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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [888,2,Mod(49,888)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// 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);
 
Copy content sage: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")
 
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

Copy content comment:select newform
 
Copy content sage:traces = [24,0,0,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content 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 algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 3 q^{5} - 3 q^{7} - 6 q^{11} - 12 q^{13} - 3 q^{15} + 15 q^{17} - 3 q^{19} - 3 q^{21} - 6 q^{23} - 9 q^{25} - 12 q^{27} - 12 q^{29} + 6 q^{31} + 3 q^{33} - 3 q^{35} - 9 q^{37} + 15 q^{39} + 9 q^{41}+ \cdots - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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 −2.88157 + 1.04881i 0 2.44080 − 0.888379i 0 0.173648 + 0.984808i 0
49.2 0 0.766044 + 0.642788i 0 −1.00558 + 0.366001i 0 −2.99075 + 1.08854i 0 0.173648 + 0.984808i 0
49.3 0 0.766044 + 0.642788i 0 0.534466 − 0.194530i 0 −3.76120 + 1.36896i 0 0.173648 + 0.984808i 0
49.4 0 0.766044 + 0.642788i 0 1.14695 − 0.417455i 0 2.87145 − 1.04512i 0 0.173648 + 0.984808i 0
145.1 0 0.766044 − 0.642788i 0 −2.88157 − 1.04881i 0 2.44080 + 0.888379i 0 0.173648 − 0.984808i 0
145.2 0 0.766044 − 0.642788i 0 −1.00558 − 0.366001i 0 −2.99075 − 1.08854i 0 0.173648 − 0.984808i 0
145.3 0 0.766044 − 0.642788i 0 0.534466 + 0.194530i 0 −3.76120 − 1.36896i 0 0.173648 − 0.984808i 0
145.4 0 0.766044 − 0.642788i 0 1.14695 + 0.417455i 0 2.87145 + 1.04512i 0 0.173648 − 0.984808i 0
601.1 0 0.173648 − 0.984808i 0 −2.30725 − 1.93602i 0 −2.09081 − 1.75440i 0 −0.939693 − 0.342020i 0
601.2 0 0.173648 − 0.984808i 0 −1.19833 − 1.00552i 0 2.01976 + 1.69478i 0 −0.939693 − 0.342020i 0
601.3 0 0.173648 − 0.984808i 0 1.68092 + 1.41046i 0 −1.85442 − 1.55605i 0 −0.939693 − 0.342020i 0
601.4 0 0.173648 − 0.984808i 0 1.91706 + 1.60860i 0 2.19152 + 1.83891i 0 −0.939693 − 0.342020i 0
625.1 0 0.173648 + 0.984808i 0 −2.30725 + 1.93602i 0 −2.09081 + 1.75440i 0 −0.939693 + 0.342020i 0
625.2 0 0.173648 + 0.984808i 0 −1.19833 + 1.00552i 0 2.01976 − 1.69478i 0 −0.939693 + 0.342020i 0
625.3 0 0.173648 + 0.984808i 0 1.68092 − 1.41046i 0 −1.85442 + 1.55605i 0 −0.939693 + 0.342020i 0
625.4 0 0.173648 + 0.984808i 0 1.91706 − 1.60860i 0 2.19152 − 1.83891i 0 −0.939693 + 0.342020i 0
673.1 0 −0.939693 − 0.342020i 0 −0.170060 − 0.964457i 0 0.0199072 + 0.112899i 0 0.766044 + 0.642788i 0
673.2 0 −0.939693 − 0.342020i 0 −0.0886516 − 0.502768i 0 0.245408 + 1.39178i 0 0.766044 + 0.642788i 0
673.3 0 −0.939693 − 0.342020i 0 0.155699 + 0.883014i 0 −0.809562 − 4.59125i 0 0.766044 + 0.642788i 0
673.4 0 −0.939693 − 0.342020i 0 0.716353 + 4.06264i 0 0.217895 + 1.23574i 0 0.766044 + 0.642788i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 49.4
Significant digits:
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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.b ✓ 24
37.f even 9 1 inner 888.2.bo.b ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
888.2.bo.b ✓ 24 1.a even 1 1 trivial
888.2.bo.b ✓ 24 37.f even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} + 3 T_{5}^{23} + 9 T_{5}^{22} + 49 T_{5}^{21} + 78 T_{5}^{20} - 240 T_{5}^{19} + 548 T_{5}^{18} + \cdots + 11881 \) acting on \(S_{2}^{\mathrm{new}}(888, [\chi])\). Copy content Toggle raw display