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Properties

Label 486.2.g.a
Level $486$
Weight $2$
Character orbit 486.g
Analytic conductor $3.881$
Analytic rank $0$
Dimension $72$
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] = [486,2,Mod(19,486)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(486, base_ring=CyclotomicField(54))
 
chi = DirichletCharacter(H, H._module([52]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("486.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 486 = 2 \cdot 3^{5} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 486.g (of order \(27\), degree \(18\), not 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: \(3.88072953823\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(4\) over \(\Q(\zeta_{27})\)
Twist minimal: no (minimal twist has level 162)
Sato-Tate group: $\mathrm{SU}(2)[C_{27}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q + 18 q^{13} + 9 q^{20} + 81 q^{23} + 18 q^{25} + 27 q^{26} + 18 q^{28} + 27 q^{29} - 54 q^{31} + 27 q^{35} - 9 q^{38} + 9 q^{41} + 36 q^{43} - 18 q^{46} + 27 q^{47} - 36 q^{52} + 27 q^{53} + 54 q^{55} + 9 q^{58} + 18 q^{59} - 9 q^{65} - 135 q^{67} + 18 q^{68} + 18 q^{70} - 72 q^{71} + 36 q^{73} - 99 q^{74} - 9 q^{76} - 144 q^{77} - 9 q^{79} - 18 q^{80} - 72 q^{82} - 99 q^{83} + 9 q^{85} - 72 q^{86} - 9 q^{88} - 126 q^{89} + 63 q^{91} - 36 q^{92} + 18 q^{94} - 45 q^{95} - 171 q^{97} - 36 q^{98}+O(q^{100}) \) 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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −0.993238 + 0.116093i 0 0.973045 − 0.230616i −2.13185 − 1.07065i 0 0.413572 + 1.38143i −0.939693 + 0.342020i 0 2.24173 + 0.815922i
19.2 −0.993238 + 0.116093i 0 0.973045 − 0.230616i −0.907364 − 0.455695i 0 −0.755228 − 2.52264i −0.939693 + 0.342020i 0 0.954132 + 0.347276i
19.3 −0.993238 + 0.116093i 0 0.973045 − 0.230616i 0.110030 + 0.0552591i 0 −0.0823094 − 0.274932i −0.939693 + 0.342020i 0 −0.115701 − 0.0421118i
19.4 −0.993238 + 0.116093i 0 0.973045 − 0.230616i 2.54814 + 1.27972i 0 0.518331 + 1.73135i −0.939693 + 0.342020i 0 −2.67948 − 0.975250i
37.1 −0.286803 + 0.957990i 0 −0.835488 − 0.549509i −0.423867 + 0.982634i 0 −0.264903 − 4.54820i 0.766044 − 0.642788i 0 −0.819786 − 0.687882i
37.2 −0.286803 + 0.957990i 0 −0.835488 − 0.549509i −0.417669 + 0.968265i 0 −0.0919975 − 1.57954i 0.766044 − 0.642788i 0 −0.807799 − 0.677824i
37.3 −0.286803 + 0.957990i 0 −0.835488 − 0.549509i −0.266390 + 0.617563i 0 0.0791338 + 1.35867i 0.766044 − 0.642788i 0 −0.515217 − 0.432318i
37.4 −0.286803 + 0.957990i 0 −0.835488 − 0.549509i 0.757906 − 1.75702i 0 0.278553 + 4.78256i 0.766044 − 0.642788i 0 1.46584 + 1.22999i
73.1 0.893633 − 0.448799i 0 0.597159 − 0.802123i −0.308002 + 1.02880i 0 −2.06499 + 4.78718i 0.173648 − 0.984808i 0 0.186483 + 1.05760i
73.2 0.893633 − 0.448799i 0 0.597159 − 0.802123i −0.195520 + 0.653084i 0 0.386069 − 0.895009i 0.173648 − 0.984808i 0 0.118380 + 0.671366i
73.3 0.893633 − 0.448799i 0 0.597159 − 0.802123i 0.267154 − 0.892356i 0 1.11188 − 2.57764i 0.173648 − 0.984808i 0 −0.161751 − 0.917337i
73.4 0.893633 − 0.448799i 0 0.597159 − 0.802123i 0.750365 − 2.50640i 0 0.318489 − 0.738341i 0.173648 − 0.984808i 0 −0.454317 − 2.57656i
91.1 −0.0581448 − 0.998308i 0 −0.993238 + 0.116093i −3.64873 − 0.864766i 0 1.03565 − 1.39113i 0.173648 + 0.984808i 0 −0.651148 + 3.69284i
91.2 −0.0581448 − 0.998308i 0 −0.993238 + 0.116093i −1.08715 − 0.257659i 0 1.38530 − 1.86078i 0.173648 + 0.984808i 0 −0.194011 + 1.10029i
91.3 −0.0581448 − 0.998308i 0 −0.993238 + 0.116093i −0.789983 − 0.187229i 0 −1.95256 + 2.62274i 0.173648 + 0.984808i 0 −0.140979 + 0.799532i
91.4 −0.0581448 − 0.998308i 0 −0.993238 + 0.116093i 3.39069 + 0.803609i 0 −1.32018 + 1.77332i 0.173648 + 0.984808i 0 0.605098 − 3.43168i
127.1 −0.686242 + 0.727374i 0 −0.0581448 − 0.998308i −1.83788 − 0.214818i 0 −0.0854199 + 0.0428995i 0.766044 + 0.642788i 0 1.41748 − 1.18941i
127.2 −0.686242 + 0.727374i 0 −0.0581448 − 0.998308i −1.08261 − 0.126540i 0 −0.387485 + 0.194602i 0.766044 + 0.642788i 0 0.834977 − 0.700629i
127.3 −0.686242 + 0.727374i 0 −0.0581448 − 0.998308i 1.97104 + 0.230382i 0 0.868864 − 0.436360i 0.766044 + 0.642788i 0 −1.52019 + 1.27559i
127.4 −0.686242 + 0.727374i 0 −0.0581448 − 0.998308i 3.71787 + 0.434557i 0 −3.25050 + 1.63246i 0.766044 + 0.642788i 0 −2.86744 + 2.40607i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.4
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
81.g even 27 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 486.2.g.a 72
3.b odd 2 1 162.2.g.a ✓ 72
81.g even 27 1 inner 486.2.g.a 72
81.h odd 54 1 162.2.g.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.2.g.a ✓ 72 3.b odd 2 1
162.2.g.a ✓ 72 81.h odd 54 1
486.2.g.a 72 1.a even 1 1 trivial
486.2.g.a 72 81.g even 27 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{72} - 9 T_{5}^{70} - 30 T_{5}^{69} + 63 T_{5}^{68} + 81 T_{5}^{67} - 1665 T_{5}^{66} + 4023 T_{5}^{65} - 18009 T_{5}^{64} - 92904 T_{5}^{63} + 832275 T_{5}^{62} + 782244 T_{5}^{61} - 3669129 T_{5}^{60} + \cdots + 18487617876729 \) acting on \(S_{2}^{\mathrm{new}}(486, [\chi])\). Copy content Toggle raw display