
// Make newform 25.2.d.a in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_25_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_25_d_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_25_2_d_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_25_2_d_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_25_d();"
function MakeCharacter_25_d()
    N := 25;
    order := 5;
    char_gens := [2];
    v := [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_25_d_Hecke();"
function MakeCharacter_25_d_Hecke(Kf)
    N := 25;
    order := 5;
    char_gens := [2];
    char_values := [[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 := [[0, -1, 1, 0], [0, 0, 0, -1], [-2, 1, -2, 0], [-1, 0, -1, 1], [2, -4, 4, -2], [3, 0, 3, 0], [0, 2, -4, 2], [0, -3, -1, -3], [-7, 5, -5, 7], [1, -1, 0, 2], [0, 0, -3, 0], [-2, -1, -2, 0], [2, 2, 2, 0], [0, 0, 3, -3], [-1, 1, 0, 1], [4, -4, 0, -3], [-3, 9, -3, 0], [1, 5, -5, -1], [0, 2, 6, 2], [1, -1, 0, 5], [9, -9, 9, -9], [5, -5, 0, 0], [0, 2, -5, 2], [-4, -4, 4, 4], [3, -3, 0, -1], [-5, 0, -4, 4], [-9, 9, 0, 6], [3, 0, 12, -12], [0, -10, 0, 0], [3, -15, 3, 0], [6, 10, -10, -6], [0, -11, 0, -11], [-8, 7, -8, 0], [-5, 5, -5, 5], [9, 0, 8, -8], [0, 0, -9, 9], [-3, 0, -10, 10], [0, -11, 0, 0], [0, -9, 9, -9], [7, 9, -9, -7], [-4, 4, 0, 7], [0, 6, 4, 6], [10, 8, 10, 0], [-2, 0, -6, 6], [-6, 6, 0, 6], [-3, 0, 9, -9], [-7, -3, 3, 7], [-16, 6, -6, 16], [-18, 16, -16, 18], [-6, 6, 0, -12], [0, 8, 2, 8], [27, -23, 23, -27], [-4, -5, -4, 0], [-13, 0, 10, -10], [18, 0, -3, 3], [5, 14, 5, 0], [0, -2, -14, -2], [0, 0, 0, 8], [15, -21, 21, -15], [0, -5, 7, -5], [0, 3, -28, 3], [-26, 0, -4, 4], [6, 0, -2, 2], [19, -2, 2, -19], [2, 18, 2, 0], [0, -14, -1, -14], [0, 18, -12, 18], [3, -6, 3, 0], [-5, 5, 0, 28], [12, 0, -6, 6], [5, -5, 0, -21], [19, -2, 19, 0], [0, 9, -20, 9], [4, -19, 19, -4], [-22, 22, 0, 21], [0, -20, 21, -20], [-15, 15, -15, 15], [-13, 13, 0, -8], [-9, 0, -22, 22], [-12, -9, -12, 0], [0, 4, -7, 4], [0, 0, 0, -32], [0, 24, -15, 24], [0, -3, 25, -3], [7, -28, 28, -7], [17, 0, -8, 8], [16, 0, 7, -7], [-2, 0, 12, -12], [-17, 7, -7, 17], [-18, 13, -18, 0], [0, -25, 12, -25], [3, -3, 0, 6], [-12, 29, -12, 0], [-36, 21, -36, 0], [-7, 0, -9, 9], [12, -12, 0, 18], [-7, 16, -7, 0], [-20, 20, 0, 3], [18, -15, 15, -18], [18, -16, 18, 0], [-6, 6, 0, 31], [8, 0, 2, -2], [-1, -8, -1, 0], [0, 4, 23, 4], [18, -18, 0, -21], [-18, -14, 14, 18], [-6, -9, -6, 0], [-19, 0, -5, 5], [-3, 0, 4, -4], [27, 0, 15, -15], [-21, 0, 9, -9], [-21, 2, -21, 0], [0, -2, 13, -2], [0, 4, 33, 4], [0, 2, -9, 2], [-5, -2, -5, 0], [8, 0, 24, -24], [30, -30, 0, -12], [-5, 5, 0, -11], [-4, -18, -4, 0], [24, 3, -3, -24], [4, 6, -6, -4], [-9, 10, -10, 9], [0, 4, -11, 4], [15, -27, 15, 0], [14, 0, 34, -34], [-30, 15, -30, 0], [0, -6, -27, -6], [-10, 19, -19, 10], [0, 21, -7, 21], [18, 3, -3, -18], [-4, 0, -20, 20], [-31, 0, -16, 16], [11, 0, -12, 12], [-22, -3, 3, 22], [0, -12, 6, -12], [-14, -17, 17, 14], [10, 18, 10, 0], [2, -2, 0, -13], [21, -18, 21, 0], [5, -11, 11, -5], [27, -27, 0, -3], [-38, 44, -44, 38], [22, -9, 9, -22], [3, -3, 0, 31], [-6, 9, -9, 6], [28, -28, 0, -30], [-24, 0, -27, 27], [12, 9, 12, 0], [9, 27, 9, 0], [-12, -18, 18, 12], [0, 20, -28, 20], [0, -12, -28, -12], [16, -20, 16, 0], [25, 0, 36, -36], [37, -39, 39, -37], [0, 23, -39, 23], [-35, 35, 0, 15], [-15, -27, -15, 0], [-12, 39, -12, 0], [-14, 14, 0, 20], [19, -19, 0, -23], [0, 16, 14, 16], [-1, 1, 0, -4], [25, -11, 11, -25], [0, -1, 9, -1], [-20, 3, -20, 0], [-16, 16, 0, 1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_25_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_25_2_d_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_25_2_d_a(:prec:=4)
    chi := MakeCharacter_25_d();
    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_25_2_d_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_25_2_d_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_25_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([],Snew);
    return Vf;
end function;