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Properties

Label 110.2.k.a
Level $110$
Weight $2$
Character orbit 110.k
Analytic conductor $0.878$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [110,2,Mod(7,110)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(110, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([5, 14]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("110.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 110 = 2 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 110.k (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: \(0.878354422234\)
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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 4 q^{3} - 8 q^{5} - 20 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 4 q^{3} - 8 q^{5} - 20 q^{7} + 12 q^{11} - 16 q^{12} - 16 q^{15} + 12 q^{16} - 20 q^{17} - 4 q^{20} - 4 q^{22} - 8 q^{23} - 20 q^{25} + 8 q^{26} + 8 q^{27} - 20 q^{28} + 16 q^{31} - 104 q^{33} - 4 q^{36} + 20 q^{37} - 36 q^{38} - 20 q^{41} - 20 q^{42} + 16 q^{45} + 40 q^{46} + 40 q^{47} + 4 q^{48} + 40 q^{50} + 40 q^{51} + 40 q^{52} + 96 q^{55} - 8 q^{56} + 48 q^{58} + 20 q^{60} + 80 q^{61} + 40 q^{62} + 100 q^{63} + 24 q^{66} + 20 q^{68} - 56 q^{70} - 56 q^{71} - 20 q^{73} - 76 q^{75} - 96 q^{77} + 16 q^{78} - 12 q^{80} - 68 q^{81} - 80 q^{85} - 56 q^{86} - 4 q^{88} - 80 q^{90} - 68 q^{91} - 12 q^{92} + 76 q^{93} + 20 q^{95} + 72 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −0.987688 + 0.156434i −1.45771 − 2.86091i 0.951057 − 0.309017i −1.70643 + 1.44502i 1.88730 + 2.59765i −1.35828 − 0.692077i −0.891007 + 0.453990i −4.29653 + 5.91367i 1.45937 − 1.69418i
7.2 −0.987688 + 0.156434i −0.176422 − 0.346247i 0.951057 − 0.309017i 0.0767418 − 2.23475i 0.228415 + 0.314386i −0.0136870 − 0.00697389i −0.891007 + 0.453990i 1.67459 − 2.30488i 0.273795 + 2.21924i
7.3 −0.987688 + 0.156434i 0.992087 + 1.94708i 0.951057 − 0.309017i 0.244349 + 2.22268i −1.28446 − 1.76791i −3.05681 − 1.55752i −0.891007 + 0.453990i −1.04353 + 1.43630i −0.589044 − 2.15709i
7.4 0.987688 − 0.156434i −1.07350 − 2.10686i 0.951057 − 0.309017i 1.91408 + 1.15599i −1.38987 − 1.91299i −3.37294 − 1.71860i 0.891007 − 0.453990i −1.52312 + 2.09639i 2.07135 + 0.842332i
7.5 0.987688 − 0.156434i −0.757609 − 1.48689i 0.951057 − 0.309017i −1.65111 − 1.50793i −0.980883 − 1.35007i 2.92930 + 1.49255i 0.891007 − 0.453990i 0.126480 − 0.174085i −1.86667 − 1.23107i
7.6 0.987688 − 0.156434i 1.18907 + 2.33368i 0.951057 − 0.309017i −1.05320 − 1.97251i 1.53950 + 2.11894i −2.88379 − 1.46937i 0.891007 − 0.453990i −2.26883 + 3.12278i −1.34880 − 1.78346i
13.1 −0.891007 + 0.453990i −2.20142 − 0.348671i 0.587785 − 0.809017i 2.18748 + 0.463629i 2.11978 − 0.688757i 0.620489 + 3.91761i −0.156434 + 0.987688i 1.87153 + 0.608097i −2.15954 + 0.579997i
13.2 −0.891007 + 0.453990i −1.23025 − 0.194853i 0.587785 − 0.809017i −1.87069 − 1.22496i 1.18463 − 0.384908i −0.466963 − 2.94829i −0.156434 + 0.987688i −1.37761 − 0.447613i 2.22292 + 0.242173i
13.3 −0.891007 + 0.453990i 2.03488 + 0.322293i 0.587785 − 0.809017i 0.667967 − 2.13397i −1.95941 + 0.636650i −0.0611439 − 0.386047i −0.156434 + 0.987688i 1.18368 + 0.384601i 0.373639 + 2.20463i
13.4 0.891007 − 0.453990i −2.79893 − 0.443308i 0.587785 − 0.809017i 0.537022 − 2.17062i −2.69513 + 0.875699i −0.516545 − 3.26134i 0.156434 − 0.987688i 4.78434 + 1.55453i −0.506953 − 2.17784i
13.5 0.891007 − 0.453990i −0.216404 − 0.0342750i 0.587785 − 0.809017i 1.60893 + 1.55285i −0.208378 + 0.0677061i −0.0165943 − 0.104772i 0.156434 − 0.987688i −2.80751 − 0.912216i 2.13855 + 0.653156i
13.6 0.891007 − 0.453990i 1.61854 + 0.256351i 0.587785 − 0.809017i −2.22860 − 0.182641i 1.55851 − 0.506390i 0.119289 + 0.753160i 0.156434 − 0.987688i −0.299227 − 0.0972249i −2.06861 + 0.849027i
17.1 −0.891007 − 0.453990i −2.20142 + 0.348671i 0.587785 + 0.809017i 2.18748 − 0.463629i 2.11978 + 0.688757i 0.620489 − 3.91761i −0.156434 − 0.987688i 1.87153 − 0.608097i −2.15954 − 0.579997i
17.2 −0.891007 − 0.453990i −1.23025 + 0.194853i 0.587785 + 0.809017i −1.87069 + 1.22496i 1.18463 + 0.384908i −0.466963 + 2.94829i −0.156434 − 0.987688i −1.37761 + 0.447613i 2.22292 − 0.242173i
17.3 −0.891007 − 0.453990i 2.03488 − 0.322293i 0.587785 + 0.809017i 0.667967 + 2.13397i −1.95941 − 0.636650i −0.0611439 + 0.386047i −0.156434 − 0.987688i 1.18368 − 0.384601i 0.373639 − 2.20463i
17.4 0.891007 + 0.453990i −2.79893 + 0.443308i 0.587785 + 0.809017i 0.537022 + 2.17062i −2.69513 − 0.875699i −0.516545 + 3.26134i 0.156434 + 0.987688i 4.78434 − 1.55453i −0.506953 + 2.17784i
17.5 0.891007 + 0.453990i −0.216404 + 0.0342750i 0.587785 + 0.809017i 1.60893 − 1.55285i −0.208378 − 0.0677061i −0.0165943 + 0.104772i 0.156434 + 0.987688i −2.80751 + 0.912216i 2.13855 − 0.653156i
17.6 0.891007 + 0.453990i 1.61854 − 0.256351i 0.587785 + 0.809017i −2.22860 + 0.182641i 1.55851 + 0.506390i 0.119289 − 0.753160i 0.156434 + 0.987688i −0.299227 + 0.0972249i −2.06861 − 0.849027i
57.1 −0.453990 − 0.891007i −0.348671 + 2.20142i −0.587785 + 0.809017i −1.49719 + 1.66085i 2.11978 − 0.688757i −3.91761 + 0.620489i 0.987688 + 0.156434i −1.87153 − 0.608097i 2.15954 + 0.579997i
57.2 −0.453990 − 0.891007i −0.194853 + 1.23025i −0.587785 + 0.809017i 0.793405 − 2.09058i 1.18463 − 0.384908i 2.94829 − 0.466963i 0.987688 + 0.156434i 1.37761 + 0.447613i −2.22292 + 0.242173i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.6
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 110.2.k.a ✓ 48
3.b odd 2 1 990.2.bh.c 48
4.b odd 2 1 880.2.cm.c 48
5.b even 2 1 550.2.bh.b 48
5.c odd 4 1 inner 110.2.k.a ✓ 48
5.c odd 4 1 550.2.bh.b 48
11.d odd 10 1 inner 110.2.k.a ✓ 48
15.e even 4 1 990.2.bh.c 48
20.e even 4 1 880.2.cm.c 48
33.f even 10 1 990.2.bh.c 48
44.g even 10 1 880.2.cm.c 48
55.h odd 10 1 550.2.bh.b 48
55.l even 20 1 inner 110.2.k.a ✓ 48
55.l even 20 1 550.2.bh.b 48
165.u odd 20 1 990.2.bh.c 48
220.w odd 20 1 880.2.cm.c 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.2.k.a ✓ 48 1.a even 1 1 trivial
110.2.k.a ✓ 48 5.c odd 4 1 inner
110.2.k.a ✓ 48 11.d odd 10 1 inner
110.2.k.a ✓ 48 55.l even 20 1 inner
550.2.bh.b 48 5.b even 2 1
550.2.bh.b 48 5.c odd 4 1
550.2.bh.b 48 55.h odd 10 1
550.2.bh.b 48 55.l even 20 1
880.2.cm.c 48 4.b odd 2 1
880.2.cm.c 48 20.e even 4 1
880.2.cm.c 48 44.g even 10 1
880.2.cm.c 48 220.w odd 20 1
990.2.bh.c 48 3.b odd 2 1
990.2.bh.c 48 15.e even 4 1
990.2.bh.c 48 33.f even 10 1
990.2.bh.c 48 165.u odd 20 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(110, [\chi])\).