Newform orbit 3479.1.d.a

• Label: 3479.1.d.a
• Level: $3479$
• Weight: $1$
• Character orbit: 3479.d
• Self dual: yes
• Analytic conductor: $1.736$
• Analytic rank: $0$
• Dimension: $1$
• Projective image: $D_{2}$
• CM/RM discs: -7, -71, 497
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3479 = 7^{2} \cdot 71 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3479.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(1.73624717895\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{2}\)
Projective field:	Galois closure of \(\Q(\sqrt{-7}, \sqrt{-71})\)
Artin image:	$D_4$
Artin field:	Galois closure of 4.0.24353.1

$q$-expansion

\(f(q)\)

\(=\)

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3479\mathbb{Z}\right)^\times\).

\(n\)	\(640\)	\(1569\)
\(\chi(n)\)	\(1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

638.1

0

2.00000

0

3.00000

0

0

0

4.00000

−1.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by \(\Q(\sqrt{-7}) \)
71.b	odd	2	1	CM by \(\Q(\sqrt{-71}) \)
497.b	even	2	1	RM by \(\Q(\sqrt{497}) \)

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3479.1.d.a	✓	1
7.b	odd	2	1	CM	3479.1.d.a	✓	1
7.c	even	3	2		3479.1.g.a		2
7.d	odd	6	2		3479.1.g.a		2
71.b	odd	2	1	CM	3479.1.d.a	✓	1
497.b	even	2	1	RM	3479.1.d.a	✓	1
497.g	odd	6	2		3479.1.g.a		2
497.i	even	6	2		3479.1.g.a		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3479.1.d.a	✓	1	1.a	even	1	1	trivial
3479.1.d.a	✓	1	7.b	odd	2	1	CM
3479.1.d.a	✓	1	71.b	odd	2	1	CM
3479.1.d.a	✓	1	497.b	even	2	1	RM
3479.1.g.a		2	7.c	even	3	2
3479.1.g.a		2	7.d	odd	6	2
3479.1.g.a		2	497.g	odd	6	2
3479.1.g.a		2	497.i	even	6	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3479, [\chi])\):

\( T_{2} - 2 \)

\( T_{3} \)

Hecke characteristic polynomials

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