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Properties

Label 798.2.cf.d
Level $798$
Weight $2$
Character orbit 798.cf
Analytic conductor $6.372$
Analytic rank $0$
Dimension $60$
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] = [798,2,Mod(29,798)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(798, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 0, 17]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("798.29");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 798.cf (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: \(6.37206208130\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(10\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 60 q + 3 q^{5} - 3 q^{6} - 30 q^{7} + 30 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q + 3 q^{5} - 3 q^{6} - 30 q^{7} + 30 q^{8} + 6 q^{9} + 3 q^{10} - 18 q^{11} + 3 q^{15} - 21 q^{17} + 6 q^{18} - 6 q^{19} + 21 q^{22} + 18 q^{23} - 3 q^{25} + 18 q^{26} - 6 q^{27} + 12 q^{29} + 36 q^{30} - 9 q^{31} + 39 q^{33} + 42 q^{34} - 6 q^{35} + 9 q^{36} - 3 q^{38} - 24 q^{39} - 3 q^{40} + 18 q^{41} - 3 q^{42} + 18 q^{43} + 21 q^{44} + 15 q^{45} - 9 q^{46} + 6 q^{47} - 6 q^{48} - 30 q^{49} - 30 q^{50} + 3 q^{51} + 12 q^{53} - 18 q^{54} - 33 q^{55} - 60 q^{56} + 60 q^{57} - 12 q^{58} + 39 q^{59} + 3 q^{60} - 24 q^{61} + 24 q^{62} - 12 q^{63} - 30 q^{64} + 6 q^{65} - 6 q^{66} + 81 q^{67} - 27 q^{68} - 15 q^{69} - 6 q^{70} - 6 q^{71} - 6 q^{72} - 30 q^{73} - 15 q^{74} + 36 q^{75} + 54 q^{78} - 3 q^{80} - 90 q^{81} - 18 q^{82} + 36 q^{83} - 21 q^{85} + 18 q^{86} + 93 q^{87} - 18 q^{88} + 36 q^{90} - 18 q^{92} + 105 q^{93} - 108 q^{95} - 51 q^{97} - 27 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −0.173648 − 0.984808i −1.71535 − 0.239961i −0.939693 + 0.342020i −1.13343 + 3.11406i 0.0615514 + 1.73096i −0.500000 + 0.866025i 0.500000 + 0.866025i 2.88484 + 0.823234i 3.26357 + 0.575456i
29.2 −0.173648 − 0.984808i −1.51926 + 0.831772i −0.939693 + 0.342020i 0.315054 − 0.865604i 1.08295 + 1.35174i −0.500000 + 0.866025i 0.500000 + 0.866025i 1.61631 − 2.52736i −0.907162 − 0.159957i
29.3 −0.173648 − 0.984808i −1.43778 − 0.965803i −0.939693 + 0.342020i 1.50984 − 4.14825i −0.701462 + 1.58365i −0.500000 + 0.866025i 0.500000 + 0.866025i 1.13445 + 2.77723i −4.34741 − 0.766566i
29.4 −0.173648 − 0.984808i −1.09976 − 1.33811i −0.939693 + 0.342020i −0.341079 + 0.937107i −1.12681 + 1.31541i −0.500000 + 0.866025i 0.500000 + 0.866025i −0.581072 + 2.94319i 0.982098 + 0.173170i
29.5 −0.173648 − 0.984808i 0.00562672 + 1.73204i −0.939693 + 0.342020i 1.31493 − 3.61273i 1.70475 − 0.306307i −0.500000 + 0.866025i 0.500000 + 0.866025i −2.99994 + 0.0194914i −3.78618 − 0.667606i
29.6 −0.173648 − 0.984808i 0.0600586 − 1.73101i −0.939693 + 0.342020i −0.563801 + 1.54903i −1.71514 + 0.241440i −0.500000 + 0.866025i 0.500000 + 0.866025i −2.99279 − 0.207924i 1.62340 + 0.286249i
29.7 −0.173648 − 0.984808i 0.475905 + 1.66539i −0.939693 + 0.342020i −1.00798 + 2.76940i 1.55745 − 0.757866i −0.500000 + 0.866025i 0.500000 + 0.866025i −2.54703 + 1.58513i 2.90236 + 0.511765i
29.8 −0.173648 − 0.984808i 1.40087 + 1.01860i −0.939693 + 0.342020i 0.245167 − 0.673592i 0.759868 − 1.55647i −0.500000 + 0.866025i 0.500000 + 0.866025i 0.924898 + 2.85387i −0.705931 − 0.124475i
29.9 −0.173648 − 0.984808i 1.51785 − 0.834342i −0.939693 + 0.342020i −1.02719 + 2.82219i −1.08524 − 1.34991i −0.500000 + 0.866025i 0.500000 + 0.866025i 1.60775 − 2.53282i 2.95769 + 0.521520i
29.10 −0.173648 − 0.984808i 1.54579 − 0.781366i −0.939693 + 0.342020i 0.248799 − 0.683569i −1.03792 − 1.38662i −0.500000 + 0.866025i 0.500000 + 0.866025i 1.77893 − 2.41566i −0.716388 − 0.126319i
71.1 0.939693 − 0.342020i −1.59678 − 0.671041i 0.766044 − 0.642788i 1.69178 − 2.01618i −1.72999 − 0.0844413i −0.500000 − 0.866025i 0.500000 − 0.866025i 2.09941 + 2.14301i 0.900175 − 2.47321i
71.2 0.939693 − 0.342020i −1.44472 + 0.955392i 0.766044 − 0.642788i −0.895992 + 1.06780i −1.03083 + 1.39190i −0.500000 − 0.866025i 0.500000 − 0.866025i 1.17445 − 2.76055i −0.476747 + 1.30985i
71.3 0.939693 − 0.342020i −1.30483 − 1.13905i 0.766044 − 0.642788i −1.87767 + 2.23772i −1.61571 − 0.624076i −0.500000 − 0.866025i 0.500000 − 0.866025i 0.405147 + 2.97252i −0.999087 + 2.74497i
71.4 0.939693 − 0.342020i −1.15465 + 1.29104i 0.766044 − 0.642788i −0.255441 + 0.304423i −0.643450 + 1.60810i −0.500000 − 0.866025i 0.500000 − 0.866025i −0.333582 − 2.98140i −0.135918 + 0.373430i
71.5 0.939693 − 0.342020i −0.574062 − 1.63415i 0.766044 − 0.642788i −0.930833 + 1.10932i −1.09835 − 1.33926i −0.500000 − 0.866025i 0.500000 − 0.866025i −2.34091 + 1.87621i −0.495286 + 1.36079i
71.6 0.939693 − 0.342020i 0.625634 − 1.61511i 0.766044 − 0.642788i 1.41873 − 1.69077i 0.0355032 − 1.73169i −0.500000 − 0.866025i 0.500000 − 0.866025i −2.21716 − 2.02094i 0.754888 − 2.07404i
71.7 0.939693 − 0.342020i 0.675133 + 1.59505i 0.766044 − 0.642788i 2.45180 − 2.92195i 1.17996 + 1.26795i −0.500000 − 0.866025i 0.500000 − 0.866025i −2.08839 + 2.15375i 1.30458 − 3.58430i
71.8 0.939693 − 0.342020i 1.37714 + 1.05047i 0.766044 − 0.642788i −0.757793 + 0.903103i 1.65337 + 0.516108i −0.500000 − 0.866025i 0.500000 − 0.866025i 0.793032 + 2.89329i −0.403213 + 1.10782i
71.9 0.939693 − 0.342020i 1.49252 − 0.878851i 0.766044 − 0.642788i −1.03816 + 1.23723i 1.10193 − 1.33632i −0.500000 − 0.866025i 0.500000 − 0.866025i 1.45524 − 2.62341i −0.552394 + 1.51769i
71.10 0.939693 − 0.342020i 1.73096 + 0.0614366i 0.766044 − 0.642788i 1.45963 − 1.73952i 1.64758 − 0.534292i −0.500000 − 0.866025i 0.500000 − 0.866025i 2.99245 + 0.212689i 0.776653 − 2.13384i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.10
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.j even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 798.2.cf.d yes 60
3.b odd 2 1 798.2.cf.a ✓ 60
19.f odd 18 1 798.2.cf.a ✓ 60
57.j even 18 1 inner 798.2.cf.d yes 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.cf.a ✓ 60 3.b odd 2 1
798.2.cf.a ✓ 60 19.f odd 18 1
798.2.cf.d yes 60 1.a even 1 1 trivial
798.2.cf.d yes 60 57.j even 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{60} - 3 T_{5}^{59} + 6 T_{5}^{58} + 84 T_{5}^{57} - 423 T_{5}^{56} + 723 T_{5}^{55} + \cdots + 48106875271744 \) acting on \(S_{2}^{\mathrm{new}}(798, [\chi])\). Copy content Toggle raw display