
// Make newform 5712.2.a.bm in Magma, downloaded from the LMFDB on 16 November 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5712_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_5712_2_a_bm();".
// 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_5712_2_a_bm();".
// 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
        poly := [-6, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [0, 2]];
    Rf_basisdens := [1, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_5712_a();"
function MakeCharacter_5712_a()
    N := 5712;
    order := 1;
    char_gens := [2143, 1429, 3809, 3265, 2689];
    v := [1, 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_5712_a_Hecke(Kf)
    return MakeCharacter_5712_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, 0], [-1, 0], [2, 0], [-1, 0], [0, -1], [-2, -1], [-1, 0], [0, 1], [4, 0], [2, -1], [0, -2], [-2, 1], [-2, 0], [-4, 0], [8, 0], [6, 0], [4, 1], [10, 0], [4, 2], [-4, -2], [-2, 0], [0, 2], [12, 1], [-2, 2], [6, 0], [-6, 1], [-4, -2], [8, -1], [-2, 3], [-2, -2], [-8, 2], [4, 0], [2, 0], [4, 2], [-2, 2], [8, -2], [-2, -3], [-4, -1], [16, 0], [10, -2], [-12, 0], [-6, 2], [0, 4], [2, -2], [18, -1], [16, 0], [-20, -1], [-12, -2], [-4, -4], [14, 1], [-2, 2], [-8, 2], [-2, -4], [12, -1], [6, -2], [8, -2], [-6, -2], [-12, 2], [14, -3], [2, -4], [12, -2], [-6, 3], [8, 1], [-8, -2], [30, 0], [10, 3], [-12, 0], [2, 4], [24, 1], [22, -1], [-18, -2], [-16, 2], [-24, 2], [22, -2], [20, 1], [-8, -2], [6, -4], [2, 6], [6, 0], [2, 2], [-4, 6], [6, -2], [-36, 0], [2, 6], [16, 2], [-20, -2], [6, 4], [34, 0], [18, -1], [-8, 4], [4, 7], [-8, -6], [32, 0], [12, -4], [-12, -1], [0, 0], [26, 1], [6, 0], [24, -1], [-10, -5], [4, -5], [14, -4], [12, 7], [-30, 0], [36, -1], [10, 4], [-4, -7], [-18, 2], [-8, -2], [22, -4], [8, 4], [-10, -2], [-18, -2], [-4, 0], [0, 2], [-2, -4], [4, 4], [8, 4], [-22, -1], [12, 2], [30, 1], [10, 6], [-14, 0], [0, 5], [4, -6], [-26, 2], [-26, 3], [-16, 2], [12, 0], [-18, 1], [36, 2], [20, -2], [-24, -4], [6, 6], [-18, 4], [10, -2], [-6, -5], [20, 4], [-30, -1], [-2, 0], [20, -6], [-14, -1], [0, -2], [-24, 1], [-10, -1], [-24, -4], [2, 8], [-2, 8], [24, -3], [0, -2], [-26, -1], [6, 4], [4, 2], [24, -2], [-20, -1], [-20, 6], [16, 2], [-18, 0], [50, 0], [10, -10], [0, 1], [-14, 4], [-24, -4], [12, 1], [-14, -4], [-24, 6], [-48, 2], [2, 0], [-38, 2], [18, -4], [-36, 4], [-26, 3], [-36, 4], [26, 0], [-20, -2], [22, 0], [-20, 7], [-6, -7], [20, 2], [30, 7], [-8, 10], [-28, 6], [-26, 5], [6, 2], [0, -8], [42, 3], [6, 7], [8, -11], [-22, 2], [24, -8], [-18, 4], [8, 7], [-12, 4], [-30, 1], [-24, -7], [-34, 4], [-6, 4], [-50, 1], [-10, 0], [16, 0], [18, -1], [-40, 0], [-2, -1], [-22, -4], [20, 7], [2, 7], [12, -2], [12, 2], [46, -2], [24, 7], [10, 6], [-14, -5], [-16, 2], [36, 7], [-24, -6], [-10, 4], [-16, -8], [-18, 0], [-20, -4], [-42, 2], [22, 6], [-16, -8], [-30, 0], [8, -6], [36, -3], [-42, -4], [-26, 4], [-28, 4], [-4, 4], [-48, 5], [-34, 0], [12, 0], [56, 2], [-22, 0], [16, -9], [-8, 10], [-2, -4], [-22, -9], [8, -1], [8, 2], [48, 1], [-16, 1], [28, 2], [-2, 4], [-2, 4], [24, 4], [32, 8], [-4, 0], [-4, 0], [24, 4], [-18, -8], [-18, 12], [-24, 10], [38, -4], [2, 3], [4, 10], [-10, 5], [28, -4], [-30, 6], [-38, 6], [8, 10], [20, -2], [-14, 8], [34, -6], [6, -8], [-8, -11], [30, 8], [22, -6], [20, -3], [6, -8], [26, 4], [12, 2], [26, 0], [-56, 0], [18, 4], [-12, 14], [-4, -4], [-50, -2], [-14, 8], [-28, 9], [-8, -6], [40, -8], [-8, -8], [-2, -7], [4, 0], [-8, -4], [26, -6], [10, 3], [28, -2], [70, 2], [-6, -7], [44, 6], [66, -4], [28, -2], [-2, 3], [42, -2], [-60, 0], [18, -1], [-4, -10], [40, 1], [50, 2], [30, 0], [8, -8], [-32, 11], [-4, 17], [34, 2], [12, -2], [-22, 4], [-32, -4], [22, -4], [8, -12], [58, -5], [30, 4], [28, 6], [16, 14], [-6, 12], [20, -3], [36, 8], [-10, -8], [-46, -8], [-36, -2], [2, -2], [-30, 3], [48, -8], [-58, -4], [18, 6], [-28, 3], [20, -6], [36, -4], [10, -15], [-26, 1], [-78, 1], [32, 4], [68, 1], [-12, 9], [60, 6], [14, -8], [-18, -4], [-50, 0], [-28, -12], [6, -10], [2, -8], [66, -4], [-16, 12], [-42, 6], [-16, 13], [10, -2], [-44, -4], [-32, -2], [-30, -10], [-32, 9], [34, -6], [-42, 6], [-40, -6], [38, 4], [-10, 10], [-32, -8], [20, -10], [-34, -4], [-64, 0], [-58, 3], [-10, 0], [-8, -10], [8, -5], [24, -7], [58, 0], [42, 10], [-8, 12], [26, -14], [20, 2], [-20, 8], [60, -2], [6, 0], [44, 2], [-10, 9], [-36, -2], [76, -2], [-30, -8], [-34, 4], [34, 2], [58, 2], [34, -4], [0, 14], [-2, -6], [-20, -6], [-24, 2], [-8, 0], [30, 11], [-4, -5], [0, -6], [34, 8], [18, -6], [36, 4], [96, -1], [48, 0], [-70, 2], [-52, 8], [38, -8], [-36, 3], [-18, -10], [10, -8], [-30, -8], [24, -4], [-42, 8], [-14, 13], [64, -2], [62, 0], [-38, 4], [0, -7], [-36, 0], [-42, 4], [22, 8], [44, -10], [-12, 17], [2, 0], [-22, 0], [-12, 6], [40, 6], [-2, 0], [-56, -2], [-74, 2], [-38, 6], [-60, -6], [-28, -11], [22, 12], [-14, 15], [-8, -15], [62, 2], [-56, 1], [-24, 12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5712_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5712_2_a_bm();".
// 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_5712_2_a_bm(:prec:=2)
    chi := MakeCharacter_5712_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_5712_2_a_bm();".
// 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_5712_2_a_bm( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5712_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-2, 1]>,<11,R![-24, 0, 1]>,<13,R![-20, 4, 1]>,<19,R![-24, 0, 1]>],Snew);
    return Vf;
end function;