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Properties

Label 1280.2.j.d
Level $1280$
Weight $2$
Character orbit 1280.j
Analytic conductor $10.221$
Analytic rank $0$
Dimension $32$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1280,2,Mod(63,1280)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1280.63"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1280, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1280.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,0,0,0,0,0,0,0,-32,0,0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content 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 algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 32 q^{9} + 16 q^{13} + 8 q^{17} + 8 q^{21} + 16 q^{25} - 16 q^{29} + 56 q^{33} - 40 q^{45} - 8 q^{57} - 8 q^{61} - 72 q^{65} + 40 q^{69} + 8 q^{73} - 64 q^{81} - 24 q^{85} - 16 q^{89} + 224 q^{93}+ \cdots + 72 q^{97}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
63.1 0 − 2.97565i 0 2.23290 + 0.119064i 0 −3.13319 + 3.13319i 0 −5.85447 0
63.2 0 − 2.59124i 0 0.0160455 − 2.23601i 0 0.639450 − 0.639450i 0 −3.71454 0
63.3 0 − 2.41944i 0 −2.21185 + 0.328182i 0 −0.988699 + 0.988699i 0 −2.85370 0
63.4 0 − 2.34468i 0 1.84823 − 1.25858i 0 3.15963 − 3.15963i 0 −2.49751 0
63.5 0 − 1.81199i 0 −2.20939 − 0.344369i 0 1.54814 − 1.54814i 0 −0.283291 0
63.6 0 − 0.939635i 0 −0.792335 + 2.09098i 0 1.01861 − 1.01861i 0 2.11709 0
63.7 0 − 0.782176i 0 1.69250 + 1.46131i 0 −1.70686 + 1.70686i 0 2.38820 0
63.8 0 − 0.549343i 0 −0.576096 − 2.16058i 0 −3.23508 + 3.23508i 0 2.69822 0
63.9 0 0.549343i 0 −0.576096 − 2.16058i 0 3.23508 − 3.23508i 0 2.69822 0
63.10 0 0.782176i 0 1.69250 + 1.46131i 0 1.70686 − 1.70686i 0 2.38820 0
63.11 0 0.939635i 0 −0.792335 + 2.09098i 0 −1.01861 + 1.01861i 0 2.11709 0
63.12 0 1.81199i 0 −2.20939 − 0.344369i 0 −1.54814 + 1.54814i 0 −0.283291 0
63.13 0 2.34468i 0 1.84823 − 1.25858i 0 −3.15963 + 3.15963i 0 −2.49751 0
63.14 0 2.41944i 0 −2.21185 + 0.328182i 0 0.988699 − 0.988699i 0 −2.85370 0
63.15 0 2.59124i 0 0.0160455 − 2.23601i 0 −0.639450 + 0.639450i 0 −3.71454 0
63.16 0 2.97565i 0 2.23290 + 0.119064i 0 3.13319 − 3.13319i 0 −5.85447 0
447.1 0 − 2.97565i 0 2.23290 − 0.119064i 0 3.13319 + 3.13319i 0 −5.85447 0
447.2 0 − 2.59124i 0 0.0160455 + 2.23601i 0 −0.639450 − 0.639450i 0 −3.71454 0
447.3 0 − 2.41944i 0 −2.21185 − 0.328182i 0 0.988699 + 0.988699i 0 −2.85370 0
447.4 0 − 2.34468i 0 1.84823 + 1.25858i 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 63.16
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
80.j even 4 1 inner
80.t odd 4 1 inner

Twists

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

Hecke kernels

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

\( T_{3}^{16} + 32T_{3}^{14} + 412T_{3}^{12} + 2728T_{3}^{10} + 9828T_{3}^{8} + 18752T_{3}^{6} + 17344T_{3}^{4} + 7168T_{3}^{2} + 1024 \) Copy content Toggle raw display
\( T_{13}^{8} - 4T_{13}^{7} - 40T_{13}^{6} + 224T_{13}^{5} - 204T_{13}^{4} - 480T_{13}^{3} + 912T_{13}^{2} - 448T_{13} + 64 \) Copy content Toggle raw display