
// Make newform 2790.2.a.be in Magma, downloaded from the LMFDB on 20 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_be();".
// 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_be();".
// 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 := [-4, -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], [0, -1], [2, 0], [-2, 2], [0, -3], [-4, 1], [-2, -2], [1, 0], [2, -2], [-2, -2], [0, 5], [0, -2], [-10, 1], [8, 2], [6, 0], [0, -6], [0, 3], [2, -5], [-12, 3], [-4, -4], [6, 3], [-6, 0], [-2, 1], [-12, 4], [0, 1], [-10, 6], [-6, 3], [0, 0], [0, 6], [-10, -4], [-12, -2], [6, -9], [0, 0], [-6, 9], [0, -6], [4, 1], [2, -4], [12, 0], [10, -9], [4, 4], [-6, 8], [-6, 0], [-4, -5], [-16, -3], [-16, -2], [0, -3], [-14, -1], [-6, 9], [-24, 0], [2, 4], [4, -8], [-6, 3], [0, 8], [-10, 4], [4, 9], [10, -6], [-10, 2], [0, 0], [-6, -8], [0, 0], [12, -4], [-10, 0], [-22, 2], [4, -4], [-10, 0], [-4, -4], [-10, 6], [-10, 4], [24, 3], [0, 6], [2, -3], [16, -9], [0, 4], [-18, 4], [-14, 9], [6, 3], [-22, -6], [-16, 4], [6, -8], [20, 4], [2, 13], [-16, 10], [0, -9], [10, -8], [6, -12], [6, -6], [0, 12], [20, -4], [-8, -1], [24, 0], [8, 7], [20, 0], [-16, -4], [6, -10], [22, 4], [16, -15], [30, -4], [32, -2], [14, 5], [4, -8], [-18, -5], [36, 2], [10, 2], [-20, 8], [30, 0], [0, -15], [2, 0], [-16, 9], [2, 0], [18, -1], [-4, 6], [-4, 13], [26, -8], [16, -9], [28, 7], [-30, -4], [0, 12], [6, 8], [22, -4], [6, -15], [-24, -3], [-8, 9], [-18, 5], [-6, 9], [40, -2], [0, 9], [14, -8], [12, 16], [12, 15], [40, 2], [10, 0], [-18, 15], [-18, -1], [14, -1], [0, -7], [-14, 16], [30, 3], [24, -19], [6, -10], [0, -10], [-12, 4], [-14, 15], [-24, 7], [-6, 21], [22, -20], [-4, 12], [4, -17], [-50, 0], [-30, 8], [-8, 15], [48, -6], [24, 0], [32, -8], [24, -10], [-10, 1], [26, 6], [-18, 6], [4, 2], [30, -18], [-32, -4], [32, 2], [-2, -16], [0, -16], [28, 3], [-50, 8], [18, 4], [-2, 13], [-20, 8], [-30, 9], [4, -20], [-22, -6], [24, -18], [-10, 5], [12, 16], [-42, 9], [-48, -1], [-46, -3], [-20, -12], [0, -3], [-18, 4], [38, -16], [24, 2], [46, 4], [-30, -7], [24, 0], [10, 6], [16, -7], [-38, 15], [-40, 11], [28, 0], [-26, -2], [20, -20], [-34, 14], [42, -8], [-30, -3], [2, -7], [0, 14], [-26, -11], [-12, -19], [-6, -15], [-2, -13], [-16, -10], [-14, -8], [-8, 8], [-28, 2], [-26, 10], [20, -24], [18, 9], [-42, 2], [-4, -12], [-32, 11], [-48, 6], [-2, 5], [-20, -12], [-2, 8], [24, -24], [18, -12], [10, 21], [-32, 8], [-34, -10], [32, 9], [-16, 5], [18, 15], [-10, 20], [0, 24], [0, 18], [-8, 26], [-30, 4], [12, 24], [0, 6], [38, -17], [20, -32], [36, -9], [30, 3], [-22, 16], [-48, -3], [-32, 8], [4, -28], [4, 18], [0, 6], [6, -24], [46, -10], [-40, 1], [48, -6], [24, -15], [28, 8], [-48, 0], [-6, -3], [-2, 16], [-4, 19], [-50, 5], [-22, 2], [8, -22], [-10, -20], [48, -11], [-38, 4], [-22, 6], [8, 15], [52, -14], [34, -9], [-30, 1], [-18, 24], [16, -1], [54, -7], [-42, 2], [16, -18], [66, 2], [6, 6], [4, 0], [34, -13], [28, -9], [42, 4], [12, -28], [-40, 23], [-58, 12], [34, -4], [48, -21], [0, -8], [-56, 14], [-16, -14], [-58, 0], [40, -18], [-56, 14], [18, -21], [-30, -12], [48, 12], [-30, 16], [54, 7], [-32, 29], [38, -14], [0, 22], [-62, 6], [-2, 2], [-56, 20], [-54, -8], [16, -11], [80, 3], [-54, -8], [38, -5], [-16, 24], [-16, 37], [-20, 6], [2, -26], [-36, 18], [14, -24], [24, 6], [22, 0], [32, -16], [14, 14], [6, 20], [-40, -3], [-40, 2], [18, 14], [28, -8], [-8, 20], [-14, 6], [-18, 21], [12, 12], [42, 9], [-18, -13], [-32, -3], [34, -5], [-70, -2], [0, -9], [-16, 30], [24, 12], [30, -23], [30, 12], [18, 16], [16, -18], [-44, 2], [40, 5], [-56, -5], [-66, 0], [82, 5], [6, -9], [0, -32], [42, 2], [-34, 4], [-26, 10], [-12, -21], [18, 0], [0, 2], [-2, 30], [24, -9], [16, -2], [18, -30], [-20, -24], [10, -15], [14, -7], [8, -2], [54, -12], [-34, 28], [-32, 10], [56, -17], [22, -26], [-24, 14], [18, -1], [-58, -1], [-60, -7], [80, 2], [24, 26], [50, -21], [2, -16], [-80, 2], [-18, 21], [0, 36], [8, -27], [-16, 14], [-6, 24], [-32, 6], [74, 0], [48, 9], [24, 15], [42, -6], [70, 10], [42, 7], [-42, -10], [-10, -28], [72, -14], [-18, -2], [-36, 6], [-8, -22], [24, -12], [-54, 18], [0, 33], [-12, -27], [18, 14], [38, -19], [16, 4], [32, -18], [72, 5], [42, -24], [-36, -13], [-66, 4], [-20, -6], [6, 34], [18, 13], [-14, 1], [56, 15], [-10, 20], [-18, -18], [-56, 2], [-18, 0], [-30, -12], [40, -1], [-8, 17], [22, 12], [-62, -2], [12, 12], [-4, -32], [-62, -10], [-10, -5], [-8, -2], [64, -25], [26, -25], [-8, -22], [70, -8], [18, -6], [12, -45], [-8, 4], [-14, -6], [-34, -17], [0, 21], [22, -25], [-4, 0], [32, -34]];
    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_be();".
// 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_be(: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_be();".
// 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_be( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2790_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-4, -1, 1]>,<11,R![-4, 1, 1]>,<13,R![-2, 1]>,<17,R![-16, 2, 1]>,<19,R![-36, 3, 1]>],Snew);
    return Vf;
end function;