Newform orbit 2704.1.d.b

• Label: 2704.1.d.b
• Level: $2704$
• Weight: $1$
• Character orbit: 2704.d
• Self dual: yes
• 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.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(1.34947179416\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{3}) \)
Defining polynomial:	\( x^{2} - 3 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
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

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\)	\(677\)	\(1185\)	\(2367\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

2367.1

1.73205
−1.73205

0

0

0

−1.73205

0

0

0

1.00000

0

2367.2

0

0

0

1.73205

0

0

0

1.00000

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.b	even	2	1	inner
52.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2704.1.d.b		2
4.b	odd	2	1	CM	2704.1.d.b		2
13.b	even	2	1	inner	2704.1.d.b		2
13.c	even	3	2		2704.1.bb.b		4
13.d	odd	4	2		2704.1.c.a		2
13.e	even	6	2		2704.1.bb.b		4
13.f	odd	12	2		208.1.y.a	✓	2
13.f	odd	12	2		2704.1.y.a		2
39.k	even	12	2		1872.1.bs.b		2
52.b	odd	2	1	inner	2704.1.d.b		2
52.f	even	4	2		2704.1.c.a		2
52.i	odd	6	2		2704.1.bb.b		4
52.j	odd	6	2		2704.1.bb.b		4
52.l	even	12	2		208.1.y.a	✓	2
52.l	even	12	2		2704.1.y.a		2
104.u	even	12	2		832.1.y.a		2
104.x	odd	12	2		832.1.y.a		2
156.v	odd	12	2		1872.1.bs.b		2
208.be	odd	12	2		3328.1.x.a		4
208.bf	even	12	2		3328.1.x.a		4
208.bk	even	12	2		3328.1.x.a		4
208.bl	odd	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.f	odd	12	2
208.1.y.a	✓	2	52.l	even	12	2
832.1.y.a		2	104.u	even	12	2
832.1.y.a		2	104.x	odd	12	2
1872.1.bs.b		2	39.k	even	12	2
1872.1.bs.b		2	156.v	odd	12	2
2704.1.c.a		2	13.d	odd	4	2
2704.1.c.a		2	52.f	even	4	2
2704.1.d.b		2	1.a	even	1	1	trivial
2704.1.d.b		2	4.b	odd	2	1	CM
2704.1.d.b		2	13.b	even	2	1	inner
2704.1.d.b		2	52.b	odd	2	1	inner
2704.1.y.a		2	13.f	odd	12	2
2704.1.y.a		2	52.l	even	12	2
2704.1.bb.b		4	13.c	even	3	2
2704.1.bb.b		4	13.e	even	6	2
2704.1.bb.b		4	52.i	odd	6	2
2704.1.bb.b		4	52.j	odd	6	2
3328.1.x.a		4	208.be	odd	12	2
3328.1.x.a		4	208.bf	even	12	2
3328.1.x.a		4	208.bk	even	12	2
3328.1.x.a		4	208.bl	odd	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

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