Properties

Label 19.3.b.a
Level $19$
Weight $3$
Character orbit 19.b
Self dual yes
Analytic conductor $0.518$
Analytic rank $0$
Dimension $1$
CM discriminant -19
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [19,3,Mod(18,19)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(19, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("19.18");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 19.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: \(0.517712502285\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 4 q^{4} - 9 q^{5} - 5 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{4} - 9 q^{5} - 5 q^{7} + 9 q^{9} + 3 q^{11} + 16 q^{16} + 15 q^{17} - 19 q^{19} - 36 q^{20} - 30 q^{23} + 56 q^{25} - 20 q^{28} + 45 q^{35} + 36 q^{36} - 85 q^{43} + 12 q^{44} - 81 q^{45} + 75 q^{47} - 24 q^{49} - 27 q^{55} + 103 q^{61} - 45 q^{63} + 64 q^{64} + 60 q^{68} - 25 q^{73} - 76 q^{76} - 15 q^{77} - 144 q^{80} + 81 q^{81} + 90 q^{83} - 135 q^{85} - 120 q^{92} + 171 q^{95} + 27 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/19\mathbb{Z}\right)^\times\).

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
18.1
0
0 0 4.00000 −9.00000 0 −5.00000 0 9.00000 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
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 19.3.b.a 1
3.b odd 2 1 171.3.c.a 1
4.b odd 2 1 304.3.e.a 1
5.b even 2 1 475.3.c.a 1
5.c odd 4 2 475.3.d.a 2
8.b even 2 1 1216.3.e.a 1
8.d odd 2 1 1216.3.e.b 1
12.b even 2 1 2736.3.o.a 1
19.b odd 2 1 CM 19.3.b.a 1
19.c even 3 2 361.3.d.a 2
19.d odd 6 2 361.3.d.a 2
19.e even 9 6 361.3.f.a 6
19.f odd 18 6 361.3.f.a 6
57.d even 2 1 171.3.c.a 1
76.d even 2 1 304.3.e.a 1
95.d odd 2 1 475.3.c.a 1
95.g even 4 2 475.3.d.a 2
152.b even 2 1 1216.3.e.b 1
152.g odd 2 1 1216.3.e.a 1
228.b odd 2 1 2736.3.o.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.3.b.a 1 1.a even 1 1 trivial
19.3.b.a 1 19.b odd 2 1 CM
171.3.c.a 1 3.b odd 2 1
171.3.c.a 1 57.d even 2 1
304.3.e.a 1 4.b odd 2 1
304.3.e.a 1 76.d even 2 1
361.3.d.a 2 19.c even 3 2
361.3.d.a 2 19.d odd 6 2
361.3.f.a 6 19.e even 9 6
361.3.f.a 6 19.f odd 18 6
475.3.c.a 1 5.b even 2 1
475.3.c.a 1 95.d odd 2 1
475.3.d.a 2 5.c odd 4 2
475.3.d.a 2 95.g even 4 2
1216.3.e.a 1 8.b even 2 1
1216.3.e.a 1 152.g odd 2 1
1216.3.e.b 1 8.d odd 2 1
1216.3.e.b 1 152.b even 2 1
2736.3.o.a 1 12.b even 2 1
2736.3.o.a 1 228.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{3}^{\mathrm{new}}(19, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 9 \) Copy content Toggle raw display
$7$ \( T + 5 \) Copy content Toggle raw display
$11$ \( T - 3 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T - 15 \) Copy content Toggle raw display
$19$ \( T + 19 \) Copy content Toggle raw display
$23$ \( T + 30 \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T + 85 \) Copy content Toggle raw display
$47$ \( T - 75 \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T - 103 \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 25 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T - 90 \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
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