Newform orbit 3328.1.h.a

• Label: 3328.1.h.a
• Level: $3328$
• Weight: $1$
• Character orbit: 3328.h
• Analytic conductor: $1.661$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{2}$
• CM/RM discs: -4, -52, 13
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3328 = 2^{8} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3328.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.66088836204\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(i)\)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 208)
Projective image:	\(D_{2}\)
Projective field:	Galois closure of \(\Q(i, \sqrt{13})\)
Artin image:	$D_4:C_2$
Artin field:	Galois closure of 8.0.2835349504.2

$q$-expansion

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

\(f(q)\)	\(=\)	\( q - q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3328\mathbb{Z}\right)^\times\).

\(n\)	\(261\)	\(769\)	\(1535\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1663.1

		1.00000i
	−	1.00000i

0

0

0

0

0

0

0

−1.00000

0

1663.2

0

0

0

0

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	RM by \(\Q(\sqrt{13}) \)
52.b	odd	2	1	CM by \(\Q(\sqrt{-13}) \)
8.b	even	2	1	inner
8.d	odd	2	1	inner
104.e	even	2	1	inner
104.h	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3328.1.h.a		2
4.b	odd	2	1	CM	3328.1.h.a		2
8.b	even	2	1	inner	3328.1.h.a		2
8.d	odd	2	1	inner	3328.1.h.a		2
13.b	even	2	1	RM	3328.1.h.a		2
16.e	even	4	1		208.1.c.a	✓	1
16.e	even	4	1		832.1.c.a		1
16.f	odd	4	1		208.1.c.a	✓	1
16.f	odd	4	1		832.1.c.a		1
48.i	odd	4	1		1872.1.i.b		1
48.k	even	4	1		1872.1.i.b		1
52.b	odd	2	1	CM	3328.1.h.a		2
104.e	even	2	1	inner	3328.1.h.a		2
104.h	odd	2	1	inner	3328.1.h.a		2
208.l	even	4	1		2704.1.d.a		1
208.m	odd	4	1		2704.1.d.a		1
208.o	odd	4	1		208.1.c.a	✓	1
208.o	odd	4	1		832.1.c.a		1
208.p	even	4	1		208.1.c.a	✓	1
208.p	even	4	1		832.1.c.a		1
208.r	odd	4	1		2704.1.d.a		1
208.s	even	4	1		2704.1.d.a		1
208.be	odd	12	2		2704.1.bb.a		2
208.bf	even	12	2		2704.1.bb.a		2
208.bg	odd	12	2		2704.1.y.b		2
208.bh	even	12	2		2704.1.y.b		2
208.bi	odd	12	2		2704.1.y.b		2
208.bj	even	12	2		2704.1.y.b		2
208.bk	even	12	2		2704.1.bb.a		2
208.bl	odd	12	2		2704.1.bb.a		2
624.v	even	4	1		1872.1.i.b		1
624.bi	odd	4	1		1872.1.i.b		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
208.1.c.a	✓	1	16.e	even	4	1
208.1.c.a	✓	1	16.f	odd	4	1
208.1.c.a	✓	1	208.o	odd	4	1
208.1.c.a	✓	1	208.p	even	4	1
832.1.c.a		1	16.e	even	4	1
832.1.c.a		1	16.f	odd	4	1
832.1.c.a		1	208.o	odd	4	1
832.1.c.a		1	208.p	even	4	1
1872.1.i.b		1	48.i	odd	4	1
1872.1.i.b		1	48.k	even	4	1
1872.1.i.b		1	624.v	even	4	1
1872.1.i.b		1	624.bi	odd	4	1
2704.1.d.a		1	208.l	even	4	1
2704.1.d.a		1	208.m	odd	4	1
2704.1.d.a		1	208.r	odd	4	1
2704.1.d.a		1	208.s	even	4	1
2704.1.y.b		2	208.bg	odd	12	2
2704.1.y.b		2	208.bh	even	12	2
2704.1.y.b		2	208.bi	odd	12	2
2704.1.y.b		2	208.bj	even	12	2
2704.1.bb.a		2	208.be	odd	12	2
2704.1.bb.a		2	208.bf	even	12	2
2704.1.bb.a		2	208.bk	even	12	2
2704.1.bb.a		2	208.bl	odd	12	2
3328.1.h.a		2	1.a	even	1	1	trivial
3328.1.h.a		2	4.b	odd	2	1	CM
3328.1.h.a		2	8.b	even	2	1	inner
3328.1.h.a		2	8.d	odd	2	1	inner
3328.1.h.a		2	13.b	even	2	1	RM
3328.1.h.a		2	52.b	odd	2	1	CM
3328.1.h.a		2	104.e	even	2	1	inner
3328.1.h.a		2	104.h	odd	2	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3328, [\chi])\).

Hecke characteristic polynomials

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