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Properties

Label 888.2.f.b
Level $888$
Weight $2$
Character orbit 888.f
Analytic conductor $7.091$
Analytic rank $0$
Dimension $44$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [888,2,Mod(445,888)] 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("888.445"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(888, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [44] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.09071569949\)
Analytic rank: \(0\)
Dimension: \(44\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 44 q + 2 q^{4} + 2 q^{6} - 6 q^{8} - 44 q^{9} - 4 q^{10} - 4 q^{12} - 2 q^{14} - 4 q^{15} + 10 q^{16} + 20 q^{17} + 8 q^{20} + 4 q^{24} - 76 q^{25} + 8 q^{26} - 10 q^{28} - 4 q^{30} - 4 q^{31} + 10 q^{32}+ \cdots + 106 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
445.1 −1.39660 − 0.222517i − 1.00000i 1.90097 + 0.621534i 4.40147i −0.222517 + 1.39660i −1.69571 −2.51659 − 1.29103i −1.00000 0.979403 − 6.14709i
445.2 −1.39660 + 0.222517i 1.00000i 1.90097 − 0.621534i − 4.40147i −0.222517 − 1.39660i −1.69571 −2.51659 + 1.29103i −1.00000 0.979403 + 6.14709i
445.3 −1.39311 − 0.243394i 1.00000i 1.88152 + 0.678149i 2.36265i 0.243394 − 1.39311i 0.174899 −2.45611 − 1.40269i −1.00000 0.575053 − 3.29143i
445.4 −1.39311 + 0.243394i − 1.00000i 1.88152 − 0.678149i − 2.36265i 0.243394 + 1.39311i 0.174899 −2.45611 + 1.40269i −1.00000 0.575053 + 3.29143i
445.5 −1.33500 − 0.466657i 1.00000i 1.56446 + 1.24598i − 1.64248i 0.466657 − 1.33500i −0.162831 −1.50712 − 2.39345i −1.00000 −0.766476 + 2.19272i
445.6 −1.33500 + 0.466657i − 1.00000i 1.56446 − 1.24598i 1.64248i 0.466657 + 1.33500i −0.162831 −1.50712 + 2.39345i −1.00000 −0.766476 − 2.19272i
445.7 −1.28658 − 0.587112i − 1.00000i 1.31060 + 1.51074i − 4.14819i −0.587112 + 1.28658i −3.69286 −0.799225 − 2.71316i −1.00000 −2.43545 + 5.33699i
445.8 −1.28658 + 0.587112i 1.00000i 1.31060 − 1.51074i 4.14819i −0.587112 − 1.28658i −3.69286 −0.799225 + 2.71316i −1.00000 −2.43545 − 5.33699i
445.9 −1.26487 − 0.632539i 1.00000i 1.19979 + 1.60016i 2.21090i 0.632539 − 1.26487i 3.34890 −0.505414 − 2.78290i −1.00000 1.39848 − 2.79650i
445.10 −1.26487 + 0.632539i − 1.00000i 1.19979 − 1.60016i − 2.21090i 0.632539 + 1.26487i 3.34890 −0.505414 + 2.78290i −1.00000 1.39848 + 2.79650i
445.11 −1.05129 − 0.945938i − 1.00000i 0.210403 + 1.98890i − 0.115431i −0.945938 + 1.05129i 0.475318 1.66018 − 2.28993i −1.00000 −0.109190 + 0.121351i
445.12 −1.05129 + 0.945938i 1.00000i 0.210403 − 1.98890i 0.115431i −0.945938 − 1.05129i 0.475318 1.66018 + 2.28993i −1.00000 −0.109190 − 0.121351i
445.13 −0.784871 − 1.17643i 1.00000i −0.767956 + 1.84668i 2.04572i 1.17643 − 0.784871i 2.83247 2.77523 − 0.545965i −1.00000 2.40664 − 1.60563i
445.14 −0.784871 + 1.17643i − 1.00000i −0.767956 − 1.84668i − 2.04572i 1.17643 + 0.784871i 2.83247 2.77523 + 0.545965i −1.00000 2.40664 + 1.60563i
445.15 −0.682430 − 1.23866i 1.00000i −1.06858 + 1.69060i − 4.04044i 1.23866 − 0.682430i 0.198224 2.82332 + 0.169889i −1.00000 −5.00475 + 2.75732i
445.16 −0.682430 + 1.23866i − 1.00000i −1.06858 − 1.69060i 4.04044i 1.23866 + 0.682430i 0.198224 2.82332 − 0.169889i −1.00000 −5.00475 − 2.75732i
445.17 −0.356656 − 1.36850i − 1.00000i −1.74559 + 0.976168i 1.37987i −1.36850 + 0.356656i −4.88478 1.95846 + 2.04069i −1.00000 1.88836 − 0.492140i
445.18 −0.356656 + 1.36850i 1.00000i −1.74559 − 0.976168i − 1.37987i −1.36850 − 0.356656i −4.88478 1.95846 − 2.04069i −1.00000 1.88836 + 0.492140i
445.19 −0.319747 − 1.37759i − 1.00000i −1.79552 + 0.880963i − 0.223291i −1.37759 + 0.319747i 3.28882 1.78772 + 2.19181i −1.00000 −0.307604 + 0.0713967i
445.20 −0.319747 + 1.37759i 1.00000i −1.79552 − 0.880963i 0.223291i −1.37759 − 0.319747i 3.28882 1.78772 − 2.19181i −1.00000 −0.307604 − 0.0713967i
See all 44 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 445.44
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 888.2.f.b ✓ 44
4.b odd 2 1 3552.2.f.b 44
8.b even 2 1 inner 888.2.f.b ✓ 44
8.d odd 2 1 3552.2.f.b 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
888.2.f.b ✓ 44 1.a even 1 1 trivial
888.2.f.b ✓ 44 8.b even 2 1 inner
3552.2.f.b 44 4.b odd 2 1
3552.2.f.b 44 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{44} + 148 T_{5}^{42} + 10044 T_{5}^{40} + 414944 T_{5}^{38} + 11685340 T_{5}^{36} + \cdots + 629407744 \) acting on \(S_{2}^{\mathrm{new}}(888, [\chi])\). Copy content Toggle raw display