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Properties

Label 810.2.k.e
Level $810$
Weight $2$
Character orbit 810.k
Analytic conductor $6.468$
Analytic rank $0$
Dimension $24$
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] = [810,2,Mod(91,810)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(810, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([8, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("810.91");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 810.k (of order \(9\), degree \(6\), 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: \(6.46788256372\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{9})\)
Twist minimal: no (minimal twist has level 270)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 3 q^{7} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 3 q^{7} - 12 q^{8} - 12 q^{10} + 9 q^{13} + 3 q^{14} - 15 q^{17} - 9 q^{19} + 12 q^{23} + 12 q^{26} + 6 q^{28} + 33 q^{29} - 21 q^{31} - 15 q^{34} - 3 q^{35} - 12 q^{37} - 12 q^{38} - 21 q^{41} + 12 q^{43} - 9 q^{44} - 15 q^{46} + 21 q^{47} + 27 q^{49} + 9 q^{52} + 60 q^{53} + 18 q^{55} + 3 q^{56} - 21 q^{58} - 36 q^{59} + 33 q^{61} - 18 q^{62} - 12 q^{64} - 12 q^{67} + 3 q^{68} - 6 q^{70} - 12 q^{71} - 6 q^{74} - 12 q^{76} + 60 q^{77} - 15 q^{79} + 24 q^{80} + 6 q^{82} + 27 q^{83} - 15 q^{85} + 12 q^{86} - 9 q^{88} - 30 q^{89} - 9 q^{91} - 6 q^{92} + 12 q^{94} + 15 q^{95} + 3 q^{97} - 27 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1 0.173648 + 0.984808i 0 −0.939693 + 0.342020i 0.766044 + 0.642788i 0 −3.30330 − 1.20230i −0.500000 − 0.866025i 0 −0.500000 + 0.866025i
91.2 0.173648 + 0.984808i 0 −0.939693 + 0.342020i 0.766044 + 0.642788i 0 −1.33944 − 0.487517i −0.500000 − 0.866025i 0 −0.500000 + 0.866025i
91.3 0.173648 + 0.984808i 0 −0.939693 + 0.342020i 0.766044 + 0.642788i 0 0.582091 + 0.211864i −0.500000 − 0.866025i 0 −0.500000 + 0.866025i
91.4 0.173648 + 0.984808i 0 −0.939693 + 0.342020i 0.766044 + 0.642788i 0 4.56065 + 1.65994i −0.500000 − 0.866025i 0 −0.500000 + 0.866025i
181.1 −0.939693 + 0.342020i 0 0.766044 − 0.642788i 0.173648 + 0.984808i 0 −3.84295 − 3.22462i −0.500000 + 0.866025i 0 −0.500000 − 0.866025i
181.2 −0.939693 + 0.342020i 0 0.766044 − 0.642788i 0.173648 + 0.984808i 0 −0.268547 − 0.225338i −0.500000 + 0.866025i 0 −0.500000 − 0.866025i
181.3 −0.939693 + 0.342020i 0 0.766044 − 0.642788i 0.173648 + 0.984808i 0 1.26626 + 1.06252i −0.500000 + 0.866025i 0 −0.500000 − 0.866025i
181.4 −0.939693 + 0.342020i 0 0.766044 − 0.642788i 0.173648 + 0.984808i 0 3.34524 + 2.80699i −0.500000 + 0.866025i 0 −0.500000 − 0.866025i
361.1 0.766044 − 0.642788i 0 0.173648 − 0.984808i −0.939693 + 0.342020i 0 −0.314431 − 1.78323i −0.500000 − 0.866025i 0 −0.500000 + 0.866025i
361.2 0.766044 − 0.642788i 0 0.173648 − 0.984808i −0.939693 + 0.342020i 0 −0.158856 − 0.900919i −0.500000 − 0.866025i 0 −0.500000 + 0.866025i
361.3 0.766044 − 0.642788i 0 0.173648 − 0.984808i −0.939693 + 0.342020i 0 0.192018 + 1.08899i −0.500000 − 0.866025i 0 −0.500000 + 0.866025i
361.4 0.766044 − 0.642788i 0 0.173648 − 0.984808i −0.939693 + 0.342020i 0 0.781269 + 4.43080i −0.500000 − 0.866025i 0 −0.500000 + 0.866025i
451.1 0.766044 + 0.642788i 0 0.173648 + 0.984808i −0.939693 − 0.342020i 0 −0.314431 + 1.78323i −0.500000 + 0.866025i 0 −0.500000 − 0.866025i
451.2 0.766044 + 0.642788i 0 0.173648 + 0.984808i −0.939693 − 0.342020i 0 −0.158856 + 0.900919i −0.500000 + 0.866025i 0 −0.500000 − 0.866025i
451.3 0.766044 + 0.642788i 0 0.173648 + 0.984808i −0.939693 − 0.342020i 0 0.192018 − 1.08899i −0.500000 + 0.866025i 0 −0.500000 − 0.866025i
451.4 0.766044 + 0.642788i 0 0.173648 + 0.984808i −0.939693 − 0.342020i 0 0.781269 − 4.43080i −0.500000 + 0.866025i 0 −0.500000 − 0.866025i
631.1 −0.939693 − 0.342020i 0 0.766044 + 0.642788i 0.173648 − 0.984808i 0 −3.84295 + 3.22462i −0.500000 − 0.866025i 0 −0.500000 + 0.866025i
631.2 −0.939693 − 0.342020i 0 0.766044 + 0.642788i 0.173648 − 0.984808i 0 −0.268547 + 0.225338i −0.500000 − 0.866025i 0 −0.500000 + 0.866025i
631.3 −0.939693 − 0.342020i 0 0.766044 + 0.642788i 0.173648 − 0.984808i 0 1.26626 − 1.06252i −0.500000 − 0.866025i 0 −0.500000 + 0.866025i
631.4 −0.939693 − 0.342020i 0 0.766044 + 0.642788i 0.173648 − 0.984808i 0 3.34524 − 2.80699i −0.500000 − 0.866025i 0 −0.500000 + 0.866025i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 91.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 810.2.k.e 24
3.b odd 2 1 270.2.k.e ✓ 24
27.e even 9 1 inner 810.2.k.e 24
27.e even 9 1 7290.2.a.v 12
27.f odd 18 1 270.2.k.e ✓ 24
27.f odd 18 1 7290.2.a.s 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.2.k.e ✓ 24 3.b odd 2 1
270.2.k.e ✓ 24 27.f odd 18 1
810.2.k.e 24 1.a even 1 1 trivial
810.2.k.e 24 27.e even 9 1 inner
7290.2.a.s 12 27.f odd 18 1
7290.2.a.v 12 27.e even 9 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{24} - 3 T_{7}^{23} - 9 T_{7}^{22} + 29 T_{7}^{21} + 303 T_{7}^{20} - 1899 T_{7}^{19} + \cdots + 2483776 \) acting on \(S_{2}^{\mathrm{new}}(810, [\chi])\). Copy content Toggle raw display