Properties

Label 8.19.d.a
Level $8$
Weight $19$
Character orbit 8.d
Self dual yes
Analytic conductor $16.431$
Analytic rank $0$
Dimension $1$
CM discriminant -8
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8,19,Mod(3,8)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 19, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8.3");
 
S:= CuspForms(chi, 19);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 19 \)
Character orbit: \([\chi]\) \(=\) 8.d (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: \(16.4308910168\)
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)
 
\(f(q)\) \(=\) \( q - 512 q^{2} - 3266 q^{3} + 262144 q^{4} + 1672192 q^{6} - 134217728 q^{8} - 376753733 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 512 q^{2} - 3266 q^{3} + 262144 q^{4} + 1672192 q^{6} - 134217728 q^{8} - 376753733 q^{9} - 354349618 q^{11} - 856162304 q^{12} + 68719476736 q^{16} + 119842447106 q^{17} + 192897911296 q^{18} + 335013705758 q^{19} + 181427004416 q^{22} + 438355099648 q^{24} + 3814697265625 q^{25} + 2495793009052 q^{27} - 35184372088832 q^{32} + 1157305852388 q^{33} - 61359332918272 q^{34} - 98763730583552 q^{36} - 171527017348096 q^{38} + 522162604887122 q^{41} + 10\!\cdots\!14 q^{43}+ \cdots + 13\!\cdots\!94 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
0
−512.000 −3266.00 262144. 0 1.67219e6 0 −1.34218e8 −3.76754e8 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8.19.d.a 1
3.b odd 2 1 72.19.b.a 1
4.b odd 2 1 32.19.d.a 1
8.b even 2 1 32.19.d.a 1
8.d odd 2 1 CM 8.19.d.a 1
24.f even 2 1 72.19.b.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.19.d.a 1 1.a even 1 1 trivial
8.19.d.a 1 8.d odd 2 1 CM
32.19.d.a 1 4.b odd 2 1
32.19.d.a 1 8.b even 2 1
72.19.b.a 1 3.b odd 2 1
72.19.b.a 1 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 3266 \) acting on \(S_{19}^{\mathrm{new}}(8, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 512 \) Copy content Toggle raw display
$3$ \( T + 3266 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 354349618 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T - 119842447106 \) Copy content Toggle raw display
$19$ \( T - 335013705758 \) Copy content Toggle raw display
$23$ \( T \) 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
$41$ \( T - 522162604887122 \) Copy content Toggle raw display
$43$ \( T - 1000250360894414 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T + 10\!\cdots\!22 \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T + 50\!\cdots\!94 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 60\!\cdots\!74 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T + 35\!\cdots\!06 \) Copy content Toggle raw display
$89$ \( T - 48\!\cdots\!18 \) Copy content Toggle raw display
$97$ \( T - 15\!\cdots\!66 \) Copy content Toggle raw display
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