Properties

Label 8.7.d.a
Level $8$
Weight $7$
Character orbit 8.d
Self dual yes
Analytic conductor $1.840$
Analytic rank $0$
Dimension $1$
CM discriminant -8
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8,7,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, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8.3");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 7 \)
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: \(1.84043266896\)
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)
 
\(f(q)\) \(=\) \( q - 8 q^{2} + 46 q^{3} + 64 q^{4} - 368 q^{6} - 512 q^{8} + 1387 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 8 q^{2} + 46 q^{3} + 64 q^{4} - 368 q^{6} - 512 q^{8} + 1387 q^{9} - 2338 q^{11} + 2944 q^{12} + 4096 q^{16} - 1726 q^{17} - 11096 q^{18} - 2482 q^{19} + 18704 q^{22} - 23552 q^{24} + 15625 q^{25} + 30268 q^{27} - 32768 q^{32} - 107548 q^{33} + 13808 q^{34} + 88768 q^{36} + 19856 q^{38} + 134642 q^{41} - 74914 q^{43} - 149632 q^{44} + 188416 q^{48} + 117649 q^{49} - 125000 q^{50} - 79396 q^{51} - 242144 q^{54} - 114172 q^{57} + 304958 q^{59} + 262144 q^{64} + 860384 q^{66} - 596626 q^{67} - 110464 q^{68} - 710144 q^{72} - 593134 q^{73} + 718750 q^{75} - 158848 q^{76} + 381205 q^{81} - 1077136 q^{82} + 678926 q^{83} + 599312 q^{86} + 1197056 q^{88} - 357262 q^{89} - 1507328 q^{96} + 1822754 q^{97} - 941192 q^{98} - 3242806 q^{99}+O(q^{100}) \) 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
−8.00000 46.0000 64.0000 0 −368.000 0 −512.000 1387.00 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 8.7.d.a 1
3.b odd 2 1 72.7.b.a 1
4.b odd 2 1 32.7.d.a 1
8.b even 2 1 32.7.d.a 1
8.d odd 2 1 CM 8.7.d.a 1
12.b even 2 1 288.7.b.a 1
16.e even 4 2 256.7.c.d 2
16.f odd 4 2 256.7.c.d 2
24.f even 2 1 72.7.b.a 1
24.h odd 2 1 288.7.b.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.7.d.a 1 1.a even 1 1 trivial
8.7.d.a 1 8.d odd 2 1 CM
32.7.d.a 1 4.b odd 2 1
32.7.d.a 1 8.b even 2 1
72.7.b.a 1 3.b odd 2 1
72.7.b.a 1 24.f even 2 1
256.7.c.d 2 16.e even 4 2
256.7.c.d 2 16.f odd 4 2
288.7.b.a 1 12.b even 2 1
288.7.b.a 1 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 46 \) acting on \(S_{7}^{\mathrm{new}}(8, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 8 \) Copy content Toggle raw display
$3$ \( T - 46 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 2338 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T + 1726 \) Copy content Toggle raw display
$19$ \( T + 2482 \) 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
$41$ \( T - 134642 \) Copy content Toggle raw display
$43$ \( T + 74914 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T - 304958 \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T + 596626 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 593134 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T - 678926 \) Copy content Toggle raw display
$89$ \( T + 357262 \) Copy content Toggle raw display
$97$ \( T - 1822754 \) Copy content Toggle raw display
show more
show less