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Properties

Label 819.2.fn.g
Level $819$
Weight $2$
Character orbit 819.fn
Analytic conductor $6.540$
Analytic rank $0$
Dimension $36$
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] = [819,2,Mod(73,819)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(819, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("819.73");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.fn (of order \(12\), degree \(4\), 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: \(6.53974792554\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(9\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 273)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 4 q^{7} - 4 q^{11} + 36 q^{14} + 12 q^{16} + 4 q^{17} - 12 q^{19} + 44 q^{20} - 8 q^{22} - 12 q^{23} + 48 q^{25} + 28 q^{26} + 12 q^{28} + 16 q^{29} + 56 q^{32} + 48 q^{34} - 8 q^{35} + 22 q^{37} + 16 q^{38} + 60 q^{40} - 32 q^{41} - 4 q^{44} - 44 q^{46} + 14 q^{47} - 6 q^{49} + 68 q^{50} - 82 q^{52} + 8 q^{53} - 8 q^{56} - 84 q^{58} - 70 q^{59} + 36 q^{61} + 48 q^{62} + 8 q^{65} + 38 q^{67} - 36 q^{68} - 40 q^{70} + 36 q^{71} - 46 q^{73} - 40 q^{74} + 60 q^{76} + 60 q^{77} + 38 q^{80} + 24 q^{83} + 44 q^{85} + 24 q^{86} - 168 q^{88} - 38 q^{89} - 14 q^{91} + 40 q^{92} - 36 q^{97} - 58 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1 −0.587020 + 2.19079i 0 −2.72291 − 1.57208i 1.78371 + 0.477944i 0 −2.58471 − 0.565048i 1.83495 − 1.83495i 0 −2.09415 + 3.62718i
73.2 −0.549668 + 2.05139i 0 −2.17401 − 1.25516i −3.81253 − 1.02157i 0 −0.231538 − 2.63560i 0.766371 − 0.766371i 0 4.19125 − 7.25946i
73.3 −0.341564 + 1.27474i 0 0.223767 + 0.129192i 0.271244 + 0.0726797i 0 2.39655 + 1.12096i −2.10746 + 2.10746i 0 −0.185295 + 0.320940i
73.4 −0.217671 + 0.812358i 0 1.11951 + 0.646347i −1.60194 − 0.429239i 0 0.0720542 + 2.64477i −1.95812 + 1.95812i 0 0.697391 − 1.20792i
73.5 −0.0608509 + 0.227099i 0 1.68418 + 0.972362i 3.32462 + 0.890829i 0 −2.22038 − 1.43872i −0.655801 + 0.655801i 0 −0.404612 + 0.700809i
73.6 0.242689 − 0.905729i 0 0.970604 + 0.560379i 3.03397 + 0.812951i 0 2.50344 + 0.856042i 2.06919 − 2.06919i 0 1.47263 − 2.55066i
73.7 0.246078 − 0.918377i 0 0.949189 + 0.548015i −0.797354 − 0.213650i 0 −2.22965 + 1.42430i 2.08146 − 2.08146i 0 −0.392423 + 0.679697i
73.8 0.590298 − 2.20302i 0 −2.77280 − 1.60088i 1.71910 + 0.460631i 0 1.30433 − 2.30189i −1.93810 + 1.93810i 0 2.02956 − 3.51530i
73.9 0.677708 − 2.52924i 0 −4.20573 − 2.42818i −3.92082 − 1.05058i 0 1.12389 + 2.39518i −5.28864 + 5.28864i 0 −5.31435 + 9.20472i
460.1 −0.587020 − 2.19079i 0 −2.72291 + 1.57208i 1.78371 − 0.477944i 0 −2.58471 + 0.565048i 1.83495 + 1.83495i 0 −2.09415 − 3.62718i
460.2 −0.549668 − 2.05139i 0 −2.17401 + 1.25516i −3.81253 + 1.02157i 0 −0.231538 + 2.63560i 0.766371 + 0.766371i 0 4.19125 + 7.25946i
460.3 −0.341564 − 1.27474i 0 0.223767 − 0.129192i 0.271244 − 0.0726797i 0 2.39655 − 1.12096i −2.10746 − 2.10746i 0 −0.185295 − 0.320940i
460.4 −0.217671 − 0.812358i 0 1.11951 − 0.646347i −1.60194 + 0.429239i 0 0.0720542 − 2.64477i −1.95812 − 1.95812i 0 0.697391 + 1.20792i
460.5 −0.0608509 − 0.227099i 0 1.68418 − 0.972362i 3.32462 − 0.890829i 0 −2.22038 + 1.43872i −0.655801 − 0.655801i 0 −0.404612 − 0.700809i
460.6 0.242689 + 0.905729i 0 0.970604 − 0.560379i 3.03397 − 0.812951i 0 2.50344 − 0.856042i 2.06919 + 2.06919i 0 1.47263 + 2.55066i
460.7 0.246078 + 0.918377i 0 0.949189 − 0.548015i −0.797354 + 0.213650i 0 −2.22965 − 1.42430i 2.08146 + 2.08146i 0 −0.392423 − 0.679697i
460.8 0.590298 + 2.20302i 0 −2.77280 + 1.60088i 1.71910 − 0.460631i 0 1.30433 + 2.30189i −1.93810 − 1.93810i 0 2.02956 + 3.51530i
460.9 0.677708 + 2.52924i 0 −4.20573 + 2.42818i −3.92082 + 1.05058i 0 1.12389 − 2.39518i −5.28864 − 5.28864i 0 −5.31435 − 9.20472i
577.1 −2.11542 − 0.566824i 0 2.42164 + 1.39814i −0.169176 + 0.631372i 0 −1.81137 + 1.92845i −1.23310 − 1.23310i 0 0.715753 − 1.23972i
577.2 −1.93173 − 0.517606i 0 1.73162 + 0.999754i 0.960415 − 3.58432i 0 −0.534135 − 2.59127i 0.000695828 0 0.000695828i 0 −3.71053 + 6.42683i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.bb even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.fn.g 36
3.b odd 2 1 273.2.bz.b yes 36
7.d odd 6 1 819.2.fn.f 36
13.d odd 4 1 819.2.fn.f 36
21.g even 6 1 273.2.bz.a ✓ 36
39.f even 4 1 273.2.bz.a ✓ 36
91.bb even 12 1 inner 819.2.fn.g 36
273.cb odd 12 1 273.2.bz.b yes 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.bz.a ✓ 36 21.g even 6 1
273.2.bz.a ✓ 36 39.f even 4 1
273.2.bz.b yes 36 3.b odd 2 1
273.2.bz.b yes 36 273.cb odd 12 1
819.2.fn.f 36 7.d odd 6 1
819.2.fn.f 36 13.d odd 4 1
819.2.fn.g 36 1.a even 1 1 trivial
819.2.fn.g 36 91.bb even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\):

\( T_{2}^{36} - 63 T_{2}^{32} - 40 T_{2}^{31} + 244 T_{2}^{29} + 3022 T_{2}^{28} + 2840 T_{2}^{27} + \cdots + 144 \) Copy content Toggle raw display
\( T_{19}^{36} + 12 T_{19}^{35} + 57 T_{19}^{34} + 202 T_{19}^{33} - 3375 T_{19}^{32} + \cdots + 85\!\cdots\!89 \) Copy content Toggle raw display