
// Make newform 124.4.d.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_124_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_124_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_124_4_d_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_124_4_d_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 := [8, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_124_d();"
function MakeCharacter_124_d()
    N := 124;
    order := 2;
    char_gens := [63, 65];
    v := [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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_124_d_Hecke();"
function MakeCharacter_124_d_Hecke(Kf)
    N := 124;
    order := 2;
    char_gens := [63, 65];
    char_values := [[-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, 1], [0, 0], [-2, 0], [6, -12], [0, 0], [0, 0], [0, 0], [10, -20], [0, 0], [0, 0], [31, -62], [0, 0], [-278, 0], [0, 0], [-34, 68], [0, 0], [-94, 188], [0, 0], [154, -308], [118, -236], [0, 0], [0, 0], [0, 0], [0, 0], [1906, 0], [1790, 0], [-298, 596], [274, -548], [-1546, 0], [2402, 0], [0, 0], [538, -1076], [0, 0], [0, 0], [3470, 0], [0, 0], [-1146, 0], [-582, 1164], [0, 0], [-4522, 0], [0, 0], [0, 0], [-50, 100], [-4542, 0], [0, 0], [0, 0], [1002, -2004], [0, 0], [1114, -2228], [0, 0], [-5398, 0], [0, 0], [0, 0], [0, 0], [-4286, 0], [0, 0], [0, 0], [0, 0], [0, 0], [650, 0], [1602, -3204], [5278, 0], [746, -1492], [710, -1420], [0, 0], [11206, 0], [0, 0], [0, 0], [0, 0], [-4410, 0], [0, 0], [-2330, 4660], [0, 0], [-14258, 0], [-126, 252], [0, 0], [0, 0], [-15626, 0], [0, 0], [0, 0], [3034, -6068], [11838, 0], [-2338, 4676], [0, 0], [3286, -6572], [1282, -2564], [0, 0], [0, 0], [0, 0], [0, 0], [-3286, 6572], [2174, -4348], [0, 0], [0, 0], [0, 0], [-122, 244], [0, 0], [-14630, 0], [0, 0], [24870, 0], [4474, -8948], [0, 0], [-4022, 8044], [0, 0], [0, 0], [-25326, 0], [0, 0], [-25342, 0], [-4810, 9620], [0, 0], [-3394, 6788], [0, 0], [30074, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [7222, 0], [-6070, 12140], [30350, 0], [0, 0], [0, 0], [-4558, 9116], [3210, -6420], [17702, 0], [0, 0], [0, 0], [4134, -8268], [27078, 0], [0, 0], [0, 0], [-1378, 2756], [0, 0], [0, 0], [25170, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-6670, 13340], [0, 0], [0, 0], [0, 0], [0, 0], [5830, -11660], [19278, 0], [42826, 0], [0, 0], [0, 0], [-38634, 0], [0, 0], [0, 0], [-2026, 4052], [-9134, 18268], [0, 0], [3430, -6860], [0, 0], [-43974, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-1870, 3740], [-23246, 0], [0, 0], [0, 0], [54686, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_124_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_124_4_d_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_124_4_d_a(:prec:=2)
    chi := MakeCharacter_124_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(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_124_4_d_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_124_4_d_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_124_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![0, 1]>,<5,R![2, 1]>],Snew);
    return Vf;
end function;