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Properties

Label 220.2.t.a
Level $220$
Weight $2$
Character orbit 220.t
Analytic conductor $1.757$
Analytic rank $0$
Dimension $24$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [220,2,Mod(9,220)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(220, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 5, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("220.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 220 = 2^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 220.t (of order \(10\), degree \(4\), 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: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + q^{5} + 14 q^{9} - 2 q^{11} - q^{15} + 8 q^{19} - 28 q^{21} + 27 q^{25} - 16 q^{29} - 26 q^{31} + 17 q^{35} + 12 q^{39} + 10 q^{41} - 40 q^{45} - 46 q^{49} - 12 q^{51} - 33 q^{55} - 48 q^{59} - 10 q^{61}+ \cdots + 156 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 0 −1.60349 − 2.20702i 0 2.23079 − 0.153527i 0 1.00600 − 1.38464i 0 −1.37268 + 4.22469i 0
9.2 0 −1.22731 − 1.68925i 0 −2.22531 − 0.219109i 0 −1.67492 + 2.30533i 0 −0.420216 + 1.29329i 0
9.3 0 −0.447653 − 0.616141i 0 −0.909494 + 2.04275i 0 2.82695 − 3.89096i 0 0.747814 − 2.30153i 0
9.4 0 0.447653 + 0.616141i 0 1.93649 + 1.11803i 0 −2.82695 + 3.89096i 0 0.747814 − 2.30153i 0
9.5 0 1.22731 + 1.68925i 0 1.67152 − 1.48527i 0 1.67492 − 2.30533i 0 −0.420216 + 1.29329i 0
9.6 0 1.60349 + 2.20702i 0 −1.89499 + 1.18702i 0 −1.00600 + 1.38464i 0 −1.37268 + 4.22469i 0
49.1 0 −1.60349 + 2.20702i 0 2.23079 + 0.153527i 0 1.00600 + 1.38464i 0 −1.37268 − 4.22469i 0
49.2 0 −1.22731 + 1.68925i 0 −2.22531 + 0.219109i 0 −1.67492 − 2.30533i 0 −0.420216 − 1.29329i 0
49.3 0 −0.447653 + 0.616141i 0 −0.909494 − 2.04275i 0 2.82695 + 3.89096i 0 0.747814 + 2.30153i 0
49.4 0 0.447653 − 0.616141i 0 1.93649 − 1.11803i 0 −2.82695 − 3.89096i 0 0.747814 + 2.30153i 0
49.5 0 1.22731 − 1.68925i 0 1.67152 + 1.48527i 0 1.67492 + 2.30533i 0 −0.420216 − 1.29329i 0
49.6 0 1.60349 − 2.20702i 0 −1.89499 − 1.18702i 0 −1.00600 − 1.38464i 0 −1.37268 − 4.22469i 0
69.1 0 −2.93917 + 0.954994i 0 −1.85993 − 1.24124i 0 2.42366 + 0.787496i 0 5.29965 − 3.85042i 0
69.2 0 −1.89864 + 0.616907i 0 2.21463 + 0.308858i 0 −0.430637 − 0.139922i 0 0.797226 − 0.579219i 0
69.3 0 −0.989226 + 0.321419i 0 −1.25698 + 1.84933i 0 −2.15207 − 0.699249i 0 −1.55179 + 1.12744i 0
69.4 0 0.989226 − 0.321419i 0 1.37039 − 1.76693i 0 2.15207 + 0.699249i 0 −1.55179 + 1.12744i 0
69.5 0 1.89864 − 0.616907i 0 0.978101 + 2.01080i 0 0.430637 + 0.139922i 0 0.797226 − 0.579219i 0
69.6 0 2.93917 − 0.954994i 0 −1.75524 − 1.38533i 0 −2.42366 − 0.787496i 0 5.29965 − 3.85042i 0
169.1 0 −2.93917 − 0.954994i 0 −1.85993 + 1.24124i 0 2.42366 − 0.787496i 0 5.29965 + 3.85042i 0
169.2 0 −1.89864 − 0.616907i 0 2.21463 − 0.308858i 0 −0.430637 + 0.139922i 0 0.797226 + 0.579219i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.c even 5 1 inner
55.j even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 220.2.t.a ✓ 24
4.b odd 2 1 880.2.cd.d 24
5.b even 2 1 inner 220.2.t.a ✓ 24
5.c odd 4 2 1100.2.n.f 24
11.c even 5 1 inner 220.2.t.a ✓ 24
11.c even 5 1 2420.2.b.i 12
11.d odd 10 1 2420.2.b.h 12
20.d odd 2 1 880.2.cd.d 24
44.h odd 10 1 880.2.cd.d 24
55.h odd 10 1 2420.2.b.h 12
55.j even 10 1 inner 220.2.t.a ✓ 24
55.j even 10 1 2420.2.b.i 12
55.k odd 20 2 1100.2.n.f 24
220.n odd 10 1 880.2.cd.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
220.2.t.a ✓ 24 1.a even 1 1 trivial
220.2.t.a ✓ 24 5.b even 2 1 inner
220.2.t.a ✓ 24 11.c even 5 1 inner
220.2.t.a ✓ 24 55.j even 10 1 inner
880.2.cd.d 24 4.b odd 2 1
880.2.cd.d 24 20.d odd 2 1
880.2.cd.d 24 44.h odd 10 1
880.2.cd.d 24 220.n odd 10 1
1100.2.n.f 24 5.c odd 4 2
1100.2.n.f 24 55.k odd 20 2
2420.2.b.h 12 11.d odd 10 1
2420.2.b.h 12 55.h odd 10 1
2420.2.b.i 12 11.c even 5 1
2420.2.b.i 12 55.j even 10 1

Hecke kernels

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