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Properties

Label 220.2.l.d
Level $220$
Weight $2$
Character orbit 220.l
Analytic conductor $1.757$
Analytic rank $0$
Dimension $28$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [220,2,Mod(23,220)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(220, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("220.23");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 220 = 2^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 220.l (of order \(4\), degree \(2\), 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: \(28\)
Relative dimension: \(14\) over \(\Q(i)\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 2 q^{2} + 4 q^{3} - 4 q^{4} + 2 q^{5} - 12 q^{6} + 2 q^{7} + 2 q^{8} - 6 q^{10} + 8 q^{12} + 6 q^{13} + 20 q^{14} - 8 q^{15} + 8 q^{16} - 14 q^{17} - 6 q^{18} - 4 q^{19} + 8 q^{20} - 16 q^{21} - 2 q^{22}+ \cdots + 20 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −1.41049 − 0.102586i 2.05029 − 2.05029i 1.97895 + 0.289393i −2.23513 + 0.0646962i −3.10225 + 2.68158i −3.00953 − 3.00953i −2.76160 − 0.611199i − 5.40741i 3.15926 + 0.138041i
23.2 −1.32532 − 0.493487i −0.130983 + 0.130983i 1.51294 + 1.30805i 2.21938 + 0.272654i 0.238233 − 0.108956i −0.450097 − 0.450097i −1.35962 − 2.48021i 2.96569i −2.80684 − 1.45659i
23.3 −1.09671 + 0.892874i −1.63523 + 1.63523i 0.405551 − 1.95845i 2.22762 + 0.194242i 0.333320 − 3.25343i 2.27323 + 2.27323i 1.30388 + 2.50996i − 2.34794i −2.61648 + 1.77595i
23.4 −0.738695 − 1.20596i 2.25953 − 2.25953i −0.908659 + 1.78167i 1.86216 + 1.23788i −4.39400 − 1.05579i 0.985925 + 0.985925i 2.81983 − 0.220306i − 7.21094i 0.117261 − 3.16010i
23.5 −0.354689 − 1.36901i −1.16576 + 1.16576i −1.74839 + 0.971149i −1.42385 − 1.72413i 2.00942 + 1.18246i 1.60129 + 1.60129i 1.94965 + 2.04911i 0.282018i −1.85534 + 2.56081i
23.6 −0.296256 + 1.38283i 0.668480 − 0.668480i −1.82447 − 0.819345i −0.500108 + 2.17942i 0.726357 + 1.12244i 1.29641 + 1.29641i 1.67353 − 2.28020i 2.10627i −2.86562 − 1.33723i
23.7 −0.171537 + 1.40377i −1.44333 + 1.44333i −1.94115 − 0.481597i 0.849085 − 2.06859i −1.77852 − 2.27368i −3.60557 − 3.60557i 1.00903 − 2.64232i − 1.16638i 2.75818 + 1.54676i
23.8 0.0339977 + 1.41380i 2.01721 − 2.01721i −1.99769 + 0.0961322i −0.571381 − 2.16183i 2.92052 + 2.78336i 0.577795 + 0.577795i −0.203829 − 2.82107i − 5.13826i 3.03699 − 0.881318i
23.9 0.513913 − 1.31753i 0.912547 − 0.912547i −1.47179 − 1.35419i 1.46492 − 1.68939i −0.733342 − 1.67128i 0.0653068 + 0.0653068i −2.54057 + 1.24319i 1.33451i −1.47298 − 2.79827i
23.10 0.857036 + 1.12494i −0.635010 + 0.635010i −0.530980 + 1.92823i −1.96236 + 1.07199i −1.25857 − 0.170122i −2.25370 − 2.25370i −2.62421 + 1.05524i 2.19352i −2.88773 − 1.28880i
23.11 1.15410 − 0.817343i −1.62701 + 1.62701i 0.663900 − 1.88659i −0.0183109 + 2.23599i −0.547909 + 3.20756i 1.77258 + 1.77258i −0.775788 − 2.71995i − 2.29432i 1.80644 + 2.59553i
23.12 1.18156 + 0.777125i −0.810938 + 0.810938i 0.792153 + 1.83644i 0.127766 − 2.23241i −1.58837 + 0.327969i 2.32765 + 2.32765i −0.491167 + 2.78545i 1.68476i 1.88583 − 2.53843i
23.13 1.23890 − 0.682007i 1.16674 − 1.16674i 1.06973 − 1.68987i −2.21837 − 0.280768i 0.649749 − 2.24120i 2.18012 + 2.18012i 0.172785 − 2.82314i 0.277416i −2.93982 + 1.16510i
23.14 1.41419 − 0.00742634i 0.373449 − 0.373449i 1.99989 − 0.0210046i 1.17859 + 1.90025i 0.525356 − 0.530903i −2.76139 − 2.76139i 2.82808 − 0.0445564i 2.72107i 1.68086 + 2.67856i
67.1 −1.41049 + 0.102586i 2.05029 + 2.05029i 1.97895 − 0.289393i −2.23513 − 0.0646962i −3.10225 − 2.68158i −3.00953 + 3.00953i −2.76160 + 0.611199i 5.40741i 3.15926 − 0.138041i
67.2 −1.32532 + 0.493487i −0.130983 − 0.130983i 1.51294 − 1.30805i 2.21938 − 0.272654i 0.238233 + 0.108956i −0.450097 + 0.450097i −1.35962 + 2.48021i − 2.96569i −2.80684 + 1.45659i
67.3 −1.09671 − 0.892874i −1.63523 − 1.63523i 0.405551 + 1.95845i 2.22762 − 0.194242i 0.333320 + 3.25343i 2.27323 − 2.27323i 1.30388 − 2.50996i 2.34794i −2.61648 − 1.77595i
67.4 −0.738695 + 1.20596i 2.25953 + 2.25953i −0.908659 − 1.78167i 1.86216 − 1.23788i −4.39400 + 1.05579i 0.985925 − 0.985925i 2.81983 + 0.220306i 7.21094i 0.117261 + 3.16010i
67.5 −0.354689 + 1.36901i −1.16576 − 1.16576i −1.74839 − 0.971149i −1.42385 + 1.72413i 2.00942 − 1.18246i 1.60129 − 1.60129i 1.94965 − 2.04911i − 0.282018i −1.85534 − 2.56081i
67.6 −0.296256 − 1.38283i 0.668480 + 0.668480i −1.82447 + 0.819345i −0.500108 − 2.17942i 0.726357 − 1.12244i 1.29641 − 1.29641i 1.67353 + 2.28020i − 2.10627i −2.86562 + 1.33723i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.14
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 220.2.l.d yes 28
4.b odd 2 1 220.2.l.c ✓ 28
5.c odd 4 1 220.2.l.c ✓ 28
20.e even 4 1 inner 220.2.l.d yes 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
220.2.l.c ✓ 28 4.b odd 2 1
220.2.l.c ✓ 28 5.c odd 4 1
220.2.l.d yes 28 1.a even 1 1 trivial
220.2.l.d yes 28 20.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{28} - 4 T_{3}^{27} + 8 T_{3}^{26} + 12 T_{3}^{25} + 88 T_{3}^{24} - 340 T_{3}^{23} + 728 T_{3}^{22} + \cdots + 9216 \) acting on \(S_{2}^{\mathrm{new}}(220, [\chi])\). Copy content Toggle raw display