Newform orbit 2997.1.v.a

• Label: 2997.1.v.a
• Level: $2997$
• Weight: $1$
• Character orbit: 2997.v
• Analytic conductor: $1.496$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{6}$
• CM discriminant: -3
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2997 = 3^{4} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2997.v (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.49569784286\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 111)
Projective image:	\(D_{6}\)
Projective field:	Galois closure of 6.2.624095613.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\)	\(=\)	q - q^4 - z * q^7 + (z^2 + z) * q^13 + q^16 - q^25 + z * q^28 + (-z^2 + 1) * q^31 + q^37 + (z + 1) * q^43 + (-z^2 - z) * q^52 - q^64 + q^67 + q^73 + (-z^2 + 1) * q^79 + (-z^2 + 1) * q^91 + (z + 1) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - q^{7} + 2 q^{16} - 2 q^{25} + q^{28} + 3 q^{31} + 2 q^{37} + 3 q^{43} - 2 q^{64} + 2 q^{67} + 2 q^{73} + 3 q^{79} + 3 q^{91} + 3 q^{97}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2997\mathbb{Z}\right)^\times\).

\(n\)	\(1297\)	\(1703\)
\(\chi(n)\)	\(-\zeta_{6}^{2}\)	\(\zeta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1565.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

0

−1.00000

0

0

−0.500000

−

0.866025i

0

0

0

2321.1

0

0

−1.00000

0

0

−0.500000

+

0.866025i

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
333.k	even	6	1	inner
333.v	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2997.1.v.a		2
3.b	odd	2	1	CM	2997.1.v.a		2
9.c	even	3	1		111.1.h.a	✓	2
9.c	even	3	1		2997.1.o.a		2
9.d	odd	6	1		111.1.h.a	✓	2
9.d	odd	6	1		2997.1.o.a		2
36.f	odd	6	1		1776.1.bq.a		2
36.h	even	6	1		1776.1.bq.a		2
37.e	even	6	1		2997.1.o.a		2
45.h	odd	6	1		2775.1.w.a		2
45.j	even	6	1		2775.1.w.a		2
45.k	odd	12	2		2775.1.bb.a		4
45.l	even	12	2		2775.1.bb.a		4
111.h	odd	6	1		2997.1.o.a		2
333.k	even	6	1	inner	2997.1.v.a		2
333.o	odd	6	1		111.1.h.a	✓	2
333.t	even	6	1		111.1.h.a	✓	2
333.v	odd	6	1	inner	2997.1.v.a		2
1332.u	even	6	1		1776.1.bq.a		2
1332.be	odd	6	1		1776.1.bq.a		2
1665.bc	even	6	1		2775.1.w.a		2
1665.bm	odd	6	1		2775.1.w.a		2
1665.db	odd	12	2		2775.1.bb.a		4
1665.dh	even	12	2		2775.1.bb.a		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
111.1.h.a	✓	2	9.c	even	3	1
111.1.h.a	✓	2	9.d	odd	6	1
111.1.h.a	✓	2	333.o	odd	6	1
111.1.h.a	✓	2	333.t	even	6	1
1776.1.bq.a		2	36.f	odd	6	1
1776.1.bq.a		2	36.h	even	6	1
1776.1.bq.a		2	1332.u	even	6	1
1776.1.bq.a		2	1332.be	odd	6	1
2775.1.w.a		2	45.h	odd	6	1
2775.1.w.a		2	45.j	even	6	1
2775.1.w.a		2	1665.bc	even	6	1
2775.1.w.a		2	1665.bm	odd	6	1
2775.1.bb.a		4	45.k	odd	12	2
2775.1.bb.a		4	45.l	even	12	2
2775.1.bb.a		4	1665.db	odd	12	2
2775.1.bb.a		4	1665.dh	even	12	2
2997.1.o.a		2	9.c	even	3	1
2997.1.o.a		2	9.d	odd	6	1
2997.1.o.a		2	37.e	even	6	1
2997.1.o.a		2	111.h	odd	6	1
2997.1.v.a		2	1.a	even	1	1	trivial
2997.1.v.a		2	3.b	odd	2	1	CM
2997.1.v.a		2	333.k	even	6	1	inner
2997.1.v.a		2	333.v	odd	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2997, [\chi])\).

Hecke characteristic polynomials

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