
// Make newform 475.2.a.e in Magma, downloaded from the LMFDB on 19 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_e();".
// 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_e();".
// 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, -4, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-3, -1, 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], [-1, 0, -1], [0, 0, 0], [-1, 0, 1], [0, 0, -1], [1, -3, 3], [-5, 0, -1], [1, 0, 0], [-1, -3, 2], [-2, 0, -1], [1, -1, 3], [-4, 2, -5], [-2, 5, -2], [-1, 6, -2], [-5, 5, -1], [-9, -2, 2], [-2, 2, 2], [1, 1, 1], [6, -4, 1], [3, -2, 0], [-2, 3, -2], [-3, -4, 5], [-2, -3, -6], [-9, 5, -2], [5, 1, 3], [3, -2, -5], [3, 4, -3], [-3, 9, -4], [2, -5, -4], [4, 3, 1], [11, -3, 0], [2, -3, -2], [-12, 1, 2], [-7, 4, -10], [2, -7, 8], [-4, 5, -3], [-1, -5, -4], [-15, -1, 3], [8, -9, 2], [-5, 6, -9], [-1, -4, -4], [7, -10, 10], [10, 0, 9], [-8, 9, 4], [-8, 0, 5], [-14, 6, -3], [-7, 0, -2], [11, 6, -6], [16, -3, 0], [0, 5, 5], [8, -6, 7], [1, -13, 11], [-18, 7, -2], [-2, -7, 5], [-11, -4, -5], [1, -5, 5], [6, 3, -5], [4, 2, 9], [-9, -4, -4], [8, -12, 10], [10, 1, 6], [-5, -3, 10], [5, 3, -4], [5, -9, 13], [6, -4, -1], [-20, -1, 0], [-2, 7, -9], [-7, 11, -3], [2, -2, -3], [-8, 17, -11], [-4, -5, 5], [-17, 1, -7], [-8, -5, -10], [-13, -1, -11], [-11, 7, -5], [13, 7, -1], [-5, -10, 0], [6, 3, -1], [-9, 8, 9], [21, -9, 12], [-10, 7, 3], [-9, 17, -5], [-18, -4, -1], [7, -5, 8], [0, 0, 5], [-6, -1, 0], [15, -5, 5], [21, -7, 19], [-16, -3, -11], [14, -9, 13], [-4, -7, 4], [20, 2, -2], [6, 7, 0], [-16, 9, -18], [-19, 12, -9], [21, -12, 17], [-14, 7, 6], [8, -6, 14], [-5, -8, -5], [22, 2, 3], [24, -7, 13], [-33, 8, -3], [20, 0, 7], [0, -6, 6], [8, -7, 0], [-3, -12, -3], [-12, 4, 9], [1, -11, 1], [-4, 8, -15], [8, -17, 0], [22, -11, 16], [8, 19, -13], [-16, 10, -14], [-2, 17, -8], [0, -3, -8], [-21, -14, 10], [-26, 4, -20], [-9, 7, -20], [-21, 13, 6], [6, -13, -9], [14, 9, -3], [29, -13, 12], [12, -10, 15], [5, 10, -13], [-14, -12, 4], [22, -9, -1], [-5, 9, 1], [15, 15, -5], [5, -6, -5], [-13, 6, 2], [8, -15, 19], [-21, 13, -14], [-1, -11, -8], [-9, 15, 2], [0, 15, -6], [10, 2, -12], [-27, -1, -8], [9, 7, 9], [18, -3, -4], [17, 13, 3], [12, 13, -3], [18, -9, 2], [-4, 17, 3], [10, 5, 9], [14, -6, -12], [3, -7, 1], [-26, -12, 11], [8, -3, 1], [12, -17, 18], [-9, -6, -4], [-4, -12, -16], [13, 1, 2], [-25, 16, 1], [17, -10, 15], [1, -17, 4], [-5, 9, -15], [14, -17, -1], [-2, -1, -10], [-1, 4, -13], [20, 1, 8], [17, -9, 4], [-4, 3, 2], [-42, 9, -1], [-24, 10, -8], [-7, 3, -15], [25, -12, 14], [-10, 7, 12], [23, -9, -8]];
    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_e();".
// 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_e(: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_e();".
// 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_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_475_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-5, -3, 2, 1]>],Snew);
    return Vf;
end function;