
// Make newform 2790.2.a.bd in Magma, downloaded from the LMFDB on 24 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_bd();".
// 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_bd();".
// 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 := [-8, -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_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], [0, 1], [-4, 1], [-2, 0], [-2, 2], [-4, 1], [0, -1], [2, 2], [-1, 0], [6, -2], [6, -2], [4, -1], [0, 2], [2, -1], [4, -2], [-2, 4], [4, 2], [0, -1], [2, -3], [-8, 1], [12, 0], [-2, 1], [2, 0], [2, -3], [-8, 0], [-4, 1], [-2, 2], [6, -1], [0, 4], [-4, 6], [-10, 4], [-4, -2], [2, 3], [-8, 0], [6, 1], [4, 6], [16, -1], [18, 0], [12, 0], [-2, 5], [-16, 0], [2, 4], [-6, -4], [-16, 1], [-4, 1], [16, -2], [4, -3], [14, -3], [-2, 5], [16, 4], [18, -4], [4, 0], [-10, -1], [-8, 8], [2, 0], [-24, 3], [22, -2], [-10, -2], [12, 0], [-6, 4], [4, 8], [-8, 0], [10, 0], [18, 2], [12, -4], [2, 8], [4, -12], [14, -2], [-2, -4], [0, -1], [0, -6], [-2, 5], [-12, -5], [0, -4], [10, -8], [22, -7], [-26, 1], [10, -6], [12, 0], [6, 4], [0, 8], [26, 3], [8, 2], [-4, -9], [-18, 8], [-6, 12], [-14, 10], [16, -4], [-20, 4], [8, -5], [-24, -4], [-4, -7], [20, 4], [8, -4], [34, -2], [-10, -8], [4, -5], [-2, -4], [4, -2], [-22, 3], [-20, 8], [-2, 1], [-20, -2], [18, -6], [-12, 0], [-18, -8], [0, -3], [26, -4], [-8, 1], [22, 4], [-18, 3], [-4, 6], [-16, -9], [-18, 0], [12, -11], [0, -15], [-14, -8], [-36, 0], [22, -8], [-14, 4], [18, 7], [-4, 5], [-36, 5], [-22, -7], [30, 3], [-32, -2], [16, 1], [-34, 0], [4, -8], [32, 1], [-32, -2], [-26, 0], [30, 5], [-6, -1], [2, -7], [-20, 3], [34, -4], [14, -7], [-44, 1], [-22, 2], [-24, 2], [4, 4], [6, -11], [0, -5], [-2, -11], [6, -4], [-4, -4], [-24, 9], [-18, -8], [30, -8], [28, 5], [24, 2], [-4, -4], [16, -4], [8, -6], [6, 3], [-38, 2], [-14, 10], [-4, 2], [22, -10], [8, 0], [-12, 18], [46, 0], [-24, 8], [-24, -7], [6, 0], [18, -12], [2, -13], [36, -8], [22, 11], [8, -16], [26, -2], [-32, -6], [14, 7], [28, 8], [34, -3], [-16, -9], [38, -1], [16, 0], [-44, -5], [6, -4], [22, -8], [32, -10], [-6, 16], [-26, 9], [20, -8], [-6, 14], [-8, 5], [-38, 9], [-20, -13], [20, 12], [34, -14], [-36, 4], [-26, 6], [-14, 4], [38, -3], [38, 5], [40, 6], [58, 1], [8, -9], [30, -15], [-6, 11], [20, -14], [2, 4], [0, 12], [-20, 10], [6, -18], [44, 0], [26, 7], [42, 6], [8, 8], [-4, -13], [8, -2], [18, -11], [0, 8], [46, -8], [8, 8], [-30, -8], [-34, -9], [24, 8], [-18, 10], [40, -7], [-44, -3], [62, -3], [6, 4], [0, 12], [-8, 6], [-44, 6], [-2, -12], [-44, 0], [16, -2], [-18, -3], [12, 8], [-8, -7], [-22, 3], [-22, -16], [-36, 3], [24, 4], [-52, -4], [-52, 10], [24, -10], [14, -12], [30, 14], [-32, 13], [-16, 2], [-12, -9], [12, 4], [48, -4], [6, -11], [30, -4], [32, -3], [-14, 5], [-30, 6], [-20, -6], [22, 16], [-12, -1], [-38, 16], [-22, -6], [-56, -1], [-20, -6], [-18, 13], [38, 9], [30, 4], [-4, -5], [-6, -11], [-10, -14], [28, 2], [-22, -10], [-2, -18], [52, 0], [-46, -3], [8, -11], [18, 8], [-24, 8], [-36, -1], [-34, 0], [10, 4], [52, -11], [0, 20], [-8, -18], [-8, 22], [-26, 4], [-20, -18], [32, -10], [-22, -3], [-22, 16], [56, 0], [-18, 0], [10, -5], [-12, -3], [-10, -10], [52, -6], [14, -18], [34, 6], [-32, 12], [-22, 12], [-4, -21], [-20, -7], [-54, 0], [26, 5], [16, 0], [20, 13], [12, -22], [18, -6], [4, 10], [14, -12], [-40, 6], [6, 16], [-48, 16], [10, -2], [-2, 24], [-4, -9], [40, -2], [-38, 6], [12, -8], [0, -4], [-14, 18], [-42, 7], [-12, 0], [-30, -1], [26, -1], [24, -19], [14, 7], [-30, -2], [-36, -5], [-4, -6], [-64, -4], [50, -1], [46, 0], [-30, 12], [16, -2], [12, -2], [-12, 1], [-36, -5], [-2, 12], [38, -7], [34, 15], [48, -4], [-10, 22], [-26, 4], [26, -22], [16, 1], [-30, 0], [-68, 10], [-10, 2], [-76, -3], [0, 18], [-38, 10], [4, -12], [-46, -1], [-6, -3], [0, 10], [-42, -8], [6, -12], [-16, -10], [60, 1], [-2, -22], [-8, 10], [62, 7], [-18, -3], [8, 15], [36, 2], [36, -10], [-30, 13], [-14, -4], [24, -18], [2, 29], [-20, 4], [20, -15], [-32, -10], [10, -8], [8, -6], [14, 0], [-28, -1], [8, 11], [-46, 6], [-2, -2], [-14, 17], [2, 18], [22, -16], [-24, -14], [-18, 2], [20, -26], [8, 2], [16, -8], [22, 18], [52, -13], [-8, -5], [18, 22], [-46, 19], [36, 12], [-12, 14], [0, 9], [-6, -28], [-24, 1], [-10, 24], [-4, -26], [10, 14], [54, 7], [-26, 21], [-40, -9], [6, -4], [-6, 6], [-8, -10], [-18, 8], [-66, -4], [28, -19], [20, 15], [-30, -20], [-6, -22], [4, -4], [36, -12], [58, -14], [42, -17], [-16, -2], [-48, 23], [6, -13], [56, -10], [50, 8], [-26, 18], [-40, 5], [-12, 0], [26, -14], [26, 17], [-44, -11], [-34, -19], [60, -12], [-8, 22]];
    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_bd();".
// 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_bd(: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_bd();".
// 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_bd( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2790_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-8, -1, 1]>,<11,R![4, 7, 1]>,<13,R![2, 1]>,<17,R![-32, 2, 1]>,<19,R![4, 7, 1]>],Snew);
    return Vf;
end function;