
// Make newform 48.3.g.a in Magma, downloaded from the LMFDB on 01 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_48_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_48_g_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_48_3_g_a();".
// 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_48_3_g_a();".
// 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 := [1, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [-1, 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_48_g();"
function MakeCharacter_48_g()
    N := 48;
    order := 2;
    char_gens := [31, 37, 17];
    v := [1, 2, 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_48_g_Hecke();"
function MakeCharacter_48_g_Hecke(Kf)
    N := 48;
    order := 2;
    char_gens := [31, 37, 17];
    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 := 3;
    raw_aps := [[0, 0], [0, -1], [6, 0], [0, -4], [0, 12], [-14, 0], [-6, 0], [0, 4], [0, 0], [30, 0], [0, -12], [26, 0], [-54, 0], [0, 12], [0, -24], [-18, 0], [0, -12], [-70, 0], [0, -68], [0, 48], [82, 0], [0, -44], [0, -12], [114, 0], [34, 0], [-18, 0], [0, -76], [0, 84], [34, 0], [-78, 0], [0, 60], [0, 60], [186, 0], [0, -28], [-186, 0], [0, -20], [170, 0], [0, 164], [0, -120], [-42, 0], [0, -84], [82, 0], [0, 192], [-94, 0], [-258, 0], [0, 68], [0, -52], [0, 204], [0, 84], [226, 0], [114, 0], [0, 192], [178, 0], [0, 60], [258, 0], [0, -216], [510, 0], [0, -260], [-14, 0], [354, 0], [0, 84], [-498, 0], [0, 108], [0, -24], [290, 0], [-210, 0], [0, 116], [-302, 0], [0, -36], [-358, 0], [-558, 0], [0, -48], [0, -124], [554, 0], [0, -308], [0, -288], [198, 0], [-646, 0], [330, 0], [130, 0], [0, -204], [-398, 0], [0, -72], [-142, 0], [0, 324], [0, 252], [-198, 0], [-446, 0], [342, 0], [0, 92], [0, 228], [0, 456], [0, -4], [0, 372], [0, -468], [0, -192], [-306, 0], [522, 0], [0, 28], [802, 0], [0, 20], [-474, 0], [0, -396], [-150, 0], [0, 388], [-46, 0], [0, -204], [114, 0], [0, -144], [626, 0], [0, -388], [-694, 0], [-30, 0], [0, 196], [0, 268], [-390, 0], [0, 468], [0, -336], [774, 0], [0, -132], [-454, 0], [434, 0], [-234, 0], [0, 156], [0, -12], [-1074, 0], [898, 0], [0, 552], [0, -468], [370, 0], [0, 492], [0, 792], [0, -44], [514, 0], [-966, 0], [-958, 0], [-546, 0], [0, -140], [-1338, 0], [-966, 0], [0, 876], [222, 0], [0, -740], [0, -828], [226, 0], [0, -288], [-70, 0], [42, 0], [0, -532], [0, 96], [-166, 0], [-702, 0], [0, -364], [0, -744], [0, -836], [0, 192], [0, -324], [-438, 0], [1826, 0], [-330, 0], [0, 108], [-1110, 0], [0, 436], [0, -228], [-918, 0], [0, 24], [0, -28], [554, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_48_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_48_3_g_a();".
// 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_48_3_g_a(:prec:=2)
    chi := MakeCharacter_48_g();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_48_3_g_a();".
// 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_48_3_g_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_48_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([],Snew);
    return Vf;
end function;