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Properties

Label 1020.3.o.b
Level $1020$
Weight $3$
Character orbit 1020.o
Analytic conductor $27.793$
Analytic rank $0$
Dimension $64$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1020,3,Mod(509,1020)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1020, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1020.509");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1020 = 2^{2} \cdot 3 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1020.o (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: no
Analytic conductor: \(27.7929869648\)
Analytic rank: \(0\)
Dimension: \(64\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 68 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 68 q^{9} + 36 q^{15} - 32 q^{19} - 32 q^{21} - 16 q^{25} + 728 q^{49} - 292 q^{51} + 320 q^{55} + 120 q^{69} - 476 q^{81} - 520 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
509.1 0 −2.97645 − 0.375123i 0 −3.62494 + 3.44380i 0 −8.39956 0 8.71857 + 2.23307i 0
509.2 0 −2.97645 − 0.375123i 0 3.62494 + 3.44380i 0 −8.39956 0 8.71857 + 2.23307i 0
509.3 0 −2.97645 + 0.375123i 0 −3.62494 − 3.44380i 0 −8.39956 0 8.71857 − 2.23307i 0
509.4 0 −2.97645 + 0.375123i 0 3.62494 − 3.44380i 0 −8.39956 0 8.71857 − 2.23307i 0
509.5 0 −2.95322 − 0.527699i 0 −0.407426 + 4.98337i 0 9.86518 0 8.44307 + 3.11683i 0
509.6 0 −2.95322 − 0.527699i 0 0.407426 + 4.98337i 0 9.86518 0 8.44307 + 3.11683i 0
509.7 0 −2.95322 + 0.527699i 0 −0.407426 − 4.98337i 0 9.86518 0 8.44307 − 3.11683i 0
509.8 0 −2.95322 + 0.527699i 0 0.407426 − 4.98337i 0 9.86518 0 8.44307 − 3.11683i 0
509.9 0 −2.64063 − 1.42376i 0 −4.96063 − 0.626251i 0 4.22273 0 4.94581 + 7.51924i 0
509.10 0 −2.64063 − 1.42376i 0 4.96063 − 0.626251i 0 4.22273 0 4.94581 + 7.51924i 0
509.11 0 −2.64063 + 1.42376i 0 −4.96063 + 0.626251i 0 4.22273 0 4.94581 − 7.51924i 0
509.12 0 −2.64063 + 1.42376i 0 4.96063 + 0.626251i 0 4.22273 0 4.94581 − 7.51924i 0
509.13 0 −2.56816 − 1.55066i 0 −4.18466 − 2.73653i 0 2.77583 0 4.19089 + 7.96470i 0
509.14 0 −2.56816 − 1.55066i 0 4.18466 − 2.73653i 0 2.77583 0 4.19089 + 7.96470i 0
509.15 0 −2.56816 + 1.55066i 0 −4.18466 + 2.73653i 0 2.77583 0 4.19089 − 7.96470i 0
509.16 0 −2.56816 + 1.55066i 0 4.18466 + 2.73653i 0 2.77583 0 4.19089 − 7.96470i 0
509.17 0 −1.93046 − 2.29637i 0 −1.64376 − 4.72208i 0 −5.93817 0 −1.54665 + 8.86611i 0
509.18 0 −1.93046 − 2.29637i 0 1.64376 − 4.72208i 0 −5.93817 0 −1.54665 + 8.86611i 0
509.19 0 −1.93046 + 2.29637i 0 −1.64376 + 4.72208i 0 −5.93817 0 −1.54665 − 8.86611i 0
509.20 0 −1.93046 + 2.29637i 0 1.64376 + 4.72208i 0 −5.93817 0 −1.54665 − 8.86611i 0
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 509.64
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner
17.b even 2 1 inner
51.c odd 2 1 inner
85.c even 2 1 inner
255.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1020.3.o.b ✓ 64
3.b odd 2 1 inner 1020.3.o.b ✓ 64
5.b even 2 1 inner 1020.3.o.b ✓ 64
15.d odd 2 1 inner 1020.3.o.b ✓ 64
17.b even 2 1 inner 1020.3.o.b ✓ 64
51.c odd 2 1 inner 1020.3.o.b ✓ 64
85.c even 2 1 inner 1020.3.o.b ✓ 64
255.h odd 2 1 inner 1020.3.o.b ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1020.3.o.b ✓ 64 1.a even 1 1 trivial
1020.3.o.b ✓ 64 3.b odd 2 1 inner
1020.3.o.b ✓ 64 5.b even 2 1 inner
1020.3.o.b ✓ 64 15.d odd 2 1 inner
1020.3.o.b ✓ 64 17.b even 2 1 inner
1020.3.o.b ✓ 64 51.c odd 2 1 inner
1020.3.o.b ✓ 64 85.c even 2 1 inner
1020.3.o.b ✓ 64 255.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} - 483 T_{7}^{14} + 93602 T_{7}^{12} - 9322676 T_{7}^{10} + 509337580 T_{7}^{8} + \cdots + 5202012718080 \) acting on \(S_{3}^{\mathrm{new}}(1020, [\chi])\). Copy content Toggle raw display