↑

Properties

Label 220.2.u.a
Level $220$
Weight $2$
Character orbit 220.u
Analytic conductor $1.757$
Analytic rank $0$
Dimension $48$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [220,2,Mod(13,220)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(220, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([0, 15, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("220.13");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 220 = 2^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 220.u (of order \(20\), degree \(8\), 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: \(1.75670884447\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(6\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 2 q^{3} + 4 q^{5} + 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 2 q^{3} + 4 q^{5} + 10 q^{7} - 16 q^{15} + 10 q^{17} + 16 q^{23} - 26 q^{25} - 10 q^{27} + 16 q^{31} + 28 q^{33} - 34 q^{37} - 20 q^{41} - 56 q^{45} - 2 q^{47} - 80 q^{51} + 6 q^{53} - 18 q^{55} - 120 q^{57} - 40 q^{61} - 50 q^{63} - 72 q^{67} + 4 q^{71} - 20 q^{73} + 20 q^{75} - 36 q^{77} + 100 q^{81} + 40 q^{85} - 8 q^{91} - 14 q^{93} + 50 q^{95} + 6 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 0 −2.95041 − 0.467300i 0 1.40317 + 1.74101i 0 −0.372695 − 2.35311i 0 5.63340 + 1.83040i 0
13.2 0 −0.834729 − 0.132208i 0 −2.08639 − 0.804354i 0 −0.228981 − 1.44573i 0 −2.17388 − 0.706335i 0
13.3 0 −0.464116 − 0.0735087i 0 −0.794013 + 2.09035i 0 0.413999 + 2.61389i 0 −2.64317 − 0.858818i 0
13.4 0 0.194131 + 0.0307474i 0 1.80035 − 1.32618i 0 −0.509098 − 3.21432i 0 −2.81643 − 0.915113i 0
13.5 0 2.50020 + 0.395993i 0 −0.797801 − 2.08890i 0 0.371734 + 2.34703i 0 3.24104 + 1.05308i 0
13.6 0 2.95172 + 0.467507i 0 0.0236267 + 2.23594i 0 −0.776647 − 4.90356i 0 5.64095 + 1.83285i 0
17.1 0 −2.95041 + 0.467300i 0 1.40317 − 1.74101i 0 −0.372695 + 2.35311i 0 5.63340 − 1.83040i 0
17.2 0 −0.834729 + 0.132208i 0 −2.08639 + 0.804354i 0 −0.228981 + 1.44573i 0 −2.17388 + 0.706335i 0
17.3 0 −0.464116 + 0.0735087i 0 −0.794013 − 2.09035i 0 0.413999 − 2.61389i 0 −2.64317 + 0.858818i 0
17.4 0 0.194131 − 0.0307474i 0 1.80035 + 1.32618i 0 −0.509098 + 3.21432i 0 −2.81643 + 0.915113i 0
17.5 0 2.50020 − 0.395993i 0 −0.797801 + 2.08890i 0 0.371734 − 2.34703i 0 3.24104 − 1.05308i 0
17.6 0 2.95172 − 0.467507i 0 0.0236267 − 2.23594i 0 −0.776647 + 4.90356i 0 5.64095 − 1.83285i 0
57.1 0 −0.467300 + 2.95041i 0 −0.111852 + 2.23327i 0 2.35311 − 0.372695i 0 −5.63340 − 1.83040i 0
57.2 0 −0.132208 + 0.834729i 0 1.21514 − 1.87708i 0 1.44573 − 0.228981i 0 2.17388 + 0.706335i 0
57.3 0 −0.0735087 + 0.464116i 0 1.87104 + 1.22442i 0 −2.61389 + 0.413999i 0 2.64317 + 0.858818i 0
57.4 0 0.0307474 − 0.194131i 0 −2.23602 − 0.0146855i 0 3.21432 − 0.509098i 0 2.81643 + 0.915113i 0
57.5 0 0.395993 − 2.50020i 0 −0.582392 − 2.15889i 0 −2.34703 + 0.371734i 0 −3.24104 − 1.05308i 0
57.6 0 0.467507 − 2.95172i 0 1.29514 + 1.82280i 0 4.90356 − 0.776647i 0 −5.64095 − 1.83285i 0
73.1 0 −2.69137 + 1.37132i 0 −0.0704176 − 2.23496i 0 −0.328865 + 0.645435i 0 3.59960 − 4.95443i 0
73.2 0 −1.23629 + 0.629920i 0 −2.10058 + 0.766533i 0 0.729926 − 1.43256i 0 −0.631750 + 0.869529i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
11.d odd 10 1 inner
55.l even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 220.2.u.a ✓ 48
4.b odd 2 1 880.2.cm.b 48
5.c odd 4 1 inner 220.2.u.a ✓ 48
11.d odd 10 1 inner 220.2.u.a ✓ 48
20.e even 4 1 880.2.cm.b 48
44.g even 10 1 880.2.cm.b 48
55.l even 20 1 inner 220.2.u.a ✓ 48
220.w odd 20 1 880.2.cm.b 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
220.2.u.a ✓ 48 1.a even 1 1 trivial
220.2.u.a ✓ 48 5.c odd 4 1 inner
220.2.u.a ✓ 48 11.d odd 10 1 inner
220.2.u.a ✓ 48 55.l even 20 1 inner
880.2.cm.b 48 4.b odd 2 1
880.2.cm.b 48 20.e even 4 1
880.2.cm.b 48 44.g even 10 1
880.2.cm.b 48 220.w odd 20 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(220, [\chi])\).