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Properties

Label 618.2.i.e
Level $618$
Weight $2$
Character orbit 618.i
Analytic conductor $4.935$
Analytic rank $0$
Dimension $80$
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] = [618,2,Mod(13,618)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(618, base_ring=CyclotomicField(34))
 
chi = DirichletCharacter(H, H._module([0, 24]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("618.13");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 618 = 2 \cdot 3 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 618.i (of order \(17\), degree \(16\), 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: \(4.93475484492\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(5\) over \(\Q(\zeta_{17})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{17}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 5 q^{2} + 5 q^{3} - 5 q^{4} + 14 q^{5} + 5 q^{6} - 2 q^{7} - 5 q^{8} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q - 5 q^{2} + 5 q^{3} - 5 q^{4} + 14 q^{5} + 5 q^{6} - 2 q^{7} - 5 q^{8} - 5 q^{9} - 3 q^{10} + 18 q^{11} + 5 q^{12} + 14 q^{13} - 2 q^{14} + 3 q^{15} - 5 q^{16} - 4 q^{17} - 5 q^{18} - 14 q^{19} - 3 q^{20} + 2 q^{21} - 16 q^{22} - 13 q^{23} + 5 q^{24} + q^{25} + 31 q^{26} + 5 q^{27} - 2 q^{28} - 22 q^{29} - 14 q^{30} + 26 q^{31} - 5 q^{32} + 33 q^{33} - 4 q^{34} - 2 q^{35} - 5 q^{36} - 25 q^{37} - 14 q^{38} + 20 q^{39} + 14 q^{40} - 35 q^{41} + 19 q^{42} + 16 q^{43} + 18 q^{44} - 3 q^{45} + 72 q^{46} - 34 q^{47} + 5 q^{48} - 5 q^{49} - 16 q^{50} + 4 q^{51} - 3 q^{52} - 72 q^{53} + 5 q^{54} + 6 q^{55} + 32 q^{56} - 54 q^{57} - 5 q^{58} - 96 q^{59} + 3 q^{60} + 4 q^{61} - 8 q^{62} - 2 q^{63} - 5 q^{64} + 31 q^{65} - q^{66} + 48 q^{67} - 4 q^{68} - 4 q^{69} + 32 q^{70} - 14 q^{71} - 5 q^{72} + 48 q^{73} - 8 q^{74} + 16 q^{75} + 3 q^{76} + 2 q^{77} + 3 q^{78} + 34 q^{79} - 3 q^{80} - 5 q^{81} - q^{82} + 24 q^{83} - 15 q^{84} + 26 q^{85} + 16 q^{86} + 5 q^{87} + q^{88} - 18 q^{89} - 3 q^{90} + 46 q^{91} - 13 q^{92} - 26 q^{93} - 34 q^{94} - 45 q^{95} + 5 q^{96} + 89 q^{97} - 39 q^{98} - 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −0.982973 − 0.183750i −0.0922684 + 0.995734i 0.932472 + 0.361242i −1.64516 + 2.17854i 0.273663 − 0.961826i −3.55337 + 2.20015i −0.850217 − 0.526432i −0.982973 − 0.183750i 2.01746 − 1.83915i
13.2 −0.982973 − 0.183750i −0.0922684 + 0.995734i 0.932472 + 0.361242i −1.33065 + 1.76206i 0.273663 − 0.961826i 3.72476 − 2.30628i −0.850217 − 0.526432i −0.982973 − 0.183750i 1.63177 − 1.48755i
13.3 −0.982973 − 0.183750i −0.0922684 + 0.995734i 0.932472 + 0.361242i −0.134939 + 0.178688i 0.273663 − 0.961826i −0.960451 + 0.594686i −0.850217 − 0.526432i −0.982973 − 0.183750i 0.165475 − 0.150851i
13.4 −0.982973 − 0.183750i −0.0922684 + 0.995734i 0.932472 + 0.361242i 0.805170 − 1.06622i 0.273663 − 0.961826i −0.0286951 + 0.0177673i −0.850217 − 0.526432i −0.982973 − 0.183750i −0.987378 + 0.900114i
13.5 −0.982973 − 0.183750i −0.0922684 + 0.995734i 0.932472 + 0.361242i 2.51067 − 3.32466i 0.273663 − 0.961826i −1.50492 + 0.931808i −0.850217 − 0.526432i −0.982973 − 0.183750i −3.07882 + 2.80672i
61.1 0.0922684 − 0.995734i −0.739009 + 0.673696i −0.982973 − 0.183750i −1.80139 − 3.61768i 0.602635 + 0.798017i −1.37164 − 4.82083i −0.273663 + 0.961826i 0.0922684 − 0.995734i −3.76846 + 1.45991i
61.2 0.0922684 − 0.995734i −0.739009 + 0.673696i −0.982973 − 0.183750i −0.724571 − 1.45513i 0.602635 + 0.798017i 0.780553 + 2.74336i −0.273663 + 0.961826i 0.0922684 − 0.995734i −1.51578 + 0.587217i
61.3 0.0922684 − 0.995734i −0.739009 + 0.673696i −0.982973 − 0.183750i −0.581356 − 1.16752i 0.602635 + 0.798017i 0.529313 + 1.86034i −0.273663 + 0.961826i 0.0922684 − 0.995734i −1.21618 + 0.471151i
61.4 0.0922684 − 0.995734i −0.739009 + 0.673696i −0.982973 − 0.183750i 0.566028 + 1.13674i 0.602635 + 0.798017i −0.812033 − 2.85400i −0.273663 + 0.961826i 0.0922684 − 0.995734i 1.18412 − 0.458729i
61.5 0.0922684 − 0.995734i −0.739009 + 0.673696i −0.982973 − 0.183750i 1.61662 + 3.24661i 0.602635 + 0.798017i −0.407831 − 1.43338i −0.273663 + 0.961826i 0.0922684 − 0.995734i 3.38192 − 1.31016i
79.1 0.739009 − 0.673696i −0.932472 + 0.361242i 0.0922684 − 0.995734i −2.35292 − 1.45687i −0.445738 + 0.895163i −0.0712563 + 0.0943586i −0.602635 − 0.798017i 0.739009 − 0.673696i −2.72031 + 0.508515i
79.2 0.739009 − 0.673696i −0.932472 + 0.361242i 0.0922684 − 0.995734i −1.32246 − 0.818833i −0.445738 + 0.895163i 0.232560 − 0.307960i −0.602635 − 0.798017i 0.739009 − 0.673696i −1.52895 + 0.285811i
79.3 0.739009 − 0.673696i −0.932472 + 0.361242i 0.0922684 − 0.995734i 0.854361 + 0.528998i −0.445738 + 0.895163i −2.00389 + 2.65357i −0.602635 − 0.798017i 0.739009 − 0.673696i 0.987763 − 0.184645i
79.4 0.739009 − 0.673696i −0.932472 + 0.361242i 0.0922684 − 0.995734i 2.09060 + 1.29444i −0.445738 + 0.895163i −2.45156 + 3.24639i −0.602635 − 0.798017i 0.739009 − 0.673696i 2.41703 − 0.451821i
79.5 0.739009 − 0.673696i −0.932472 + 0.361242i 0.0922684 − 0.995734i 2.86948 + 1.77671i −0.445738 + 0.895163i 2.50222 − 3.31348i −0.602635 − 0.798017i 0.739009 − 0.673696i 3.31754 − 0.620155i
133.1 0.739009 + 0.673696i −0.932472 − 0.361242i 0.0922684 + 0.995734i −2.35292 + 1.45687i −0.445738 − 0.895163i −0.0712563 − 0.0943586i −0.602635 + 0.798017i 0.739009 + 0.673696i −2.72031 − 0.508515i
133.2 0.739009 + 0.673696i −0.932472 − 0.361242i 0.0922684 + 0.995734i −1.32246 + 0.818833i −0.445738 − 0.895163i 0.232560 + 0.307960i −0.602635 + 0.798017i 0.739009 + 0.673696i −1.52895 − 0.285811i
133.3 0.739009 + 0.673696i −0.932472 − 0.361242i 0.0922684 + 0.995734i 0.854361 − 0.528998i −0.445738 − 0.895163i −2.00389 − 2.65357i −0.602635 + 0.798017i 0.739009 + 0.673696i 0.987763 + 0.184645i
133.4 0.739009 + 0.673696i −0.932472 − 0.361242i 0.0922684 + 0.995734i 2.09060 − 1.29444i −0.445738 − 0.895163i −2.45156 − 3.24639i −0.602635 + 0.798017i 0.739009 + 0.673696i 2.41703 + 0.451821i
133.5 0.739009 + 0.673696i −0.932472 − 0.361242i 0.0922684 + 0.995734i 2.86948 − 1.77671i −0.445738 − 0.895163i 2.50222 + 3.31348i −0.602635 + 0.798017i 0.739009 + 0.673696i 3.31754 + 0.620155i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.5
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
103.e even 17 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 618.2.i.e ✓ 80
103.e even 17 1 inner 618.2.i.e ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
618.2.i.e ✓ 80 1.a even 1 1 trivial
618.2.i.e ✓ 80 103.e even 17 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{80} - 14 T_{5}^{79} + 110 T_{5}^{78} - 701 T_{5}^{77} + 3946 T_{5}^{76} - 19532 T_{5}^{75} + \cdots + 47\!\cdots\!44 \) acting on \(S_{2}^{\mathrm{new}}(618, [\chi])\). Copy content Toggle raw display