Embedded newform 6400.2.a.by.1.2

• Label: 6400.2.a.by.1.2
• Level: $6400$
• Weight: $2$
• Character: 6400.1
• Self dual: yes
• Analytic conductor: $51.104$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -8
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 6400 = 2^{8} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6400.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(51.1042572936\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{8})^+\)
Defining polynomial:	\( x^{2} - 2 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 128)
Fricke sign:	\(-1\)
Sato-Tate group:	$N(\mathrm{U}(1))$

Embedding invariants

Embedding label			1.2
Root			\(1.41421\) of defining polynomial
Character	\(\chi\)	\(=\)	6400.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+2.82843 q^{3} +5.00000 q^{9} +2.82843 q^{11} -6.00000 q^{17} +8.48528 q^{19} +5.65685 q^{27} +8.00000 q^{33} +6.00000 q^{41} +8.48528 q^{43} -7.00000 q^{49} -16.9706 q^{51} +24.0000 q^{57} +14.1421 q^{59} -8.48528 q^{67} -2.00000 q^{73} +1.00000 q^{81} +2.82843 q^{83} +18.0000 q^{89} +10.0000 q^{97} +14.1421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 10 q^{9} - 12 q^{17} + 16 q^{33} + 12 q^{41} - 14 q^{49} + 48 q^{57} - 4 q^{73} + 2 q^{81} + 36 q^{89} + 20 q^{97}+O(q^{100}) \)

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
			\(3\)	2.82843	1.63299	0.816497	−	0.577350i	\(-0.195913\pi\)
0.816497	+	0.577350i				\(0.195913\pi\)
\(5\)	0	0
			\(7\)	0	0		−	1.00000i	\(-0.5\pi\)
		1.00000i				\(0.5\pi\)
\(11\)	2.82843	0.852803	0.426401	−	0.904534i	\(-0.359781\pi\)
			0.426401	+	0.904534i	\(0.359781\pi\)
\(13\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(17\)	−6.00000	−1.45521	−0.727607	−	0.685994i	\(-0.759367\pi\)
			−0.727607	+	0.685994i	\(0.759367\pi\)
\(19\)	8.48528	1.94666	0.973329	−	0.229416i	\(-0.0736815\pi\)
			0.973329	+	0.229416i	\(0.0736815\pi\)
\(23\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(29\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(31\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(37\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(41\)	6.00000	0.937043	0.468521	−	0.883452i	\(-0.344787\pi\)
			0.468521	+	0.883452i	\(0.344787\pi\)
\(43\)	8.48528	1.29399	0.646997	−	0.762493i	\(-0.276025\pi\)
			0.646997	+	0.762493i	\(0.276025\pi\)
\(47\)	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	6400.2.a.by.1.2		2
4.3	odd	2	inner	6400.2.a.by.1.1		2
5.4	even	2		256.2.a.e.1.1		2
8.3	odd	2	CM	6400.2.a.by.1.2		2
8.5	even	2	inner	6400.2.a.by.1.1		2
15.14	odd	2		2304.2.a.t.1.1		2
16.3	odd	4		3200.2.d.c.1601.1		2
16.5	even	4		3200.2.d.c.1601.1		2
16.11	odd	4		3200.2.d.c.1601.2		2
16.13	even	4		3200.2.d.c.1601.2		2
20.19	odd	2		256.2.a.e.1.2		2
40.19	odd	2		256.2.a.e.1.1		2
40.29	even	2		256.2.a.e.1.2		2
60.59	even	2		2304.2.a.t.1.2		2
80.3	even	4		3200.2.f.o.449.4		4
80.13	odd	4		3200.2.f.o.449.1		4
80.19	odd	4		128.2.b.a.65.2	yes	2
80.27	even	4		3200.2.f.o.449.3		4
80.29	even	4		128.2.b.a.65.1	✓	2
80.37	odd	4		3200.2.f.o.449.2		4
80.43	even	4		3200.2.f.o.449.1		4
80.53	odd	4		3200.2.f.o.449.4		4
80.59	odd	4		128.2.b.a.65.1	✓	2
80.67	even	4		3200.2.f.o.449.2		4
80.69	even	4		128.2.b.a.65.2	yes	2
80.77	odd	4		3200.2.f.o.449.3		4
120.29	odd	2		2304.2.a.t.1.2		2
120.59	even	2		2304.2.a.t.1.1		2
160.19	odd	8		1024.2.e.a.257.1		2
160.29	even	8		1024.2.e.a.257.1		2
160.59	odd	8		1024.2.e.a.769.1		2
160.69	even	8		1024.2.e.f.769.1		2
160.99	odd	8		1024.2.e.f.257.1		2
160.109	even	8		1024.2.e.f.257.1		2
160.139	odd	8		1024.2.e.f.769.1		2
160.149	even	8		1024.2.e.a.769.1		2
240.29	odd	4		1152.2.d.c.577.2		2
240.59	even	4		1152.2.d.c.577.2		2
240.149	odd	4		1152.2.d.c.577.1		2
240.179	even	4		1152.2.d.c.577.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.2.b.a.65.1	✓	2	80.29	even	4
128.2.b.a.65.1	✓	2	80.59	odd	4
128.2.b.a.65.2	yes	2	80.19	odd	4
128.2.b.a.65.2	yes	2	80.69	even	4
256.2.a.e.1.1		2	5.4	even	2
256.2.a.e.1.1		2	40.19	odd	2
256.2.a.e.1.2		2	20.19	odd	2
256.2.a.e.1.2		2	40.29	even	2
1024.2.e.a.257.1		2	160.19	odd	8
1024.2.e.a.257.1		2	160.29	even	8
1024.2.e.a.769.1		2	160.59	odd	8
1024.2.e.a.769.1		2	160.149	even	8
1024.2.e.f.257.1		2	160.99	odd	8
1024.2.e.f.257.1		2	160.109	even	8
1024.2.e.f.769.1		2	160.69	even	8
1024.2.e.f.769.1		2	160.139	odd	8
1152.2.d.c.577.1		2	240.149	odd	4
1152.2.d.c.577.1		2	240.179	even	4
1152.2.d.c.577.2		2	240.29	odd	4
1152.2.d.c.577.2		2	240.59	even	4
2304.2.a.t.1.1		2	15.14	odd	2
2304.2.a.t.1.1		2	120.59	even	2
2304.2.a.t.1.2		2	60.59	even	2
2304.2.a.t.1.2		2	120.29	odd	2
3200.2.d.c.1601.1		2	16.3	odd	4
3200.2.d.c.1601.1		2	16.5	even	4
3200.2.d.c.1601.2		2	16.11	odd	4
3200.2.d.c.1601.2		2	16.13	even	4
3200.2.f.o.449.1		4	80.13	odd	4
3200.2.f.o.449.1		4	80.43	even	4
3200.2.f.o.449.2		4	80.37	odd	4
3200.2.f.o.449.2		4	80.67	even	4
3200.2.f.o.449.3		4	80.27	even	4
3200.2.f.o.449.3		4	80.77	odd	4
3200.2.f.o.449.4		4	80.3	even	4
3200.2.f.o.449.4		4	80.53	odd	4
6400.2.a.by.1.1		2	4.3	odd	2	inner
6400.2.a.by.1.1		2	8.5	even	2	inner
6400.2.a.by.1.2		2	1.1	even	1	trivial
6400.2.a.by.1.2		2	8.3	odd	2	CM