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Properties

Label 1280.2.q.c
Level $1280$
Weight $2$
Character orbit 1280.q
Analytic conductor $10.221$
Analytic rank $0$
Dimension $32$
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] = [1280,2,Mod(449,1280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1280, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1280.449");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1280.q (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: \(10.2208514587\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 8 q^{5} + 48 q^{21} + 16 q^{29} - 24 q^{45} + 16 q^{49} - 32 q^{61} + 48 q^{65} - 96 q^{69} - 144 q^{81} - 96 q^{85}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1 0 −2.29023 − 2.29023i 0 2.11151 − 0.735889i 0 −2.23020 0 7.49031i 0
449.2 0 −2.29023 − 2.29023i 0 −0.735889 + 2.11151i 0 −2.23020 0 7.49031i 0
449.3 0 −1.66858 − 1.66858i 0 −1.74837 − 1.39398i 0 1.72267 0 2.56831i 0
449.4 0 −1.66858 − 1.66858i 0 −1.39398 − 1.74837i 0 1.72267 0 2.56831i 0
449.5 0 −0.921855 − 0.921855i 0 −0.0918247 + 2.23418i 0 −2.61462 0 − 1.30037i 0
449.6 0 −0.921855 − 0.921855i 0 2.23418 − 0.0918247i 0 −2.61462 0 − 1.30037i 0
449.7 0 −0.347662 − 0.347662i 0 −0.144207 − 2.23141i 0 −3.90158 0 − 2.75826i 0
449.8 0 −0.347662 − 0.347662i 0 −2.23141 − 0.144207i 0 −3.90158 0 − 2.75826i 0
449.9 0 0.347662 + 0.347662i 0 −2.23141 − 0.144207i 0 3.90158 0 − 2.75826i 0
449.10 0 0.347662 + 0.347662i 0 −0.144207 − 2.23141i 0 3.90158 0 − 2.75826i 0
449.11 0 0.921855 + 0.921855i 0 2.23418 − 0.0918247i 0 2.61462 0 − 1.30037i 0
449.12 0 0.921855 + 0.921855i 0 −0.0918247 + 2.23418i 0 2.61462 0 − 1.30037i 0
449.13 0 1.66858 + 1.66858i 0 −1.39398 − 1.74837i 0 −1.72267 0 2.56831i 0
449.14 0 1.66858 + 1.66858i 0 −1.74837 − 1.39398i 0 −1.72267 0 2.56831i 0
449.15 0 2.29023 + 2.29023i 0 −0.735889 + 2.11151i 0 2.23020 0 7.49031i 0
449.16 0 2.29023 + 2.29023i 0 2.11151 − 0.735889i 0 2.23020 0 7.49031i 0
1089.1 0 −2.29023 + 2.29023i 0 2.11151 + 0.735889i 0 −2.23020 0 − 7.49031i 0
1089.2 0 −2.29023 + 2.29023i 0 −0.735889 − 2.11151i 0 −2.23020 0 − 7.49031i 0
1089.3 0 −1.66858 + 1.66858i 0 −1.74837 + 1.39398i 0 1.72267 0 − 2.56831i 0
1089.4 0 −1.66858 + 1.66858i 0 −1.39398 + 1.74837i 0 1.72267 0 − 2.56831i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 449.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
16.e even 4 1 inner
16.f odd 4 1 inner
20.d odd 2 1 inner
80.k odd 4 1 inner
80.q even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1280.2.q.c ✓ 32
4.b odd 2 1 inner 1280.2.q.c ✓ 32
5.b even 2 1 inner 1280.2.q.c ✓ 32
8.b even 2 1 1280.2.q.d yes 32
8.d odd 2 1 1280.2.q.d yes 32
16.e even 4 1 inner 1280.2.q.c ✓ 32
16.e even 4 1 1280.2.q.d yes 32
16.f odd 4 1 inner 1280.2.q.c ✓ 32
16.f odd 4 1 1280.2.q.d yes 32
20.d odd 2 1 inner 1280.2.q.c ✓ 32
40.e odd 2 1 1280.2.q.d yes 32
40.f even 2 1 1280.2.q.d yes 32
80.k odd 4 1 inner 1280.2.q.c ✓ 32
80.k odd 4 1 1280.2.q.d yes 32
80.q even 4 1 inner 1280.2.q.c ✓ 32
80.q even 4 1 1280.2.q.d yes 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1280.2.q.c ✓ 32 1.a even 1 1 trivial
1280.2.q.c ✓ 32 4.b odd 2 1 inner
1280.2.q.c ✓ 32 5.b even 2 1 inner
1280.2.q.c ✓ 32 16.e even 4 1 inner
1280.2.q.c ✓ 32 16.f odd 4 1 inner
1280.2.q.c ✓ 32 20.d odd 2 1 inner
1280.2.q.c ✓ 32 80.k odd 4 1 inner
1280.2.q.c ✓ 32 80.q even 4 1 inner
1280.2.q.d yes 32 8.b even 2 1
1280.2.q.d yes 32 8.d odd 2 1
1280.2.q.d yes 32 16.e even 4 1
1280.2.q.d yes 32 16.f odd 4 1
1280.2.q.d yes 32 40.e odd 2 1
1280.2.q.d yes 32 40.f even 2 1
1280.2.q.d yes 32 80.k odd 4 1
1280.2.q.d yes 32 80.q even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1280, [\chi])\):

\( T_{3}^{16} + 144T_{3}^{12} + 3828T_{3}^{8} + 10080T_{3}^{4} + 576 \) Copy content Toggle raw display
\( T_{29}^{8} - 4T_{29}^{7} + 8T_{29}^{6} - 40T_{29}^{5} + 3208T_{29}^{4} - 16016T_{29}^{3} + 39200T_{29}^{2} + 54880T_{29} + 38416 \) Copy content Toggle raw display