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Properties

Label 618.2.i.b
Level $618$
Weight $2$
Character orbit 618.i
Analytic conductor $4.935$
Analytic rank $0$
Dimension $32$
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: \(32\)
Relative dimension: \(2\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 2 q^{2} - 2 q^{3} - 2 q^{4} + q^{5} - 2 q^{6} + 6 q^{7} - 2 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 2 q^{2} - 2 q^{3} - 2 q^{4} + q^{5} - 2 q^{6} + 6 q^{7} - 2 q^{8} - 2 q^{9} + q^{10} + 5 q^{11} - 2 q^{12} - 14 q^{13} + 6 q^{14} + q^{15} - 2 q^{16} + 32 q^{17} - 2 q^{18} - 3 q^{19} + q^{20} - 11 q^{21} - 12 q^{22} - 15 q^{23} - 2 q^{24} - 15 q^{25} - 31 q^{26} - 2 q^{27} + 6 q^{28} + 7 q^{29} + q^{30} + 6 q^{31} - 2 q^{32} + 5 q^{33} - 2 q^{34} + 41 q^{35} - 2 q^{36} - 13 q^{37} - 3 q^{38} - 14 q^{39} - 16 q^{40} - 5 q^{41} - 11 q^{42} - 3 q^{43} - 12 q^{44} + q^{45} - 66 q^{46} - 14 q^{47} - 2 q^{48} - 6 q^{49} - 15 q^{50} - 2 q^{51} + 20 q^{52} - 6 q^{53} - 2 q^{54} + 11 q^{55} + 6 q^{56} + 14 q^{57} + 24 q^{58} - 43 q^{59} + 18 q^{60} + 48 q^{61} + 6 q^{62} + 6 q^{63} - 2 q^{64} - 84 q^{65} + 5 q^{66} + 7 q^{67} - 2 q^{68} + 19 q^{69} - 10 q^{70} - 59 q^{71} - 2 q^{72} - 9 q^{73} + 21 q^{74} + 19 q^{75} - 3 q^{76} - 33 q^{77} + 20 q^{78} + 33 q^{79} + q^{80} - 2 q^{81} + 46 q^{82} + 71 q^{83} - 11 q^{84} + 42 q^{85} - 20 q^{86} + 24 q^{87} - 12 q^{88} - q^{89} + q^{90} + 50 q^{91} - 15 q^{92} + 6 q^{93} + 20 q^{94} - 5 q^{95} - 2 q^{96} - 61 q^{97} - 57 q^{98} - 12 q^{99}+O(q^{100}) \) 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 −0.227530 + 0.301298i −0.273663 + 0.961826i −2.62063 + 1.62263i −0.850217 − 0.526432i −0.982973 − 0.183750i 0.279019 − 0.254360i
13.2 −0.982973 − 0.183750i 0.0922684 − 0.995734i 0.932472 + 0.361242i 1.41681 − 1.87616i −0.273663 + 0.961826i 3.07013 − 1.90094i −0.850217 − 0.526432i −0.982973 − 0.183750i −1.73743 + 1.58388i
61.1 0.0922684 − 0.995734i 0.739009 − 0.673696i −0.982973 − 0.183750i −0.987359 − 1.98288i −0.602635 − 0.798017i −0.265584 − 0.933430i −0.273663 + 0.961826i 0.0922684 − 0.995734i −2.06553 + 0.800190i
61.2 0.0922684 − 0.995734i 0.739009 − 0.673696i −0.982973 − 0.183750i 0.659970 + 1.32540i −0.602635 − 0.798017i −0.125121 − 0.439755i −0.273663 + 0.961826i 0.0922684 − 0.995734i 1.38064 − 0.534862i
79.1 0.739009 − 0.673696i 0.932472 − 0.361242i 0.0922684 − 0.995734i −0.573838 − 0.355306i 0.445738 − 0.895163i −0.276149 + 0.365681i −0.602635 − 0.798017i 0.739009 − 0.673696i −0.663439 + 0.124018i
79.2 0.739009 − 0.673696i 0.932472 − 0.361242i 0.0922684 − 0.995734i 2.04630 + 1.26701i 0.445738 − 0.895163i 1.60743 − 2.12858i −0.602635 − 0.798017i 0.739009 − 0.673696i 2.36581 − 0.442247i
133.1 0.739009 + 0.673696i 0.932472 + 0.361242i 0.0922684 + 0.995734i −0.573838 + 0.355306i 0.445738 + 0.895163i −0.276149 − 0.365681i −0.602635 + 0.798017i 0.739009 + 0.673696i −0.663439 − 0.124018i
133.2 0.739009 + 0.673696i 0.932472 + 0.361242i 0.0922684 + 0.995734i 2.04630 − 1.26701i 0.445738 + 0.895163i 1.60743 + 2.12858i −0.602635 + 0.798017i 0.739009 + 0.673696i 2.36581 + 0.442247i
169.1 0.932472 + 0.361242i −0.982973 − 0.183750i 0.739009 + 0.673696i −0.117737 − 0.413803i −0.850217 − 0.526432i 1.44653 − 2.90503i 0.445738 + 0.895163i 0.932472 + 0.361242i 0.0396963 − 0.428391i
169.2 0.932472 + 0.361242i −0.982973 − 0.183750i 0.739009 + 0.673696i 1.12572 + 3.95648i −0.850217 − 0.526432i 0.724947 − 1.45589i 0.445738 + 0.895163i 0.932472 + 0.361242i −0.379547 + 4.09597i
175.1 −0.273663 + 0.961826i −0.602635 − 0.798017i −0.850217 − 0.526432i −2.13672 − 0.399423i 0.932472 − 0.361242i −3.35312 − 3.05677i 0.739009 − 0.673696i −0.273663 + 0.961826i 0.968918 − 1.94585i
175.2 −0.273663 + 0.961826i −0.602635 − 0.798017i −0.850217 − 0.526432i 1.26682 + 0.236810i 0.932472 − 0.361242i 2.72924 + 2.48803i 0.739009 − 0.673696i −0.273663 + 0.961826i −0.574452 + 1.15366i
229.1 0.445738 + 0.895163i −0.850217 − 0.526432i −0.602635 + 0.798017i −0.173955 − 0.158581i 0.0922684 − 0.995734i −0.698417 + 0.130557i −0.982973 − 0.183750i 0.445738 + 0.895163i 0.0644174 − 0.226404i
229.2 0.445738 + 0.895163i −0.850217 − 0.526432i −0.602635 + 0.798017i 1.70386 + 1.55327i 0.0922684 − 0.995734i 1.04152 − 0.194694i −0.982973 − 0.183750i 0.445738 + 0.895163i −0.630958 + 2.21759i
343.1 −0.850217 − 0.526432i −0.273663 + 0.961826i 0.445738 + 0.895163i −2.44819 − 0.948433i 0.739009 − 0.673696i 0.338833 + 3.65659i 0.0922684 − 0.995734i −0.850217 − 0.526432i 1.58221 + 2.09518i
343.2 −0.850217 − 0.526432i −0.273663 + 0.961826i 0.445738 + 0.895163i −1.14875 − 0.445028i 0.739009 − 0.673696i −0.243312 − 2.62575i 0.0922684 − 0.995734i −0.850217 − 0.526432i 0.742410 + 0.983110i
373.1 0.932472 − 0.361242i −0.982973 + 0.183750i 0.739009 − 0.673696i −0.117737 + 0.413803i −0.850217 + 0.526432i 1.44653 + 2.90503i 0.445738 − 0.895163i 0.932472 − 0.361242i 0.0396963 + 0.428391i
373.2 0.932472 − 0.361242i −0.982973 + 0.183750i 0.739009 − 0.673696i 1.12572 − 3.95648i −0.850217 + 0.526432i 0.724947 + 1.45589i 0.445738 − 0.895163i 0.932472 − 0.361242i −0.379547 − 4.09597i
385.1 0.0922684 + 0.995734i 0.739009 + 0.673696i −0.982973 + 0.183750i −0.987359 + 1.98288i −0.602635 + 0.798017i −0.265584 + 0.933430i −0.273663 − 0.961826i 0.0922684 + 0.995734i −2.06553 − 0.800190i
385.2 0.0922684 + 0.995734i 0.739009 + 0.673696i −0.982973 + 0.183750i 0.659970 − 1.32540i −0.602635 + 0.798017i −0.125121 + 0.439755i −0.273663 − 0.961826i 0.0922684 + 0.995734i 1.38064 + 0.534862i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.2
Significant digits:
Format:

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.b ✓ 32
103.e even 17 1 inner 618.2.i.b ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
618.2.i.b ✓ 32 1.a even 1 1 trivial
618.2.i.b ✓ 32 103.e even 17 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{32} - T_{5}^{31} + 13 T_{5}^{30} + 43 T_{5}^{29} - 23 T_{5}^{28} + 284 T_{5}^{27} + 902 T_{5}^{26} + \cdots + 18769 \) acting on \(S_{2}^{\mathrm{new}}(618, [\chi])\). Copy content Toggle raw display