
// Make newform 225.4.b.a in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_225_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_225_b_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_225_4_b_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_225_4_b_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, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [0, 5]];
    Rf_basisdens := [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_225_b();"
function MakeCharacter_225_b()
    N := 225;
    order := 2;
    char_gens := [101, 127];
    v := [2, 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_225_b_Hecke();"
function MakeCharacter_225_b_Hecke(Kf)
    N := 225;
    order := 2;
    char_gens := [101, 127];
    char_values := [[1, 0], [-1, 0]];
    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 := 4;
    raw_aps := [[0, 1], [0, 0], [0, 0], [0, 6], [-50, 0], [0, -4], [0, -2], [44, 0], [0, -24], [-50, 0], [108, 0], [0, 8], [-400, 0], [0, 56], [0, -56], [0, 122], [50, 0], [-518, 0], [0, 36], [-700, 0], [0, -82], [516, 0], [0, -132], [-1500, 0], [0, 326], [450, 0], [0, 154], [0, 132], [-1754, 0], [0, 62], [0, 214], [-1950, 0], [0, 210], [-1676, 0], [2050, 0], [448, 0], [0, 20], [0, -380], [0, 384], [0, -510], [3650, 0], [-4342, 0], [3500, 0], [0, 670], [0, 18], [-3664, 0], [-268, 0], [0, -734], [0, 752], [1434, 0], [0, 690], [-4900, 0], [4822, 0], [4650, 0], [0, 1026], [0, 256], [3350, 0], [5512, 0], [0, -984], [-4500, 0], [0, -1380], [0, -306], [0, -608], [5700, 0], [0, 622], [0, -190], [2292, 0], [0, 1546], [0, -224], [-1186, 0], [0, -726], [-1800, 0], [0, -1698], [0, 20], [8084, 0], [0, 1896], [-10950, 0], [0, -2768], [-9300, 0], [2854, 0], [1150, 0], [-11162, 0], [1200, 0], [0, 302], [-424, 0], [0, -2472], [-1300, 0], [0, 1438], [150, 0], [0, 534], [0, 236], [-14100, 0], [0, 1970], [2450, 0], [17036, 0], [0, 4120], [5750, 0], [15500, 0], [0, -2788], [-20478, 0], [0, -2408], [0, 4710], [0, 1224], [-11700, 0], [-8188, 0], [0, -2338], [0, 4212], [0, 4582], [-1400, 0], [-11002, 0], [0, -926], [0, 4808], [0, -378], [-19244, 0], [15892, 0], [12600, 0], [0, 1452], [0, 1480], [0, 958], [-1450, 0], [11818, 0], [0, 1110], [0, 2586], [0, -6516], [10228, 0], [-8350, 0], [14954, 0], [29400, 0], [0, 3266], [0, 6160], [9524, 0], [0, 5720], [-8252, 0], [0, 4984], [-27900, 0], [11506, 0], [0, 2502], [0, 220], [0, -898], [28600, 0], [10068, 0], [14250, 0], [0, 1366], [0, -1784], [3534, 0], [-8000, 0], [0, 1032], [0, 1534], [-25804, 0], [0, -80], [0, 7020], [18700, 0], [0, -596], [0, -7176], [0, -9048], [-33200, 0], [-35356, 0], [-25700, 0], [0, -10578], [-38050, 0], [0, -5928], [0, -3034], [0, 1094], [-15150, 0], [0, 6238], [0, 1512], [32672, 0], [0, 948]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_225_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_225_4_b_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_225_4_b_a(:prec:=2)
    chi := MakeCharacter_225_b();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_225_4_b_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_225_4_b_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_225_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![25, 0, 1]>,<7,R![900, 0, 1]>,<11,R![50, 1]>],Snew);
    return Vf;
end function;