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Properties

Label 1161.2.e.a
Level $1161$
Weight $2$
Character orbit 1161.e
Analytic conductor $9.271$
Analytic rank $0$
Dimension $84$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1161,2,Mod(208,1161)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1161, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1161.208");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1161 = 3^{3} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1161.e (of order \(3\), degree \(2\), not 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: \(9.27063167467\)
Analytic rank: \(0\)
Dimension: \(84\)
Relative dimension: \(42\) over \(\Q(\zeta_{3})\)
Twist minimal: no (minimal twist has level 387)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 84 q - 42 q^{4} - 5 q^{5} + 3 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 84 q - 42 q^{4} - 5 q^{5} + 3 q^{7} + 12 q^{8} + q^{11} - 14 q^{14} - 42 q^{16} - q^{17} - 14 q^{20} + 9 q^{22} - q^{23} - 33 q^{25} - 12 q^{26} + 12 q^{28} - 19 q^{29} - 3 q^{31} - 22 q^{32} + 6 q^{34} + 4 q^{35} - 44 q^{38} + 6 q^{40} + q^{41} - 10 q^{44} - 12 q^{46} - 3 q^{47} - 39 q^{49} + 108 q^{50} - 9 q^{52} - 18 q^{53} + 3 q^{55} + 24 q^{56} + 6 q^{58} - 2 q^{59} - 18 q^{61} - 4 q^{62} + 72 q^{64} + 18 q^{65} + 18 q^{67} + 68 q^{68} - 21 q^{70} - 34 q^{71} + 21 q^{73} + 36 q^{74} + 6 q^{76} - 14 q^{77} + 7 q^{80} - 6 q^{82} + 18 q^{83} + 3 q^{85} - 89 q^{86} - 9 q^{88} - 40 q^{89} - 27 q^{91} - 66 q^{92} + 36 q^{94} - 14 q^{95} + 6 q^{97} - 41 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
208.1 −1.37404 + 2.37990i 0 −2.77595 − 4.80808i −1.60064 + 2.77239i 0 0.964248 + 1.67013i 9.76087 0 −4.39867 − 7.61871i
208.2 −1.32377 + 2.29284i 0 −2.50475 − 4.33836i 1.71703 − 2.97399i 0 2.36721 + 4.10013i 7.96781 0 4.54593 + 7.87377i
208.3 −1.30221 + 2.25550i 0 −2.39152 − 4.14224i 1.82902 − 3.16796i 0 −1.30133 − 2.25397i 7.24823 0 4.76355 + 8.25071i
208.4 −1.27354 + 2.20584i 0 −2.24383 − 3.88643i −0.741421 + 1.28418i 0 −2.19591 − 3.80343i 6.33630 0 −1.88847 − 3.27092i
208.5 −1.26092 + 2.18398i 0 −2.17984 − 3.77559i −0.0431929 + 0.0748123i 0 0.0774291 + 0.134111i 5.95071 0 −0.108926 − 0.188665i
208.6 −1.06621 + 1.84672i 0 −1.27360 − 2.20593i −1.09642 + 1.89905i 0 1.08248 + 1.87491i 1.16684 0 −2.33802 − 4.04956i
208.7 −0.987846 + 1.71100i 0 −0.951679 − 1.64836i −0.499189 + 0.864620i 0 −0.0111031 − 0.0192311i −0.190934 0 −0.986243 − 1.70822i
208.8 −0.980054 + 1.69750i 0 −0.921010 − 1.59524i −0.342846 + 0.593826i 0 −0.864414 − 1.49721i −0.309657 0 −0.672015 − 1.16396i
208.9 −0.943102 + 1.63350i 0 −0.778882 − 1.34906i −2.01140 + 3.48385i 0 2.10616 + 3.64797i −0.834146 0 −3.79391 − 6.57125i
208.10 −0.838521 + 1.45236i 0 −0.406235 − 0.703619i 1.68953 − 2.92635i 0 −0.169732 − 0.293985i −1.99154 0 2.83341 + 4.90761i
208.11 −0.802341 + 1.38970i 0 −0.287504 − 0.497971i 1.17921 − 2.04245i 0 0.575775 + 0.997272i −2.28666 0 1.89226 + 3.27749i
208.12 −0.727761 + 1.26052i 0 −0.0592724 − 0.102663i 1.47541 − 2.55549i 0 2.35432 + 4.07780i −2.73850 0 2.14749 + 3.71957i
208.13 −0.719445 + 1.24612i 0 −0.0352032 − 0.0609738i 0.000595936 − 0.00103219i 0 −1.91773 − 3.32160i −2.77647 0 0.000857487 0.00148521i
208.14 −0.583082 + 1.00993i 0 0.320030 + 0.554308i 0.190265 − 0.329549i 0 0.0777829 + 0.134724i −3.07874 0 0.221880 + 0.384308i
208.15 −0.566083 + 0.980484i 0 0.359100 + 0.621980i −2.16437 + 3.74879i 0 −1.68506 − 2.91861i −3.07745 0 −2.45042 − 4.24425i
208.16 −0.427838 + 0.741038i 0 0.633909 + 1.09796i −0.990795 + 1.71611i 0 −1.79315 − 3.10582i −2.79619 0 −0.847800 − 1.46843i
208.17 −0.276499 + 0.478910i 0 0.847097 + 1.46721i −0.163630 + 0.283415i 0 0.722165 + 1.25083i −2.04288 0 −0.0904869 − 0.156728i
208.18 −0.249829 + 0.432716i 0 0.875171 + 1.51584i −0.591808 + 1.02504i 0 2.30832 + 3.99813i −1.87389 0 −0.295701 − 0.512169i
208.19 −0.154285 + 0.267230i 0 0.952392 + 1.64959i −0.112687 + 0.195179i 0 −0.712798 − 1.23460i −1.20490 0 −0.0347719 − 0.0602266i
208.20 −0.130747 + 0.226460i 0 0.965811 + 1.67283i 1.49605 − 2.59123i 0 −1.47947 − 2.56251i −1.02809 0 0.391206 + 0.677589i
See all 84 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 208.42
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
387.e even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1161.2.e.a 84
3.b odd 2 1 387.2.e.a ✓ 84
9.c even 3 1 1161.2.g.a 84
9.d odd 6 1 387.2.g.a yes 84
43.c even 3 1 1161.2.g.a 84
129.f odd 6 1 387.2.g.a yes 84
387.e even 3 1 inner 1161.2.e.a 84
387.p odd 6 1 387.2.e.a ✓ 84
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
387.2.e.a ✓ 84 3.b odd 2 1
387.2.e.a ✓ 84 387.p odd 6 1
387.2.g.a yes 84 9.d odd 6 1
387.2.g.a yes 84 129.f odd 6 1
1161.2.e.a 84 1.a even 1 1 trivial
1161.2.e.a 84 387.e even 3 1 inner
1161.2.g.a 84 9.c even 3 1
1161.2.g.a 84 43.c even 3 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1161, [\chi])\).