
// Make newform 288.3.e.d in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_288_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_288_e_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_288_3_e_d();".
// 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_288_3_e_d();".
// 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 := [2, 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_288_e();"
function MakeCharacter_288_e()
    N := 288;
    order := 2;
    char_gens := [127, 37, 65];
    v := [2, 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_288_e_Hecke();"
function MakeCharacter_288_e_Hecke(Kf)
    N := 288;
    order := 2;
    char_gens := [127, 37, 65];
    char_values := [[1, 0], [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 := 3;
    raw_aps := [[0, 0], [0, 0], [0, 1], [8, 0], [0, 8], [-8, 0], [0, -9], [32, 0], [0, 24], [0, 31], [40, 0], [-26, 0], [0, -47], [16, 0], [0, -8], [0, 23], [0, 16], [-54, 0], [-80, 0], [0, -56], [96, 0], [-104, 0], [0, -72], [0, -55], [-80, 0], [0, -87], [-72, 0], [0, -128], [-88, 0], [0, 97], [-56, 0], [0, 176], [0, -33], [16, 0], [0, -129], [168, 0], [-10, 0], [80, 0], [0, 208], [0, -39], [0, -96], [8, 0], [0, 48], [-258, 0], [0, -263], [88, 0], [368, 0], [104, 0], [0, -120], [344, 0], [0, -57], [0, 192], [-272, 0], [0, 24], [0, 31], [0, -144], [0, 33], [264, 0], [40, 0], [0, -135], [-224, 0], [0, -249], [-432, 0], [0, 144], [-14, 0], [0, 417], [16, 0], [128, 0], [0, -248], [10, 0], [0, -175], [0, -232], [-696, 0], [-454, 0], [-64, 0], [0, 256], [0, -273], [-662, 0], [0, -121], [-176, 0], [0, 56], [488, 0], [0, 216], [478, 0], [392, 0], [0, 24], [0, 89], [-16, 0], [0, -201], [568, 0], [0, 296], [0, 24], [-424, 0], [0, 512], [-192, 0], [0, 312], [0, -177], [0, 111], [576, 0], [-536, 0], [144, 0], [0, -377], [0, -408], [0, 703], [416, 0], [834, 0], [0, 48], [0, 159], [0, -488], [-626, 0], [232, 0], [666, 0], [0, 201], [-768, 0], [472, 0], [0, 263], [-240, 0], [0, -504], [0, -823], [0, 144], [-1018, 0], [-382, 0], [0, -175], [0, 120], [1040, 0], [0, -465], [952, 0], [0, 136], [-648, 0], [1208, 0], [-1312, 0], [0, -432], [-632, 0], [-840, 0], [0, 369], [130, 0], [0, -201], [-864, 0], [0, 537], [0, 641], [1472, 0], [0, 841], [-664, 0], [0, 176], [-280, 0], [0, -392], [-762, 0], [0, -1087], [1552, 0], [0, 16], [-86, 0], [0, -465], [496, 0], [0, -720], [-880, 0], [0, 1104], [264, 0], [0, -201], [1070, 0], [0, 407], [0, -432], [0, 735], [696, 0], [0, 1112], [0, -359], [0, 336], [520, 0], [-486, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_288_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_288_3_e_d();".
// 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_288_3_e_d(:prec:=2)
    chi := MakeCharacter_288_e();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    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_288_3_e_d();".
// 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_288_3_e_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_288_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![2, 0, 1]>,<7,R![-8, 1]>],Snew);
    return Vf;
end function;