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Properties

Label 220.2.l.c
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 / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [220,2,Mod(23,220)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://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);
 
Copy content sage: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")
 
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

Copy content comment:select newform
 
Copy content sage:traces = [28,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content 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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 2 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{5} - 12 q^{6} - 2 q^{7} - 14 q^{8} + 6 q^{10} - 8 q^{12} + 6 q^{13} - 20 q^{14} + 8 q^{15} + 8 q^{16} - 14 q^{17} + 34 q^{18} + 4 q^{19} + 8 q^{20} - 16 q^{21}+ \cdots - 20 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −1.41380 − 0.0339977i −2.01721 + 2.01721i 1.99769 + 0.0961322i −0.571381 − 2.16183i 2.92052 − 2.78336i −0.577795 − 0.577795i −2.82107 − 0.203829i − 5.13826i 0.734323 + 3.07584i
23.2 −1.40377 + 0.171537i 1.44333 − 1.44333i 1.94115 − 0.481597i 0.849085 − 2.06859i −1.77852 + 2.27368i 3.60557 + 3.60557i −2.64232 + 1.00903i − 1.16638i −0.837083 + 3.04947i
23.3 −1.38283 + 0.296256i −0.668480 + 0.668480i 1.82447 − 0.819345i −0.500108 + 2.17942i 0.726357 − 1.12244i −1.29641 − 1.29641i −2.28020 + 1.67353i 2.10627i 0.0459001 − 3.16194i
23.4 −1.12494 − 0.857036i 0.635010 − 0.635010i 0.530980 + 1.92823i −1.96236 + 1.07199i −1.25857 + 0.170122i 2.25370 + 2.25370i 1.05524 − 2.62421i 2.19352i 3.12626 + 0.475891i
23.5 −0.892874 + 1.09671i 1.63523 − 1.63523i −0.405551 − 1.95845i 2.22762 + 0.194242i 0.333320 + 3.25343i −2.27323 − 2.27323i 2.50996 + 1.30388i − 2.34794i −2.20201 + 2.26962i
23.6 −0.777125 − 1.18156i 0.810938 − 0.810938i −0.792153 + 1.83644i 0.127766 − 2.23241i −1.58837 − 0.327969i −2.32765 − 2.32765i 2.78545 − 0.491167i 1.68476i −2.73701 + 1.58390i
23.7 0.00742634 − 1.41419i −0.373449 + 0.373449i −1.99989 − 0.0210046i 1.17859 + 1.90025i 0.525356 + 0.530903i 2.76139 + 2.76139i −0.0445564 + 2.82808i 2.72107i 2.69607 − 1.65264i
23.8 0.102586 + 1.41049i −2.05029 + 2.05029i −1.97895 + 0.289393i −2.23513 + 0.0646962i −3.10225 − 2.68158i 3.00953 + 3.00953i −0.611199 − 2.76160i − 5.40741i −0.320547 − 3.14599i
23.9 0.493487 + 1.32532i 0.130983 − 0.130983i −1.51294 + 1.30805i 2.21938 + 0.272654i 0.238233 + 0.108956i 0.450097 + 0.450097i −2.48021 − 1.35962i 2.96569i 0.733881 + 3.07594i
23.10 0.682007 − 1.23890i −1.16674 + 1.16674i −1.06973 − 1.68987i −2.21837 − 0.280768i 0.649749 + 2.24120i −2.18012 − 2.18012i −2.82314 + 0.172785i 0.277416i −1.86079 + 2.55685i
23.11 0.817343 − 1.15410i 1.62701 − 1.62701i −0.663900 − 1.88659i −0.0183109 + 2.23599i −0.547909 − 3.20756i −1.77258 − 1.77258i −2.71995 − 0.775788i − 2.29432i 2.56560 + 1.84871i
23.12 1.20596 + 0.738695i −2.25953 + 2.25953i 0.908659 + 1.78167i 1.86216 + 1.23788i −4.39400 + 1.05579i −0.985925 − 0.985925i −0.220306 + 2.81983i − 7.21094i 1.33127 + 2.86840i
23.13 1.31753 − 0.513913i −0.912547 + 0.912547i 1.47179 − 1.35419i 1.46492 − 1.68939i −0.733342 + 1.67128i −0.0653068 − 0.0653068i 1.24319 − 2.54057i 1.33451i 1.06188 − 2.97866i
23.14 1.36901 + 0.354689i 1.16576 − 1.16576i 1.74839 + 0.971149i −1.42385 − 1.72413i 2.00942 − 1.18246i −1.60129 − 1.60129i 2.04911 + 1.94965i 0.282018i −1.33774 − 2.86539i
67.1 −1.41380 + 0.0339977i −2.01721 − 2.01721i 1.99769 − 0.0961322i −0.571381 + 2.16183i 2.92052 + 2.78336i −0.577795 + 0.577795i −2.82107 + 0.203829i 5.13826i 0.734323 − 3.07584i
67.2 −1.40377 − 0.171537i 1.44333 + 1.44333i 1.94115 + 0.481597i 0.849085 + 2.06859i −1.77852 − 2.27368i 3.60557 − 3.60557i −2.64232 − 1.00903i 1.16638i −0.837083 − 3.04947i
67.3 −1.38283 − 0.296256i −0.668480 − 0.668480i 1.82447 + 0.819345i −0.500108 − 2.17942i 0.726357 + 1.12244i −1.29641 + 1.29641i −2.28020 − 1.67353i − 2.10627i 0.0459001 + 3.16194i
67.4 −1.12494 + 0.857036i 0.635010 + 0.635010i 0.530980 − 1.92823i −1.96236 − 1.07199i −1.25857 − 0.170122i 2.25370 − 2.25370i 1.05524 + 2.62421i − 2.19352i 3.12626 − 0.475891i
67.5 −0.892874 − 1.09671i 1.63523 + 1.63523i −0.405551 + 1.95845i 2.22762 − 0.194242i 0.333320 − 3.25343i −2.27323 + 2.27323i 2.50996 − 1.30388i 2.34794i −2.20201 − 2.26962i
67.6 −0.777125 + 1.18156i 0.810938 + 0.810938i −0.792153 − 1.83644i 0.127766 + 2.23241i −1.58837 + 0.327969i −2.32765 + 2.32765i 2.78545 + 0.491167i − 1.68476i −2.73701 − 1.58390i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.14
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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.c ✓ 28
4.b odd 2 1 220.2.l.d yes 28
5.c odd 4 1 220.2.l.d yes 28
20.e even 4 1 inner 220.2.l.c ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
220.2.l.c ✓ 28 1.a even 1 1 trivial
220.2.l.c ✓ 28 20.e even 4 1 inner
220.2.l.d yes 28 4.b odd 2 1
220.2.l.d yes 28 5.c odd 4 1

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