
// Make newform 2790.2.a.bg in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2790_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_2790_2_a_bg();".
// 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_2790_2_a_bg();".
// 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 := [-6, 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_2790_a();"
function MakeCharacter_2790_a()
    N := 2790;
    order := 1;
    char_gens := [2171, 1117, 1801];
    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_2790_a_Hecke(Kf)
    return MakeCharacter_2790_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+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 := 2;
    raw_aps := [[1, 0], [0, 0], [-1, 0], [-2, 0], [-2, 1], [2, -1], [0, -2], [0, 2], [-2, 0], [-8, -1], [-1, 0], [2, 1], [0, 0], [-8, 1], [-6, 0], [2, -3], [-4, 2], [-4, -1], [-8, 2], [0, 4], [-4, 0], [0, 2], [-4, -5], [-6, -4], [4, -2], [-6, 4], [-8, 4], [0, -8], [6, 2], [0, 6], [-14, -2], [12, -2], [-8, 2], [2, -3], [6, 2], [8, 6], [18, 0], [16, -2], [-6, 4], [2, -4], [-2, -3], [16, -3], [-2, -6], [2, -4], [2, 3], [4, -8], [-12, -4], [6, 6], [-12, -2], [-8, 5], [0, 10], [8, 6], [14, 6], [-10, -5], [-22, 0], [18, 2], [0, 7], [20, -2], [-10, 3], [6, 8], [8, -6], [6, 10], [-12, 0], [8, -8], [0, -10], [-6, -2], [10, -7], [4, 4], [-12, 1], [10, 10], [-8, 4], [2, -2], [-18, 0], [6, -2], [0, 6], [6, -10], [-20, -7], [-2, -2], [-2, -2], [-26, 2], [32, -2], [-2, -6], [-6, 6], [4, 10], [2, -2], [16, 0], [34, -2], [-16, 2], [0, -5], [22, -8], [0, 0], [-24, 4], [-6, -6], [6, 7], [2, 7], [8, 0], [4, -5], [-10, 12], [-8, 3], [-2, -6], [-16, -2], [-14, 3], [8, 2], [18, 10], [18, 5], [-32, 6], [-12, 9], [0, -10], [-10, -10], [2, -8], [4, -8], [2, -9], [18, 0], [6, -11], [4, 14], [42, 2], [8, -11], [-6, -2], [46, 0], [-8, -10], [10, 6], [-24, 2], [6, 13], [8, 8], [-36, 4], [10, -12], [4, -11], [-28, -8], [-28, -4], [34, 2], [10, 5], [30, 6], [26, 10], [2, -1], [38, 2], [-20, -4], [-14, 15], [-12, -15], [-34, -1], [-18, 10], [-12, 6], [24, -5], [-34, 2], [4, 15], [-16, 5], [10, -2], [-18, -8], [6, -16], [-22, 7], [-10, -6], [10, 18], [-34, 4], [-24, -11], [-18, 4], [0, 0], [-12, -12], [32, -8], [-10, -2], [-14, -8], [-12, 5], [-8, 5], [-32, -10], [-6, 6], [-32, 4], [-10, -4], [-34, -6], [-8, -8], [-14, -18], [50, 2], [38, -3], [-2, 3], [24, -1], [50, 2], [-8, -6], [14, -6], [-26, -4], [8, -20], [2, 4], [-12, -16], [8, 7], [12, -4], [30, 3], [-2, 22], [-40, 8], [-38, -8], [-12, 9], [18, 16], [-20, 0], [-42, 6], [-24, -4], [4, 16], [-24, -2], [-14, 7], [0, -5], [52, 0], [-40, -4], [-18, 8], [-2, 14], [-2, -12], [22, 12], [26, -10], [-52, 4], [6, -4], [4, 12], [8, -8], [14, -13], [-26, 6], [-8, -7], [-20, -12], [0, -12], [48, -6], [-36, 3], [10, -24], [-20, -2], [-12, 12], [28, 16], [-22, 0], [4, 20], [-54, 6], [34, -4], [28, -15], [24, -4], [-40, -4], [-8, -12], [-32, -6], [-12, 1], [-18, 8], [4, 18], [-22, 0], [-48, -4], [-50, -3], [-20, -16], [-46, -10], [2, -8], [-36, 13], [-10, -8], [-22, 0], [-10, -4], [-18, -1], [-4, -8], [12, 16], [-58, -3], [-14, 12], [28, -7], [-8, 16], [0, 0], [-2, 8], [18, -1], [-26, 5], [-16, -12], [34, 10], [24, 16], [-26, -12], [8, -20], [-30, -4], [36, 8], [22, -2], [-56, 3], [-14, -12], [34, 4], [6, 0], [0, 7], [8, 19], [30, 16], [-24, 10], [12, 20], [42, -10], [34, -8], [-4, 12], [-46, 10], [-14, 4], [12, -7], [0, -4], [-20, -16], [-8, -6], [-12, 28], [48, -10], [-44, 5], [-30, -8], [-22, 17], [42, 4], [-8, 12], [-24, -8], [30, -18], [44, -8], [36, -14], [-20, -26], [6, 3], [-26, -18], [18, -12], [30, 8], [-16, 0], [4, -8], [-4, 28], [-10, -15], [-12, 15], [-24, -6], [-42, 10], [18, 19], [-20, -13], [-52, 6], [42, -15], [36, -8], [60, -2], [2, -1], [-68, 6], [-24, -11], [14, -6], [-8, -10], [14, 8], [-26, 6], [4, -5], [22, -4], [36, -17], [-20, 12], [38, -4], [-6, 3], [36, 14], [24, 2], [-18, -8], [-14, -9], [8, 22], [-6, 14], [22, 4], [4, 10], [-30, -6], [64, -10], [0, 6], [2, 14], [18, 7], [14, 14], [22, 18], [56, 0], [32, 7], [-4, 12], [8, -2], [44, -13], [18, -28], [-42, 8], [-10, 8], [-58, -11], [4, 22], [-56, -1], [36, 16], [46, -4], [12, 26], [-2, -34], [-60, 13], [-36, -6], [14, -4], [22, -3], [-8, 6], [-6, -34], [-10, -20], [-10, 26], [-16, -16], [-48, -2], [-14, -5], [-40, -2], [6, 4], [-14, 8], [18, 0], [34, 12], [-48, -6], [8, -10], [44, 8], [-26, -14], [42, 16], [14, 4], [44, -14], [16, 10], [44, -8], [26, -26], [-58, -6], [-14, 5], [42, 9], [-24, 8], [42, -4], [18, 8], [28, -6], [32, -15], [56, 2], [34, 22], [-12, -4], [10, -5], [-4, -8], [40, -8], [26, 21], [-44, 3], [18, -8], [38, 24], [18, 11], [16, -2], [56, 12], [32, -20], [4, 18], [-84, -6], [38, 0], [6, 11], [-36, -5], [-32, -11], [30, 16], [16, -4], [12, -10], [36, 7], [14, -14], [54, 16], [30, 10], [4, 35], [6, -5], [-40, -4], [54, 14], [0, 17], [-18, -11], [18, 4], [30, 12], [56, 4], [38, -20], [-12, 2], [88, -4], [-52, 11], [2, -7], [-54, -14], [-24, -14], [14, 20], [10, -1], [-32, 26], [18, 28], [50, 7], [48, -2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2790_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2790_2_a_bg();".
// 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_2790_2_a_bg(:prec:=2)
    chi := MakeCharacter_2790_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_2790_2_a_bg();".
// 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_2790_2_a_bg( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2790_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![2, 1]>,<11,R![-2, 4, 1]>,<13,R![-2, -4, 1]>,<17,R![-24, 0, 1]>,<19,R![-24, 0, 1]>],Snew);
    return Vf;
end function;