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Properties

Label 160.6.o.a
Level $160$
Weight $6$
Character orbit 160.o
Analytic conductor $25.661$
Analytic rank $0$
Dimension $56$
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] = [160,6,Mod(47,160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(160, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("160.47");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 160.o (of order \(4\), degree \(2\), not 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: \(25.6614111701\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(28\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 40)
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.

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 4 q^{3}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q + 4 q^{3} + 8 q^{11} - 408 q^{17} - 3120 q^{25} - 968 q^{27} - 976 q^{33} + 4780 q^{35} - 8 q^{41} - 1308 q^{43} - 20872 q^{51} + 968 q^{57} + 17680 q^{65} - 89252 q^{67} - 25184 q^{73} + 127740 q^{75} - 67792 q^{81} + 126444 q^{83} - 329432 q^{91} + 212576 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 0 −20.2357 + 20.2357i 0 −6.76469 + 55.4909i 0 37.6042 − 37.6042i 0 − 575.965i 0
47.2 0 −20.2357 + 20.2357i 0 6.76469 − 55.4909i 0 −37.6042 + 37.6042i 0 − 575.965i 0
47.3 0 −17.2238 + 17.2238i 0 −54.8737 − 10.6715i 0 −14.7411 + 14.7411i 0 − 350.316i 0
47.4 0 −17.2238 + 17.2238i 0 54.8737 + 10.6715i 0 14.7411 − 14.7411i 0 − 350.316i 0
47.5 0 −11.5932 + 11.5932i 0 48.9682 + 26.9650i 0 −75.8554 + 75.8554i 0 − 25.8041i 0
47.6 0 −11.5932 + 11.5932i 0 −48.9682 − 26.9650i 0 75.8554 − 75.8554i 0 − 25.8041i 0
47.7 0 −10.6033 + 10.6033i 0 15.3629 − 53.7493i 0 159.941 − 159.941i 0 18.1393i 0
47.8 0 −10.6033 + 10.6033i 0 −15.3629 + 53.7493i 0 −159.941 + 159.941i 0 18.1393i 0
47.9 0 −10.0657 + 10.0657i 0 −23.9125 + 50.5291i 0 96.6855 − 96.6855i 0 40.3635i 0
47.10 0 −10.0657 + 10.0657i 0 23.9125 − 50.5291i 0 −96.6855 + 96.6855i 0 40.3635i 0
47.11 0 −4.45554 + 4.45554i 0 49.9881 + 25.0237i 0 169.279 − 169.279i 0 203.296i 0
47.12 0 −4.45554 + 4.45554i 0 −49.9881 − 25.0237i 0 −169.279 + 169.279i 0 203.296i 0
47.13 0 −2.84747 + 2.84747i 0 51.8149 − 20.9812i 0 −44.8183 + 44.8183i 0 226.784i 0
47.14 0 −2.84747 + 2.84747i 0 −51.8149 + 20.9812i 0 44.8183 − 44.8183i 0 226.784i 0
47.15 0 3.12794 − 3.12794i 0 34.4559 + 44.0204i 0 −19.1929 + 19.1929i 0 223.432i 0
47.16 0 3.12794 − 3.12794i 0 −34.4559 − 44.0204i 0 19.1929 − 19.1929i 0 223.432i 0
47.17 0 5.85746 − 5.85746i 0 13.4330 − 54.2637i 0 100.487 − 100.487i 0 174.380i 0
47.18 0 5.85746 − 5.85746i 0 −13.4330 + 54.2637i 0 −100.487 + 100.487i 0 174.380i 0
47.19 0 7.52871 − 7.52871i 0 50.5551 − 23.8576i 0 −51.0903 + 51.0903i 0 129.637i 0
47.20 0 7.52871 − 7.52871i 0 −50.5551 + 23.8576i 0 51.0903 − 51.0903i 0 129.637i 0
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.28
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 160.6.o.a 56
4.b odd 2 1 40.6.k.a ✓ 56
5.c odd 4 1 inner 160.6.o.a 56
8.b even 2 1 40.6.k.a ✓ 56
8.d odd 2 1 inner 160.6.o.a 56
20.e even 4 1 40.6.k.a ✓ 56
40.i odd 4 1 40.6.k.a ✓ 56
40.k even 4 1 inner 160.6.o.a 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.6.k.a ✓ 56 4.b odd 2 1
40.6.k.a ✓ 56 8.b even 2 1
40.6.k.a ✓ 56 20.e even 4 1
40.6.k.a ✓ 56 40.i odd 4 1
160.6.o.a 56 1.a even 1 1 trivial
160.6.o.a 56 5.c odd 4 1 inner
160.6.o.a 56 8.d odd 2 1 inner
160.6.o.a 56 40.k even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(160, [\chi])\).