
// Make newform 8112.2.a.z in Magma, downloaded from the LMFDB on 19 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_8112_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_8112_2_a_z();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_8112_2_a_z();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_8112_a();"
function MakeCharacter_8112_a()
    N := 8112;
    order := 1;
    char_gens := [5071, 6085, 2705, 3889];
    v := [1, 1, 1, 1];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

function MakeCharacter_8112_a_Hecke( : Kf := Kf)
    return MakeCharacter_8112_a();
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..prec]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 2;
    raw_aps := [[0], [1], [0], [-2], [-2], [0], [-2], [6], [0], [-2], [2], [4], [4], [-4], [10], [-10], [6], [-2], [-2], [6], [-16], [-8], [-14], [12], [-8], [10], [-8], [-12], [-4], [2], [8], [-20], [-12], [4], [-24], [22], [-10], [10], [-14], [2], [20], [2], [8], [-16], [-16], [0], [-12], [-14], [-6], [-20], [-26], [-2], [-8], [-12], [14], [24], [30], [22], [10], [-12], [-28], [-32], [18], [-32], [-14], [24], [6], [-26], [28], [-20], [12], [-30], [-32], [30], [14], [6], [-6], [28], [20], [-16], [-20], [-28], [6], [-2], [24], [4], [20], [-16], [24], [6], [12], [2], [-6], [20], [30], [40], [-8], [-6], [-28], [4], [-20], [-8], [-12], [38], [-28], [-32], [-22], [36], [-24], [-46], [-16], [44], [20], [10], [-26], [-2], [-18], [8], [30], [20], [-28], [6], [-42], [-18], [-34], [18], [12], [24], [0], [-4], [50], [6], [-16], [-10], [36], [32], [40], [-46], [18], [10], [-2], [-8], [-24], [-34], [-30], [10], [4], [-42], [28], [38], [20], [46], [12], [16], [-12], [-24], [16], [-36], [26], [0], [-2], [-42], [26], [-20], [-12], [-38], [16], [6], [-8], [42], [34], [12], [0], [32], [0], [-22], [10], [48], [-24], [-10], [-10], [36], [6], [44], [-6], [26], [58], [22], [-48], [10], [-54], [-22], [-20], [8], [-44], [-10], [0], [42], [-12], [-64], [-48], [-48], [-60], [10], [70], [30], [-30], [44], [-20], [52], [-34], [-42], [-32], [-34], [14], [40], [40], [-30], [-26], [-40], [6], [-18], [-12], [50], [28], [42], [42], [-16], [16], [6], [-22], [44], [10], [-42], [-28], [-18], [-24], [-16], [-36], [48], [66], [-28], [-16], [20], [-60], [48], [-42], [-50], [14], [-72], [68], [12], [58], [46], [30], [-10], [14], [62], [42], [40], [24], [52], [20], [62], [-44], [4], [24], [-12], [58], [-6], [50], [-50], [64], [-80], [-38], [-14], [66], [-68], [-40], [-36], [56], [-34], [-32], [44], [58], [-32], [2], [24], [70], [14], [78], [68], [60], [38], [6], [18], [48], [-54], [12], [-62], [54], [-16], [-64], [36], [-28], [-56], [12], [46], [10], [18], [-32], [32], [50], [36], [-62], [-46], [74], [70], [8], [78], [-76], [72], [30], [-50], [-4], [34], [-6], [6], [-80], [22], [20], [62], [-88], [6], [-34], [-54], [-68], [60], [-80], [-16], [12], [-6], [-48], [88], [-72], [36], [30], [2], [-6], [42], [-82], [48], [-48], [24], [-22], [-54], [74], [-46], [14], [30], [44], [6], [-48], [-62], [52], [-62], [64], [54], [-18], [12], [44], [42], [6], [32], [-2], [34], [-32], [0], [-78], [-34], [-72], [-92], [62], [36], [42], [90], [-6], [-22], [-18], [-48], [40], [24], [-66], [68], [34], [-22], [74], [54], [-4], [-24], [4], [-34], [-10], [-52], [24], [64], [-36], [68], [-46], [70], [-50], [-58], [-36], [-44], [22], [-2], [-50], [-64], [-44], [48], [18], [76], [78], [52], [-56], [0], [-4], [-68], [-30], [48]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8112_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8112_2_a_z();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_8112_2_a_z(:prec:=1)
    chi := MakeCharacter_8112_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_8112_2_a_z();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_8112_2_a_z( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8112_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![0, 1]>,<7,R![2, 1]>,<11,R![2, 1]>],Snew);
    return Vf;
end function;