
// Make newform 1152.4.d.e in Magma, downloaded from the LMFDB on 20 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1152_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1152_d_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_1152_4_d_e();".
// 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_1152_4_d_e();".
// 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 := [2, 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_1152_d();"
function MakeCharacter_1152_d()
    N := 1152;
    order := 2;
    char_gens := [127, 901, 641];
    v := [2, 1, 2];
    // 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_1152_d_Hecke();"
function MakeCharacter_1152_d_Hecke(Kf)
    N := 1152;
    order := 2;
    char_gens := [127, 901, 641];
    char_values := [[1, 0], [-1, 0], [1, 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+1)]];
    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 := 4;
    raw_aps := [[0, 0], [0, 0], [0, 0], [0, 0], [0, -25], [0, 0], [90, 0], [0, 45], [0, 0], [0, 0], [0, 0], [0, 0], [-522, 0], [0, -171], [0, 0], [0, 0], [0, 115], [0, 0], [0, -387], [0, 0], [430, 0], [0, 0], [0, -241], [1026, 0], [1910, 0], [0, 0], [0, 0], [0, -497], [0, 0], [270, 0], [0, 0], [0, -965], [-2250, 0], [0, -1035], [0, 0], [0, 0], [0, 0], [0, -1431], [0, 0], [0, 0], [0, 1015], [0, 0], [0, 0], [-2090, 0], [0, 0], [0, 0], [0, -135], [0, 0], [0, -673], [0, 0], [-6030, 0], [0, 0], [1222, 0], [0, -2365], [3870, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-9342, 0], [0, 1809], [0, 0], [0, -2547], [0, 0], [-8390, 0], [0, 0], [0, 3105], [11410, 0], [0, 4043], [0, 0], [-4770, 0], [0, 0], [0, 0], [0, 0], [0, -3195], [0, 0], [0, 0], [0, 0], [7002, 0], [16346, 0], [0, 1235], [0, 0], [0, 0], [5510, 0], [0, 0], [0, -1321], [17514, 0], [-18070, 0], [0, 0], [0, 0], [0, 4763], [0, 0], [0, 0], [0, -6365], [0, 4545], [0, 0], [0, 0], [-9162, 0], [0, 8289], [0, 0], [0, -4707], [0, 0], [0, -4381], [-2394, 0], [0, 1305], [-19550, 0], [0, 9367], [26190, 0], [0, 0], [14398, 0], [0, 0], [0, 0], [28530, 0], [0, -855], [0, 0], [-6678, 0], [0, -5571], [0, 0], [0, 0], [0, 5695], [0, 0], [19190, 0], [0, 0], [0, 11891], [0, 12825], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, -6255], [0, 0], [0, 0], [0, 0], [34182, 0], [-40106, 0], [0, 0], [0, 15417], [0, 0], [-14346, 0], [0, -9495], [0, 0], [0, 0], [0, 14623], [0, 0], [0, 0], [0, 0], [18630, 0], [0, -7515], [0, 0], [0, 0], [41742, 0], [0, -18459], [0, 0], [0, -16947], [0, 0], [0, 0], [-6966, 0], [56270, 0], [0, 0], [0, -893], [45990, 0], [0, 0], [0, 21395], [17370, 0], [0, 0], [0, 0], [0, 0], [-60154, 0], [0, 0], [0, -12665], [0, 0], [0, 0], [46190, 0], [0, 0], [-50814, 0], [0, -7875], [0, 0], [0, 0], [0, 0], [0, 0], [0, -3265], [0, 0], [70290, 0], [0, 0], [0, 0], [0, 0], [0, 24489], [-1334, 0], [0, 0], [-6910, 0], [0, 11419], [0, -315], [0, 0], [0, 22183], [80370, 0], [47698, 0], [0, 0], [-46530, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-6014, 0], [0, 30575], [0, 0], [0, 0], [0, -8809], [32274, 0], [0, -31095], [70630, 0], [0, 0], [0, 0], [0, -27353], [0, 0], [-85462, 0], [0, 0], [24282, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-103446, 0], [0, 0], [0, -38077], [0, 0], [-71550, 0], [0, 0], [0, 0], [0, 27835], [0, 0], [0, 39285], [0, 0], [-5742, 0], [0, 37341], [0, 0], [63974, 0], [0, 0], [0, 6095], [0, 0], [0, -4561], [0, 35505], [0, 0], [0, 0], [111150, 0], [0, 0], [0, 0], [0, -22565], [0, 29565], [0, 0], [0, 0], [29502, 0], [0, 0], [-60946, 0], [0, 0], [0, 31255], [0, 0], [0, 25677], [0, 0], [37370, 0], [0, 0], [0, -45713], [0, 0], [0, 0], [117450, 0], [0, -15075], [0, 0], [-71838, 0], [0, 34461], [0, 0], [0, 0], [0, 15957], [-120130, 0], [0, 0], [-141550, 0], [0, 0], [0, 25967], [0, 0], [10798, 0], [0, 48355], [0, 0], [0, 0], [0, 0], [0, 0], [0, -16587], [0, 0], [-50330, 0], [0, 0], [0, 0], [86526, 0], [0, 0], [0, 29203], [-26910, 0], [0, -58745], [0, 0], [0, 0], [0, 0], [0, 0], [0, -23425], [0, -3483], [-151670, 0], [0, 0], [0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1152_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1152_4_d_e();".
// 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_1152_4_d_e(:prec:=2)
    chi := MakeCharacter_1152_d();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_1152_4_d_e();".
// 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_1152_4_d_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1152_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![0, 1]>,<7,R![0, 1]>,<17,R![-90, 1]>],Snew);
    return Vf;
end function;