Properties

Label 8.13.d.a
Level $8$
Weight $13$
Character orbit 8.d
Self dual yes
Analytic conductor $7.312$
Analytic rank $0$
Dimension $1$
CM discriminant -8
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8,13,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, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8.3");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 13 \)
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: \(7.31195053821\)
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)
 
\(f(q)\) \(=\) \( q + 64 q^{2} + 658 q^{3} + 4096 q^{4} + 42112 q^{6} + 262144 q^{8} - 98477 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 64 q^{2} + 658 q^{3} + 4096 q^{4} + 42112 q^{6} + 262144 q^{8} - 98477 q^{9} + 1923122 q^{11} + 2695168 q^{12} + 16777216 q^{16} - 45296062 q^{17} - 6302528 q^{18} - 87931438 q^{19} + 123079808 q^{22} + 172490752 q^{24} + 244140625 q^{25} - 414486044 q^{27} + 1073741824 q^{32} + 1265414276 q^{33} - 2898947968 q^{34} - 403361792 q^{36} - 5627612032 q^{38} + 8628259682 q^{41} - 7030618702 q^{43} + 7877107712 q^{44} + 11039408128 q^{48} + 13841287201 q^{49} + 15625000000 q^{50} - 29804808796 q^{51} - 26527106816 q^{54} - 57858886204 q^{57} + 8638314482 q^{59} + 68719476736 q^{64} + 80986513664 q^{66} + 175045819538 q^{67} - 185532669952 q^{68} - 25815154688 q^{72} + 49139489378 q^{73} + 160644531250 q^{75} - 360167170048 q^{76} - 220397101595 q^{81} + 552208619648 q^{82} - 192940233262 q^{83} - 449959596928 q^{86} + 504134893568 q^{88} - 866326445278 q^{89} + 706522120192 q^{96} + 1656488134658 q^{97} + 885842380864 q^{98} - 189383285194 q^{99}+O(q^{100}) \) 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
64.0000 658.000 4096.00 0 42112.0 0 262144. −98477.0 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
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 8.13.d.a 1
3.b odd 2 1 72.13.b.a 1
4.b odd 2 1 32.13.d.a 1
8.b even 2 1 32.13.d.a 1
8.d odd 2 1 CM 8.13.d.a 1
24.f even 2 1 72.13.b.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.13.d.a 1 1.a even 1 1 trivial
8.13.d.a 1 8.d odd 2 1 CM
32.13.d.a 1 4.b odd 2 1
32.13.d.a 1 8.b even 2 1
72.13.b.a 1 3.b odd 2 1
72.13.b.a 1 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 658 \) acting on \(S_{13}^{\mathrm{new}}(8, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 64 \) Copy content Toggle raw display
$3$ \( T - 658 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 1923122 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T + 45296062 \) Copy content Toggle raw display
$19$ \( T + 87931438 \) 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
$41$ \( T - 8628259682 \) Copy content Toggle raw display
$43$ \( T + 7030618702 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T - 8638314482 \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T - 175045819538 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 49139489378 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T + 192940233262 \) Copy content Toggle raw display
$89$ \( T + 866326445278 \) Copy content Toggle raw display
$97$ \( T - 1656488134658 \) Copy content Toggle raw display
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