
// Make newform 294.4.e.f in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_294_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_294_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_294_4_e_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_294_4_e_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, -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_294_e();"
function MakeCharacter_294_e()
    N := 294;
    order := 3;
    char_gens := [197, 199];
    v := [3, 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_294_e_Hecke();"
function MakeCharacter_294_e_Hecke(Kf)
    N := 294;
    order := 3;
    char_gens := [197, 199];
    char_values := [[1, 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 := 4;
    raw_aps := [[0, 2], [-3, 3], [0, -8], [0, 0], [-40, 40], [4, 0], [84, -84], [0, -148], [0, -84], [58, 0], [136, -136], [0, 222], [420, 0], [-164, 0], [0, -488], [-478, 478], [-548, 548], [0, -692], [908, -908], [-524, 0], [-440, 440], [0, -1216], [-684, 0], [0, -604], [-832, 0], [-464, 464], [0, 632], [0, 160], [2198, -2198], [770, 0], [-184, 0], [0, 1452], [-646, 646], [-3012, 0], [0, 3170], [1880, -1880], [-604, 604], [0, -1116], [-1784, 0], [0, 344], [-1392, 1392], [4052, 0], [0, 3108], [-50, 50], [-162, 0], [-1544, 1544], [-1204, 0], [2000, 0], [-388, 388], [0, -4180], [0, 1322], [2412, 0], [4336, -4336], [764, 0], [0, -4300], [3860, -3860], [2800, -2800], [0, 4880], [6674, -6674], [-9402, 0], [9100, -9100], [5952, 0], [-3004, 0], [-688, 688], [0, -5592], [0, 2922], [0, 7492], [10766, 0], [3984, -3984], [-180, 0], [10428, -10428], [0, -8684], [-5648, 5648], [0, 2546], [8268, 0], [0, 10872], [-10434, 10434], [0, 3044], [0, -8910], [-5616, 5616], [-8932, 0], [-5538, 0], [6700, -6700], [-5048, 0], [0, 1344], [0, 4392], [3666, 0], [0, -26], [7656, 0], [12608, 0], [0, 3068], [-6456, 6456], [-11896, 11896], [-264, 0], [0, 2628], [-13568, 0], [0, -20656], [-3628, 3628], [0, -4852], [0, 7130], [-12788, 0], [-2406, 2406], [25412, -25412], [0, 9690], [-5604, 5604], [21568, -21568], [20300, 0], [0, -13812], [21996, -21996], [8368, 0], [0, -21504], [10270, -10270], [28358, 0], [-16292, 16292], [11256, 0], [-15518, 15518], [-10452, 0], [-72, 72], [0, -11962], [-6016, 0], [-26068, 26068], [-20530, 0], [0, 10056], [-6152, 6152], [0, 14716], [28202, 0], [0, -22114], [0, 9288], [23848, 0], [0, 34756], [-26044, 26044], [36204, 0], [0, 11424], [-16622, 0], [0, 38524], [-18440, 0], [13968, -13968], [-10916, 10916], [12360, 0], [-3402, 3402], [292, 0], [0, -6910], [-568, 568], [-12144, 0], [-14828, 14828], [-22824, 0], [-41780, 0], [21420, -21420], [0, 18132], [0, -24036], [0, 4374], [-46348, 0], [-20660, 0], [0, 1800], [41996, -41996], [-41308, 0], [0, -3936], [0, 7212], [-38976, 0], [-53544, 53544], [0, 21392], [21162, 0], [8224, 0], [0, 8140], [-32158, 32158], [41416, -41416], [-12296, 12296], [-57652, 57652]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_294_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_294_4_e_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_294_4_e_f(:prec:=2)
    chi := MakeCharacter_294_e();
    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_294_4_e_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_294_4_e_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_294_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![64, 8, 1]>,<11,R![1600, 40, 1]>],Snew);
    return Vf;
end function;