
// Make newform 1584.3.j.i in Magma, downloaded from the LMFDB on 16 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1584_j();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1584_j_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_1584_3_j_i();".
// 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_1584_3_j_i();".
// 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 := [2116, 0, 46, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 0, 0, 1], [23, 0, 1, 0], [0, 92, 0, 1]];
    Rf_basisdens := [1, 46, 23, 46];
    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_1584_j();"
function MakeCharacter_1584_j()
    N := 1584;
    order := 2;
    char_gens := [991, 1189, 353, 145];
    v := [2, 2, 2, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1584_j_Hecke();"
function MakeCharacter_1584_j_Hecke(Kf)
    N := 1584;
    order := 2;
    char_gens := [991, 1189, 353, 145];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [1, 0, 0, 0], [-1, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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 := 3;
    raw_aps := [[0, 0, 0, 0], [0, 0, 0, 0], [0, -1, 0, 0], [0, 0, 0, -1], [0, -1, 5, 0], [0, 0, 0, -1], [0, 0, 6, 0], [0, 0, 0, 0], [0, -5, 0, 0], [0, 0, 20, 0], [10, 0, 0, 0], [50, 0, 0, 0], [0, 0, -20, 0], [0, 0, 0, -4], [0, -5, 0, 0], [0, 5, 0, 0], [0, 10, 0, 0], [0, 0, 0, -5], [10, 0, 0, 0], [0, -5, 0, 0], [0, 0, 0, 6], [0, 0, 0, 5], [0, 0, -44, 0], [0, 2, 0, 0], [-40, 0, 0, 0], [0, 0, -40, 0], [-170, 0, 0, 0], [0, 0, -54, 0], [0, 0, 0, -15], [0, -10, 0, 0], [0, 0, 0, -9], [0, 0, 20, 0], [0, -40, 0, 0], [0, 0, 0, 10], [0, 0, -80, 0], [0, 0, 0, 5], [170, 0, 0, 0], [-140, 0, 0, 0], [0, 0, 116, 0], [0, 0, 176, 0], [0, 22, 0, 0], [-130, 0, 0, 0], [0, 13, 0, 0], [0, 0, 0, -16], [0, 0, -24, 0], [202, 0, 0, 0], [0, 0, 0, 10], [250, 0, 0, 0], [0, 0, -26, 0], [158, 0, 0, 0], [0, 0, -36, 0], [0, 0, -220, 0], [0, 0, 0, -20], [0, -20, 0, 0], [0, 50, 0, 0], [0, 0, -56, 0], [0, 5, 0, 0], [0, 0, 0, -15], [0, 0, 0, 1], [0, 0, 110, 0], [0, 0, 0, 34], [0, 0, 244, 0], [0, 0, 0, -6], [0, 55, 0, 0], [-70, 0, 0, 0], [0, -25, 0, 0], [40, 0, 0, 0], [0, 0, 0, 14], [0, 0, 284, 0], [0, 0, 0, 55], [0, 20, 0, 0], [0, 0, 60, 0], [130, 0, 0, 0], [0, 0, 0, 41], [-308, 0, 0, 0], [0, -35, 0, 0], [0, 95, 0, 0], [-310, 0, 0, 0], [0, -82, 0, 0], [0, 0, 0, 50], [0, -8, 0, 0], [-310, 0, 0, 0], [0, 0, -60, 0], [-610, 0, 0, 0], [0, 0, 0, 35], [0, 40, 0, 0], [0, 32, 0, 0], [0, 0, 0, -14], [0, 0, 380, 0], [670, 0, 0, 0], [0, -50, 0, 0], [0, 0, 280, 0], [-230, 0, 0, 0], [0, 0, -10, 0], [-152, 0, 0, 0], [0, 0, 444, 0], [0, 95, 0, 0], [0, -22, 0, 0], [0, 0, 0, 24], [0, 0, 0, -75], [0, 0, 0, -16], [0, 0, -276, 0], [0, 0, 626, 0], [0, 0, 350, 0], [0, 0, 0, 10], [-400, 0, 0, 0], [0, 130, 0, 0], [0, 0, 354, 0], [0, 157, 0, 0], [0, 0, 0, 10], [0, 0, 0, -59], [0, 0, 0, 9], [0, -10, 0, 0], [490, 0, 0, 0], [-338, 0, 0, 0], [0, 38, 0, 0], [130, 0, 0, 0], [0, -155, 0, 0], [0, 35, 0, 0], [0, 0, -230, 0], [278, 0, 0, 0], [0, 0, 0, 46], [0, 0, 284, 0], [0, -140, 0, 0], [-482, 0, 0, 0], [0, 0, 120, 0], [-1210, 0, 0, 0], [0, -167, 0, 0], [-1130, 0, 0, 0], [0, 0, 0, -79], [0, 0, 0, -90], [0, 0, -76, 0], [298, 0, 0, 0], [-490, 0, 0, 0], [0, 0, -290, 0], [0, 0, 0, 20], [0, -205, 0, 0], [0, 0, 0, 16], [0, 35, 0, 0], [0, 0, -660, 0], [0, 0, 0, 0], [0, 0, 560, 0], [310, 0, 0, 0], [0, 0, -516, 0], [-850, 0, 0, 0], [0, -65, 0, 0], [0, 0, 0, 91], [0, 0, -556, 0], [820, 0, 0, 0], [0, 115, 0, 0], [0, 0, 0, 59], [0, -220, 0, 0], [-230, 0, 0, 0], [0, 0, -316, 0], [430, 0, 0, 0], [0, -23, 0, 0], [0, 0, 0, 105], [0, 140, 0, 0], [0, 0, 0, 84], [0, 0, -20, 0], [0, 40, 0, 0], [0, 0, -84, 0], [0, 0, 0, -59], [0, -128, 0, 0], [0, -130, 0, 0], [0, -5, 0, 0], [-830, 0, 0, 0], [0, 0, 0, -101], [0, 0, 0, -150], [0, -85, 0, 0], [0, 0, -140, 0], [-1090, 0, 0, 0], [0, 0, -40, 0], [0, 0, 0, 4], [-1250, 0, 0, 0], [0, -190, 0, 0], [0, 0, 0, 110], [0, -115, 0, 0], [0, 0, 0, 51], [0, 0, 0, -35], [-1190, 0, 0, 0], [0, 0, 820, 0], [230, 0, 0, 0], [0, 0, -756, 0], [0, -125, 0, 0], [0, -13, 0, 0], [0, 0, 0, -99], [1210, 0, 0, 0], [0, 0, 0, 140], [0, 0, -1200, 0], [-1060, 0, 0, 0], [0, 0, -116, 0], [-2192, 0, 0, 0], [0, -115, 0, 0], [0, 0, -354, 0], [0, 260, 0, 0], [0, 0, 0, -10], [-1930, 0, 0, 0], [0, 0, -1084, 0], [0, 0, 996, 0], [0, 0, 1120, 0], [0, 0, 0, 185], [1070, 0, 0, 0], [0, 0, 0, -110], [0, -212, 0, 0], [0, 95, 0, 0], [2158, 0, 0, 0], [0, 0, 164, 0], [0, 0, 950, 0], [1570, 0, 0, 0], [0, 0, 0, -24], [0, -193, 0, 0], [-1310, 0, 0, 0], [0, 160, 0, 0], [0, 0, 1100, 0], [2408, 0, 0, 0], [0, 0, 0, 79], [0, 0, 950, 0], [0, -365, 0, 0], [0, 125, 0, 0], [0, 0, 0, 125], [0, 0, 0, -5], [0, -10, 0, 0], [-1670, 0, 0, 0], [0, 0, 134, 0], [0, 0, 0, 235], [0, 200, 0, 0], [0, -275, 0, 0], [0, 0, 0, -211], [0, 0, -530, 0], [-1270, 0, 0, 0], [0, 0, 0, 180], [0, 0, 0, 85], [0, 0, -1190, 0], [1780, 0, 0, 0], [0, 0, 1264, 0], [-370, 0, 0, 0], [0, 0, 624, 0], [0, 40, 0, 0], [0, 373, 0, 0], [0, 130, 0, 0], [0, 0, 0, 150], [2530, 0, 0, 0], [-130, 0, 0, 0], [0, 0, 514, 0], [0, 0, 520, 0], [1150, 0, 0, 0], [0, 400, 0, 0], [0, 0, 0, -170], [0, 0, -1104, 0], [0, 0, 0, 191], [0, 0, 180, 0], [0, -125, 0, 0], [1868, 0, 0, 0], [0, 0, 1244, 0], [0, 0, -420, 0], [-2290, 0, 0, 0], [0, 0, 0, -106], [0, -205, 0, 0], [0, 0, 0, -66], [0, 0, 0, 179], [0, 0, 196, 0], [0, 0, 0, 35], [0, 0, 0, -189], [0, -10, 0, 0], [-698, 0, 0, 0], [0, -187, 0, 0], [0, -220, 0, 0], [0, 0, 0, -164], [0, 0, 424, 0], [1718, 0, 0, 0], [-1130, 0, 0, 0], [1940, 0, 0, 0], [0, 0, 0, 145], [0, 0, 0, -64], [610, 0, 0, 0], [0, 130, 0, 0], [0, 0, 0, -35], [0, 0, 0, 30], [0, 0, 730, 0], [0, 0, 1676, 0], [2350, 0, 0, 0], [0, 0, 1036, 0], [0, 0, 0, 115], [0, 0, 0, -44], [0, 247, 0, 0], [1580, 0, 0, 0], [0, 0, -644, 0], [670, 0, 0, 0], [0, 0, 930, 0], [0, -385, 0, 0], [0, 310, 0, 0], [0, 0, 826, 0], [0, 0, -670, 0], [0, 0, 0, -119], [0, 0, 520, 0], [-1598, 0, 0, 0], [0, 515, 0, 0], [0, 0, 180, 0], [0, 0, 0, -124], [0, 0, 0, 56], [0, 0, -144, 0], [0, 0, 0, -135]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1584_j_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1584_3_j_i();".
// 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_1584_3_j_i(:prec:=4)
    chi := MakeCharacter_1584_j();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_1584_3_j_i();".
// 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_1584_3_j_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1584_j();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![-46, 0, 1]>,<7,R![138, 0, 1]>],Snew);
    return Vf;
end function;