
// Make newform 48.4.c.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_48_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_48_c_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_4_c_b();".
// 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_4_c_b();".
// 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 := [4, 0, -2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [-2, 0, 2, 0], [1, 4, -1, -1], [0, 0, 0, 6]];
    Rf_basisdens := [1, 1, 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_c();"
function MakeCharacter_48_c()
    N := 48;
    order := 2;
    char_gens := [31, 37, 17];
    v := [1, 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_48_c_Hecke();"
function MakeCharacter_48_c_Hecke(Kf)
    N := 48;
    order := 2;
    char_gens := [31, 37, 17];
    char_values := [[-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+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, -1, 0], [0, 0, 0, -1], [0, -5, 0, 0], [0, -3, -6, 0], [-26, 0, 0, 0], [0, 0, 0, 4], [0, -31, 0, 0], [0, 18, 36, 0], [0, 0, 0, 1], [0, -9, 0, 0], [206, 0, 0, 0], [0, 0, 0, -18], [0, -27, 0, 0], [0, -12, -24, 0], [0, 0, 0, 3], [0, 57, 114, 0], [278, 0, 0, 0], [0, 257, 0, 0], [0, 6, 12, 0], [-422, 0, 0, 0], [0, -193, 0, 0], [0, 3, 6, 0], [0, 0, 0, -22], [-1070, 0, 0, 0], [0, 0, 0, -105], [0, -5, 0, 0], [0, 141, 282, 0], [358, 0, 0, 0], [0, 0, 0, 4], [0, -9, 0, 0], [0, -177, -354, 0], [0, 0, 0, 134], [0, 445, 0, 0], [0, 0, 0, 139], [0, -589, 0, 0], [-106, 0, 0, 0], [0, -23, 0, 0], [0, 318, 636, 0], [0, 0, 0, -131], [0, -285, -570, 0], [4510, 0, 0, 0], [0, -192, -384, 0], [4994, 0, 0, 0], [0, 0, 0, 127], [0, 1147, 0, 0], [0, 1417, 0, 0], [0, -1161, 0, 0], [0, 303, 606, 0], [430, 0, 0, 0], [0, 0, 0, 374], [0, 420, 840, 0], [4690, 0, 0, 0], [0, 129, 258, 0], [0, 0, 0, -16], [0, -522, -1044, 0], [0, 0, 0, -187], [0, -193, 0, 0], [-2018, 0, 0, 0], [0, 0, 0, -278], [0, -1467, 0, 0], [0, 0, 0, -25], [0, 1017, 0, 0], [0, -966, -1932, 0], [-3718, 0, 0, 0], [0, 0, 0, 425], [0, 445, 0, 0], [2530, 0, 0, 0], [0, -567, -1134, 0], [-7786, 0, 0, 0], [0, 0, 0, -24], [0, 894, 1788, 0], [0, 3631, 0, 0], [-4258, 0, 0, 0], [0, -787, 0, 0], [0, -432, -864, 0], [0, 0, 0, -321], [13574, 0, 0, 0], [0, 0, 0, 692], [2890, 0, 0, 0], [0, -849, -1698, 0], [10414, 0, 0, 0], [0, 1332, 2664, 0], [-9214, 0, 0, 0], [0, 2115, 0, 0], [0, 225, 450, 0], [0, 0, 0, 168], [-38, 0, 0, 0], [0, 0, 0, 933], [0, -3281, 0, 0], [0, 51, 102, 0], [0, -888, -1776, 0], [0, -1157, 0, 0], [0, 1725, 3450, 0], [0, -3591, 0, 0], [0, -1302, -2604, 0], [0, 0, 0, -327], [0, 0, 0, 678], [0, -5035, 0, 0], [-6794, 0, 0, 0], [0, 1129, 0, 0], [0, 0, 0, -1331], [0, 891, 1782, 0], [0, 0, 0, -950], [0, 2741, 0, 0], [-8974, 0, 0, 0], [0, 501, 1002, 0], [0, 0, 0, -868], [0, 1026, 2052, 0], [23690, 0, 0, 0], [0, -6521, 0, 0], [9422, 0, 0, 0], [0, 0, 0, -58], [0, 3533, 0, 0], [0, 4019, 0, 0], [0, 0, 0, 968], [0, -423, 0, 0], [0, 1254, 2508, 0], [0, 0, 0, 501], [0, -1749, -3498, 0], [-10018, 0, 0, 0], [1682, 0, 0, 0], [0, 0, 0, -537], [0, 2373, 4746, 0], [0, 7281, 0, 0], [0, 0, 0, -967], [-27602, 0, 0, 0], [0, 3228, 6456, 0], [0, 7875, 0, 0], [-15914, 0, 0, 0], [0, 657, 0, 0], [0, -882, -1764, 0], [0, -4801, 0, 0], [11422, 0, 0, 0], [0, 0, 0, 746], [18818, 0, 0, 0], [0, 0, 0, -113], [0, 5729, 0, 0], [0, 0, 0, 1825], [0, 0, 0, -562], [0, 1125, 0, 0], [0, 0, 0, -781], [0, -2893, 0, 0], [0, 3177, 6354, 0], [-12554, 0, 0, 0], [0, 2214, 4428, 0], [-4546, 0, 0, 0], [0, 0, 0, 1922], [0, -6179, 0, 0], [0, -1032, -2064, 0], [20870, 0, 0, 0], [0, 0, 0, -1692], [0, -815, 0, 0], [0, -246, -492, 0], [0, 5333, 0, 0], [0, -4548, -9096, 0], [0, -11709, 0, 0], [0, 0, 0, 3328], [-7030, 0, 0, 0], [0, 0, 0, -1619], [0, -333, -666, 0], [0, 0, 0, -502], [0, -8437, 0, 0], [0, -4803, -9606, 0], [0, 0, 0, 1620], [0, -3822, -7644, 0], [0, 10759, 0, 0], [-45106, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_48_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_48_4_c_b();".
// 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_4_c_b(:prec:=4)
    chi := MakeCharacter_48_c();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_4_c_b();".
// 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_4_c_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_48_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![288, 0, 1]>],Snew);
    return Vf;
end function;