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Properties

Label 108.2.l.a
Level $108$
Weight $2$
Character orbit 108.l
Analytic conductor $0.862$
Analytic rank $0$
Dimension $96$
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] = [108,2,Mod(11,108)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(108, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 13]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("108.11");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 108 = 2^{2} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 108.l (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: \(0.862384341830\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 6 q^{2} - 6 q^{4} - 12 q^{5} - 6 q^{6} - 9 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q - 6 q^{2} - 6 q^{4} - 12 q^{5} - 6 q^{6} - 9 q^{8} - 12 q^{9} - 3 q^{10} - 15 q^{12} - 12 q^{13} - 21 q^{14} - 6 q^{16} - 18 q^{17} - 27 q^{18} - 27 q^{20} - 12 q^{21} - 6 q^{22} - 12 q^{24} - 12 q^{25} - 12 q^{28} - 24 q^{29} + 9 q^{30} + 24 q^{32} - 42 q^{33} - 12 q^{34} + 24 q^{36} - 6 q^{37} + 18 q^{38} - 21 q^{40} - 42 q^{41} + 54 q^{42} + 63 q^{44} - 24 q^{45} - 3 q^{46} + 69 q^{48} - 12 q^{49} + 87 q^{50} - 33 q^{52} + 78 q^{54} + 99 q^{56} - 24 q^{57} - 33 q^{58} + 102 q^{60} - 12 q^{61} + 90 q^{62} - 3 q^{64} + 12 q^{65} + 87 q^{66} + 51 q^{68} + 12 q^{69} - 21 q^{70} + 12 q^{72} - 6 q^{73} + 21 q^{74} - 18 q^{76} + 12 q^{77} - 24 q^{78} + 12 q^{81} - 12 q^{82} - 12 q^{84} - 42 q^{85} - 30 q^{86} + 18 q^{88} - 78 q^{90} - 123 q^{92} + 60 q^{93} + 21 q^{94} - 138 q^{96} - 30 q^{97} - 180 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −1.37686 − 0.322907i 1.59825 + 0.667539i 1.79146 + 0.889191i −1.27150 + 3.49343i −1.98500 − 1.43519i −1.63150 − 0.287677i −2.17946 − 1.80276i 2.10878 + 2.13378i 2.87873 − 4.39937i
11.2 −1.36287 − 0.377622i −1.69735 + 0.344959i 1.71480 + 1.02930i 0.420820 − 1.15619i 2.44353 + 0.170825i 1.81474 + 0.319988i −1.94836 − 2.05034i 2.76201 − 1.17103i −1.01013 + 1.41682i
11.3 −1.27725 + 0.607153i 0.971698 + 1.43381i 1.26273 − 1.55097i 1.29726 − 3.56419i −2.11164 − 1.24136i 2.58045 + 0.455003i −0.671144 + 2.74765i −1.11160 + 2.78646i 0.507086 + 5.33999i
11.4 −1.06666 − 0.928568i 0.745195 − 1.56355i 0.275524 + 1.98093i 0.605021 − 1.66228i −2.24673 + 0.975810i 0.748045 + 0.131901i 1.54554 − 2.36882i −1.88937 − 2.33030i −2.18889 + 1.21129i
11.5 −1.00492 + 0.995056i −1.23220 + 1.21724i 0.0197253 − 1.99990i −0.710267 + 1.95144i 0.0270364 − 2.44934i −3.83975 − 0.677051i 1.97019 + 2.02937i 0.0366374 − 2.99978i −1.22804 − 2.66780i
11.6 −0.805437 + 1.16244i 1.23220 − 1.21724i −0.702543 − 1.87255i −0.710267 + 1.95144i 0.422515 + 2.41277i 3.83975 + 0.677051i 2.74258 + 0.691554i 0.0366374 − 2.99978i −1.69636 − 2.39741i
11.7 −0.419899 − 1.35044i −0.537194 + 1.64664i −1.64737 + 1.13410i −0.847966 + 2.32977i 2.44925 + 0.0340252i 4.59553 + 0.810316i 2.22326 + 1.74846i −2.42284 − 1.76913i 3.50227 + 0.166858i
11.8 −0.376137 + 1.36328i −0.971698 − 1.43381i −1.71704 − 1.02556i 1.29726 − 3.56419i 2.32017 − 0.785385i −2.58045 − 0.455003i 2.04396 − 1.95505i −1.11160 + 2.78646i 4.37103 + 3.10915i
11.9 0.0793838 − 1.41198i 1.70299 + 0.315971i −1.98740 − 0.224177i 0.470103 − 1.29160i 0.581335 − 2.37951i −1.57428 − 0.277589i −0.474302 + 2.78838i 2.80032 + 1.07619i −1.78640 − 0.766310i
11.10 0.554410 − 1.30101i −0.490974 − 1.66101i −1.38526 − 1.44259i −0.197421 + 0.542409i −2.43319 − 0.282117i 1.70706 + 0.301001i −2.64482 + 1.00245i −2.51789 + 1.63102i 0.596228 + 0.557564i
11.11 0.557089 + 1.29987i −1.59825 − 0.667539i −1.37930 + 1.44828i −1.27150 + 3.49343i −0.0226541 − 2.44938i 1.63150 + 0.287677i −2.65097 − 0.986086i 2.10878 + 2.13378i −5.24933 + 0.293367i
11.12 0.608544 + 1.27659i 1.69735 − 0.344959i −1.25935 + 1.55372i 0.420820 − 1.15619i 1.47328 + 1.95689i −1.81474 − 0.319988i −2.74983 − 0.662160i 2.76201 − 1.17103i 1.73207 − 0.166382i
11.13 1.09968 + 0.889210i −0.745195 + 1.56355i 0.418610 + 1.95570i 0.605021 − 1.66228i −2.20980 + 1.05677i −0.748045 − 0.131901i −1.27869 + 2.52289i −1.88937 − 2.33030i 2.14345 − 1.28999i
11.14 1.18497 − 0.771906i 0.490974 + 1.66101i 0.808323 − 1.82938i −0.197421 + 0.542409i 1.86393 + 1.58926i −1.70706 − 0.301001i −0.454264 − 2.79171i −2.51789 + 1.63102i 0.184750 + 0.795131i
11.15 1.37675 − 0.323366i −1.70299 − 0.315971i 1.79087 − 0.890387i 0.470103 − 1.29160i −2.44676 + 0.115676i 1.57428 + 0.277589i 2.17765 − 1.80495i 2.80032 + 1.07619i 0.229554 − 1.93022i
11.16 1.40284 + 0.179019i 0.537194 − 1.64664i 1.93590 + 0.502269i −0.847966 + 2.32977i 1.04838 − 2.21380i −4.59553 − 0.810316i 2.62584 + 1.05116i −2.42284 − 1.76913i −1.60663 + 3.11648i
23.1 −1.40830 − 0.129170i 1.33417 − 1.10453i 1.96663 + 0.363822i 0.297855 + 0.0525198i −2.02159 + 1.38318i 0.312070 + 0.371910i −2.72261 − 0.766402i 0.560024 − 2.94727i −0.412685 − 0.112438i
23.2 −1.33631 − 0.462897i −0.808457 + 1.53180i 1.57145 + 1.23715i −3.39370 − 0.598401i 1.78941 − 1.67272i −1.88319 − 2.24430i −1.52728 − 2.38064i −1.69280 − 2.47678i 4.25804 + 2.37058i
23.3 −1.30706 + 0.540000i −1.71606 − 0.234845i 1.41680 − 1.41162i 0.137245 + 0.0242000i 2.36980 − 0.619715i 2.98823 + 3.56123i −1.08956 + 2.61015i 2.88970 + 0.806015i −0.192456 + 0.0424817i
23.4 −1.20024 + 0.747945i 0.644352 + 1.60773i 0.881156 − 1.79543i 3.32285 + 0.585909i −1.97587 − 1.44773i −1.72640 − 2.05745i 0.285283 + 2.81400i −2.16962 + 2.07189i −4.42645 + 1.78208i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
27.f odd 18 1 inner
108.l even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.2.l.a ✓ 96
3.b odd 2 1 324.2.l.a 96
4.b odd 2 1 inner 108.2.l.a ✓ 96
9.c even 3 1 972.2.l.c 96
9.c even 3 1 972.2.l.d 96
9.d odd 6 1 972.2.l.a 96
9.d odd 6 1 972.2.l.b 96
12.b even 2 1 324.2.l.a 96
27.e even 9 1 324.2.l.a 96
27.e even 9 1 972.2.l.a 96
27.e even 9 1 972.2.l.b 96
27.f odd 18 1 inner 108.2.l.a ✓ 96
27.f odd 18 1 972.2.l.c 96
27.f odd 18 1 972.2.l.d 96
36.f odd 6 1 972.2.l.c 96
36.f odd 6 1 972.2.l.d 96
36.h even 6 1 972.2.l.a 96
36.h even 6 1 972.2.l.b 96
108.j odd 18 1 324.2.l.a 96
108.j odd 18 1 972.2.l.a 96
108.j odd 18 1 972.2.l.b 96
108.l even 18 1 inner 108.2.l.a ✓ 96
108.l even 18 1 972.2.l.c 96
108.l even 18 1 972.2.l.d 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.2.l.a ✓ 96 1.a even 1 1 trivial
108.2.l.a ✓ 96 4.b odd 2 1 inner
108.2.l.a ✓ 96 27.f odd 18 1 inner
108.2.l.a ✓ 96 108.l even 18 1 inner
324.2.l.a 96 3.b odd 2 1
324.2.l.a 96 12.b even 2 1
324.2.l.a 96 27.e even 9 1
324.2.l.a 96 108.j odd 18 1
972.2.l.a 96 9.d odd 6 1
972.2.l.a 96 27.e even 9 1
972.2.l.a 96 36.h even 6 1
972.2.l.a 96 108.j odd 18 1
972.2.l.b 96 9.d odd 6 1
972.2.l.b 96 27.e even 9 1
972.2.l.b 96 36.h even 6 1
972.2.l.b 96 108.j odd 18 1
972.2.l.c 96 9.c even 3 1
972.2.l.c 96 27.f odd 18 1
972.2.l.c 96 36.f odd 6 1
972.2.l.c 96 108.l even 18 1
972.2.l.d 96 9.c even 3 1
972.2.l.d 96 27.f odd 18 1
972.2.l.d 96 36.f odd 6 1
972.2.l.d 96 108.l even 18 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(108, [\chi])\).