
// Make newform 625.2.d.g in Magma, downloaded from the LMFDB on 04 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_625_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_625_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_625_2_d_g();".
// 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_625_2_d_g();".
// 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_625_d();"
function MakeCharacter_625_d()
    N := 625;
    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_625_d_Hecke();"
function MakeCharacter_625_d_Hecke(Kf)
    N := 625;
    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], [1, -1, 0, -2], [0, 0, 0, 0], [3, 0, 0, 0], [3, -3, 3, -3], [3, -3, 3, 0], [0, 2, -3, 2], [0, 1, -3, 1], [2, 0, 0, -2], [-6, 6, 0, 3], [0, -5, 2, -5], [-6, 0, -6, 0], [0, 3, 0, 0], [9, 0, 0, 0], [-7, 7, 0, 3], [1, -1, 0, 3], [-3, -6, -3, 0], [-2, -3, 3, 2], [0, -3, 12, -3], [0, 0, 0, 3], [-3, 0, 0, 3], [4, -4, 0, -7], [0, 4, 2, 4], [-6, -6, 6, 6], [3, -3, 0, 3], [-3, 0, 0, 0], [6, -6, 0, 3], [8, 0, 5, -5], [7, -1, 7, 0], [8, 7, 8, 0], [0, 6, -6, 0], [0, 15, -3, 15], [-1, 5, -1, 0], [9, -11, 11, -9], [-15, 0, 0, 0], [-13, 0, 5, -5], [-12, 0, -15, 15], [3, -18, 3, 0], [0, 2, -8, 2], [-13, 5, -5, 13], [-6, 6, 0, 3], [0, -10, 12, -10], [0, -12, 0, 0], [9, 0, 15, -15], [8, -8, 0, -17], [0, 0, -10, 10], [-12, 2, -2, 12], [-3, 0, 0, 3], [-5, 6, -6, 5], [-1, 1, 0, 3], [0, -16, 17, -16], [-21, 24, -24, 21], [-10, 13, -10, 0], [-18, 0, -15, 15], [3, 0, 20, -20], [-7, 12, -7, 0], [0, -9, -3, -9], [-5, 5, 0, -12], [15, -9, 9, -15], [0, 0, 12, 0], [0, -6, 12, -6], [-11, 0, -20, 20], [18, 0, 15, -15], [3, 12, -12, -3], [-12, 12, -12, 0], [0, -8, 22, -8], [0, 5, -18, 5], [24, -15, 24, 0], [8, -8, 0, 3], [15, 0, 20, -20], [16, -16, 0, -22], [-3, 9, -3, 0], [0, -3, -3, -3], [-3, 0, 0, 3], [9, -9, 0, -7], [0, 4, -33, 4], [-21, 9, -9, 21], [-12, 12, 0, 3], [12, 0, 0, 0], [2, 24, 2, 0], [0, 6, -3, 6], [5, -5, 0, -17], [0, 15, -3, 15], [0, -6, -18, -6], [-1, 19, -19, 1], [9, 0, 10, -10], [-15, 0, 15, -15], [18, 0, 0, 0], [-12, 27, -27, 12], [18, -3, 18, 0], [0, 2, -38, 2], [9, -9, 0, 18], [-6, 0, -6, 0], [15, -27, 15, 0], [10, 0, -20, 20], [11, -11, 0, 8], [12, -6, 12, 0], [-15, 15, 0, 18], [-18, 30, -30, 18], [-10, 28, -10, 0], [3, -3, 0, -27], [8, 0, -20, 20], [8, 12, 8, 0], [0, -24, 12, -24], [0, 0, 0, 13], [15, -24, 24, -15], [4, 0, 4, 0], [9, 0, -20, 20], [0, 0, 15, -15], [-33, 0, -20, 20], [18, 0, 30, -30], [-12, -3, -12, 0], [0, 2, 27, 2], [0, -4, -23, -4], [0, 0, 2, 0], [15, 18, 15, 0], [-6, 0, -15, 15], [-2, 2, 0, -17], [-14, 14, 0, 18], [-3, 24, -3, 0], [18, 2, -2, -18], [12, 15, -15, -12], [30, -14, 14, -30], [0, -26, 27, -26], [-10, -32, -10, 0], [27, 0, 15, -15], [-13, 9, -13, 0], [0, -9, 42, -9], [0, -24, 24, 0], [0, 9, -18, 9], [-21, 19, -19, 21], [34, 0, 5, -5], [17, 0, 20, -20], [-27, 0, 0, 0], [18, -18, 18, -18], [0, 16, -3, 16], [-18, 20, -20, 18], [-6, -15, -6, 0], [23, -23, 0, -2], [-18, -6, -18, 0], [28, -18, 18, -28], [-15, 15, 0, 3], [-3, 15, -15, 3], [-30, 31, -31, 30], [19, -19, 0, -37], [24, 9, -9, -24], [-24, 24, 0, 33], [18, 0, -10, 10], [-18, 14, -18, 0], [-2, -8, -2, 0], [15, 6, -6, -15], [0, -30, 27, -30], [0, 24, -3, 24], [4, -40, 4, 0], [18, 0, -15, 15], [-12, -18, 18, 12], [0, 26, -8, 26], [-6, 6, 0, -27], [9, 15, 9, 0], [-15, 18, -15, 0], [-22, 22, 0, 13], [31, -31, 0, -37], [0, -3, -18, -3], [30, -30, 0, 3], [25, -24, 24, -25], [0, 19, 17, 19], [10, 33, 10, 0], [3, -3, 0, 48]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_625_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_625_2_d_g();".
// 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_625_2_d_g(:prec:=4)
    chi := MakeCharacter_625_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_625_2_d_g();".
// 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_625_2_d_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_625_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, -3, 4, -2, 1]>,<3,R![1, 4, 6, -1, 1]>],Snew);
    return Vf;
end function;