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Properties

Label 1998.2.k.a
Level $1998$
Weight $2$
Character orbit 1998.k
Analytic conductor $15.954$
Analytic rank $0$
Dimension $76$
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] = [1998,2,Mod(1063,1998)] 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("1998.1063"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1998, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1998 = 2 \cdot 3^{3} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1998.k (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(15.9541103239\)
Analytic rank: \(0\)
Dimension: \(76\)
Relative dimension: \(38\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 666)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 76 q - 76 q^{4} - 2 q^{7} + 4 q^{11} + 76 q^{16} - 12 q^{23} - 100 q^{25} + 24 q^{26} + 2 q^{28} - 18 q^{29} + 6 q^{31} + 18 q^{35} + 10 q^{37} - 12 q^{38} - 72 q^{41} + 6 q^{43} - 4 q^{44} + 20 q^{47}+ \cdots + 24 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1063.1 − 1.00000i 0 −1.00000 4.19531i 0 −2.14037 + 3.70723i 1.00000i 0 4.19531
1063.2 − 1.00000i 0 −1.00000 4.04494i 0 0.760459 − 1.31715i 1.00000i 0 4.04494
1063.3 − 1.00000i 0 −1.00000 − 3.29656i 0 −1.41549 + 2.45170i 1.00000i 0 −3.29656
1063.4 − 1.00000i 0 −1.00000 − 3.13787i 0 −2.21825 + 3.84212i 1.00000i 0 −3.13787
1063.5 − 1.00000i 0 −1.00000 2.61464i 0 1.13430 − 1.96467i 1.00000i 0 2.61464
1063.6 − 1.00000i 0 −1.00000 − 2.56050i 0 1.18303 − 2.04907i 1.00000i 0 −2.56050
1063.7 − 1.00000i 0 −1.00000 2.31571i 0 −1.28307 + 2.22235i 1.00000i 0 2.31571
1063.8 − 1.00000i 0 −1.00000 − 2.28972i 0 1.54492 − 2.67588i 1.00000i 0 −2.28972
1063.9 − 1.00000i 0 −1.00000 2.11387i 0 2.44421 − 4.23350i 1.00000i 0 2.11387
1063.10 − 1.00000i 0 −1.00000 1.23967i 0 −1.37235 + 2.37697i 1.00000i 0 1.23967
1063.11 − 1.00000i 0 −1.00000 − 0.804249i 0 1.72960 − 2.99575i 1.00000i 0 −0.804249
1063.12 − 1.00000i 0 −1.00000 0.464281i 0 −0.261306 + 0.452595i 1.00000i 0 0.464281
1063.13 − 1.00000i 0 −1.00000 − 0.112170i 0 −1.28303 + 2.22227i 1.00000i 0 −0.112170
1063.14 − 1.00000i 0 −1.00000 0.0139965i 0 1.38428 − 2.39764i 1.00000i 0 0.0139965
1063.15 − 1.00000i 0 −1.00000 1.11767i 0 −1.39318 + 2.41305i 1.00000i 0 1.11767
1063.16 − 1.00000i 0 −1.00000 − 2.04307i 0 −0.0196697 + 0.0340689i 1.00000i 0 −2.04307
1063.17 − 1.00000i 0 −1.00000 2.69693i 0 0.847015 − 1.46707i 1.00000i 0 2.69693
1063.18 − 1.00000i 0 −1.00000 − 3.01453i 0 0.111912 − 0.193837i 1.00000i 0 −3.01453
1063.19 − 1.00000i 0 −1.00000 − 3.55836i 0 −0.253017 + 0.438238i 1.00000i 0 −3.55836
1063.20 1.00000i 0 −1.00000 4.41362i 0 1.98016 − 3.42973i − 1.00000i 0 −4.41362
See all 76 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1063.38
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
333.k even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1998.2.k.a 76
3.b odd 2 1 666.2.k.a ✓ 76
9.c even 3 1 1998.2.t.a 76
9.d odd 6 1 666.2.t.a yes 76
37.e even 6 1 1998.2.t.a 76
111.h odd 6 1 666.2.t.a yes 76
333.k even 6 1 inner 1998.2.k.a 76
333.v odd 6 1 666.2.k.a ✓ 76
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
666.2.k.a ✓ 76 3.b odd 2 1
666.2.k.a ✓ 76 333.v odd 6 1
666.2.t.a yes 76 9.d odd 6 1
666.2.t.a yes 76 111.h odd 6 1
1998.2.k.a 76 1.a even 1 1 trivial
1998.2.k.a 76 333.k even 6 1 inner
1998.2.t.a 76 9.c even 3 1
1998.2.t.a 76 37.e even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1998, [\chi])\).