Newform orbit 8.3.d.a

• Label: 8.3.d.a
• Level: $8$
• Weight: $3$
• Character orbit: 8.d
• Self dual: yes
• Analytic conductor: $0.218$
• Analytic rank: $0$
• Dimension: $1$
• CM discriminant: -8
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 8 = 2^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	8.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.217984211488\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\)

\(=\)

\( q - 2 q^{2} - 2 q^{3} + 4 q^{4} + 4 q^{6} - 8 q^{8} - 5 q^{9}+O(q^{10}) \)

Expression as an eta quotient

\(f(z) = \eta(z)^{2}\eta(2z)\eta(4z)\eta(8z)^{2}=q\prod_{n=1}^\infty(1 - q^{n})^{2}(1 - q^{2n})^{}(1 - q^{4n})^{}(1 - q^{8n})^{2}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(7\)
\(\chi(n)\)	\(-1\)	\(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

3.1

0

−2.00000

−2.00000

4.00000

0

4.00000

0

−8.00000

−5.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by \(\Q(\sqrt{-2}) \)

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	8.3.d.a	✓	1
3.b	odd	2	1		72.3.b.a		1
4.b	odd	2	1		32.3.d.a		1
5.b	even	2	1		200.3.g.a		1
5.c	odd	4	2		200.3.e.a		2
7.b	odd	2	1		392.3.g.a		1
7.c	even	3	2		392.3.k.d		2
7.d	odd	6	2		392.3.k.b		2
8.b	even	2	1		32.3.d.a		1
8.d	odd	2	1	CM	8.3.d.a	✓	1
12.b	even	2	1		288.3.b.a		1
16.e	even	4	2		256.3.c.e		2
16.f	odd	4	2		256.3.c.e		2
20.d	odd	2	1		800.3.g.a		1
20.e	even	4	2		800.3.e.a		2
24.f	even	2	1		72.3.b.a		1
24.h	odd	2	1		288.3.b.a		1
28.d	even	2	1		1568.3.g.a		1
40.e	odd	2	1		200.3.g.a		1
40.f	even	2	1		800.3.g.a		1
40.i	odd	4	2		800.3.e.a		2
40.k	even	4	2		200.3.e.a		2
48.i	odd	4	2		2304.3.g.j		2
48.k	even	4	2		2304.3.g.j		2
56.e	even	2	1		392.3.g.a		1
56.h	odd	2	1		1568.3.g.a		1
56.k	odd	6	2		392.3.k.d		2
56.m	even	6	2		392.3.k.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
8.3.d.a	✓	1	1.a	even	1	1	trivial
8.3.d.a	✓	1	8.d	odd	2	1	CM
32.3.d.a		1	4.b	odd	2	1
32.3.d.a		1	8.b	even	2	1
72.3.b.a		1	3.b	odd	2	1
72.3.b.a		1	24.f	even	2	1
200.3.e.a		2	5.c	odd	4	2
200.3.e.a		2	40.k	even	4	2
200.3.g.a		1	5.b	even	2	1
200.3.g.a		1	40.e	odd	2	1
256.3.c.e		2	16.e	even	4	2
256.3.c.e		2	16.f	odd	4	2
288.3.b.a		1	12.b	even	2	1
288.3.b.a		1	24.h	odd	2	1
392.3.g.a		1	7.b	odd	2	1
392.3.g.a		1	56.e	even	2	1
392.3.k.b		2	7.d	odd	6	2
392.3.k.b		2	56.m	even	6	2
392.3.k.d		2	7.c	even	3	2
392.3.k.d		2	56.k	odd	6	2
800.3.e.a		2	20.e	even	4	2
800.3.e.a		2	40.i	odd	4	2
800.3.g.a		1	20.d	odd	2	1
800.3.g.a		1	40.f	even	2	1
1568.3.g.a		1	28.d	even	2	1
1568.3.g.a		1	56.h	odd	2	1
2304.3.g.j		2	48.i	odd	4	2
2304.3.g.j		2	48.k	even	4	2

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(8, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T + 2 \)
$3$	\( T + 2 \)
$5$	\( T \)
$7$	\( T \)
$11$	\( T - 14 \)