
// Make newform 700.2.c.f in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_700_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_700_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_700_2_c_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_2_c_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, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 1, 1], [0, 1, -1, -1], [-1, 1, 1, -1]];
    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_700_c();"
function MakeCharacter_700_c()
    N := 700;
    order := 2;
    char_gens := [351, 477, 101];
    v := [1, 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_700_c_Hecke();"
function MakeCharacter_700_c_Hecke(Kf)
    N := 700;
    order := 2;
    char_gens := [351, 477, 101];
    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 := 2;
    raw_aps := [[1, 0, 0, 1], [0, 0, 1, -1], [0, 0, 0, 0], [2, 0, -1, 1], [1, -1, -1, 2], [-2, 1, 2, 1], [-4, -1, -2, -1], [6, 0, 0, 0], [1, -1, -2, -1], [-3, -2, -4, -2], [6, 0, 0, 0], [3, -3, 2, 1], [0, 0, -2, 2], [-2, 0, 0, 0], [0, 0, 1, -1], [1, -1, 0, 1], [-1, -1, -2, -1], [3, -3, 2, 1], [-1, -1, -2, -1], [2, -2, 2, 0], [-8, -2, -4, -2], [-3, 3, -5, 2], [-6, 6, -2, -4], [-3, 3, 2, -5], [-6, -3, -6, -3], [-6, 6, -2, -4], [6, -6, -3, 9], [5, -3, -6, -3], [-7, 2, 4, 2], [1, -1, 2, -1], [-13, -1, -2, -1], [-5, 1, 2, 1], [-5, 5, -6, 1], [2, 2, 4, 2], [-8, 2, 4, 2], [5, -5, 3, 2], [-2, 4, 8, 4], [6, 6, 12, 6], [0, 0, -3, 3], [2, 5, 10, 5], [-2, 2, 6, -8], [3, -3, -4, 7], [1, -1, 3, -2], [-3, 3, -2, -1], [-2, 2, 10, -12], [1, 1, 2, 1], [-1, 1, 3, -4], [6, -6, 1, 5], [6, -6, 5, 1], [6, -6, -2, 8], [-8, 8, 4, -12], [-1, 1, 15, -16], [3, -3, 6, -3], [16, 4, 8, 4], [6, 0, 0, 0], [9, 1, 2, 1], [6, -6, 0, 6], [-7, -1, -2, -1], [2, -2, -12, 14], [-3, -2, -4, -2], [0, 0, -7, 7], [-8, -5, -10, -5], [0, 0, 1, -1], [2, -4, -8, -4], [20, -1, -2, -1], [5, -5, -4, 9], [-8, 8, -6, -2], [0, 0, 0, 0], [17, 7, 14, 7], [-6, 6, 10, -16], [0, 3, 6, 3], [-4, 4, -10, 6], [12, -12, -7, 19], [-7, 7, 6, -13], [-8, 8, 6, -14], [-12, 12, -2, -10], [11, 4, 8, 4], [10, 1, 2, 1], [-15, 2, 4, 2], [-9, 9, 8, -17], [-7, -7, -14, -7], [19, 0, 0, 0], [1, -1, -9, 10], [22, 4, 8, 4], [-20, -8, -16, -8], [26, 0, 0, 0], [-7, -2, -4, -2], [-12, 12, 2, -14], [0, 0, 16, -16], [9, 3, 6, 3], [0, 0, 13, -13], [3, 9, 18, 9], [2, -6, -12, -6], [-11, 11, 7, -18], [-5, 5, -9, 4], [0, 0, 9, -9], [-6, 6, -8, 2], [12, -12, -6, 18], [0, 0, 14, -14], [-7, 6, 12, 6], [19, 1, 2, 1], [4, -4, 8, -4], [6, -6, -18, 24], [-12, 2, 4, 2], [-12, 12, 10, -22], [12, 3, 6, 3], [-12, 12, 2, -14], [-28, -1, -2, -1], [-1, 1, 7, -8], [3, -3, -12, 15], [-12, 12, -3, -9], [-5, 5, 0, -5], [8, -8, 4, 4], [13, -5, -10, -5], [3, -3, 19, -16], [4, 2, 4, 2], [6, -6, -11, 17], [0, 0, -6, 6], [14, -14, -6, 20], [13, -13, 3, 10], [6, -6, 16, -10], [9, -9, -18, 27], [-18, -3, -6, -3], [5, -5, -10, -5], [-2, 10, 20, 10], [-23, -6, -12, -6], [-21, 0, 0, 0], [41, -1, -2, -1], [-12, 12, 6, -18], [-14, 1, 2, 1], [3, -3, -1, 4], [-23, -3, -6, -3], [-5, 5, -9, 4], [-12, 12, -8, -4], [-9, 9, -14, 5], [-9, 9, 0, -9], [8, -1, -2, -1], [12, -12, 5, 7], [0, -9, -18, -9], [-3, -2, -4, -2], [-34, 2, 4, 2], [13, 6, 12, 6], [-16, 2, 4, 2], [24, -4, -8, -4], [12, -12, -14, 26], [-3, 3, 6, 3], [-12, 6, 12, 6], [38, 8, 16, 8], [20, 8, 16, 8], [30, 4, 8, 4], [4, -4, 6, -2], [-18, 18, 4, -22], [31, -3, -6, -3], [-6, 6, 2, -8], [-25, -3, -6, -3], [2, -2, -14, 16], [11, -11, -9, 20], [3, -3, 28, -25], [-18, -3, -6, -3], [-3, 3, 26, -29], [8, 6, 12, 6], [2, -2, 10, -8], [9, 9, 18, 9], [0, -6, -12, -6], [-10, 10, 14, -24], [6, -6, 9, -3], [8, -8, 18, -10], [8, -1, -2, -1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_700_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_700_2_c_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_2_c_f(:prec:=4)
    chi := MakeCharacter_700_c();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    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_2_c_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_2_c_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_700_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![3, 0, 1]>,<11,R![1, 0, 14, 0, 1]>,<13,R![-3, 6, 1]>,<19,R![-6, 1]>,<23,R![-8, -4, 1]>],Snew);
    return Vf;
end function;