
// Make newform 225.4.b.c in Magma, downloaded from the LMFDB on 25 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_c();".
// 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_c();".
// 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, 2]];
    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, 2], [0, 0], [0, 0], [0, 3], [-32, 0], [0, 19], [0, -13], [-100, 0], [0, -39], [-50, 0], [-108, 0], [0, 133], [-22, 0], [0, -221], [0, 257], [0, 1], [500, 0], [-518, 0], [0, 63], [-412, 0], [0, 439], [-600, 0], [0, 141], [-150, 0], [0, 193], [-702, 0], [0, 299], [0, 597], [550, 0], [0, 781], [0, 923], [2208, 0], [0, 1167], [700, 0], [2050, 0], [1852, 0], [0, -1247], [0, -1381], [0, -1563], [0, -39], [-1300, 0], [1742, 0], [-3772, 0], [0, 179], [0, 1107], [2600, 0], [-1168, 0], [0, 3239], [0, -323], [-3750, 0], [0, 741], [1400, 0], [3022, 0], [1248, 0], [0, -1053], [0, -1819], [-6550, 0], [-4388, 0], [0, 273], [6858, 0], [0, -4641], [0, 2421], [0, -1297], [-7332, 0], [0, -781], [0, -713], [-4008, 0], [0, 4433], [0, 857], [-1150, 0], [0, -2199], [1800, 0], [0, -2937], [0, 1039], [-7900, 0], [0, -3759], [-1950, 0], [0, 6893], [-6402, 0], [-11150, 0], [-13700, 0], [-5438, 0], [-7692, 0], [0, 559], [2600, 0], [0, -5979], [-17050, 0], [0, -4747], [11418, 0], [0, -3981], [0, -3263], [17400, 0], [0, 583], [-7072, 0], [-100, 0], [0, 1301], [11150, 0], [3638, 0], [0, 1039], [5622, 0], [0, 8243], [0, -5853], [0, -12519], [17550, 0], [10712, 0], [0, -6827], [0, -7083], [0, 8921], [-17600, 0], [27302, 0], [0, -1897], [0, 6619], [0, 5787], [-8300, 0], [-7508, 0], [27378, 0], [0, -921], [0, 5057], [0, 5201], [7100, 0], [-7118, 0], [0, 15639], [0, 15027], [0, -2259], [29272, 0], [5798, 0], [-8950, 0], [7800, 0], [0, -4277], [0, -1441], [-18700, 0], [0, 6121], [-31148, 0], [0, -3847], [4518, 0], [39550, 0], [0, 11061], [0, -8317], [0, -13793], [3850, 0], [10032, 0], [-20562, 0], [0, -5161], [0, -4423], [25350, 0], [46000, 0], [0, 8499], [0, 13247], [21500, 0], [0, 12881], [0, 15273], [-32942, 0], [0, 13559], [0, 19317], [0, -897], [-41732, 0], [-29200, 0], [-48650, 0], [0, -5667], [31178, 0], [0, -2343], [0, -299], [0, 20863], [-24312, 0], [0, -20473], [0, 21141], [1172, 0], [0, -15807]];
    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_c();".
// 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_c(: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_c();".
// 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_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_225_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![16, 0, 1]>,<7,R![36, 0, 1]>,<11,R![32, 1]>],Snew);
    return Vf;
end function;