Newform orbit 2704.1.y.a

• Label: 2704.1.y.a
• Level: $2704$
• Weight: $1$
• Character orbit: 2704.y
• Analytic conductor: $1.349$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{6}$
• CM discriminant: -4
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2704 = 2^{4} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2704.y (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.34947179416\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 208)
Projective image:	\(D_{6}\)
Projective field:	Galois closure of 6.0.23762752.2

$q$-expansion

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

\(f(q)\)	\(=\)	q + (z^2 + z) * q^5 + z^2 * q^9 + z^2 * q^17 + (z^2 - z - 1) * q^25 - z * q^29 + (-z^2 + 1) * q^37 + (z^2 - 1) * q^41 + (-z - 1) * q^45 + z * q^49 - q^53 - z^2 * q^61 + (z^2 + z) * q^73 - z * q^81 + (-z - 1) * q^85
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{9} - q^{17} - 4 q^{25} - q^{29} + 3 q^{37} - 3 q^{41} - 3 q^{45} + q^{49} - 2 q^{53} + q^{61} - q^{81} - 3 q^{85}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2704\mathbb{Z}\right)^\times\).

\(n\)	\(677\)	\(1185\)	\(2367\)
\(\chi(n)\)	\(1\)	\(-\zeta_{6}^{2}\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1375.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

0

0

1.73205i

0

0

0

−0.500000

+

0.866025i

0

2175.1

0

0

0

−

1.73205i

0

0

0

−0.500000

−

0.866025i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)
13.e	even	6	1	inner
52.i	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2704.1.y.a		2
4.b	odd	2	1	CM	2704.1.y.a		2
13.b	even	2	1		208.1.y.a	✓	2
13.c	even	3	1		208.1.y.a	✓	2
13.c	even	3	1		2704.1.c.a		2
13.d	odd	4	2		2704.1.bb.b		4
13.e	even	6	1		2704.1.c.a		2
13.e	even	6	1	inner	2704.1.y.a		2
13.f	odd	12	2		2704.1.d.b		2
13.f	odd	12	2		2704.1.bb.b		4
39.d	odd	2	1		1872.1.bs.b		2
39.i	odd	6	1		1872.1.bs.b		2
52.b	odd	2	1		208.1.y.a	✓	2
52.f	even	4	2		2704.1.bb.b		4
52.i	odd	6	1		2704.1.c.a		2
52.i	odd	6	1	inner	2704.1.y.a		2
52.j	odd	6	1		208.1.y.a	✓	2
52.j	odd	6	1		2704.1.c.a		2
52.l	even	12	2		2704.1.d.b		2
52.l	even	12	2		2704.1.bb.b		4
104.e	even	2	1		832.1.y.a		2
104.h	odd	2	1		832.1.y.a		2
104.n	odd	6	1		832.1.y.a		2
104.r	even	6	1		832.1.y.a		2
156.h	even	2	1		1872.1.bs.b		2
156.p	even	6	1		1872.1.bs.b		2
208.o	odd	4	2		3328.1.x.a		4
208.p	even	4	2		3328.1.x.a		4
208.bg	odd	12	2		3328.1.x.a		4
208.bj	even	12	2		3328.1.x.a		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
208.1.y.a	✓	2	13.b	even	2	1
208.1.y.a	✓	2	13.c	even	3	1
208.1.y.a	✓	2	52.b	odd	2	1
208.1.y.a	✓	2	52.j	odd	6	1
832.1.y.a		2	104.e	even	2	1
832.1.y.a		2	104.h	odd	2	1
832.1.y.a		2	104.n	odd	6	1
832.1.y.a		2	104.r	even	6	1
1872.1.bs.b		2	39.d	odd	2	1
1872.1.bs.b		2	39.i	odd	6	1
1872.1.bs.b		2	156.h	even	2	1
1872.1.bs.b		2	156.p	even	6	1
2704.1.c.a		2	13.c	even	3	1
2704.1.c.a		2	13.e	even	6	1
2704.1.c.a		2	52.i	odd	6	1
2704.1.c.a		2	52.j	odd	6	1
2704.1.d.b		2	13.f	odd	12	2
2704.1.d.b		2	52.l	even	12	2
2704.1.y.a		2	1.a	even	1	1	trivial
2704.1.y.a		2	4.b	odd	2	1	CM
2704.1.y.a		2	13.e	even	6	1	inner
2704.1.y.a		2	52.i	odd	6	1	inner
2704.1.bb.b		4	13.d	odd	4	2
2704.1.bb.b		4	13.f	odd	12	2
2704.1.bb.b		4	52.f	even	4	2
2704.1.bb.b		4	52.l	even	12	2
3328.1.x.a		4	208.o	odd	4	2
3328.1.x.a		4	208.p	even	4	2
3328.1.x.a		4	208.bg	odd	12	2
3328.1.x.a		4	208.bj	even	12	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 3 \) acting on \(S_{1}^{\mathrm{new}}(2704, [\chi])\).

Hecke characteristic polynomials

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