
// Make newform 1827.2.a.d in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1827_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_1827_2_a_d();".
// 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_1827_2_a_d();".
// 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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_1827_a();"
function MakeCharacter_1827_a()
    N := 1827;
    order := 1;
    char_gens := [407, 262, 379];
    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_1827_a_Hecke(Kf)
    return MakeCharacter_1827_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], [2], [1], [-4], [-2], [-2], [-4], [0], [-1], [-8], [-10], [6], [12], [8], [-6], [-12], [-10], [-12], [16], [2], [0], [-4], [6], [-6], [-6], [-8], [12], [-18], [-2], [16], [4], [6], [-4], [-6], [-8], [22], [4], [24], [-6], [4], [6], [-8], [2], [-22], [-24], [4], [-16], [12], [14], [-26], [24], [18], [12], [6], [0], [18], [8], [-10], [-10], [-4], [10], [12], [32], [10], [2], [-20], [34], [-20], [14], [6], [16], [24], [22], [-4], [-16], [-6], [-18], [-2], [18], [12], [-26], [-24], [-22], [-8], [-20], [-18], [10], [-30], [-16], [4], [-8], [24], [-20], [4], [-16], [-22], [30], [12], [30], [-28], [2], [20], [-42], [-20], [-22], [-44], [-42], [0], [-30], [40], [6], [38], [4], [8], [-18], [36], [-24], [-14], [20], [-26], [2], [-38], [12], [-12], [34], [-26], [16], [0], [-26], [-12], [-32], [-32], [38], [-18], [-22], [10], [-28], [2], [38], [-36], [-6], [-8], [28], [6], [0], [-2], [-18], [36], [40], [14], [-50], [-44], [32], [44], [-24], [-56], [6], [-54], [-6], [-28], [-26], [-56], [-4], [-2], [32], [32], [-2], [-14], [-6], [12], [46], [32], [-38], [0], [14], [-20], [-6], [48], [14], [-64], [-28], [-58], [14], [24], [50], [-34], [-20], [58], [-64], [34], [-36], [-28], [-46], [-44], [46], [-62], [-34], [-26], [-56], [18], [64], [14], [26], [-68], [42], [-8], [36], [-42], [36], [2], [50], [-24], [28], [-48], [26], [-32], [-18], [64], [-30], [-10], [-72], [-18], [16], [60], [6], [6], [-56], [32], [60], [-2], [68], [16], [-26], [28], [24], [-22], [-54], [44], [0], [4], [28], [-24], [46], [-26], [40], [0], [28], [44], [72], [-2], [70], [64], [-14], [-46], [20], [38], [52], [18], [42], [-48], [-12], [-74], [-58], [30], [76], [18], [-26], [-4], [26], [-34], [52], [26], [-32], [-70], [-80], [12], [-34], [10], [76], [64], [-56], [-24], [38], [-76], [-24], [50], [10], [-40], [-26], [-30], [28], [-42], [60], [-82], [10], [56], [-14], [-28], [28], [-14], [-14], [-16], [-28], [-36], [34], [-36], [62], [-48], [6], [-32], [10], [14], [52], [-48], [58], [-60], [24], [-62], [30], [-36], [-70], [42], [80], [22], [26], [4], [-20], [-24], [-70], [14], [-62], [-16], [-60], [76], [-12], [-2], [62], [82], [48], [22], [54], [26], [88], [34], [-76], [-2], [28], [88], [-6], [52], [74], [-78], [-16], [-74], [-26], [-72], [-60], [-66], [-48], [-58], [30], [72], [60], [-36], [-38], [-30], [8], [26], [36], [-100], [-88], [-22], [56], [30], [-76], [88], [-86], [-42], [58], [-78], [-42], [-8], [-18], [52], [-56], [-32], [-58], [-84], [56], [18], [-62], [-4], [44], [0], [-102], [0], [54], [-36], [42], [14], [-50], [32], [78], [18], [56], [-50], [30], [28], [76], [2], [42], [-36], [-76], [26], [82], [24], [-88], [78], [-96], [2], [-2], [72], [76], [10], [-22], [-28], [-10], [60], [48]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1827_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1827_2_a_d();".
// 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_1827_2_a_d(:prec:=1)
    chi := MakeCharacter_1827_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_1827_2_a_d();".
// 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_1827_2_a_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1827_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 1]>,<5,R![-2, 1]>],Snew);
    return Vf;
end function;