↑

Properties

Label 160.6.f.a
Level $160$
Weight $6$
Character orbit 160.f
Analytic conductor $25.661$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [160,6,Mod(49,160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(160, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("160.49");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 160.f (of order \(2\), degree \(1\), 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: \(28\)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 1940 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 1940 q^{9} + 488 q^{15} + 1556 q^{25} - 4368 q^{31} - 23360 q^{39} - 2480 q^{41} - 38420 q^{49} + 48776 q^{55} + 37200 q^{65} + 69232 q^{71} + 35984 q^{79} + 122596 q^{81} - 178744 q^{89} - 89416 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 0 −28.9041 0 −40.5064 − 38.5257i 0 128.100i 0 592.448 0
49.2 0 −28.9041 0 −40.5064 + 38.5257i 0 − 128.100i 0 592.448 0
49.3 0 −21.6833 0 36.8367 − 42.0483i 0 − 236.448i 0 227.167 0
49.4 0 −21.6833 0 36.8367 + 42.0483i 0 236.448i 0 227.167 0
49.5 0 −21.4357 0 48.3867 + 27.9951i 0 − 39.9262i 0 216.491 0
49.6 0 −21.4357 0 48.3867 − 27.9951i 0 39.9262i 0 216.491 0
49.7 0 −16.0077 0 −19.1184 + 52.5308i 0 20.5525i 0 13.2465 0
49.8 0 −16.0077 0 −19.1184 − 52.5308i 0 − 20.5525i 0 13.2465 0
49.9 0 −10.5561 0 −52.7686 − 18.4521i 0 − 47.9937i 0 −131.568 0
49.10 0 −10.5561 0 −52.7686 + 18.4521i 0 47.9937i 0 −131.568 0
49.11 0 −7.17847 0 1.28331 + 55.8870i 0 − 146.905i 0 −191.470 0
49.12 0 −7.17847 0 1.28331 − 55.8870i 0 146.905i 0 −191.470 0
49.13 0 −1.29818 0 51.3939 − 21.9924i 0 170.399i 0 −241.315 0
49.14 0 −1.29818 0 51.3939 + 21.9924i 0 − 170.399i 0 −241.315 0
49.15 0 1.29818 0 −51.3939 + 21.9924i 0 170.399i 0 −241.315 0
49.16 0 1.29818 0 −51.3939 − 21.9924i 0 − 170.399i 0 −241.315 0
49.17 0 7.17847 0 −1.28331 − 55.8870i 0 − 146.905i 0 −191.470 0
49.18 0 7.17847 0 −1.28331 + 55.8870i 0 146.905i 0 −191.470 0
49.19 0 10.5561 0 52.7686 + 18.4521i 0 − 47.9937i 0 −131.568 0
49.20 0 10.5561 0 52.7686 − 18.4521i 0 47.9937i 0 −131.568 0
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.b even 2 1 inner
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 160.6.f.a 28
4.b odd 2 1 40.6.f.a ✓ 28
5.b even 2 1 inner 160.6.f.a 28
5.c odd 4 2 800.6.d.e 28
8.b even 2 1 inner 160.6.f.a 28
8.d odd 2 1 40.6.f.a ✓ 28
20.d odd 2 1 40.6.f.a ✓ 28
20.e even 4 2 200.6.d.e 28
40.e odd 2 1 40.6.f.a ✓ 28
40.f even 2 1 inner 160.6.f.a 28
40.i odd 4 2 800.6.d.e 28
40.k even 4 2 200.6.d.e 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.6.f.a ✓ 28 4.b odd 2 1
40.6.f.a ✓ 28 8.d odd 2 1
40.6.f.a ✓ 28 20.d odd 2 1
40.6.f.a ✓ 28 40.e odd 2 1
160.6.f.a 28 1.a even 1 1 trivial
160.6.f.a 28 5.b even 2 1 inner
160.6.f.a 28 8.b even 2 1 inner
160.6.f.a 28 40.f even 2 1 inner
200.6.d.e 28 20.e even 4 2
200.6.d.e 28 40.k even 4 2
800.6.d.e 28 5.c odd 4 2
800.6.d.e 28 40.i odd 4 2

Hecke kernels

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