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Properties

Label 798.2.cf.a
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.

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 3 q^{3} - 3 q^{5} - 30 q^{7} - 30 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q - 3 q^{3} - 3 q^{5} - 30 q^{7} - 30 q^{8} + 9 q^{9} + 3 q^{10} + 18 q^{11} - 21 q^{15} + 21 q^{17} - 6 q^{18} - 6 q^{19} + 6 q^{21} + 21 q^{22} - 18 q^{23} - 3 q^{24} - 3 q^{25} - 18 q^{26} + 15 q^{27} - 12 q^{29} - 36 q^{30} - 9 q^{31} + 12 q^{33} + 42 q^{34} + 6 q^{35} + 6 q^{36} + 3 q^{38} - 24 q^{39} - 3 q^{40} - 18 q^{41} + 18 q^{43} - 21 q^{44} + 15 q^{45} - 9 q^{46} - 6 q^{47} - 30 q^{49} + 30 q^{50} - 24 q^{51} - 12 q^{53} - 36 q^{54} - 33 q^{55} + 60 q^{56} - 12 q^{57} - 12 q^{58} - 39 q^{59} - 21 q^{60} - 24 q^{61} - 24 q^{62} - 9 q^{63} - 30 q^{64} - 6 q^{65} + 51 q^{66} + 81 q^{67} + 27 q^{68} - 39 q^{69} - 6 q^{70} + 6 q^{71} - 30 q^{73} + 15 q^{74} - 36 q^{75} + 78 q^{78} + 3 q^{80} + 9 q^{81} - 18 q^{82} - 36 q^{83} - 21 q^{85} - 18 q^{86} - 69 q^{87} - 18 q^{88} + 18 q^{92} + 21 q^{93} + 108 q^{95} - 51 q^{97} + 21 q^{99}+O(q^{100}) \) 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 −1.71514 − 0.241440i −0.939693 + 0.342020i 0.563801 − 1.54903i −0.0600586 − 1.73101i −0.500000 + 0.866025i −0.500000 − 0.866025i 2.88341 + 0.828208i 1.62340 + 0.286249i
29.2 0.173648 + 0.984808i −1.12681 − 1.31541i −0.939693 + 0.342020i 0.341079 − 0.937107i 1.09976 − 1.33811i −0.500000 + 0.866025i −0.500000 − 0.866025i −0.460601 + 2.96443i 0.982098 + 0.173170i
29.3 0.173648 + 0.984808i −1.08524 + 1.34991i −0.939693 + 0.342020i 1.02719 − 2.82219i −1.51785 − 0.834342i −0.500000 + 0.866025i −0.500000 − 0.866025i −0.644513 − 2.92995i 2.95769 + 0.521520i
29.4 0.173648 + 0.984808i −1.03792 + 1.38662i −0.939693 + 0.342020i −0.248799 + 0.683569i −1.54579 − 0.781366i −0.500000 + 0.866025i −0.500000 − 0.866025i −0.845449 − 2.87841i −0.716388 − 0.126319i
29.5 0.173648 + 0.984808i −0.701462 − 1.58365i −0.939693 + 0.342020i −1.50984 + 4.14825i 1.43778 − 0.965803i −0.500000 + 0.866025i −0.500000 − 0.866025i −2.01590 + 2.22174i −4.34741 − 0.766566i
29.6 0.173648 + 0.984808i 0.0615514 − 1.73096i −0.939693 + 0.342020i 1.13343 − 3.11406i 1.71535 − 0.239961i −0.500000 + 0.866025i −0.500000 − 0.866025i −2.99242 − 0.213086i 3.26357 + 0.575456i
29.7 0.173648 + 0.984808i 0.759868 + 1.55647i −0.939693 + 0.342020i −0.245167 + 0.673592i −1.40087 + 1.01860i −0.500000 + 0.866025i −0.500000 − 0.866025i −1.84520 + 2.36543i −0.705931 − 0.124475i
29.8 0.173648 + 0.984808i 1.08295 − 1.35174i −0.939693 + 0.342020i −0.315054 + 0.865604i 1.51926 + 0.831772i −0.500000 + 0.866025i −0.500000 − 0.866025i −0.654428 − 2.92775i −0.907162 − 0.159957i
29.9 0.173648 + 0.984808i 1.55745 + 0.757866i −0.939693 + 0.342020i 1.00798 − 2.76940i −0.475905 + 1.66539i −0.500000 + 0.866025i −0.500000 − 0.866025i 1.85128 + 2.36067i 2.90236 + 0.511765i
29.10 0.173648 + 0.984808i 1.70475 + 0.306307i −0.939693 + 0.342020i −1.31493 + 3.61273i −0.00562672 + 1.73204i −0.500000 + 0.866025i −0.500000 − 0.866025i 2.81235 + 1.04435i −3.78618 − 0.667606i
71.1 −0.939693 + 0.342020i −1.72999 + 0.0844413i 0.766044 − 0.642788i −1.69178 + 2.01618i 1.59678 − 0.671041i −0.500000 − 0.866025i −0.500000 + 0.866025i 2.98574 − 0.292166i 0.900175 − 2.47321i
71.2 −0.939693 + 0.342020i −1.61571 + 0.624076i 0.766044 − 0.642788i 1.87767 − 2.23772i 1.30483 − 1.13905i −0.500000 − 0.866025i −0.500000 + 0.866025i 2.22106 − 2.01666i −0.999087 + 2.74497i
71.3 −0.939693 + 0.342020i −1.09835 + 1.33926i 0.766044 − 0.642788i 0.930833 − 1.10932i 0.574062 − 1.63415i −0.500000 − 0.866025i −0.500000 + 0.866025i −0.587235 − 2.94196i −0.495286 + 1.36079i
71.4 −0.939693 + 0.342020i −1.03083 − 1.39190i 0.766044 − 0.642788i 0.895992 − 1.06780i 1.44472 + 0.955392i −0.500000 − 0.866025i −0.500000 + 0.866025i −0.874766 + 2.86963i −0.476747 + 1.30985i
71.5 −0.939693 + 0.342020i −0.643450 − 1.60810i 0.766044 − 0.642788i 0.255441 − 0.304423i 1.15465 + 1.29104i −0.500000 − 0.866025i −0.500000 + 0.866025i −2.17194 + 2.06946i −0.135918 + 0.373430i
71.6 −0.939693 + 0.342020i 0.0355032 + 1.73169i 0.766044 − 0.642788i −1.41873 + 1.69077i −0.625634 − 1.61511i −0.500000 − 0.866025i −0.500000 + 0.866025i −2.99748 + 0.122961i 0.754888 − 2.07404i
71.7 −0.939693 + 0.342020i 1.10193 + 1.33632i 0.766044 − 0.642788i 1.03816 − 1.23723i −1.49252 − 0.878851i −0.500000 − 0.866025i −0.500000 + 0.866025i −0.571513 + 2.94506i −0.552394 + 1.51769i
71.8 −0.939693 + 0.342020i 1.17996 − 1.26795i 0.766044 − 0.642788i −2.45180 + 2.92195i −0.675133 + 1.59505i −0.500000 − 0.866025i −0.500000 + 0.866025i −0.215398 − 2.99226i 1.30458 − 3.58430i
71.9 −0.939693 + 0.342020i 1.64758 + 0.534292i 0.766044 − 0.642788i −1.45963 + 1.73952i −1.73096 + 0.0614366i −0.500000 − 0.866025i −0.500000 + 0.866025i 2.42906 + 1.76058i 0.776653 − 2.13384i
71.10 −0.939693 + 0.342020i 1.65337 − 0.516108i 0.766044 − 0.642788i 0.757793 − 0.903103i −1.37714 + 1.05047i −0.500000 − 0.866025i −0.500000 + 0.866025i 2.46727 − 1.70663i −0.403213 + 1.10782i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.10
Significant digits:
Format:

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

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