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Properties

Label 1287.2.m.b
Level $1287$
Weight $2$
Character orbit 1287.m
Analytic conductor $10.277$
Analytic rank $0$
Dimension $40$
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] = [1287,2,Mod(980,1287)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1287, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1287.980");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1287 = 3^{2} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1287.m (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: \(10.2767467401\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 8 q^{7} - 24 q^{13} - 56 q^{16} + 16 q^{19} - 24 q^{28} - 104 q^{31} + 8 q^{34} + 24 q^{37} - 168 q^{40} + 56 q^{46} - 24 q^{52} - 8 q^{55} + 96 q^{58} + 72 q^{61} - 16 q^{67} - 152 q^{70} - 32 q^{73} + 216 q^{76} - 8 q^{79} - 104 q^{85} + 72 q^{91} + 8 q^{94} + 8 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
980.1 −1.94535 − 1.94535i 0 5.56878i −0.741736 − 0.741736i 0 −0.186699 − 0.186699i 6.94253 − 6.94253i 0 2.88587i
980.2 −1.86449 − 1.86449i 0 4.95268i −2.11701 − 2.11701i 0 1.36448 + 1.36448i 5.50526 − 5.50526i 0 7.89429i
980.3 −1.71335 − 1.71335i 0 3.87115i 1.97826 + 1.97826i 0 −2.27804 − 2.27804i 3.20595 − 3.20595i 0 − 6.77891i
980.4 −1.34359 − 1.34359i 0 1.61047i −0.0868569 − 0.0868569i 0 3.15397 + 3.15397i −0.523374 + 0.523374i 0 0.233400i
980.5 −1.33292 − 1.33292i 0 1.55337i −1.30614 − 1.30614i 0 3.09306 + 3.09306i −0.595322 + 0.595322i 0 3.48197i
980.6 −1.10878 − 1.10878i 0 0.458795i −0.848638 − 0.848638i 0 −1.01620 − 1.01620i −1.70886 + 1.70886i 0 1.88191i
980.7 −1.02205 − 1.02205i 0 0.0891614i 1.31964 + 1.31964i 0 −2.36956 − 2.36956i −1.95297 + 1.95297i 0 − 2.69746i
980.8 −0.783358 − 0.783358i 0 − 0.772700i 3.00807 + 3.00807i 0 −0.875801 − 0.875801i −2.17202 + 2.17202i 0 − 4.71280i
980.9 −0.528915 − 0.528915i 0 − 1.44050i 2.33686 + 2.33686i 0 2.94103 + 2.94103i −1.81973 + 1.81973i 0 − 2.47200i
980.10 −0.233224 − 0.233224i 0 − 1.89121i −0.608271 − 0.608271i 0 −1.82624 − 1.82624i −0.907524 + 0.907524i 0 0.283727i
980.11 0.233224 + 0.233224i 0 − 1.89121i 0.608271 + 0.608271i 0 −1.82624 − 1.82624i 0.907524 − 0.907524i 0 0.283727i
980.12 0.528915 + 0.528915i 0 − 1.44050i −2.33686 − 2.33686i 0 2.94103 + 2.94103i 1.81973 − 1.81973i 0 − 2.47200i
980.13 0.783358 + 0.783358i 0 − 0.772700i −3.00807 − 3.00807i 0 −0.875801 − 0.875801i 2.17202 − 2.17202i 0 − 4.71280i
980.14 1.02205 + 1.02205i 0 0.0891614i −1.31964 − 1.31964i 0 −2.36956 − 2.36956i 1.95297 − 1.95297i 0 − 2.69746i
980.15 1.10878 + 1.10878i 0 0.458795i 0.848638 + 0.848638i 0 −1.01620 − 1.01620i 1.70886 − 1.70886i 0 1.88191i
980.16 1.33292 + 1.33292i 0 1.55337i 1.30614 + 1.30614i 0 3.09306 + 3.09306i 0.595322 − 0.595322i 0 3.48197i
980.17 1.34359 + 1.34359i 0 1.61047i 0.0868569 + 0.0868569i 0 3.15397 + 3.15397i 0.523374 − 0.523374i 0 0.233400i
980.18 1.71335 + 1.71335i 0 3.87115i −1.97826 − 1.97826i 0 −2.27804 − 2.27804i −3.20595 + 3.20595i 0 − 6.77891i
980.19 1.86449 + 1.86449i 0 4.95268i 2.11701 + 2.11701i 0 1.36448 + 1.36448i −5.50526 + 5.50526i 0 7.89429i
980.20 1.94535 + 1.94535i 0 5.56878i 0.741736 + 0.741736i 0 −0.186699 − 0.186699i −6.94253 + 6.94253i 0 2.88587i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 980.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.d odd 4 1 inner
39.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1287.2.m.b ✓ 40
3.b odd 2 1 inner 1287.2.m.b ✓ 40
13.d odd 4 1 inner 1287.2.m.b ✓ 40
39.f even 4 1 inner 1287.2.m.b ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1287.2.m.b ✓ 40 1.a even 1 1 trivial
1287.2.m.b ✓ 40 3.b odd 2 1 inner
1287.2.m.b ✓ 40 13.d odd 4 1 inner
1287.2.m.b ✓ 40 39.f even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} + 178 T_{2}^{36} + 12245 T_{2}^{32} + 415176 T_{2}^{28} + 7445234 T_{2}^{24} + 72161316 T_{2}^{20} + \cdots + 2313441 \) acting on \(S_{2}^{\mathrm{new}}(1287, [\chi])\). Copy content Toggle raw display