Embedded newform 1024.2.e.j.769.2

• Label: 1024.2.e.j.769.2
• Level: $1024$
• Weight: $2$
• Character: 1024.769
• Analytic conductor: $8.177$
• Analytic rank: $0$
• Dimension: $4$
• CM discriminant: -4
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1024 = 2^{10} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1024.e (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.17668116698\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(i)\)
Coefficient field:	\(\Q(\zeta_{8})\)
Defining polynomial:	\( x^{4} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	no (minimal twist has level 32)
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			769.2
Root			\(-0.707107 - 0.707107i\) of defining polynomial
Character	\(\chi\)	\(=\)	1024.769
Dual form			1024.2.e.j.257.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.41421 + 1.41421i) q^{5} +3.00000i q^{9} +(-4.24264 + 4.24264i) q^{13} -2.00000 q^{17} -1.00000i q^{25} +(-7.07107 + 7.07107i) q^{29} +(1.41421 + 1.41421i) q^{37} -10.0000i q^{41} +(-4.24264 + 4.24264i) q^{45} +7.00000 q^{49} +(9.89949 + 9.89949i) q^{53} +(-7.07107 + 7.07107i) q^{61} -12.0000 q^{65} +6.00000i q^{73} -9.00000 q^{81} +(-2.82843 - 2.82843i) q^{85} +10.0000i q^{89} +18.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{17} + 28 q^{49} - 48 q^{65} - 36 q^{81} + 72 q^{97}+O(q^{100}) \)

Character values

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

\(n\)	\(5\)	\(1023\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(1\)

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			0		−0.707107	−	0.707107i	\(-0.750000\pi\)
0.707107	+	0.707107i	\(0.250000\pi\)
\(5\)	1.41421	+	1.41421i	0.632456	+	0.632456i	0.948683	−	0.316228i	\(-0.102416\pi\)
							−0.316228	+	0.948683i	\(0.602416\pi\)
\(7\)		0			0		1.00000			\(0\)
							−1.00000			\(\pi\)
\(11\)		0			0		0.707107	−	0.707107i	\(-0.250000\pi\)
							−0.707107	+	0.707107i	\(0.750000\pi\)
\(13\)	−4.24264	+	4.24264i	−1.17670	+	1.17670i	−0.196116	+	0.980581i	\(0.562833\pi\)
							−0.980581	+	0.196116i	\(0.937167\pi\)
\(17\)	−2.00000			−0.485071			−0.242536	−	0.970143i	\(-0.577979\pi\)
							−0.242536	+	0.970143i	\(0.577979\pi\)
\(19\)		0			0		−0.707107	−	0.707107i	\(-0.750000\pi\)
							0.707107	+	0.707107i	\(0.250000\pi\)
\(23\)		0			0		1.00000			\(0\)
							−1.00000			\(\pi\)
\(29\)	−7.07107	+	7.07107i	−1.31306	+	1.31306i	−0.393919	+	0.919145i	\(0.628881\pi\)
							−0.919145	+	0.393919i	\(0.871119\pi\)
\(31\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(37\)	1.41421	+	1.41421i	0.232495	+	0.232495i	0.813733	−	0.581238i	\(-0.197432\pi\)
							−0.581238	+	0.813733i	\(0.697432\pi\)
\(41\)		−	10.0000i		−	1.56174i	−0.624695	−	0.780869i	\(-0.714777\pi\)
							0.624695	−	0.780869i	\(-0.285223\pi\)
\(43\)		0			0		0.707107	−	0.707107i	\(-0.250000\pi\)
							−0.707107	+	0.707107i	\(0.750000\pi\)
\(47\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1024.2.e.j.769.2		4
4.3	odd	2	CM	1024.2.e.j.769.2		4
8.3	odd	2	inner	1024.2.e.j.769.1		4
8.5	even	2	inner	1024.2.e.j.769.1		4
16.3	odd	4	inner	1024.2.e.j.257.2		4
16.5	even	4	inner	1024.2.e.j.257.1		4
16.11	odd	4	inner	1024.2.e.j.257.1		4
16.13	even	4	inner	1024.2.e.j.257.2		4
32.3	odd	8		32.2.a.a.1.1	✓	1
32.5	even	8		256.2.b.b.129.2		2
32.11	odd	8		256.2.b.b.129.1		2
32.13	even	8		64.2.a.a.1.1		1
32.19	odd	8		64.2.a.a.1.1		1
32.21	even	8		256.2.b.b.129.1		2
32.27	odd	8		256.2.b.b.129.2		2
32.29	even	8		32.2.a.a.1.1	✓	1
96.5	odd	8		2304.2.d.j.1153.1		2
96.11	even	8		2304.2.d.j.1153.2		2
96.29	odd	8		288.2.a.d.1.1		1
96.35	even	8		288.2.a.d.1.1		1
96.53	odd	8		2304.2.d.j.1153.2		2
96.59	even	8		2304.2.d.j.1153.1		2
96.77	odd	8		576.2.a.c.1.1		1
96.83	even	8		576.2.a.c.1.1		1
160.3	even	8		800.2.c.e.449.2		2
160.13	odd	8		1600.2.c.l.449.1		2
160.19	odd	8		1600.2.a.n.1.1		1
160.29	even	8		800.2.a.d.1.1		1
160.67	even	8		800.2.c.e.449.1		2
160.77	odd	8		1600.2.c.l.449.2		2
160.83	even	8		1600.2.c.l.449.1		2
160.93	odd	8		800.2.c.e.449.2		2
160.99	odd	8		800.2.a.d.1.1		1
160.109	even	8		1600.2.a.n.1.1		1
160.147	even	8		1600.2.c.l.449.2		2
160.157	odd	8		800.2.c.e.449.1		2
224.3	even	24		1568.2.i.f.961.1		2
224.13	odd	8		3136.2.a.m.1.1		1
224.61	odd	24		1568.2.i.f.1537.1		2
224.67	odd	24		1568.2.i.g.961.1		2
224.83	even	8		3136.2.a.m.1.1		1
224.93	even	24		1568.2.i.g.1537.1		2
224.125	odd	8		1568.2.a.e.1.1		1
224.131	even	24		1568.2.i.f.1537.1		2
224.157	odd	24		1568.2.i.f.961.1		2
224.163	odd	24		1568.2.i.g.1537.1		2
224.195	even	8		1568.2.a.e.1.1		1
224.221	even	24		1568.2.i.g.961.1		2
288.29	odd	24		2592.2.i.e.1729.1		2
288.61	even	24		2592.2.i.t.1729.1		2
288.67	odd	24		2592.2.i.t.865.1		2
288.131	even	24		2592.2.i.e.865.1		2
288.157	even	24		2592.2.i.t.865.1		2
288.221	odd	24		2592.2.i.e.865.1		2
288.227	even	24		2592.2.i.e.1729.1		2
288.259	odd	24		2592.2.i.t.1729.1		2
352.109	odd	8		7744.2.a.v.1.1		1
352.131	even	8		3872.2.a.f.1.1		1
352.285	odd	8		3872.2.a.f.1.1		1
352.307	even	8		7744.2.a.v.1.1		1
416.259	odd	8		5408.2.a.g.1.1		1
416.285	even	8		5408.2.a.g.1.1		1
480.29	odd	8		7200.2.a.v.1.1		1
480.227	odd	8		7200.2.f.m.6049.1		2
480.317	even	8		7200.2.f.m.6049.1		2
480.323	odd	8		7200.2.f.m.6049.2		2
480.413	even	8		7200.2.f.m.6049.2		2
480.419	even	8		7200.2.a.v.1.1		1
544.67	odd	8		9248.2.a.f.1.1		1
544.509	even	8		9248.2.a.f.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.2.a.a.1.1	✓	1	32.3	odd	8
32.2.a.a.1.1	✓	1	32.29	even	8
64.2.a.a.1.1		1	32.13	even	8
64.2.a.a.1.1		1	32.19	odd	8
256.2.b.b.129.1		2	32.11	odd	8
256.2.b.b.129.1		2	32.21	even	8
256.2.b.b.129.2		2	32.5	even	8
256.2.b.b.129.2		2	32.27	odd	8
288.2.a.d.1.1		1	96.29	odd	8
288.2.a.d.1.1		1	96.35	even	8
576.2.a.c.1.1		1	96.77	odd	8
576.2.a.c.1.1		1	96.83	even	8
800.2.a.d.1.1		1	160.29	even	8
800.2.a.d.1.1		1	160.99	odd	8
800.2.c.e.449.1		2	160.67	even	8
800.2.c.e.449.1		2	160.157	odd	8
800.2.c.e.449.2		2	160.3	even	8
800.2.c.e.449.2		2	160.93	odd	8
1024.2.e.j.257.1		4	16.5	even	4	inner
1024.2.e.j.257.1		4	16.11	odd	4	inner
1024.2.e.j.257.2		4	16.3	odd	4	inner
1024.2.e.j.257.2		4	16.13	even	4	inner
1024.2.e.j.769.1		4	8.3	odd	2	inner
1024.2.e.j.769.1		4	8.5	even	2	inner
1024.2.e.j.769.2		4	1.1	even	1	trivial
1024.2.e.j.769.2		4	4.3	odd	2	CM
1568.2.a.e.1.1		1	224.125	odd	8
1568.2.a.e.1.1		1	224.195	even	8
1568.2.i.f.961.1		2	224.3	even	24
1568.2.i.f.961.1		2	224.157	odd	24
1568.2.i.f.1537.1		2	224.61	odd	24
1568.2.i.f.1537.1		2	224.131	even	24
1568.2.i.g.961.1		2	224.67	odd	24
1568.2.i.g.961.1		2	224.221	even	24
1568.2.i.g.1537.1		2	224.93	even	24
1568.2.i.g.1537.1		2	224.163	odd	24
1600.2.a.n.1.1		1	160.19	odd	8
1600.2.a.n.1.1		1	160.109	even	8
1600.2.c.l.449.1		2	160.13	odd	8
1600.2.c.l.449.1		2	160.83	even	8
1600.2.c.l.449.2		2	160.77	odd	8
1600.2.c.l.449.2		2	160.147	even	8
2304.2.d.j.1153.1		2	96.5	odd	8
2304.2.d.j.1153.1		2	96.59	even	8
2304.2.d.j.1153.2		2	96.11	even	8
2304.2.d.j.1153.2		2	96.53	odd	8
2592.2.i.e.865.1		2	288.131	even	24
2592.2.i.e.865.1		2	288.221	odd	24
2592.2.i.e.1729.1		2	288.29	odd	24
2592.2.i.e.1729.1		2	288.227	even	24
2592.2.i.t.865.1		2	288.67	odd	24
2592.2.i.t.865.1		2	288.157	even	24
2592.2.i.t.1729.1		2	288.61	even	24
2592.2.i.t.1729.1		2	288.259	odd	24
3136.2.a.m.1.1		1	224.13	odd	8
3136.2.a.m.1.1		1	224.83	even	8
3872.2.a.f.1.1		1	352.131	even	8
3872.2.a.f.1.1		1	352.285	odd	8
5408.2.a.g.1.1		1	416.259	odd	8
5408.2.a.g.1.1		1	416.285	even	8
7200.2.a.v.1.1		1	480.29	odd	8
7200.2.a.v.1.1		1	480.419	even	8
7200.2.f.m.6049.1		2	480.227	odd	8
7200.2.f.m.6049.1		2	480.317	even	8
7200.2.f.m.6049.2		2	480.323	odd	8
7200.2.f.m.6049.2		2	480.413	even	8
7744.2.a.v.1.1		1	352.109	odd	8
7744.2.a.v.1.1		1	352.307	even	8
9248.2.a.f.1.1		1	544.67	odd	8
9248.2.a.f.1.1		1	544.509	even	8