Properties

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

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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

\(f(q)\) \(=\) \( q - q^{3} - q^{5} + O(q^{10}) \) \( q - q^{3} - q^{5} + q^{11} + q^{15} - q^{23} + q^{27} - q^{31} - q^{33} - q^{37} + 2q^{47} + q^{49} + 2q^{53} - q^{55} - q^{59} - q^{67} + q^{69} - q^{71} - q^{81} - q^{89} + q^{93} - q^{97} + O(q^{100}) \)

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
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + T + T^{2} \)
$5$ \( 1 + T + T^{2} \)
$7$ \( ( 1 - T )( 1 + T ) \)
$11$ \( 1 - T \)
$13$ \( ( 1 - T )( 1 + T ) \)
$17$ \( ( 1 - T )( 1 + T ) \)
$19$ \( ( 1 - T )( 1 + T ) \)
$23$ \( 1 + T + T^{2} \)
$29$ \( ( 1 - T )( 1 + T ) \)
$31$ \( 1 + T + T^{2} \)
$37$ \( 1 + T + T^{2} \)
$41$ \( ( 1 - T )( 1 + T ) \)
$43$ \( ( 1 - T )( 1 + T ) \)
$47$ \( ( 1 - T )^{2} \)
$53$ \( ( 1 - T )^{2} \)
$59$ \( 1 + T + T^{2} \)
$61$ \( ( 1 - T )( 1 + T ) \)
$67$ \( 1 + T + T^{2} \)
$71$ \( 1 + T + T^{2} \)
$73$ \( ( 1 - T )( 1 + T ) \)
$79$ \( ( 1 - T )( 1 + T ) \)
$83$ \( ( 1 - T )( 1 + T ) \)
$89$ \( 1 + T + T^{2} \)
$97$ \( 1 + T + T^{2} \)
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