
// Make newform 912.3.o.a in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_912_o();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_912_o_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_912_3_o_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_912_3_o_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, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [-1, 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_912_o();"
function MakeCharacter_912_o()
    N := 912;
    order := 2;
    char_gens := [799, 229, 305, 97];
    v := [2, 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_912_o_Hecke();"
function MakeCharacter_912_o_Hecke(Kf)
    N := 912;
    order := 2;
    char_gens := [799, 229, 305, 97];
    char_values := [[1, 0], [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, -1], [4, 0], [10, 0], [-10, 0], [0, -14], [10, 0], [-19, 0], [20, 0], [0, -20], [0, -10], [0, -6], [0, 20], [10, 0], [80, 0], [0, 24], [0, -20], [-10, 0], [0, 44], [0, -60], [-10, 0], [0, 10], [-70, 0], [0, -60], [0, 44], [100, 0], [0, -106], [0, -36], [0, -90], [0, -4], [0, 66], [38, 0], [190, 0], [-50, 0], [-20, 0], [0, 130], [230, 0], [-170, 0], [0, 76], [0, 136], [0, 60], [0, 150], [332, 0], [0, -56], [160, 0], [-98, 0], [0, -100], [0, 46], [0, 44], [110, 0], [190, 0], [128, 0], [0, 80], [2, 0], [0, 284], [200, 0], [0, -240], [-170, 0], [-10, 0], [0, 220], [70, 0], [0, 104], [0, 84], [-580, 0], [-370, 0], [0, 16], [0, 100], [0, 196], [590, 0], [98, 0], [190, 0], [200, 0], [-170, 0], [0, -206], [0, -120], [0, 364], [-128, 0], [650, 0], [0, -100], [0, 100], [38, 0], [0, -10], [0, 0], [0, -204], [0, 70], [110, 0], [0, 180], [290, 0], [-728, 0], [790, 0], [530, 0], [80, 0], [0, 294], [-418, 0], [-470, 0], [-100, 0], [0, 260], [0, 180], [0, 456], [650, 0], [0, -344], [-80, 0], [0, 196], [0, -380], [610, 0], [170, 0], [650, 0], [910, 0], [0, -20], [0, -100], [0, -406], [350, 0], [610, 0], [10, 0], [-350, 0], [0, 340], [-650, 0], [-820, 0], [-560, 0], [0, 260], [0, -230], [0, -364], [0, 304], [0, 276], [-470, 0], [-560, 0], [-982, 0], [-520, 0], [790, 0], [-1150, 0], [-578, 0], [0, 136], [0, -550], [-250, 0], [-770, 0], [110, 0], [0, -84], [0, 56], [0, -196], [-182, 0], [0, 480], [-8, 0], [-950, 0], [0, 276], [0, 690], [0, 680], [890, 0], [0, 724], [-182, 0], [0, -624], [0, -686], [550, 0], [1450, 0], [0, 724], [0, 64], [0, -460], [-62, 0], [-242, 0], [110, 0], [0, -460], [-1450, 0], [0, 204], [-470, 0], [0, 300], [0, -964], [0, -976], [0, -330], [1550, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_912_o_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_912_3_o_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_912_3_o_a(:prec:=2)
    chi := MakeCharacter_912_o();
    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_912_3_o_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_912_3_o_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_912_o();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![-4, 1]>],Snew);
    return Vf;
end function;