Properties

Label 59.1.b.a
Level $59$
Weight $1$
Character orbit 59.b
Self dual yes
Analytic conductor $0.029$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -59
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [59,1,Mod(58,59)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(59, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("59.58");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 59 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 59.b (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(0.0294448357453\)
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 3.1.59.1
Artin image: $S_3$
Artin field: Galois closure of 3.1.59.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - q^{3} + q^{4} - q^{5} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + q^{4} - q^{5} - q^{7} - q^{12} + q^{15} + q^{16} + 2 q^{17} - q^{19} - q^{20} + q^{21} + q^{27} - q^{28} - q^{29} + q^{35} - q^{41} - q^{48} - 2 q^{51} - q^{53} + q^{57} + q^{59} + q^{60} + q^{64} + 2 q^{68} + 2 q^{71} - q^{76} - q^{79} - q^{80} - q^{81} + q^{84} - 2 q^{85} + q^{87} + q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/59\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
58.1
0
0 −1.00000 1.00000 −1.00000 0 −1.00000 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
59.b odd 2 1 CM by \(\Q(\sqrt{-59}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 59.1.b.a 1
3.b odd 2 1 531.1.c.a 1
4.b odd 2 1 944.1.h.a 1
5.b even 2 1 1475.1.c.b 1
5.c odd 4 2 1475.1.d.a 2
7.b odd 2 1 2891.1.c.e 1
7.c even 3 2 2891.1.g.d 2
7.d odd 6 2 2891.1.g.b 2
8.b even 2 1 3776.1.h.b 1
8.d odd 2 1 3776.1.h.a 1
59.b odd 2 1 CM 59.1.b.a 1
59.c even 29 28 3481.1.d.a 28
59.d odd 58 28 3481.1.d.a 28
177.d even 2 1 531.1.c.a 1
236.c even 2 1 944.1.h.a 1
295.d odd 2 1 1475.1.c.b 1
295.e even 4 2 1475.1.d.a 2
413.b even 2 1 2891.1.c.e 1
413.g odd 6 2 2891.1.g.d 2
413.h even 6 2 2891.1.g.b 2
472.c odd 2 1 3776.1.h.b 1
472.f even 2 1 3776.1.h.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
59.1.b.a 1 1.a even 1 1 trivial
59.1.b.a 1 59.b odd 2 1 CM
531.1.c.a 1 3.b odd 2 1
531.1.c.a 1 177.d even 2 1
944.1.h.a 1 4.b odd 2 1
944.1.h.a 1 236.c even 2 1
1475.1.c.b 1 5.b even 2 1
1475.1.c.b 1 295.d odd 2 1
1475.1.d.a 2 5.c odd 4 2
1475.1.d.a 2 295.e even 4 2
2891.1.c.e 1 7.b odd 2 1
2891.1.c.e 1 413.b even 2 1
2891.1.g.b 2 7.d odd 6 2
2891.1.g.b 2 413.h even 6 2
2891.1.g.d 2 7.c even 3 2
2891.1.g.d 2 413.g odd 6 2
3481.1.d.a 28 59.c even 29 28
3481.1.d.a 28 59.d odd 58 28
3776.1.h.a 1 8.d odd 2 1
3776.1.h.a 1 472.f even 2 1
3776.1.h.b 1 8.b even 2 1
3776.1.h.b 1 472.c odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(59, [\chi])\).

Hecke characteristic polynomials

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