
			0.316228	−	0.948683i	\(-0.397584\pi\)
\(7\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(11\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(13\)	1.41421i	0.392232i	0.980581	+	0.196116i	\(0.0628330\pi\)
			−0.980581	+	0.196116i	\(0.937167\pi\)
\(17\)	8.00000	1.94029	0.970143	−	0.242536i	\(-0.0779791\pi\)
			0.970143	+	0.242536i	\(0.0779791\pi\)
\(19\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(23\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(29\)	− 9.89949i	− 1.83829i	−0.393919	−	0.919145i	\(-0.628881\pi\)
			0.393919	−	0.919145i	\(-0.371119\pi\)
\(31\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(37\)	− 7.07107i	− 1.16248i	−0.813733	−	0.581238i	\(-0.802568\pi\)
			0.813733	−	0.581238i	\(-0.197432\pi\)
\(41\)	−8.00000	−1.24939	−0.624695	−	0.780869i	\(-0.714777\pi\)
			−0.624695	+	0.780869i	\(0.714777\pi\)
\(43\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(47\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(53\)	7.07107i	0.971286i	0.874157	+	0.485643i	\(0.161414\pi\)
			−0.874157	+	0.485643i	\(0.838586\pi\)
\(59\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(61\)	− 1.41421i	− 0.181071i	−0.995893	−	0.0905357i	\(-0.971142\pi\)
			0.995893	−	0.0905357i	\(-0.0288579\pi\)
\(67\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(71\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(73\)	−6.00000	−0.702247	−0.351123	−	0.936329i	\(-0.614200\pi\)
			−0.351123	+	0.936329i	\(0.614200\pi\)
\(79\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(83\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(89\)	10.0000	1.06000	0.529999	−	0.847998i	\(-0.322192\pi\)
			0.529999	+	0.847998i	\(0.322192\pi\)
\(97\)	8.00000	0.812277	0.406138	−	0.913812i	\(-0.366875\pi\)
			0.406138	+	0.913812i	\(0.366875\pi\)

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.b.d.513.1		2
4.3	odd	2	CM	1024.2.b.d.513.1		2
8.3	odd	2	inner	1024.2.b.d.513.2		2
8.5	even	2	inner	1024.2.b.d.513.2		2
16.3	odd	4		1024.2.a.d.1.1		2
16.5	even	4		1024.2.a.d.1.2		2
16.11	odd	4		1024.2.a.d.1.2		2
16.13	even	4		1024.2.a.d.1.1		2
32.3	odd	8		512.2.e.f.385.1	yes	2
32.5	even	8		512.2.e.f.129.1	yes	2
32.11	odd	8		512.2.e.c.129.1	✓	2
32.13	even	8		512.2.e.c.385.1	yes	2
32.19	odd	8		512.2.e.c.385.1	yes	2
32.21	even	8		512.2.e.c.129.1	✓	2
32.27	odd	8		512.2.e.f.129.1	yes	2
32.29	even	8		512.2.e.f.385.1	yes	2
48.5	odd	4		9216.2.a.g.1.1		2
48.11	even	4		9216.2.a.g.1.1		2
48.29	odd	4		9216.2.a.g.1.2		2
48.35	even	4		9216.2.a.g.1.2		2
96.5	odd	8		4608.2.k.b.1153.1		2
96.11	even	8		4608.2.k.w.1153.1		2
96.29	odd	8		4608.2.k.b.3457.1		2
96.35	even	8		4608.2.k.b.3457.1		2
96.53	odd	8		4608.2.k.w.1153.1		2
96.59	even	8		4608.2.k.b.1153.1		2
96.77	odd	8		4608.2.k.w.3457.1		2
96.83	even	8		4608.2.k.w.3457.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
512.2.e.c.129.1	✓	2	32.11	odd	8
512.2.e.c.129.1	✓	2	32.21	even	8
512.2.e.c.385.1	yes	2	32.13	even	8
512.2.e.c.385.1	yes	2	32.19	odd	8
512.2.e.f.129.1	yes	2	32.5	even	8
512.2.e.f.129.1	yes	2	32.27	odd	8
512.2.e.f.385.1	yes	2	32.3	odd	8
512.2.e.f.385.1	yes	2	32.29	even	8
1024.2.a.d.1.1		2	16.3	odd	4
1024.2.a.d.1.1		2	16.13	even	4
1024.2.a.d.1.2		2	16.5	even	4
1024.2.a.d.1.2		2	16.11	odd	4
1024.2.b.d.513.1		2	1.1	even	1	trivial
1024.2.b.d.513.1		2	4.3	odd	2	CM
1024.2.b.d.513.2		2	8.3	odd	2	inner
1024.2.b.d.513.2		2	8.5	even	2	inner
4608.2.k.b.1153.1		2	96.5	odd	8
4608.2.k.b.1153.1		2	96.59	even	8
4608.2.k.b.3457.1		2	96.29	odd	8
4608.2.k.b.3457.1		2	96.35	even	8
4608.2.k.w.1153.1		2	96.11	even	8
4608.2.k.w.1153.1		2	96.53	odd	8
4608.2.k.w.3457.1		2	96.77	odd	8
4608.2.k.w.3457.1		2	96.83	even	8
9216.2.a.g.1.1		2	48.5	odd	4
9216.2.a.g.1.1		2	48.11	even	4
9216.2.a.g.1.2		2	48.29	odd	4
9216.2.a.g.1.2		2	48.35	even	4