Newform orbit 2704.2.f.g

• Label: 2704.2.f.g
• Level: $2704$
• Weight: $2$
• Character orbit: 2704.f
• Analytic conductor: $21.592$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(21.5915487066\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-1}) \)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 26)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + i q^{5} - 4 i q^{7} - 3 q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{9}+O(q^{10}) \)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

337.1

	−	1.00000i
		1.00000i

0

0

0

−

1.00000i

0

4.00000i

0

−3.00000

0

337.2

0

0

0

1.00000i

0

−

4.00000i

0

−3.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2704.2.f.g		2
4.b	odd	2	1		338.2.b.b		2
12.b	even	2	1		3042.2.b.e		2
13.b	even	2	1	inner	2704.2.f.g		2
13.d	odd	4	1		2704.2.a.h		1
13.d	odd	4	1		2704.2.a.i		1
13.f	odd	12	2		208.2.i.b		2
39.k	even	12	2		1872.2.t.k		2
52.b	odd	2	1		338.2.b.b		2
52.f	even	4	1		338.2.a.c		1
52.f	even	4	1		338.2.a.e		1
52.i	odd	6	2		338.2.e.b		4
52.j	odd	6	2		338.2.e.b		4
52.l	even	12	2		26.2.c.a	✓	2
52.l	even	12	2		338.2.c.e		2
104.u	even	12	2		832.2.i.e		2
104.x	odd	12	2		832.2.i.f		2
156.h	even	2	1		3042.2.b.e		2
156.l	odd	4	1		3042.2.a.e		1
156.l	odd	4	1		3042.2.a.k		1
156.v	odd	12	2		234.2.h.c		2
260.u	even	4	1		8450.2.a.f		1
260.u	even	4	1		8450.2.a.s		1
260.bc	even	12	2		650.2.e.c		2
260.be	odd	12	2		650.2.o.c		4
260.bl	odd	12	2		650.2.o.c		4
364.bt	even	12	2		1274.2.h.b		2
364.bv	odd	12	2		1274.2.g.a		2
364.bz	odd	12	2		1274.2.e.m		2
364.ca	even	12	2		1274.2.e.n		2
364.cg	odd	12	2		1274.2.h.a		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
26.2.c.a	✓	2	52.l	even	12	2
208.2.i.b		2	13.f	odd	12	2
234.2.h.c		2	156.v	odd	12	2
338.2.a.c		1	52.f	even	4	1
338.2.a.e		1	52.f	even	4	1
338.2.b.b		2	4.b	odd	2	1
338.2.b.b		2	52.b	odd	2	1
338.2.c.e		2	52.l	even	12	2
338.2.e.b		4	52.i	odd	6	2
338.2.e.b		4	52.j	odd	6	2
650.2.e.c		2	260.bc	even	12	2
650.2.o.c		4	260.be	odd	12	2
650.2.o.c		4	260.bl	odd	12	2
832.2.i.e		2	104.u	even	12	2
832.2.i.f		2	104.x	odd	12	2
1274.2.e.m		2	364.bz	odd	12	2
1274.2.e.n		2	364.ca	even	12	2
1274.2.g.a		2	364.bv	odd	12	2
1274.2.h.a		2	364.cg	odd	12	2
1274.2.h.b		2	364.bt	even	12	2
1872.2.t.k		2	39.k	even	12	2
2704.2.a.h		1	13.d	odd	4	1
2704.2.a.i		1	13.d	odd	4	1
2704.2.f.g		2	1.a	even	1	1	trivial
2704.2.f.g		2	13.b	even	2	1	inner
3042.2.a.e		1	156.l	odd	4	1
3042.2.a.k		1	156.l	odd	4	1
3042.2.b.e		2	12.b	even	2	1
3042.2.b.e		2	156.h	even	2	1
8450.2.a.f		1	260.u	even	4	1
8450.2.a.s		1	260.u	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2704, [\chi])\):

\( T_{3} \)

\( T_{5}^{2} + 1 \)

\( T_{11}^{2} + 16 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} + 1 \)
$7$	\( T^{2} + 16 \)
$11$	\( T^{2} + 16 \)