Newform orbit 3380.1.h.d

• Label: 3380.1.h.d
• Level: $3380$
• Weight: $1$
• Character orbit: 3380.h
• Self dual: yes
• Analytic conductor: $1.687$
• Analytic rank: $0$
• Dimension: $3$
• Projective image: $D_{7}$
• CM discriminant: -20
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3380.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(1.68683974270\)
Analytic rank:	\(0\)
Dimension:	\(3\)
Coefficient field:	\(\Q(\zeta_{14})^+\)
Defining polynomial:	\( x^{3} - x^{2} - 2x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{7}\)
Projective field:	Galois closure of 7.1.38614472000.1
Artin image:	$D_7$
Artin field:	Galois closure of 7.1.38614472000.1

$q$-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.

\(f(q)\)	\(=\)	\( q + q^{2} - \beta_1 q^{3} + q^{4} + q^{5} - \beta_1 q^{6} + ( - \beta_{2} + \beta_1 - 1) q^{7} + q^{8} + (\beta_{2} + 1) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - q^{3} + 3 q^{4} + 3 q^{5} - q^{6} - q^{7} + 3 q^{8} + 2 q^{9}+O(q^{10}) \)

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 \(\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times\).

\(n\)	\(677\)	\(1691\)	\(1861\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

339.1

1.80194
0.445042
−1.24698

1.00000

−1.80194

1.00000

1.00000

−1.80194

−0.445042

1.00000

2.24698

1.00000

339.2

1.00000

−0.445042

1.00000

1.00000

−0.445042

1.24698

1.00000

−0.801938

1.00000

339.3

1.00000

1.24698

1.00000

1.00000

1.24698

−1.80194

1.00000

0.554958

1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
20.d	odd	2	1	CM by \(\Q(\sqrt{-5}) \)

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3380.1.h.d	yes	3
4.b	odd	2	1		3380.1.h.c	yes	3
5.b	even	2	1		3380.1.h.c	yes	3
13.b	even	2	1		3380.1.h.b	✓	3
13.c	even	3	2		3380.1.v.e		6
13.d	odd	4	2		3380.1.g.c		6
13.e	even	6	2		3380.1.v.g		6
13.f	odd	12	4		3380.1.w.f		12
20.d	odd	2	1	CM	3380.1.h.d	yes	3
52.b	odd	2	1		3380.1.h.e	yes	3
52.f	even	4	2		3380.1.g.d		6
52.i	odd	6	2		3380.1.v.d		6
52.j	odd	6	2		3380.1.v.f		6
52.l	even	12	4		3380.1.w.e		12
65.d	even	2	1		3380.1.h.e	yes	3
65.g	odd	4	2		3380.1.g.d		6
65.l	even	6	2		3380.1.v.d		6
65.n	even	6	2		3380.1.v.f		6
65.s	odd	12	4		3380.1.w.e		12
260.g	odd	2	1		3380.1.h.b	✓	3
260.u	even	4	2		3380.1.g.c		6
260.v	odd	6	2		3380.1.v.e		6
260.w	odd	6	2		3380.1.v.g		6
260.bc	even	12	4		3380.1.w.f		12

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3380.1.g.c		6	13.d	odd	4	2
3380.1.g.c		6	260.u	even	4	2
3380.1.g.d		6	52.f	even	4	2
3380.1.g.d		6	65.g	odd	4	2
3380.1.h.b	✓	3	13.b	even	2	1
3380.1.h.b	✓	3	260.g	odd	2	1
3380.1.h.c	yes	3	4.b	odd	2	1
3380.1.h.c	yes	3	5.b	even	2	1
3380.1.h.d	yes	3	1.a	even	1	1	trivial
3380.1.h.d	yes	3	20.d	odd	2	1	CM
3380.1.h.e	yes	3	52.b	odd	2	1
3380.1.h.e	yes	3	65.d	even	2	1
3380.1.v.d		6	52.i	odd	6	2
3380.1.v.d		6	65.l	even	6	2
3380.1.v.e		6	13.c	even	3	2
3380.1.v.e		6	260.v	odd	6	2
3380.1.v.f		6	52.j	odd	6	2
3380.1.v.f		6	65.n	even	6	2
3380.1.v.g		6	13.e	even	6	2
3380.1.v.g		6	260.w	odd	6	2
3380.1.w.e		12	52.l	even	12	4
3380.1.w.e		12	65.s	odd	12	4
3380.1.w.f		12	13.f	odd	12	4
3380.1.w.f		12	260.bc	even	12	4

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3380, [\chi])\):

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

\( T_{7}^{3} + T_{7}^{2} - 2T_{7} - 1 \)

\( T_{17} \)

Hecke characteristic polynomials

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