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Properties

Label 111.2.q.b
Level $111$
Weight $2$
Character orbit 111.q
Analytic conductor $0.886$
Analytic rank $0$
Dimension $120$
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] = [111,2,Mod(2,111)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(111, base_ring=CyclotomicField(36))
 
chi = DirichletCharacter(H, H._module([18, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("111.2");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 111 = 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 111.q (of order \(36\), degree \(12\), 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: \(0.886339462436\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(10\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 12 q^{3} - 24 q^{4} - 12 q^{6} - 24 q^{7} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q - 12 q^{3} - 24 q^{4} - 12 q^{6} - 24 q^{7} - 18 q^{9} - 12 q^{10} + 12 q^{12} - 24 q^{13} - 18 q^{15} - 72 q^{16} - 12 q^{18} - 24 q^{19} + 6 q^{21} - 24 q^{24} - 24 q^{25} + 36 q^{27} - 36 q^{30} + 72 q^{31} + 6 q^{33} + 48 q^{34} - 36 q^{37} - 30 q^{39} + 168 q^{40} - 24 q^{42} - 36 q^{45} + 84 q^{46} - 90 q^{48} - 132 q^{49} + 48 q^{51} + 12 q^{52} - 84 q^{54} + 36 q^{55} + 24 q^{57} - 72 q^{58} + 108 q^{60} - 24 q^{61} + 30 q^{63} - 36 q^{64} + 144 q^{66} - 36 q^{67} + 102 q^{69} + 36 q^{70} + 210 q^{72} + 24 q^{76} + 186 q^{78} - 96 q^{79} + 114 q^{81} - 60 q^{82} + 78 q^{84} - 36 q^{85} - 24 q^{87} + 240 q^{90} + 192 q^{91} + 12 q^{93} - 192 q^{94} - 288 q^{96} - 96 q^{97} - 36 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −0.236515 + 2.70338i −0.628301 + 1.61407i −5.28271 − 0.931484i 2.06833 + 0.964480i −4.21485 − 2.08029i 1.79445 − 0.653127i 2.36288 − 8.81837i −2.21048 − 2.02825i −3.09655 + 5.36338i
2.2 −0.172040 + 1.96643i 1.71685 + 0.228992i −1.86764 − 0.329315i −0.601631 − 0.280545i −0.745664 + 3.33666i −0.136259 + 0.0495942i −0.0529044 + 0.197442i 2.89513 + 0.786289i 0.655177 − 1.13480i
2.3 −0.164646 + 1.88192i −1.72734 − 0.127621i −1.54488 − 0.272405i −3.45527 − 1.61122i 0.524573 − 3.22970i −3.09664 + 1.12708i −0.210870 + 0.786979i 2.96743 + 0.440892i 3.60108 − 6.23725i
2.4 −0.147794 + 1.68929i −0.916647 − 1.46961i −0.862251 − 0.152038i 2.79247 + 1.30215i 2.61808 − 1.33128i 0.542408 − 0.197420i −0.493510 + 1.84180i −1.31952 + 2.69423i −2.61242 + 4.52485i
2.5 −0.0448407 + 0.512531i 0.0999558 + 1.72916i 1.70894 + 0.301332i −1.23725 − 0.576941i −0.890732 − 0.0263064i 0.980827 − 0.356992i −0.497391 + 1.85629i −2.98002 + 0.345680i 0.351180 − 0.608261i
2.6 0.0448407 − 0.512531i −1.72025 + 0.201829i 1.70894 + 0.301332i 1.23725 + 0.576941i 0.0263064 + 0.890732i 0.980827 − 0.356992i 0.497391 − 1.85629i 2.91853 − 0.694393i 0.351180 − 0.608261i
2.7 0.147794 − 1.68929i 1.60646 + 0.647525i −0.862251 − 0.152038i −2.79247 − 1.30215i 1.33128 − 2.61808i 0.542408 − 0.197420i 0.493510 − 1.84180i 2.16142 + 2.08045i −2.61242 + 4.52485i
2.8 0.164646 − 1.88192i 0.425632 + 1.67894i −1.54488 − 0.272405i 3.45527 + 1.61122i 3.22970 − 0.524573i −3.09664 + 1.12708i 0.210870 − 0.786979i −2.63767 + 1.42922i 3.60108 − 6.23725i
2.9 0.172040 − 1.96643i −0.523640 − 1.65100i −1.86764 − 0.329315i 0.601631 + 0.280545i −3.33666 + 0.745664i −0.136259 + 0.0495942i 0.0529044 − 0.197442i −2.45160 + 1.72906i 0.655177 − 1.13480i
2.10 0.236515 − 2.70338i −1.48045 + 0.899037i −5.28271 − 0.931484i −2.06833 − 0.964480i 2.08029 + 4.21485i 1.79445 − 0.653127i −2.36288 + 8.81837i 1.38347 − 2.66196i −3.09655 + 5.36338i
5.1 −2.22435 − 1.03723i 1.24469 − 1.20447i 2.58629 + 3.08222i 2.78522 + 1.95024i −4.01793 + 1.38814i 0.634954 + 3.60100i −1.28540 − 4.79718i 0.0984907 − 2.99838i −4.17246 − 7.22691i
5.2 −1.72972 − 0.806582i −1.63427 + 0.573716i 1.05578 + 1.25823i 2.08232 + 1.45806i 3.28958 + 0.325808i −0.592946 − 3.36277i 0.176590 + 0.659045i 2.34170 − 1.87522i −2.42579 − 4.20160i
5.3 −1.46048 − 0.681032i 1.71250 − 0.259487i 0.383615 + 0.457175i −3.20425 − 2.24364i −2.67779 − 0.787293i −0.297594 − 1.68774i 0.585242 + 2.18415i 2.86533 − 0.888746i 3.15174 + 5.45898i
5.4 −1.39629 − 0.651102i −0.280388 + 1.70921i 0.240122 + 0.286167i −0.711439 − 0.498155i 1.50437 − 2.20399i 0.635947 + 3.60664i 0.648535 + 2.42037i −2.84276 − 0.958482i 0.669027 + 1.15879i
5.5 −0.253117 − 0.118031i 1.30672 + 1.13688i −1.23544 − 1.47234i 2.20666 + 1.54512i −0.196567 − 0.441996i −0.217431 − 1.23311i 0.283498 + 1.05803i 0.415019 + 2.97115i −0.376172 − 0.651549i
5.6 0.253117 + 0.118031i −1.73177 + 0.0309570i −1.23544 − 1.47234i −2.20666 − 1.54512i −0.441996 − 0.196567i −0.217431 − 1.23311i −0.283498 − 1.05803i 2.99808 − 0.107221i −0.376172 − 0.651549i
5.7 1.39629 + 0.651102i −0.883866 + 1.48956i 0.240122 + 0.286167i 0.711439 + 0.498155i −2.20399 + 1.50437i 0.635947 + 3.60664i −0.648535 − 2.42037i −1.43756 − 2.63314i 0.669027 + 1.15879i
5.8 1.46048 + 0.681032i −1.14506 − 1.29955i 0.383615 + 0.457175i 3.20425 + 2.24364i −0.787293 − 2.67779i −0.297594 − 1.68774i −0.585242 − 2.18415i −0.377684 + 2.97613i 3.15174 + 5.45898i
5.9 1.72972 + 0.806582i 0.883149 + 1.48998i 1.05578 + 1.25823i −2.08232 − 1.45806i 0.325808 + 3.28958i −0.592946 − 3.36277i −0.176590 − 0.659045i −1.44010 + 2.63175i −2.42579 − 4.20160i
5.10 2.22435 + 1.03723i −0.179265 − 1.72275i 2.58629 + 3.08222i −2.78522 − 1.95024i 1.38814 − 4.01793i 0.634954 + 3.60100i 1.28540 + 4.79718i −2.93573 + 0.617658i −4.17246 − 7.22691i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
37.i odd 36 1 inner
111.q even 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 111.2.q.b ✓ 120
3.b odd 2 1 inner 111.2.q.b ✓ 120
37.i odd 36 1 inner 111.2.q.b ✓ 120
111.q even 36 1 inner 111.2.q.b ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
111.2.q.b ✓ 120 1.a even 1 1 trivial
111.2.q.b ✓ 120 3.b odd 2 1 inner
111.2.q.b ✓ 120 37.i odd 36 1 inner
111.2.q.b ✓ 120 111.q even 36 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{120} + 12 T_{2}^{118} + 90 T_{2}^{116} + 510 T_{2}^{114} + 1599 T_{2}^{112} + \cdots + 28\!\cdots\!41 \) acting on \(S_{2}^{\mathrm{new}}(111, [\chi])\). Copy content Toggle raw display