
// Make newform 5082.2.a.n in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5082_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_5082_2_a_n();".
// 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_5082_2_a_n();".
// 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_5082_a();"
function MakeCharacter_5082_a()
    N := 5082;
    order := 1;
    char_gens := [3389, 4357, 2059];
    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_5082_a_Hecke(Kf)
    return MakeCharacter_5082_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], [1], [2], [1], [0], [2], [2], [0], [0], [2], [4], [-2], [10], [-4], [4], [-2], [-12], [2], [12], [8], [-6], [8], [8], [-14], [-14], [10], [4], [12], [-14], [18], [-8], [0], [10], [0], [10], [-8], [18], [4], [0], [-6], [12], [2], [-8], [-2], [-6], [20], [-12], [20], [0], [-14], [6], [-16], [-30], [4], [-6], [-32], [10], [8], [10], [-10], [-16], [-14], [-8], [20], [-30], [22], [20], [22], [12], [26], [-6], [32], [-28], [-22], [-20], [12], [22], [-30], [18], [-14], [-12], [6], [40], [18], [-16], [-12], [-30], [38], [18], [16], [4], [32], [-32], [36], [28], [-16], [10], [-30], [16], [-30], [44], [-14], [0], [22], [-44], [10], [-28], [34], [-24], [18], [-32], [-38], [26], [-20], [0], [18], [44], [-28], [30], [20], [18], [-26], [-14], [-28], [28], [-30], [-2], [28], [28], [-46], [-20], [40], [-8], [-18], [2], [26], [10], [-32], [34], [14], [0], [-22], [-48], [36], [-6], [-12], [-6], [34], [4], [24], [2], [-14], [-28], [-24], [-12], [-40], [40], [-54], [-22], [-6], [4], [-18], [40], [-52], [2], [-4], [16], [-14], [46], [-14], [-36], [-14], [-24], [-18], [-4], [-54], [-28], [-42], [-32], [42], [16], [0], [30], [-46], [-16], [-30], [2], [20], [-2], [-24], [10], [28], [-20], [-30], [-52], [34], [-58], [-10], [-30], [32], [-14], [24], [-6], [-46], [44], [-22], [-52], [-52], [-18], [-52], [22], [-6], [56], [20], [-48], [-46], [40], [42], [8], [-18], [10], [0], [-46], [-48], [16], [-70], [-70], [-48], [48], [-36], [-58], [-16], [32], [-66], [44], [-56], [50], [10], [-36], [-36], [-36], [40], [-28], [-50], [-38], [-32], [76], [12], [-44], [-24], [-62], [-14], [-56], [26], [-6], [36], [46], [16], [66], [26], [32], [28], [-6], [58], [66], [12], [6], [-54], [-76], [58], [18], [68], [18], [-56], [18], [-36], [12], [34], [-74], [-8], [-32], [-72], [24], [34], [72], [24], [-30], [26], [-12], [42], [54], [68], [-10], [64], [18], [-78], [28], [-78], [-56], [-8], [66], [-78], [-72], [12], [60], [-62], [60], [-70], [24], [-6], [64], [-10], [14], [-4], [24], [-22], [44], [-72], [74], [-66], [0], [26], [58], [24], [30], [-22], [36], [28], [-8], [90], [18], [30], [8], [0], [20], [12], [54], [-54], [-38], [-32], [-10], [-22], [-62], [32], [-66], [-20], [34], [-68], [-80], [10], [-40], [90], [-90], [48], [-22], [-14], [36], [40], [6], [8], [-6], [50], [-48], [-60], [-60], [90], [-22], [16], [6], [-12], [76], [24], [74], [88], [14], [36], [-56], [-70], [-78], [74], [-30], [-70], [32], [-26], [24], [36], [72], [-30], [76], [68], [-46], [-54], [60], [-52], [24], [62], [-32], [58], [20], [-54], [50], [-30], [64], [42], [-30], [80], [-66], [38], [-44], [68], [-14], [-22], [-52], [60], [30], [82], [24], [28], [2], [40], [-34], [-22], [48], [0], [18], [34], [76], [6], [-4], [-40]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5082_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5082_2_a_n();".
// 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_5082_2_a_n(:prec:=1)
    chi := MakeCharacter_5082_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_5082_2_a_n();".
// 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_5082_2_a_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5082_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-2, 1]>,<13,R![-2, 1]>,<17,R![-2, 1]>,<19,R![0, 1]>],Snew);
    return Vf;
end function;