Newform orbit 3479.1.d.e

• Label: 3479.1.d.e
• Level: $3479$
• Weight: $1$
• Character orbit: 3479.d
• Self dual: yes
• Analytic conductor: $1.736$
• Analytic rank: $0$
• Dimension: $3$
• Projective image: $D_{7}$
• CM discriminant: -71
• Inner twists: $2$

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:	\(3\)
Coefficient field:	\(\Q(\zeta_{14})^+\)
Defining polynomial:	\( x^{3} - x^{2} - 2x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 71)
Projective image:	\(D_{7}\)
Projective field:	Galois closure of 7.1.357911.1
Artin image:	$D_{14}$
Artin field:	Galois closure of 14.0.105496092121152103.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \beta_{2} q^{2} + \beta_1 q^{3} + ( - \beta_{2} + \beta_1) q^{4} + (\beta_{2} - \beta_1 + 1) q^{5} + (\beta_{2} + 1) q^{6} + (\beta_{2} - \beta_1) q^{8} + (\beta_{2} + 1) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + q^{3} + 2 q^{4} + q^{5} + 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of \(\nu = \zeta_{14} + \zeta_{14}^{-1}\):

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( \nu^{2} - 2 \)

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

0.445042
−1.24698
1.80194

−1.80194

0.445042

2.24698

−1.24698

−0.801938

0

−2.24698

−0.801938

2.24698

638.2

−0.445042

−1.24698

−0.801938

1.80194

0.554958

0

0.801938

0.554958

−0.801938

638.3

1.24698

1.80194

0.554958

0.445042

2.24698

0

−0.554958

2.24698

0.554958

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3479.1.d.e		3
7.b	odd	2	1		71.1.b.a	✓	3
7.c	even	3	2		3479.1.g.d		6
7.d	odd	6	2		3479.1.g.e		6
21.c	even	2	1		639.1.d.a		3
28.d	even	2	1		1136.1.h.a		3
35.c	odd	2	1		1775.1.d.b		3
35.f	even	4	2		1775.1.c.a		6
71.b	odd	2	1	CM	3479.1.d.e		3
497.b	even	2	1		71.1.b.a	✓	3
497.g	odd	6	2		3479.1.g.d		6
497.i	even	6	2		3479.1.g.e		6
1491.e	odd	2	1		639.1.d.a		3
1988.g	odd	2	1		1136.1.h.a		3
2485.d	even	2	1		1775.1.d.b		3
2485.j	odd	4	2		1775.1.c.a		6

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
71.1.b.a	✓	3	7.b	odd	2	1
71.1.b.a	✓	3	497.b	even	2	1
639.1.d.a		3	21.c	even	2	1
639.1.d.a		3	1491.e	odd	2	1
1136.1.h.a		3	28.d	even	2	1
1136.1.h.a		3	1988.g	odd	2	1
1775.1.c.a		6	35.f	even	4	2
1775.1.c.a		6	2485.j	odd	4	2
1775.1.d.b		3	35.c	odd	2	1
1775.1.d.b		3	2485.d	even	2	1
3479.1.d.e		3	1.a	even	1	1	trivial
3479.1.d.e		3	71.b	odd	2	1	CM
3479.1.g.d		6	7.c	even	3	2
3479.1.g.d		6	497.g	odd	6	2
3479.1.g.e		6	7.d	odd	6	2
3479.1.g.e		6	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 \(S_{1}^{\mathrm{new}}(3479, [\chi])\):

\( T_{2}^{3} + T_{2}^{2} - 2T_{2} - 1 \)

\( T_{3}^{3} - T_{3}^{2} - 2T_{3} + 1 \)

Hecke characteristic polynomials

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