
// Make newform 475.2.a.d in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_475_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_475_2_a_d();".
// 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_475_2_a_d();".
// 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, -2, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-2, 0, 1]];
    Rf_basisdens := [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_475_a();"
function MakeCharacter_475_a()
    N := 475;
    order := 1;
    char_gens := [77, 401];
    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;

function MakeCharacter_475_a_Hecke(Kf)
    return MakeCharacter_475_a();
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, -1, 0], [0, -1, 1], [0, 0, 0], [0, 1, 1], [-1, 1, -3], [-2, 0, -1], [-2, 3, -1], [-1, 0, 0], [-3, 1, 0], [1, -5, 5], [-2, -2, -3], [-3, 3, -1], [4, -3, 8], [-5, 4, -6], [0, -2, 1], [-3, -6, 4], [4, -4, -2], [-4, -4, 1], [5, -9, 5], [-7, 4, 2], [-2, -1, -8], [4, 3, -3], [-6, 5, 0], [-1, -1, -6], [-2, 0, -1], [-6, 11, -7], [6, -3, 11], [5, -1, 4], [6, -11, 2], [-13, 6, -1], [7, -13, 12], [-2, -9, 4], [-8, 1, -8], [7, 2, 2], [4, -3, 10], [-3, 4, 5], [-9, 5, -2], [10, -8, 1], [-6, 9, -12], [-12, 1, 5], [-3, -2, -6], [3, -14, 8], [11, 3, 3], [8, 5, -6], [-13, 1, 5], [-9, 17, -13], [1, 0, 12], [-7, 14, 4], [-4, 5, 10], [-13, 8, -3], [-11, 9, 11], [-2, 8, -1], [12, 3, -8], [-11, 0, -3], [-10, 3, 1], [16, -10, 15], [-7, 16, -5], [-11, 5, -9], [-3, 0, -8], [26, -2, 2], [14, -3, 4], [-15, 9, -2], [21, 9, -4], [-4, 14, -7], [19, -1, 3], [-16, 7, -10], [-7, 0, -5], [-4, 16, 1], [1, -9, 5], [11, -12, 9], [9, -22, -1], [2, -12, 1], [12, 7, -8], [2, -4, 5], [2, -6, -11], [6, -18, -5], [-9, 6, -8], [5, -16, -11], [-6, 15, 9], [-17, 3, -16], [-9, -8, 7], [0, 16, -11], [5, 3, -7], [23, -11, 8], [17, -19, 19], [0, 21, -10], [-14, 12, -9], [-2, 10, -3], [-5, 24, 1], [-13, 6, -13], [30, -9, 18], [2, -16, -6], [26, -5, 6], [24, -17, 20], [-4, -7, -7], [10, -3, -15], [-16, -1, 0], [22, -20, 22], [12, -1, -7], [-15, -5, -15], [-5, -8, 7], [4, 5, 17], [21, -19, 15], [-10, 0, -6], [14, -27, 22], [14, -15, 9], [-1, -7, 9], [28, -8, 23], [-11, 9, 3], [-12, -3, -2], [-22, 13, -22], [-21, -12, 1], [-6, -28, 10], [-8, 5, 18], [-22, -3, -6], [-5, -2, -24], [2, 12, -4], [-13, 9, -18], [-15, 5, 8], [1, 16, 3], [23, -4, 17], [-15, 21, -28], [-35, 19, -33], [-10, -3, 5], [6, -20, 24], [-21, 8, -17], [34, 4, -3], [-32, 4, 3], [12, 5, 19], [-17, 8, -18], [13, -6, -21], [-7, 5, -10], [-19, 5, -16], [-11, -9, 0], [8, 17, -16], [-10, -2, 16], [-15, -11, -12], [-12, 0, 17], [-30, 5, 6], [-2, 14, 7], [-11, 28, 1], [-18, -3, 8], [23, -18, 29], [-31, 16, -25], [22, -22, 24], [18, 14, -5], [-13, 15, -17], [7, 12, -25], [20, 5, 20], [-9, 0, -10], [24, 0, 12], [5, -13, 4], [-4, 17, 1], [22, -19, 33], [-7, -27, 20], [42, 2, -5], [15, -12, -1], [2, 29, -12], [42, -5, -5], [-16, -13, -2], [-11, -15, -14], [18, -23, 12], [5, -2, 5], [-8, 2, -12], [-20, -4, -23], [19, -8, 0], [-10, 25, -6], [-13, -3, 2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_475_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_475_2_a_d();".
// 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_475_2_a_d(:prec:=3)
    chi := MakeCharacter_475_a();
    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_475_2_a_d();".
// 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_475_2_a_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_475_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 3, 4, 1]>],Snew);
    return Vf;
end function;