
// Make newform 5082.2.a.a in Magma, downloaded from the LMFDB on 19 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_a();".
// 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_a();".
// 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], [-4], [-1], [0], [6], [4], [2], [-8], [6], [6], [-6], [-12], [-4], [6], [2], [0], [-10], [4], [-12], [0], [16], [14], [-14], [-14], [-14], [-6], [8], [-6], [14], [0], [2], [14], [-2], [10], [8], [-4], [4], [12], [-2], [-4], [-8], [0], [14], [-10], [2], [4], [6], [-6], [8], [14], [16], [8], [-16], [26], [0], [-4], [4], [10], [-26], [14], [14], [14], [10], [-34], [-6], [20], [-6], [-8], [-6], [-6], [-24], [-34], [14], [20], [-22], [6], [-20], [-26], [-4], [12], [38], [24], [26], [20], [28], [14], [-10], [-14], [-24], [36], [-24], [20], [28], [-28], [-20], [20], [18], [22], [14], [-8], [30], [26], [-10], [32], [-34], [24], [12], [12], [-4], [8], [6], [26], [-4], [28], [18], [32], [26], [30], [24], [0], [10], [-34], [-36], [24], [18], [-30], [18], [-42], [-50], [16], [-16], [-28], [2], [-24], [-28], [-24], [50], [24], [2], [-26], [26], [44], [28], [16], [-54], [6], [40], [40], [-16], [18], [-50], [28], [24], [52], [24], [-32], [18], [32], [30], [36], [-30], [32], [-36], [2], [-42], [8], [26], [-58], [44], [-32], [-4], [0], [10], [-30], [46], [40], [6], [12], [34], [-8], [-30], [18], [-24], [-36], [20], [-26], [-60], [-6], [-4], [38], [-48], [12], [-24], [52], [50], [-30], [26], [12], [36], [50], [-4], [-20], [0], [-60], [-28], [-70], [44], [66], [68], [-30], [48], [16], [48], [-60], [-10], [8], [8], [28], [22], [-58], [-4], [-50], [-56], [34], [-54], [22], [12], [-28], [48], [-22], [-70], [16], [-70], [44], [-36], [50], [-6], [-36], [-26], [-60], [10], [10], [-34], [64], [28], [-30], [-32], [-8], [64], [38], [60], [52], [42], [-50], [48], [74], [38], [-12], [-24], [8], [-36], [30], [18], [-42], [56], [-62], [-42], [24], [-26], [-76], [12], [46], [-16], [-40], [38], [-20], [46], [-34], [46], [24], [-56], [-24], [-38], [66], [0], [-26], [18], [-30], [32], [50], [12], [54], [54], [-46], [10], [-38], [20], [-54], [70], [52], [50], [32], [12], [4], [78], [36], [8], [36], [-38], [0], [30], [54], [-60], [32], [56], [-16], [-8], [54], [-18], [-6], [-38], [-58], [76], [74], [-34], [-60], [20], [-24], [-70], [-42], [-18], [28], [34], [-24], [-36], [-78], [64], [66], [8], [46], [-6], [14], [-68], [-18], [92], [16], [44], [16], [64], [22], [-10], [18], [-24], [-6], [-16], [14], [34], [18], [52], [-86], [-16], [52], [-68], [20], [2], [-74], [96], [22], [36], [-36], [72], [6], [-72], [62], [-76], [80], [28], [44], [-56], [-40], [18], [-32], [58], [70], [-6], [-44], [-32], [-36], [54], [-38], [48], [4], [48], [44], [-38], [-24], [70], [-12], [42], [-58], [-78], [-88], [-98], [38], [28], [86], [-78], [76], [-68], [16], [-34], [12], [48], [50], [80], [-32], [-26], [-10], [24], [30], [78], [68], [54], [-10], [-24], [-36], [-50], [100], [24]];
    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_a();".
// 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_a(: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_a();".
// 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_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5082_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![4, 1]>,<13,R![-6, 1]>,<17,R![-4, 1]>,<19,R![-2, 1]>],Snew);
    return Vf;
end function;