Newform orbit 3249.1.p.a

• Label: 3249.1.p.a
• Level: $3249$
• Weight: $1$
• Character orbit: 3249.p
• Analytic conductor: $1.621$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{2}$
• CM/RM discs: -3, -19, 57
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3249 = 3^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3249.p (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.62146222604\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 171)
Projective image:	\(D_{2}\)
Projective field:	Galois closure of \(\Q(\sqrt{-3}, \sqrt{-19})\)
Artin image:	$C_3\times D_4$
Artin field:	Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-expansion

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

\(f(q)\)	\(=\)	\( q - \zeta_{6} q^{4} - q^{7} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{4} - 4 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3249\mathbb{Z}\right)^\times\).

\(n\)	\(362\)	\(2890\)
\(\chi(n)\)	\(1\)	\(\zeta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1513.1

0.500000	−	0.866025i
0.500000	+	0.866025i

0

0

−0.500000

+

0.866025i

0

0

−2.00000

0

0

0

2098.1

0

0

−0.500000

−

0.866025i

0

0

−2.00000

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
19.b	odd	2	1	CM by \(\Q(\sqrt{-19}) \)
57.d	even	2	1	RM by \(\Q(\sqrt{57}) \)
19.c	even	3	1	inner
19.d	odd	6	1	inner
57.f	even	6	1	inner
57.h	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3249.1.p.a		2
3.b	odd	2	1	CM	3249.1.p.a		2
19.b	odd	2	1	CM	3249.1.p.a		2
19.c	even	3	1		171.1.c.a	✓	1
19.c	even	3	1	inner	3249.1.p.a		2
19.d	odd	6	1		171.1.c.a	✓	1
19.d	odd	6	1	inner	3249.1.p.a		2
19.e	even	9	6		3249.1.ba.c		6
19.f	odd	18	6		3249.1.ba.c		6
57.d	even	2	1	RM	3249.1.p.a		2
57.f	even	6	1		171.1.c.a	✓	1
57.f	even	6	1	inner	3249.1.p.a		2
57.h	odd	6	1		171.1.c.a	✓	1
57.h	odd	6	1	inner	3249.1.p.a		2
57.j	even	18	6		3249.1.ba.c		6
57.l	odd	18	6		3249.1.ba.c		6
76.f	even	6	1		2736.1.o.a		1
76.g	odd	6	1		2736.1.o.a		1
171.g	even	3	1		1539.1.o.b		2
171.h	even	3	1		1539.1.o.b		2
171.i	odd	6	1		1539.1.o.b		2
171.j	odd	6	1		1539.1.o.b		2
171.k	even	6	1		1539.1.o.b		2
171.n	odd	6	1		1539.1.o.b		2
171.s	odd	6	1		1539.1.o.b		2
171.t	even	6	1		1539.1.o.b		2
228.m	even	6	1		2736.1.o.a		1
228.n	odd	6	1		2736.1.o.a		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
171.1.c.a	✓	1	19.c	even	3	1
171.1.c.a	✓	1	19.d	odd	6	1
171.1.c.a	✓	1	57.f	even	6	1
171.1.c.a	✓	1	57.h	odd	6	1
1539.1.o.b		2	171.g	even	3	1
1539.1.o.b		2	171.h	even	3	1
1539.1.o.b		2	171.i	odd	6	1
1539.1.o.b		2	171.j	odd	6	1
1539.1.o.b		2	171.k	even	6	1
1539.1.o.b		2	171.n	odd	6	1
1539.1.o.b		2	171.s	odd	6	1
1539.1.o.b		2	171.t	even	6	1
2736.1.o.a		1	76.f	even	6	1
2736.1.o.a		1	76.g	odd	6	1
2736.1.o.a		1	228.m	even	6	1
2736.1.o.a		1	228.n	odd	6	1
3249.1.p.a		2	1.a	even	1	1	trivial
3249.1.p.a		2	3.b	odd	2	1	CM
3249.1.p.a		2	19.b	odd	2	1	CM
3249.1.p.a		2	19.c	even	3	1	inner
3249.1.p.a		2	19.d	odd	6	1	inner
3249.1.p.a		2	57.d	even	2	1	RM
3249.1.p.a		2	57.f	even	6	1	inner
3249.1.p.a		2	57.h	odd	6	1	inner
3249.1.ba.c		6	19.e	even	9	6
3249.1.ba.c		6	19.f	odd	18	6
3249.1.ba.c		6	57.j	even	18	6
3249.1.ba.c		6	57.l	odd	18	6

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} + 2 \) acting on \(S_{1}^{\mathrm{new}}(3249, [\chi])\).

Hecke characteristic polynomials

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