
// Make newform 700.4.e.f in Magma, downloaded from the LMFDB on 28 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_700_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_700_e_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_700_4_e_f();".
// 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_700_4_e_f();".
// 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, 0, 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_700_e();"
function MakeCharacter_700_e()
    N := 700;
    order := 2;
    char_gens := [351, 477, 101];
    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_700_e_Hecke();"
function MakeCharacter_700_e_Hecke(Kf)
    N := 700;
    order := 2;
    char_gens := [351, 477, 101];
    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, 4], [0, 0], [0, -7], [-12, 0], [0, -82], [0, 30], [-68, 0], [0, 216], [-246, 0], [-112, 0], [0, -110], [-246, 0], [0, -172], [0, -192], [0, 558], [-540, 0], [110, 0], [0, -140], [-840, 0], [0, -550], [208, 0], [0, 516], [1398, 0], [0, -1586], [-1242, 0], [0, 680], [0, -996], [-1382, 0], [0, -750], [0, -176], [-1548, 0], [0, -378], [2500, 0], [-846, 0], [-2536, 0], [0, 1186], [0, 2108], [0, 1944], [0, -1362], [-1596, 0], [-1690, 0], [3552, 0], [0, -2686], [0, 1410], [2968, 0], [-1348, 0], [0, 3872], [0, -5364], [874, 0], [0, 378], [-1920, 0], [4322, 0], [5292, 0], [0, 5118], [0, 3768], [-3918, 0], [4880, 0], [0, 3538], [-5430, 0], [0, -6436], [0, 1350], [0, -3332], [-4728, 0], [0, 5114], [0, -7206], [6260, 0], [0, 5326], [0, -36], [-3134, 0], [0, 1218], [10008, 0], [0, 1072], [0, -274], [-7652, 0], [0, 2160], [1074, 0], [0, -6926], [1938, 0], [9574, 0], [5052, 0], [3422, 0], [2208, 0], [0, -6814], [-12584, 0], [0, 6996], [-9474, 0], [0, -5786], [3438, 0], [0, 9392], [0, 4956], [20592, 0], [0, 13432], [-14172, 0], [5956, 0], [0, 16968], [-5214, 0], [-1398, 0], [0, -18580], [-18970, 0], [0, 16036], [0, -8310], [0, 7092], [7158, 0], [6500, 0], [0, -21794], [0, -9756], [0, 5586], [24, 0], [4298, 0], [0, -8480], [0, -1906], [0, -7482], [7348, 0], [4520, 0], [-19806, 0], [0, -5020], [0, -28392], [0, -17562], [-4716, 0], [-22762, 0], [0, 4802], [0, -21558], [0, 3780], [-5500, 0], [10230, 0], [-10190, 0], [-9408, 0], [0, 33064], [0, -6322], [20740, 0], [0, 32040], [-12832, 0], [0, 19906], [10842, 0], [-28274, 0], [0, -32346], [0, -30116], [0, 6594], [43014, 0], [-14164, 0], [34830, 0], [0, 31016], [0, -9876], [3154, 0], [-36936, 0], [0, 9638], [0, -10266], [4084, 0], [0, -192], [0, -19910], [14802, 0], [0, -32548], [0, -1464], [0, 49564], [8448, 0], [-14600, 0], [21102, 0], [0, 20806], [24510, 0], [0, 44148], [0, 27114], [0, 10264], [51468, 0], [0, 23790], [0, 26424], [39488, 0], [0, -30854]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_700_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_700_4_e_f();".
// 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_700_4_e_f(:prec:=2)
    chi := MakeCharacter_700_e();
    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_700_4_e_f();".
// 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_700_4_e_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_700_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![16, 0, 1]>,<11,R![12, 1]>],Snew);
    return Vf;
end function;