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Properties

Label 1280.2.s.d
Level $1280$
Weight $2$
Character orbit 1280.s
Analytic conductor $10.221$
Analytic rank $0$
Dimension $32$
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] = [1280,2,Mod(703,1280)]
 
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([2, 3, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1280.703");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1280.s (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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 8 q^{5} + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 8 q^{5} + 32 q^{9} + 8 q^{17} - 8 q^{21} - 16 q^{25} - 16 q^{29} + 56 q^{33} + 48 q^{45} - 112 q^{53} + 8 q^{57} + 8 q^{61} - 72 q^{65} + 40 q^{69} - 8 q^{73} - 144 q^{77} - 64 q^{81} - 40 q^{85} + 16 q^{89} + 72 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
703.1 0 −2.97565 0 −0.119064 + 2.23290i 0 3.13319 − 3.13319i 0 5.85447 0
703.2 0 −2.59124 0 2.23601 + 0.0160455i 0 −0.639450 + 0.639450i 0 3.71454 0
703.3 0 −2.41944 0 −0.328182 − 2.21185i 0 0.988699 − 0.988699i 0 2.85370 0
703.4 0 −2.34468 0 1.25858 + 1.84823i 0 −3.15963 + 3.15963i 0 2.49751 0
703.5 0 −1.81199 0 0.344369 − 2.20939i 0 −1.54814 + 1.54814i 0 0.283291 0
703.6 0 −0.939635 0 −2.09098 − 0.792335i 0 −1.01861 + 1.01861i 0 −2.11709 0
703.7 0 −0.782176 0 −1.46131 + 1.69250i 0 1.70686 − 1.70686i 0 −2.38820 0
703.8 0 −0.549343 0 2.16058 − 0.576096i 0 3.23508 − 3.23508i 0 −2.69822 0
703.9 0 0.549343 0 2.16058 − 0.576096i 0 −3.23508 + 3.23508i 0 −2.69822 0
703.10 0 0.782176 0 −1.46131 + 1.69250i 0 −1.70686 + 1.70686i 0 −2.38820 0
703.11 0 0.939635 0 −2.09098 − 0.792335i 0 1.01861 − 1.01861i 0 −2.11709 0
703.12 0 1.81199 0 0.344369 − 2.20939i 0 1.54814 − 1.54814i 0 0.283291 0
703.13 0 2.34468 0 1.25858 + 1.84823i 0 3.15963 − 3.15963i 0 2.49751 0
703.14 0 2.41944 0 −0.328182 − 2.21185i 0 −0.988699 + 0.988699i 0 2.85370 0
703.15 0 2.59124 0 2.23601 + 0.0160455i 0 0.639450 − 0.639450i 0 3.71454 0
703.16 0 2.97565 0 −0.119064 + 2.23290i 0 −3.13319 + 3.13319i 0 5.85447 0
1087.1 0 −2.97565 0 −0.119064 − 2.23290i 0 3.13319 + 3.13319i 0 5.85447 0
1087.2 0 −2.59124 0 2.23601 − 0.0160455i 0 −0.639450 − 0.639450i 0 3.71454 0
1087.3 0 −2.41944 0 −0.328182 + 2.21185i 0 0.988699 + 0.988699i 0 2.85370 0
1087.4 0 −2.34468 0 1.25858 − 1.84823i 0 −3.15963 − 3.15963i 0 2.49751 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 703.16
Significant digits:
Format:

Inner twists

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

Hecke kernels

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

\( T_{3}^{16} - 32T_{3}^{14} + 412T_{3}^{12} - 2728T_{3}^{10} + 9828T_{3}^{8} - 18752T_{3}^{6} + 17344T_{3}^{4} - 7168T_{3}^{2} + 1024 \) Copy content Toggle raw display
\( T_{29}^{16} + 8 T_{29}^{15} + 32 T_{29}^{14} - 176 T_{29}^{13} + 2768 T_{29}^{12} + 15456 T_{29}^{11} + \cdots + 4726837504 \) Copy content Toggle raw display