Properties

Label 1020.3.bm.a
Level $1020$
Weight $3$
Character orbit 1020.bm
Analytic conductor $27.793$
Analytic rank $0$
Dimension $72$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1020,3,Mod(217,1020)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1020, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("1020.217");
 
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.bm (of order \(4\), degree \(2\), 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: \(72\)
Relative dimension: \(36\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 8 q^{5} - 24 q^{7} + 216 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - 8 q^{5} - 24 q^{7} + 216 q^{9} + 4 q^{13} - 12 q^{15} - 16 q^{17} - 28 q^{25} + 72 q^{29} + 8 q^{31} + 36 q^{33} - 88 q^{35} - 48 q^{39} + 216 q^{41} + 60 q^{43} - 24 q^{45} + 360 q^{49} - 32 q^{55} - 144 q^{57} + 184 q^{61} - 72 q^{63} + 104 q^{65} + 48 q^{67} - 72 q^{71} + 112 q^{73} + 48 q^{75} + 136 q^{77} + 120 q^{79} + 648 q^{81} + 224 q^{83} + 344 q^{85} + 96 q^{87} + 56 q^{91} - 48 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
217.1 0 −1.73205 0 1.12936 + 4.87079i 0 −12.6614 0 3.00000 0
217.2 0 −1.73205 0 0.197503 4.99610i 0 11.2230 0 3.00000 0
217.3 0 −1.73205 0 3.95434 + 3.05994i 0 9.26802 0 3.00000 0
217.4 0 −1.73205 0 −1.91114 + 4.62034i 0 8.52539 0 3.00000 0
217.5 0 −1.73205 0 3.36666 3.69670i 0 −7.60049 0 3.00000 0
217.6 0 −1.73205 0 0.207571 4.99569i 0 −7.23740 0 3.00000 0
217.7 0 −1.73205 0 4.98759 + 0.351991i 0 6.16656 0 3.00000 0
217.8 0 −1.73205 0 −1.48152 + 4.77547i 0 −6.51708 0 3.00000 0
217.9 0 −1.73205 0 2.10409 4.53573i 0 6.12467 0 3.00000 0
217.10 0 −1.73205 0 1.79402 + 4.66706i 0 3.77059 0 3.00000 0
217.11 0 −1.73205 0 −2.43367 4.36775i 0 1.56608 0 3.00000 0
217.12 0 −1.73205 0 −4.74243 + 1.58410i 0 0.985444 0 3.00000 0
217.13 0 −1.73205 0 −4.23232 2.66222i 0 1.92496 0 3.00000 0
217.14 0 −1.73205 0 −4.30375 2.54514i 0 −4.70111 0 3.00000 0
217.15 0 −1.73205 0 4.84913 1.21901i 0 −4.97931 0 3.00000 0
217.16 0 −1.73205 0 4.95782 + 0.648100i 0 −6.22839 0 3.00000 0
217.17 0 −1.73205 0 −3.87944 + 3.15435i 0 4.49402 0 3.00000 0
217.18 0 −1.73205 0 −4.83174 + 1.28619i 0 −10.1235 0 3.00000 0
217.19 0 1.73205 0 −4.13313 2.81375i 0 12.4732 0 3.00000 0
217.20 0 1.73205 0 −3.15569 + 3.87835i 0 11.5811 0 3.00000 0
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 217.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
85.i odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1020.3.bm.a yes 72
5.c odd 4 1 1020.3.t.a 72
17.c even 4 1 1020.3.t.a 72
85.i odd 4 1 inner 1020.3.bm.a yes 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1020.3.t.a 72 5.c odd 4 1
1020.3.t.a 72 17.c even 4 1
1020.3.bm.a yes 72 1.a even 1 1 trivial
1020.3.bm.a yes 72 85.i odd 4 1 inner

Hecke kernels

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