
// Make newform 2793.2.a.m in Magma, downloaded from the LMFDB on 03 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2793_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_2793_2_a_m();".
// 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_2793_2_a_m();".
// 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_2793_a();"
function MakeCharacter_2793_a()
    N := 2793;
    order := 1;
    char_gens := [932, 2110, 2206];
    v := [1, 1, 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;

function MakeCharacter_2793_a_Hecke(Kf)
    return MakeCharacter_2793_a();
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]];
    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 := 2;
    raw_aps := [[-1, 1], [-1, 0], [0, 2], [0, 0], [-2, 2], [2, -2], [-4, -2], [1, 0], [-2, 2], [0, -6], [0, 0], [-6, 0], [0, 4], [0, 0], [6, -2], [8, -2], [0, -4], [4, 0], [-10, 2], [-10, -4], [8, 0], [-6, -6], [-2, 2], [12, -4], [-6, 6], [0, -6], [0, -12], [-2, 4], [-6, 0], [12, -2], [-10, -2], [-6, 6], [-18, 0], [12, 0], [10, -4], [-2, 2], [0, -12], [-16, 0], [0, -4], [-12, 4], [10, -4], [10, 6], [-6, -6], [14, 4], [-18, 0], [8, 4], [2, 10], [-24, 0], [-12, 4], [28, 0], [-6, 12], [6, -6], [-10, -6], [-6, -2], [-12, 8], [-18, -2], [-8, 4], [4, 4], [-6, 0], [-8, 14], [0, -12], [0, -4], [-20, 4], [-18, -2], [4, 16], [-16, 10], [-2, -2], [-14, -12], [-6, 14], [4, 20], [4, -2], [-6, -10], [-4, 20], [10, 12], [-10, -2], [28, -4], [18, 0], [-4, 4], [0, 10], [-2, 18], [-2, -22], [2, -12], [-18, -12], [10, 6], [16, 8], [26, 10], [0, 2], [26, 8], [-32, 2], [8, 8], [26, -2], [2, 26], [-22, 6], [10, 22], [8, -12], [6, 6], [-12, -4], [0, 28], [-20, 12], [10, -8], [30, -6], [-18, -8], [-28, -8], [28, -2], [4, -16], [20, 8], [-6, -26], [36, -6], [-6, 20], [26, 6], [16, -12], [6, -24], [6, 0], [0, 0], [20, 16], [12, -14], [-12, 12], [-10, -2], [-22, -16], [2, -20], [6, -6], [-6, 12], [-12, 8], [-2, -16], [0, 12], [18, 8], [42, 0], [-14, -14], [-4, -12], [4, 4], [4, 4], [-6, 8], [2, -10], [-14, 24], [-12, 18], [-12, 0], [4, -20], [4, -12], [-24, -12], [-18, 4], [-28, 8], [14, -8], [-16, -12], [-2, -20], [14, -6], [4, 8], [40, -8], [-12, -28], [16, 24], [-18, 16], [30, 12], [-12, 6], [-16, 28], [-32, -8], [-26, 6], [18, -20], [-8, -16], [-20, -10], [-20, 4], [12, -20], [14, 14], [0, 26], [-28, 8], [-36, -4], [-8, 26], [0, 20], [-10, 26], [-24, -12], [-54, 0], [48, -6], [-14, 16], [-2, -30], [-30, 6], [2, 24], [-16, -12], [36, 10], [28, 20], [18, 32], [-16, 8], [8, 24], [-28, -4], [36, -12], [30, -12], [-36, 8], [6, -10], [0, 18], [38, 0], [-4, -28], [6, -24], [42, 10], [-6, -18], [-18, 6], [-14, -6], [-16, -16], [-10, -34], [-12, 4], [10, 8], [-6, -24], [-4, -2], [-6, -14], [8, -10], [8, 20], [-38, 14], [2, -18], [18, 30], [-16, 34], [-4, -28], [-38, 8], [-6, 8], [52, -12], [22, 0], [8, 2], [-8, 28], [-16, -8], [-36, 16], [18, 30], [40, 12], [-28, -28], [-54, -4], [-10, -4], [18, 24], [32, 12], [0, -34], [8, 24], [44, 4], [2, -24], [-24, 8], [14, -40], [16, 40], [-30, -10], [-18, 0], [20, -12], [10, 38], [8, 14], [-36, 0], [-54, 10], [-32, -24], [-6, 12], [18, -38], [-12, 28], [6, -8], [24, 36], [-32, -12], [-2, 16], [-48, 8], [18, 18], [-16, 8], [-60, 4], [2, 22], [6, -26], [2, 0], [-4, -38], [18, -6], [2, -2], [40, 22], [-42, -22], [14, 0], [36, 12], [-32, -8], [40, 28], [-34, 14], [58, 8], [-28, -36], [10, -42], [28, 2], [4, 36], [-16, 22], [4, -22], [-2, -6], [2, -16], [42, 6], [-18, -18], [-36, 0], [36, -12], [-46, -10], [-36, -24], [-42, -6], [-10, 16], [-2, -28], [42, 14], [28, 8], [0, 12], [30, -10], [62, 2], [32, 12], [18, -18], [46, 16], [-12, -22], [-4, -16], [48, -24], [46, 16], [-18, -22], [16, 14], [16, 28], [10, -4], [0, 14], [-32, -24], [16, -22], [-20, 8], [-8, 24], [32, -36], [-24, -6], [4, -12], [42, 0], [-64, 0], [18, -12], [-30, -8], [-14, 38], [-38, 34], [22, 48], [-18, 22], [18, -16], [-4, 10], [-22, -30], [-34, -14], [2, 14], [-6, 6], [30, 4], [20, -4], [-54, -4], [-28, -32], [22, -8], [-28, 32], [-42, -6], [54, 20], [-42, -30], [-22, 46], [20, 16], [30, 20], [34, -16], [18, 0], [-12, -38], [-8, 24], [14, 50], [32, -4], [66, -18], [30, 36], [-76, -4], [-32, -12], [-4, -36], [-46, 4], [2, 20], [4, -32], [-22, 30], [-36, -30], [-30, -4], [16, -4], [42, -6], [0, 36], [-24, 26], [-12, 24], [-6, -48], [-6, -36], [8, -52], [-22, 4], [-24, -20], [-42, -2], [30, -38], [30, -36], [-42, 12], [2, -8], [-36, 18], [-34, -4], [-30, 24], [20, -32], [10, 4], [16, -34], [-26, 14], [46, 4], [58, -14], [-20, -36], [-46, -2], [36, 38], [-20, 36], [26, 0], [52, 20], [-10, 26], [8, -24], [-60, 10], [14, 18], [4, -16], [-18, -36], [20, 0], [-14, 56], [-36, 12], [-60, -12], [-52, 24], [-4, 0], [16, -24], [24, -12], [14, 48], [40, 16], [-2, -10], [-40, 8], [-70, -16], [10, -44], [-48, -12], [-4, 8], [66, 6], [-66, 4], [-14, 6], [2, -8], [-62, -14], [-48, 40], [-56, -4], [-20, -28], [30, 48], [36, -6], [28, 8], [-18, -6], [26, -26], [-70, -8], [54, 28], [-16, -16], [-38, 8], [16, -2], [-54, 0], [-8, -16], [36, 6], [76, -20], [20, 34], [-22, 14], [-54, -10], [-28, -8], [38, -30], [-68, -4], [-6, 44], [-10, 8], [0, 24], [-74, -10]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2793_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2793_2_a_m();".
// 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_2793_2_a_m(:prec:=2)
    chi := MakeCharacter_2793_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 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_2793_2_a_m();".
// 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_2793_2_a_m( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2793_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 2, 1]>,<5,R![-8, 0, 1]>,<11,R![-4, 4, 1]>,<13,R![-4, -4, 1]>,<17,R![8, 8, 1]>],Snew);
    return Vf;
end function;