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Properties

Label 1020.3.i.a
Level $1020$
Weight $3$
Character orbit 1020.i
Analytic conductor $27.793$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1020,3,Mod(101,1020)]
 
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, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1020.101");
 
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.i (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: \(48\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 4 q^{9} - 20 q^{15} + 48 q^{19} - 72 q^{21} + 240 q^{25} + 100 q^{33} + 152 q^{43} - 216 q^{49} + 80 q^{51} - 40 q^{55} + 496 q^{67} - 264 q^{69} + 68 q^{81} - 60 q^{85} + 280 q^{87} + 276 q^{93}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1 0 −2.97047 − 0.419866i 0 2.23607 0 12.8627i 0 8.64743 + 2.49440i 0
101.2 0 −2.97047 + 0.419866i 0 2.23607 0 − 12.8627i 0 8.64743 − 2.49440i 0
101.3 0 −2.87972 − 0.840971i 0 −2.23607 0 − 7.66100i 0 7.58553 + 4.84352i 0
101.4 0 −2.87972 + 0.840971i 0 −2.23607 0 7.66100i 0 7.58553 − 4.84352i 0
101.5 0 −2.71633 − 1.27341i 0 2.23607 0 6.56816i 0 5.75684 + 6.91800i 0
101.6 0 −2.71633 + 1.27341i 0 2.23607 0 − 6.56816i 0 5.75684 − 6.91800i 0
101.7 0 −2.69343 − 1.32115i 0 −2.23607 0 7.85817i 0 5.50913 + 7.11685i 0
101.8 0 −2.69343 + 1.32115i 0 −2.23607 0 − 7.85817i 0 5.50913 − 7.11685i 0
101.9 0 −2.63974 − 1.42541i 0 2.23607 0 − 1.62479i 0 4.93643 + 7.52540i 0
101.10 0 −2.63974 + 1.42541i 0 2.23607 0 1.62479i 0 4.93643 − 7.52540i 0
101.11 0 −2.57119 − 1.54563i 0 2.23607 0 − 7.02679i 0 4.22205 + 7.94823i 0
101.12 0 −2.57119 + 1.54563i 0 2.23607 0 7.02679i 0 4.22205 − 7.94823i 0
101.13 0 −2.03669 − 2.20270i 0 −2.23607 0 3.99365i 0 −0.703778 + 8.97244i 0
101.14 0 −2.03669 + 2.20270i 0 −2.23607 0 − 3.99365i 0 −0.703778 − 8.97244i 0
101.15 0 −1.52517 − 2.58338i 0 −2.23607 0 − 13.2593i 0 −4.34768 + 7.88021i 0
101.16 0 −1.52517 + 2.58338i 0 −2.23607 0 13.2593i 0 −4.34768 − 7.88021i 0
101.17 0 −1.08262 − 2.79784i 0 −2.23607 0 2.07478i 0 −6.65586 + 6.05802i 0
101.18 0 −1.08262 + 2.79784i 0 −2.23607 0 − 2.07478i 0 −6.65586 − 6.05802i 0
101.19 0 −0.953671 − 2.84438i 0 2.23607 0 2.94216i 0 −7.18102 + 5.42521i 0
101.20 0 −0.953671 + 2.84438i 0 2.23607 0 − 2.94216i 0 −7.18102 − 5.42521i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.48
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
17.b even 2 1 inner
51.c 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.i.a ✓ 48
3.b odd 2 1 inner 1020.3.i.a ✓ 48
17.b even 2 1 inner 1020.3.i.a ✓ 48
51.c odd 2 1 inner 1020.3.i.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1020.3.i.a ✓ 48 1.a even 1 1 trivial
1020.3.i.a ✓ 48 3.b odd 2 1 inner
1020.3.i.a ✓ 48 17.b even 2 1 inner
1020.3.i.a ✓ 48 51.c odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(1020, [\chi])\).