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Properties

Label 111.3.n.b
Level $111$
Weight $3$
Character orbit 111.n
Analytic conductor $3.025$
Analytic rank $0$
Dimension $132$
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] = [111,3,Mod(41,111)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(111, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("111.41");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 111 = 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 111.n (of order \(18\), degree \(6\), 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: \(3.02453093440\)
Analytic rank: \(0\)
Dimension: \(132\)
Relative dimension: \(22\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 132 q - 9 q^{3} - 12 q^{4} - 54 q^{7} - 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 132 q - 9 q^{3} - 12 q^{4} - 54 q^{7} - 21 q^{9} - 6 q^{10} - 69 q^{12} - 66 q^{13} - 33 q^{15} - 12 q^{16} - 27 q^{18} - 12 q^{19} + 120 q^{21} - 156 q^{22} + 117 q^{24} - 204 q^{25} - 180 q^{27} + 12 q^{28} + 138 q^{30} + 126 q^{33} + 12 q^{34} + 204 q^{36} + 102 q^{37} + 126 q^{39} + 552 q^{40} - 27 q^{42} - 189 q^{45} + 930 q^{46} - 255 q^{48} + 282 q^{49} + 351 q^{51} - 786 q^{52} - 585 q^{54} - 30 q^{55} + 357 q^{57} + 144 q^{58} - 504 q^{60} - 822 q^{61} - 231 q^{63} - 246 q^{64} - 1215 q^{66} - 276 q^{67} + 108 q^{69} - 234 q^{70} + 15 q^{72} - 408 q^{73} - 294 q^{75} + 318 q^{76} + 231 q^{78} + 240 q^{79} + 531 q^{81} + 432 q^{82} + 96 q^{84} + 498 q^{85} - 39 q^{87} - 450 q^{88} + 1356 q^{90} + 48 q^{91} - 654 q^{93} - 336 q^{94} + 2889 q^{96} - 126 q^{97} + 600 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1 −0.654560 + 3.71219i −2.94630 − 0.565104i −9.59315 − 3.49162i 0.514987 − 0.432126i 4.02630 − 10.5673i −2.28215 + 1.91495i 11.7019 − 20.2683i 8.36132 + 3.32993i 1.26704 + 2.19458i
41.2 −0.606426 + 3.43921i 0.412624 + 2.97149i −7.70167 − 2.80318i −4.56066 + 3.82685i −10.4698 − 0.382886i 7.42985 − 6.23439i 7.32668 − 12.6902i −8.65948 + 2.45222i −10.3956 − 18.0058i
41.3 −0.561673 + 3.18541i 1.47130 − 2.61444i −6.07256 − 2.21023i 4.54790 − 3.81614i 7.50165 + 6.15514i 3.85106 − 3.23143i 3.98218 − 6.89734i −4.67057 − 7.69323i 9.60151 + 16.6303i
41.4 −0.542840 + 3.07860i 2.23755 + 1.99834i −5.42431 − 1.97429i 3.90816 − 3.27933i −7.36672 + 5.80373i −9.92923 + 8.33162i 2.77038 − 4.79844i 1.01324 + 8.94278i 7.97424 + 13.8118i
41.5 −0.465075 + 2.63757i −0.482755 − 2.96090i −2.98171 − 1.08526i −4.28145 + 3.59256i 8.03411 + 0.103741i −2.24101 + 1.88043i −1.10736 + 1.91800i −8.53390 + 2.85878i −7.48445 − 12.9634i
41.6 −0.363231 + 2.05998i −2.03693 + 2.20248i −0.352824 − 0.128418i −1.34675 + 1.13006i −3.79720 − 4.99604i −3.23869 + 2.71759i −3.79083 + 6.56591i −0.701865 − 8.97259i −1.83872 − 3.18475i
41.7 −0.330184 + 1.87256i 2.82960 + 0.996666i 0.361294 + 0.131500i 1.69997 − 1.42645i −2.80061 + 4.96953i 8.66379 − 7.26978i −4.16844 + 7.21995i 7.01331 + 5.64034i 2.10981 + 3.65430i
41.8 −0.312999 + 1.77511i −2.90808 − 0.736925i 0.705741 + 0.256869i 3.91512 − 3.28518i 2.21835 − 4.93149i 6.01627 − 5.04825i −4.28184 + 7.41637i 7.91388 + 4.28608i 4.60610 + 7.97801i
41.9 −0.284368 + 1.61273i 2.89866 − 0.773171i 1.23874 + 0.450865i −5.24557 + 4.40156i 0.422631 + 4.89461i −2.68509 + 2.25306i −4.35460 + 7.54238i 7.80441 − 4.48231i −5.60685 − 9.71134i
41.10 −0.0549311 + 0.311530i −0.180286 + 2.99458i 3.66474 + 1.33386i 5.14703 − 4.31887i −0.922997 − 0.220660i 0.0621182 − 0.0521233i −1.24951 + 2.16422i −8.93499 − 1.07976i 1.06272 + 1.84069i
41.11 −0.0368612 + 0.209050i 2.22751 − 2.00953i 3.71643 + 1.35267i 4.14135 − 3.47500i 0.337983 + 0.539735i −6.66398 + 5.59174i −0.844317 + 1.46240i 0.923611 − 8.95248i 0.573795 + 0.993842i
41.12 0.0368612 − 0.209050i −2.78047 − 1.12648i 3.71643 + 1.35267i −4.14135 + 3.47500i −0.337983 + 0.539735i −6.66398 + 5.59174i 0.844317 − 1.46240i 6.46207 + 6.26431i 0.573795 + 0.993842i
41.13 0.0549311 − 0.311530i 1.19362 + 2.75232i 3.66474 + 1.33386i −5.14703 + 4.31887i 0.922997 − 0.220660i 0.0621182 − 0.0521233i 1.24951 − 2.16422i −6.15055 + 6.57045i 1.06272 + 1.84069i
41.14 0.284368 − 1.61273i −2.98829 + 0.264856i 1.23874 + 0.450865i 5.24557 − 4.40156i −0.422631 + 4.89461i −2.68509 + 2.25306i 4.35460 − 7.54238i 8.85970 − 1.58293i −5.60685 − 9.71134i
41.15 0.312999 − 1.77511i 2.48066 − 1.68711i 0.705741 + 0.256869i −3.91512 + 3.28518i −2.21835 − 4.93149i 6.01627 − 5.04825i 4.28184 − 7.41637i 3.30735 − 8.37027i 4.60610 + 7.97801i
41.16 0.330184 − 1.87256i −2.31808 + 1.90434i 0.361294 + 0.131500i −1.69997 + 1.42645i 2.80061 + 4.96953i 8.66379 − 7.26978i 4.16844 − 7.21995i 1.74697 − 8.82882i 2.10981 + 3.65430i
41.17 0.363231 − 2.05998i 2.66738 + 1.37299i −0.352824 − 0.128418i 1.34675 − 1.13006i 3.79720 − 4.99604i −3.23869 + 2.71759i 3.79083 − 6.56591i 5.22981 + 7.32455i −1.83872 − 3.18475i
41.18 0.465075 − 2.63757i −0.559047 − 2.94745i −2.98171 − 1.08526i 4.28145 − 3.59256i −8.03411 + 0.103741i −2.24101 + 1.88043i 1.10736 − 1.91800i −8.37493 + 3.29553i −7.48445 − 12.9634i
41.19 0.542840 − 3.07860i −1.41913 + 2.64312i −5.42431 − 1.97429i −3.90816 + 3.27933i 7.36672 + 5.80373i −9.92923 + 8.33162i −2.77038 + 4.79844i −4.97212 − 7.50187i 7.97424 + 13.8118i
41.20 0.561673 − 3.18541i −2.27676 − 1.95355i −6.07256 − 2.21023i −4.54790 + 3.81614i −7.50165 + 6.15514i 3.85106 − 3.23143i −3.98218 + 6.89734i 1.36725 + 8.89554i 9.60151 + 16.6303i
See next 80 embeddings (of 132 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
37.h even 18 1 inner
111.n odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 111.3.n.b ✓ 132
3.b odd 2 1 inner 111.3.n.b ✓ 132
37.h even 18 1 inner 111.3.n.b ✓ 132
111.n odd 18 1 inner 111.3.n.b ✓ 132
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
111.3.n.b ✓ 132 1.a even 1 1 trivial
111.3.n.b ✓ 132 3.b odd 2 1 inner
111.3.n.b ✓ 132 37.h even 18 1 inner
111.3.n.b ✓ 132 111.n odd 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{132} + 6 T_{2}^{130} + 3 T_{2}^{128} + 7075 T_{2}^{126} + 41121 T_{2}^{124} + \cdots + 72\!\cdots\!41 \) acting on \(S_{3}^{\mathrm{new}}(111, [\chi])\). Copy content Toggle raw display