
// Make newform 225.4.b.f in Magma, downloaded from the LMFDB on 19 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_f();".
// 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_f();".
// 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;
    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_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], [43, 0], [0, -28], [0, 91], [35, 0], [0, -162], [160, 0], [42, 0], [0, 314], [203, 0], [0, 92], [0, 196], [0, -82], [-280, 0], [-518, 0], [0, -141], [-412, 0], [0, -763], [-510, 0], [0, -777], [-945, 0], [0, -1246], [-1302, 0], [0, 532], [0, -1269], [-1070, 0], [0, 503], [0, 874], [-1092, 0], [0, 411], [595, 0], [-3200, 0], [202, 0], [0, -406], [0, -3803], [0, 4116], [0, -1512], [2585, 0], [-2758, 0], [2378, 0], [0, 3067], [0, 2346], [-4900, 0], [4307, 0], [0, 2212], [0, 476], [2940, 0], [0, -1002], [2480, 0], [1897, 0], [2373, 0], [0, -4494], [0, -722], [-6160, 0], [-7238, 0], [0, -1776], [-4542, 0], [0, 7077], [0, 4158], [0, 2569], [-2982, 0], [0, 2422], [0, -9484], [-183, 0], [0, -2861], [0, -629], [-5950, 0], [0, 11718], [8070, 0], [0, -8316], [0, 12062], [-1735, 0], [0, -7602], [3030, 0], [0, 1204], [-1077, 0], [3955, 0], [6265, 0], [-3788, 0], [15258, 0], [0, -13573], [8120, 0], [0, 6183], [-1975, 0], [0, -11831], [-1932, 0], [0, -9228], [0, 13916], [2310, 0], [0, 17114], [17228, 0], [12500, 0], [0, 868], [13370, 0], [-21637, 0], [0, 287], [-5328, 0], [0, -71], [0, -18444], [0, -672], [-10935, 0], [-13588, 0], [0, -8701], [0, 11361], [0, -11417], [-21050, 0], [7427, 0], [0, 4144], [0, 30122], [0, -11934], [-8540, 0], [-3158, 0], [4278, 0], [0, -11508], [0, -8204], [0, 5518], [13295, 0], [-9968, 0], [0, -15738], [0, -19824], [0, 11073], [-6503, 0], [10148, 0], [9980, 0], [-27510, 0], [0, 17024], [0, -34748], [12020, 0], [0, -28642], [8752, 0], [0, -10256], [-33957, 0], [-27965, 0], [0, -9912], [0, 25564], [0, 12446], [33970, 0], [18732, 0], [-6162, 0], [0, -25388], [0, 25201], [19740, 0], [29680, 0], [0, -1218], [0, 38731], [23555, 0], [0, -24872], [0, 17124], [658, 0], [0, 33727], [0, 36036], [0, -39156], [-43532, 0], [28610, 0], [-24290, 0], [0, -34461], [40628, 0], [0, -20904], [0, -1807], [0, 57584], [-27237, 0], [0, -13649], [0, -16002], [37022, 0], [0, -18396]];
    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_f();".
// 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_f(: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_f();".
// 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_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_225_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![1, 0, 1]>,<7,R![36, 0, 1]>,<11,R![-43, 1]>],Snew);
    return Vf;
end function;