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Properties

Label 888.2.bo.c
Level $888$
Weight $2$
Character orbit 888.bo
Analytic conductor $7.091$
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] = [888,2,Mod(49,888)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(888, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 14]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("888.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.bo (of order \(9\), 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: \(7.09071569949\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 3 q^{5} + 15 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 3 q^{5} + 15 q^{7} + 12 q^{13} + 3 q^{15} + 3 q^{17} + 9 q^{19} + 15 q^{21} + 27 q^{25} - 12 q^{27} - 6 q^{29} - 30 q^{31} + 9 q^{33} + 15 q^{35} + 9 q^{37} + 3 q^{39} + 15 q^{41} - 54 q^{43} + 6 q^{45} - 12 q^{47} + 27 q^{49} + 18 q^{51} + 39 q^{53} - 6 q^{55} - 3 q^{59} + 12 q^{61} + 36 q^{65} + 48 q^{67} - 15 q^{69} + 33 q^{71} - 48 q^{73} + 60 q^{75} + 36 q^{77} + 18 q^{79} - 42 q^{83} + 15 q^{87} + 36 q^{89} - 36 q^{91} - 18 q^{93} + 27 q^{95} + 9 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 0 0.766044 + 0.642788i 0 −2.56352 + 0.933046i 0 −1.16825 + 0.425209i 0 0.173648 + 0.984808i 0
49.2 0 0.766044 + 0.642788i 0 −2.00718 + 0.730552i 0 1.90041 − 0.691691i 0 0.173648 + 0.984808i 0
49.3 0 0.766044 + 0.642788i 0 3.36351 − 1.22422i 0 4.08013 − 1.48504i 0 0.173648 + 0.984808i 0
49.4 0 0.766044 + 0.642788i 0 3.41293 − 1.24220i 0 −2.06718 + 0.752391i 0 0.173648 + 0.984808i 0
145.1 0 0.766044 − 0.642788i 0 −2.56352 − 0.933046i 0 −1.16825 − 0.425209i 0 0.173648 − 0.984808i 0
145.2 0 0.766044 − 0.642788i 0 −2.00718 − 0.730552i 0 1.90041 + 0.691691i 0 0.173648 − 0.984808i 0
145.3 0 0.766044 − 0.642788i 0 3.36351 + 1.22422i 0 4.08013 + 1.48504i 0 0.173648 − 0.984808i 0
145.4 0 0.766044 − 0.642788i 0 3.41293 + 1.24220i 0 −2.06718 − 0.752391i 0 0.173648 − 0.984808i 0
601.1 0 0.173648 − 0.984808i 0 −2.76055 − 2.31638i 0 3.79688 + 3.18596i 0 −0.939693 − 0.342020i 0
601.2 0 0.173648 − 0.984808i 0 −1.02288 − 0.858301i 0 −0.439492 − 0.368778i 0 −0.939693 − 0.342020i 0
601.3 0 0.173648 − 0.984808i 0 0.835055 + 0.700694i 0 2.72050 + 2.28277i 0 −0.939693 − 0.342020i 0
601.4 0 0.173648 − 0.984808i 0 2.85598 + 2.39645i 0 −0.585166 − 0.491013i 0 −0.939693 − 0.342020i 0
625.1 0 0.173648 + 0.984808i 0 −2.76055 + 2.31638i 0 3.79688 − 3.18596i 0 −0.939693 + 0.342020i 0
625.2 0 0.173648 + 0.984808i 0 −1.02288 + 0.858301i 0 −0.439492 + 0.368778i 0 −0.939693 + 0.342020i 0
625.3 0 0.173648 + 0.984808i 0 0.835055 − 0.700694i 0 2.72050 − 2.28277i 0 −0.939693 + 0.342020i 0
625.4 0 0.173648 + 0.984808i 0 2.85598 − 2.39645i 0 −0.585166 + 0.491013i 0 −0.939693 + 0.342020i 0
673.1 0 −0.939693 − 0.342020i 0 −0.570089 − 3.23313i 0 −0.196685 − 1.11546i 0 0.766044 + 0.642788i 0
673.2 0 −0.939693 − 0.342020i 0 −0.482804 − 2.73812i 0 0.268451 + 1.52246i 0 0.766044 + 0.642788i 0
673.3 0 −0.939693 − 0.342020i 0 0.139923 + 0.793545i 0 −0.580155 − 3.29022i 0 0.766044 + 0.642788i 0
673.4 0 −0.939693 − 0.342020i 0 0.299629 + 1.69928i 0 −0.229436 − 1.30120i 0 0.766044 + 0.642788i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.f even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 888.2.bo.c ✓ 24
37.f even 9 1 inner 888.2.bo.c ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
888.2.bo.c ✓ 24 1.a even 1 1 trivial
888.2.bo.c ✓ 24 37.f even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} - 3 T_{5}^{23} - 9 T_{5}^{22} - T_{5}^{21} + 210 T_{5}^{20} + 198 T_{5}^{19} + \cdots + 353477601 \) acting on \(S_{2}^{\mathrm{new}}(888, [\chi])\). Copy content Toggle raw display