Newform orbit 44.1.d.a

• Label: 44.1.d.a
• Level: $44$
• Weight: $1$
• Character orbit: 44.d
• Self dual: yes
• Analytic conductor: $0.022$
• Analytic rank: $0$
• Dimension: $1$
• Projective image: $D_{3}$
• CM discriminant: -11
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 44 = 2^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	44.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.0219588605559\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{3}\)
Projective field:	Galois closure of 3.1.44.1
Artin image:	$S_3$
Artin field:	Galois closure of 3.1.44.1

$q$-expansion

\(f(q)\)

\(=\)

\( q - q^{3} - q^{5}+O(q^{10}) \)

Expression as an eta quotient

\(f(z) = \eta(2z)\eta(22z)=q\prod_{n=1}^\infty(1 - q^{2n})^{}(1 - q^{22n})^{}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/44\mathbb{Z}\right)^\times\).

\(n\)	\(13\)	\(23\)
\(\chi(n)\)	\(-1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

21.1

0

0

−1.00000

0

−1.00000

0

0

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.b	odd	2	1	CM by \(\Q(\sqrt{-11}) \)

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	44.1.d.a	✓	1
3.b	odd	2	1		396.1.f.a		1
4.b	odd	2	1		176.1.h.a		1
5.b	even	2	1		1100.1.f.a		1
5.c	odd	4	2		1100.1.e.a		2
7.b	odd	2	1		2156.1.h.a		1
7.c	even	3	2		2156.1.k.b		2
7.d	odd	6	2		2156.1.k.a		2
8.b	even	2	1		704.1.h.b		1
8.d	odd	2	1		704.1.h.a		1
9.c	even	3	2		3564.1.m.b		2
9.d	odd	6	2		3564.1.m.a		2
11.b	odd	2	1	CM	44.1.d.a	✓	1
11.c	even	5	4		484.1.f.a		4
11.d	odd	10	4		484.1.f.a		4
12.b	even	2	1		1584.1.j.a		1
16.e	even	4	2		2816.1.b.b		2
16.f	odd	4	2		2816.1.b.a		2
33.d	even	2	1		396.1.f.a		1
44.c	even	2	1		176.1.h.a		1
44.g	even	10	4		1936.1.n.a		4
44.h	odd	10	4		1936.1.n.a		4
55.d	odd	2	1		1100.1.f.a		1
55.e	even	4	2		1100.1.e.a		2
77.b	even	2	1		2156.1.h.a		1
77.h	odd	6	2		2156.1.k.b		2
77.i	even	6	2		2156.1.k.a		2
88.b	odd	2	1		704.1.h.b		1
88.g	even	2	1		704.1.h.a		1
99.g	even	6	2		3564.1.m.a		2
99.h	odd	6	2		3564.1.m.b		2
132.d	odd	2	1		1584.1.j.a		1
176.i	even	4	2		2816.1.b.a		2
176.l	odd	4	2		2816.1.b.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
44.1.d.a	✓	1	1.a	even	1	1	trivial
44.1.d.a	✓	1	11.b	odd	2	1	CM
176.1.h.a		1	4.b	odd	2	1
176.1.h.a		1	44.c	even	2	1
396.1.f.a		1	3.b	odd	2	1
396.1.f.a		1	33.d	even	2	1
484.1.f.a		4	11.c	even	5	4
484.1.f.a		4	11.d	odd	10	4
704.1.h.a		1	8.d	odd	2	1
704.1.h.a		1	88.g	even	2	1
704.1.h.b		1	8.b	even	2	1
704.1.h.b		1	88.b	odd	2	1
1100.1.e.a		2	5.c	odd	4	2
1100.1.e.a		2	55.e	even	4	2
1100.1.f.a		1	5.b	even	2	1
1100.1.f.a		1	55.d	odd	2	1
1584.1.j.a		1	12.b	even	2	1
1584.1.j.a		1	132.d	odd	2	1
1936.1.n.a		4	44.g	even	10	4
1936.1.n.a		4	44.h	odd	10	4
2156.1.h.a		1	7.b	odd	2	1
2156.1.h.a		1	77.b	even	2	1
2156.1.k.a		2	7.d	odd	6	2
2156.1.k.a		2	77.i	even	6	2
2156.1.k.b		2	7.c	even	3	2
2156.1.k.b		2	77.h	odd	6	2
2816.1.b.a		2	16.f	odd	4	2
2816.1.b.a		2	176.i	even	4	2
2816.1.b.b		2	16.e	even	4	2
2816.1.b.b		2	176.l	odd	4	2
3564.1.m.a		2	9.d	odd	6	2
3564.1.m.a		2	99.g	even	6	2
3564.1.m.b		2	9.c	even	3	2
3564.1.m.b		2	99.h	odd	6	2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(44, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T \)
$3$	\( T + 1 \)
$5$	\( T + 1 \)
$7$	\( T \)
$11$	\( T - 1 \)