Newform orbit 3008.1.g.a

• Label: 3008.1.g.a
• Level: $3008$
• Weight: $1$
• Character orbit: 3008.g
• Self dual: yes
• Analytic conductor: $1.501$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{5}$
• CM discriminant: -47
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3008 = 2^{6} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3008.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(1.50118755800\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{5}) \)
Defining polynomial:	\( x^{2} - x - 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 47)
Projective image:	\(D_{5}\)
Projective field:	Galois closure of 5.1.2209.1
Artin image:	$D_{10}$
Artin field:	Galois closure of 10.0.159897387008.4

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - \beta q^{3} + ( - \beta + 1) q^{7} + \beta q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} + q^{7} + q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3008\mathbb{Z}\right)^\times\).

\(n\)	\(193\)	\(1599\)	\(2821\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1409.1

1.61803
−0.618034

0

−1.61803

0

0

0

−0.618034

0

1.61803

0

1409.2

0

0.618034

0

0

0

1.61803

0

−0.618034

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
47.b	odd	2	1	CM by \(\Q(\sqrt{-47}) \)

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3008.1.g.a		2
4.b	odd	2	1		3008.1.g.b		2
8.b	even	2	1		752.1.g.a		2
8.d	odd	2	1		47.1.b.a	✓	2
24.f	even	2	1		423.1.d.a		2
40.e	odd	2	1		1175.1.d.c		2
40.k	even	4	2		1175.1.b.b		4
47.b	odd	2	1	CM	3008.1.g.a		2
56.e	even	2	1		2303.1.d.c		2
56.k	odd	6	2		2303.1.f.c		4
56.m	even	6	2		2303.1.f.b		4
72.l	even	6	2		3807.1.f.a		4
72.p	odd	6	2		3807.1.f.b		4
188.b	even	2	1		3008.1.g.b		2
376.e	odd	2	1		752.1.g.a		2
376.h	even	2	1		47.1.b.a	✓	2
376.j	even	46	22		2209.1.d.a		44
376.l	odd	46	22		2209.1.d.a		44
1128.d	odd	2	1		423.1.d.a		2
1880.f	even	2	1		1175.1.d.c		2
1880.q	odd	4	2		1175.1.b.b		4
2632.b	odd	2	1		2303.1.d.c		2
2632.s	even	6	2		2303.1.f.c		4
2632.bf	odd	6	2		2303.1.f.b		4
3384.u	even	6	2		3807.1.f.b		4
3384.bd	odd	6	2		3807.1.f.a		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
47.1.b.a	✓	2	8.d	odd	2	1
47.1.b.a	✓	2	376.h	even	2	1
423.1.d.a		2	24.f	even	2	1
423.1.d.a		2	1128.d	odd	2	1
752.1.g.a		2	8.b	even	2	1
752.1.g.a		2	376.e	odd	2	1
1175.1.b.b		4	40.k	even	4	2
1175.1.b.b		4	1880.q	odd	4	2
1175.1.d.c		2	40.e	odd	2	1
1175.1.d.c		2	1880.f	even	2	1
2209.1.d.a		44	376.j	even	46	22
2209.1.d.a		44	376.l	odd	46	22
2303.1.d.c		2	56.e	even	2	1
2303.1.d.c		2	2632.b	odd	2	1
2303.1.f.b		4	56.m	even	6	2
2303.1.f.b		4	2632.bf	odd	6	2
2303.1.f.c		4	56.k	odd	6	2
2303.1.f.c		4	2632.s	even	6	2
3008.1.g.a		2	1.a	even	1	1	trivial
3008.1.g.a		2	47.b	odd	2	1	CM
3008.1.g.b		2	4.b	odd	2	1
3008.1.g.b		2	188.b	even	2	1
3807.1.f.a		4	72.l	even	6	2
3807.1.f.a		4	3384.bd	odd	6	2
3807.1.f.b		4	72.p	odd	6	2
3807.1.f.b		4	3384.u	even	6	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + T_{3} - 1 \) acting on \(S_{1}^{\mathrm{new}}(3008, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + T - 1 \)
$5$	\( T^{2} \)
$7$	\( T^{2} - T - 1 \)
$11$	\( T^{2} \)