
// Make newform 450.2.h.a in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_450_h();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_450_h_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_450_2_h_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_450_2_h_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 := [1, -1, 1, -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_450_h();"
function MakeCharacter_450_h()
    N := 450;
    order := 5;
    char_gens := [101, 127];
    v := [5, 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_450_h_Hecke();"
function MakeCharacter_450_h_Hecke(Kf)
    N := 450;
    order := 5;
    char_gens := [101, 127];
    char_values := [[1, 0, 0, 0], [0, 0, 0, -1]];
    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, 1, -1, 1], [0, 0, 0, 0], [-2, 2, 0, 1], [-3, 0, 0, 0], [1, -3, 3, -1], [0, -1, 0, 0], [0, -3, -3, -3], [0, -3, 4, -3], [-5, 3, -3, 5], [-4, 4, 0, -3], [0, 2, 1, 2], [-7, 4, -7, 0], [-4, -1, -4, 0], [-3, 0, 2, -2], [-8, 8, 0, 1], [4, -4, 0, 4], [4, -2, 4, 0], [4, -7, 7, -4], [0, 2, -9, 2], [0, 0, 0, -3], [2, 4, -4, -2], [2, -2, 0, 4], [0, 4, 7, 4], [-2, -2, 2, 2], [-9, 9, 0, 5], [1, 0, 1, -1], [-3, 3, 0, 18], [-7, 0, -5, 5], [0, 15, 0, 0], [2, 5, 2, 0], [10, -6, 6, -10], [0, -12, 9, -12], [-5, 2, -5, 0], [-6, -1, 1, 6], [1, 0, 2, -2], [0, 0, 6, -6], [-10, 0, -9, 9], [3, 0, 3, 0], [0, -1, -14, -1], [2, 0, 0, -2], [10, -10, 0, -5], [0, -12, -2, -12], [14, -5, 14, 0], [8, 0, 14, -14], [0, 0, 0, 12], [-12, 0, -9, 9], [-2, -13, 13, 2], [9, -9, 9, -9], [-3, -17, 17, 3], [9, -9, 0, -12], [0, -6, -18, -6], [-15, 10, -10, 15], [1, -15, 1, 0], [-17, 0, -10, 10], [5, 0, 19, -19], [-1, 4, -1, 0], [0, -5, -5, -5], [-17, 17, 0, -1], [-10, 4, -4, 10], [0, 10, 3, 10], [0, -6, -16, -6], [6, 0, 9, -9], [6, 0, 18, -18], [-18, 13, -13, 18], [-9, 11, -9, 0], [0, -1, -9, -1], [0, 9, -10, 9], [12, 12, 12, 0], [-5, 5, 0, -13], [-21, 0, -2, 2], [-2, 2, 0, -18], [5, -5, 5, 0], [0, -30, 17, -30], [15, -13, 13, -15], [-4, 4, 0, -3], [0, 7, -7, 7], [-19, -4, 4, 19], [2, -2, 0, 7], [-2, 0, -10, 10], [-6, 3, -6, 0], [0, -2, -24, -2], [-15, 15, 0, 3], [0, 26, -10, 26], [0, 20, -9, 20], [-3, 17, -17, 3], [0, 0, 12, -12], [-9, 0, -8, 8], [3, 0, -23, 23], [22, -12, 12, -22], [-21, 22, -21, 0], [0, -10, 13, -10], [11, -11, 0, -33], [18, -11, 18, 0], [-2, 18, -2, 0], [39, 0, 3, -3], [-17, 17, 0, -3], [-5, -5, -5, 0], [-25, 25, 0, 17], [-18, 4, -4, 18], [-6, -19, -6, 0], [1, -1, 0, -10], [1, 0, 16, -16], [-3, -30, -3, 0], [0, -2, -19, -2], [-7, 7, 0, -6], [43, -43, 43, -43], [10, -28, 10, 0], [26, 0, 34, -34], [4, 0, 8, -8], [0, 0, -24, 24], [-29, 0, 3, -3], [9, 27, 9, 0], [0, 20, -2, 20], [0, -42, 21, -42], [0, 24, -30, 24], [20, -18, 20, 0], [-35, 0, -2, 2], [-14, 14, 0, -1], [-22, 22, 0, 27], [14, -17, 14, 0], [3, -18, 18, -3], [0, 2, -2, 0], [10, -9, 9, -10], [0, -20, -1, -20], [-15, 23, -15, 0], [-2, 0, 10, -10], [2, -11, 2, 0], [0, 17, -1, 17], [20, -6, 6, -20], [0, -32, 12, -32], [-30, 30, -30, 30], [9, 0, 15, -15], [3, 0, -3, 3], [3, 0, 12, -12], [-20, -9, 9, 20], [0, -12, -4, -12], [13, -24, 24, -13], [-12, 4, -12, 0], [1, -1, 0, 4], [2, -1, 2, 0], [-28, 36, -36, 28], [31, -31, 0, -11], [-18, -6, 6, 18], [16, -3, 3, -16], [-12, 12, 0, -34], [3, 33, -33, -3], [-3, 3, 0, 23], [-10, 0, -41, 41], [33, -14, 33, 0], [2, -10, 2, 0], [-19, 10, -10, 19], [0, 10, -32, 10], [0, 27, -25, 27], [-10, 27, -10, 0], [24, 0, -1, 1], [3, 9, -9, -3], [0, 14, 3, 14], [-16, 16, 0, -27], [29, -24, 29, 0], [-1, -10, -1, 0], [1, -1, 0, -41], [1, -1, 0, 43], [0, 18, -2, 18], [26, -26, 0, -36], [-5, -29, 29, 5], [0, 38, -15, 38], [-25, -12, -25, 0], [-36, 36, 0, 31]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_450_h_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_450_2_h_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_450_2_h_a(:prec:=4)
    chi := MakeCharacter_450_h();
    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_450_2_h_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_450_2_h_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_450_h();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![3, 1]>,<11,R![1, 7, 19, 3, 1]>],Snew);
    return Vf;
end function;