Newform orbit 343.1.b.a

• Label: 343.1.b.a
• Level: $343$
• Weight: $1$
• Character orbit: 343.b
• Self dual: yes
• Analytic conductor: $0.171$
• Analytic rank: $0$
• Dimension: $3$
• Projective image: $D_{7}$
• CM discriminant: -7
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 343 = 7^{3} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	343.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.171179299333\)
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:	yes
Projective image:	\(D_{7}\)
Projective field:	Galois closure of 7.1.40353607.1
Artin image:	$D_7$
Artin field:	Galois closure of 7.1.40353607.1

$q$-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.

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

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 \(\left(\mathbb{Z}/343\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

342.1

1.80194
0.445042
−1.24698

−1.80194

0

2.24698

0

0

0

−2.24698

1.00000

0

342.2

−0.445042

0

−0.801938

0

0

0

0.801938

1.00000

0

342.3

1.24698

0

0.554958

0

0

0

−0.554958

1.00000

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	343.1.b.a	✓	3
3.b	odd	2	1		3087.1.d.a		3
7.b	odd	2	1	CM	343.1.b.a	✓	3
7.c	even	3	2		343.1.d.a		6
7.d	odd	6	2		343.1.d.a		6
21.c	even	2	1		3087.1.d.a		3
21.g	even	6	2		3087.1.m.a		6
21.h	odd	6	2		3087.1.m.a		6
49.e	even	7	2		2401.1.f.a		6
49.e	even	7	2		2401.1.f.b		6
49.e	even	7	2		2401.1.f.c		6
49.f	odd	14	2		2401.1.f.a		6
49.f	odd	14	2		2401.1.f.b		6
49.f	odd	14	2		2401.1.f.c		6
49.g	even	21	4		2401.1.h.a		12
49.g	even	21	4		2401.1.h.b		12
49.g	even	21	4		2401.1.h.c		12
49.h	odd	42	4		2401.1.h.a		12
49.h	odd	42	4		2401.1.h.b		12
49.h	odd	42	4		2401.1.h.c		12

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
343.1.b.a	✓	3	1.a	even	1	1	trivial
343.1.b.a	✓	3	7.b	odd	2	1	CM
343.1.d.a		6	7.c	even	3	2
343.1.d.a		6	7.d	odd	6	2
2401.1.f.a		6	49.e	even	7	2
2401.1.f.a		6	49.f	odd	14	2
2401.1.f.b		6	49.e	even	7	2
2401.1.f.b		6	49.f	odd	14	2
2401.1.f.c		6	49.e	even	7	2
2401.1.f.c		6	49.f	odd	14	2
2401.1.h.a		12	49.g	even	21	4
2401.1.h.a		12	49.h	odd	42	4
2401.1.h.b		12	49.g	even	21	4
2401.1.h.b		12	49.h	odd	42	4
2401.1.h.c		12	49.g	even	21	4
2401.1.h.c		12	49.h	odd	42	4
3087.1.d.a		3	3.b	odd	2	1
3087.1.d.a		3	21.c	even	2	1
3087.1.m.a		6	21.g	even	6	2
3087.1.m.a		6	21.h	odd	6	2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(343, [\chi])\).

Hecke characteristic polynomials

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