
// Make newform 2736.3.o.e in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2736_o();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2736_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_2736_3_o_e();".
// 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_2736_3_o_e();".
// 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], [-2, 4]];
    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_2736_o();"
function MakeCharacter_2736_o()
    N := 2736;
    order := 2;
    char_gens := [1711, 2053, 1217, 1009];
    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_2736_o_Hecke();"
function MakeCharacter_2736_o_Hecke(Kf)
    N := 2736;
    order := 2;
    char_gens := [1711, 2053, 1217, 1009];
    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, 0], [-4, 0], [10, 0], [10, 0], [0, -7], [-10, 0], [-19, 0], [-20, 0], [0, 10], [0, -5], [0, -3], [0, -10], [10, 0], [-80, 0], [0, -12], [0, 10], [-10, 0], [0, 22], [0, 30], [-10, 0], [0, 5], [70, 0], [0, 30], [0, 22], [-100, 0], [0, -53], [0, 18], [0, -45], [0, 2], [0, 33], [-38, 0], [-190, 0], [-50, 0], [20, 0], [0, 65], [230, 0], [-170, 0], [0, -38], [0, -68], [0, -30], [0, 75], [-332, 0], [0, -28], [-160, 0], [-98, 0], [0, -50], [0, 23], [0, -22], [110, 0], [-190, 0], [-128, 0], [0, 40], [-2, 0], [0, -142], [-200, 0], [0, 120], [-170, 0], [-10, 0], [0, -110], [70, 0], [0, -52], [0, 42], [580, 0], [-370, 0], [0, -8], [0, 50], [0, 98], [-590, 0], [98, 0], [-190, 0], [-200, 0], [-170, 0], [0, -103], [0, -60], [0, -182], [128, 0], [650, 0], [0, 50], [0, 50], [-38, 0], [0, -5], [0, 0], [0, -102], [0, 35], [-110, 0], [0, -90], [290, 0], [728, 0], [790, 0], [-530, 0], [-80, 0], [0, 147], [418, 0], [-470, 0], [100, 0], [0, -130], [0, -90], [0, 228], [650, 0], [0, -172], [80, 0], [0, -98], [0, 190], [610, 0], [170, 0], [-650, 0], [-910, 0], [0, 10], [0, -50], [0, -203], [350, 0], [-610, 0], [10, 0], [-350, 0], [0, -170], [-650, 0], [820, 0], [560, 0], [0, -130], [0, -115], [0, -182], [0, -152], [0, -138], [-470, 0], [560, 0], [-982, 0], [520, 0], [790, 0], [-1150, 0], [-578, 0], [0, -68], [0, -275], [-250, 0], [770, 0], [110, 0], [0, 42], [0, 28], [0, 98], [182, 0], [0, 240], [8, 0], [-950, 0], [0, -138], [0, 345], [0, -340], [890, 0], [0, -362], [-182, 0], [0, 312], [0, -343], [-550, 0], [1450, 0], [0, -362], [0, 32], [0, 230], [-62, 0], [242, 0], [110, 0], [0, 230], [1450, 0], [0, -102], [-470, 0], [0, -150], [0, 482], [0, 488], [0, -165], [1550, 0], [0, -210], [740, 0], [0, -550], [0, -475], [-1460, 0], [-1330, 0], [0, 55], [-250, 0], [1198, 0], [80, 0], [0, -147], [170, 0], [790, 0], [0, 290], [0, 593], [0, -258], [1060, 0], [2168, 0], [0, -27], [0, 458], [0, 200], [-152, 0], [0, -408], [970, 0], [0, 270], [0, 550], [-1370, 0], [0, 182], [-1402, 0], [530, 0], [470, 0], [-1460, 0], [0, -520], [0, 305], [0, -367], [0, -410], [-710, 0], [-2440, 0], [-890, 0], [0, -122], [-1078, 0], [0, 40], [-1870, 0], [1280, 0], [-170, 0], [0, 182], [0, 640], [0, 670], [-650, 0], [0, 490], [0, 732], [-1300, 0], [0, 425], [0, -5], [0, -430], [2530, 0], [0, 58], [1358, 0], [0, 622], [0, 360], [0, -523], [1198, 0], [-1690, 0], [0, 60], [0, -455], [0, 410], [550, 0], [100, 0], [-2410, 0], [-2440, 0], [-770, 0], [0, 150], [0, 278], [-62, 0], [-410, 0], [0, -555], [0, 518], [-332, 0], [1330, 0], [0, 70], [0, 10], [2440, 0], [-1690, 0], [1298, 0], [1240, 0], [0, 280], [1160, 0], [-2762, 0], [2258, 0], [0, -658], [0, -582], [1790, 0], [0, 767], [0, 2], [-70, 0], [0, -513], [-430, 0], [0, -550], [0, 60], [710, 0], [0, 472], [560, 0], [0, -445], [0, -282], [-1810, 0], [1930, 0], [0, -208], [730, 0], [1030, 0], [0, -665], [0, -90], [1270, 0], [0, -2], [970, 0], [1900, 0], [0, -635], [-1130, 0], [640, 0], [1010, 0], [0, -298], [-2342, 0], [0, -150], [-52, 0], [-1070, 0], [0, 158], [0, 370], [0, 733], [-160, 0], [0, -385], [-2800, 0], [0, 750], [-3770, 0], [1010, 0], [0, 102], [1270, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2736_o_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2736_3_o_e();".
// 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_2736_3_o_e(:prec:=2)
    chi := MakeCharacter_2736_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(1999) - 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_2736_3_o_e();".
// 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_2736_3_o_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2736_o();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![4, 1]>,<7,R![-10, 1]>],Snew);
    return Vf;
end function;