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Properties

Label 160.4.o.a
Level $160$
Weight $4$
Character orbit 160.o
Analytic conductor $9.440$
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] = [160,4,Mod(47,160)] 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("160.47"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(160, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 160.o (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.44030560092\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 40)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 4 q^{3} + 8 q^{11} + 48 q^{17} + 40 q^{25} - 104 q^{27} - 112 q^{33} + 460 q^{35} - 8 q^{41} + 868 q^{43} - 1480 q^{51} + 104 q^{57} + 520 q^{65} + 1852 q^{67} - 744 q^{73} - 3300 q^{75} - 1240 q^{81}+ \cdots - 584 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 0 −6.15076 + 6.15076i 0 −11.1441 + 0.899439i 0 −16.5614 + 16.5614i 0 − 48.6638i 0
47.2 0 −6.15076 + 6.15076i 0 11.1441 − 0.899439i 0 16.5614 − 16.5614i 0 − 48.6638i 0
47.3 0 −4.00710 + 4.00710i 0 −6.85881 − 8.82931i 0 10.5258 − 10.5258i 0 − 5.11376i 0
47.4 0 −4.00710 + 4.00710i 0 6.85881 + 8.82931i 0 −10.5258 + 10.5258i 0 − 5.11376i 0
47.5 0 −3.49003 + 3.49003i 0 −4.99441 + 10.0028i 0 4.97302 − 4.97302i 0 2.63942i 0
47.6 0 −3.49003 + 3.49003i 0 4.99441 − 10.0028i 0 −4.97302 + 4.97302i 0 2.63942i 0
47.7 0 0.102537 − 0.102537i 0 −2.04880 − 10.9910i 0 −15.5472 + 15.5472i 0 26.9790i 0
47.8 0 0.102537 − 0.102537i 0 2.04880 + 10.9910i 0 15.5472 − 15.5472i 0 26.9790i 0
47.9 0 1.56085 − 1.56085i 0 −10.5634 − 3.66270i 0 18.5221 − 18.5221i 0 22.1275i 0
47.10 0 1.56085 − 1.56085i 0 10.5634 + 3.66270i 0 −18.5221 + 18.5221i 0 22.1275i 0
47.11 0 2.02737 − 2.02737i 0 −10.8122 + 2.84559i 0 −1.63237 + 1.63237i 0 18.7795i 0
47.12 0 2.02737 − 2.02737i 0 10.8122 − 2.84559i 0 1.63237 − 1.63237i 0 18.7795i 0
47.13 0 4.39586 − 4.39586i 0 −8.23152 + 7.56585i 0 −18.5943 + 18.5943i 0 − 11.6471i 0
47.14 0 4.39586 − 4.39586i 0 8.23152 − 7.56585i 0 18.5943 − 18.5943i 0 − 11.6471i 0
47.15 0 6.56128 − 6.56128i 0 −2.89463 − 10.7991i 0 −8.83176 + 8.83176i 0 − 59.1008i 0
47.16 0 6.56128 − 6.56128i 0 2.89463 + 10.7991i 0 8.83176 − 8.83176i 0 − 59.1008i 0
143.1 0 −6.15076 − 6.15076i 0 −11.1441 − 0.899439i 0 −16.5614 − 16.5614i 0 48.6638i 0
143.2 0 −6.15076 − 6.15076i 0 11.1441 + 0.899439i 0 16.5614 + 16.5614i 0 48.6638i 0
143.3 0 −4.00710 − 4.00710i 0 −6.85881 + 8.82931i 0 10.5258 + 10.5258i 0 5.11376i 0
143.4 0 −4.00710 − 4.00710i 0 6.85881 − 8.82931i 0 −10.5258 − 10.5258i 0 5.11376i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
8.d odd 2 1 inner
40.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 160.4.o.a 32
4.b odd 2 1 40.4.k.a ✓ 32
5.c odd 4 1 inner 160.4.o.a 32
8.b even 2 1 40.4.k.a ✓ 32
8.d odd 2 1 inner 160.4.o.a 32
20.d odd 2 1 200.4.k.j 32
20.e even 4 1 40.4.k.a ✓ 32
20.e even 4 1 200.4.k.j 32
40.f even 2 1 200.4.k.j 32
40.i odd 4 1 40.4.k.a ✓ 32
40.i odd 4 1 200.4.k.j 32
40.k even 4 1 inner 160.4.o.a 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.4.k.a ✓ 32 4.b odd 2 1
40.4.k.a ✓ 32 8.b even 2 1
40.4.k.a ✓ 32 20.e even 4 1
40.4.k.a ✓ 32 40.i odd 4 1
160.4.o.a 32 1.a even 1 1 trivial
160.4.o.a 32 5.c odd 4 1 inner
160.4.o.a 32 8.d odd 2 1 inner
160.4.o.a 32 40.k even 4 1 inner
200.4.k.j 32 20.d odd 2 1
200.4.k.j 32 20.e even 4 1
200.4.k.j 32 40.f even 2 1
200.4.k.j 32 40.i odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(160, [\chi])\).