
// Make newform 6498.2.a.e in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6498_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_6498_2_a_e();".
// 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_6498_2_a_e();".
// 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_6498_a();"
function MakeCharacter_6498_a()
    N := 6498;
    order := 1;
    char_gens := [3611, 6139];
    v := [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_6498_a_Hecke(Kf)
    return MakeCharacter_6498_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], [-4], [-3], [-2], [6], [0], [6], [0], [-2], [10], [9], [-4], [0], [6], [-9], [-4], [7], [-6], [-1], [4], [-3], [6], [-17], [0], [-2], [0], [16], [15], [-2], [-9], [-9], [11], [-18], [10], [-16], [-19], [-24], [6], [9], [-2], [-12], [-2], [-18], [-10], [-20], [-14], [3], [-16], [-3], [12], [-5], [3], [3], [12], [12], [-16], [8], [27], [5], [24], [7], [30], [-19], [-18], [-5], [-11], [9], [-4], [-3], [6], [-22], [4], [28], [-36], [36], [-10], [-27], [-5], [12], [10], [-30], [-26], [-14], [-9], [9], [5], [-6], [-34], [27], [-36], [-2], [0], [-25], [-6], [-24], [9], [28], [44], [4], [-24], [-21], [-6], [-7], [11], [12], [21], [-6], [13], [-20], [2], [-3], [-4], [-28], [-39], [-43], [-18], [12], [36], [40], [-14], [-42], [-36], [44], [24], [14], [-30], [32], [-22], [35], [-18], [-38], [-10], [39], [2], [-48], [7], [6], [-45], [16], [-18], [14], [-39], [-44], [-12], [-22], [-3], [-43], [18], [-20], [9], [-19], [48], [-17], [30], [-10], [3], [35], [-42], [-60], [-15], [-34], [21], [33], [24], [-8], [-4], [34], [6], [57], [-38], [18], [-49], [34], [-39], [23], [-12], [-44], [50], [26], [12], [16], [39], [-24], [24], [-20], [40], [31], [-18], [-35], [36], [37], [-48], [-33], [-6], [-13], [-40], [69], [-54], [-6], [-8], [-2], [-59], [51], [-42], [-34], [-69], [21], [-65], [17], [42], [-4], [9], [-66], [22], [50], [30], [-18], [-54], [70], [34], [45], [44], [63], [-22], [39], [-24], [64], [-12], [56], [-53], [-14], [51], [-1], [30], [65], [18], [-45], [42], [51], [-7], [20], [10], [18], [42], [-16], [21], [-29], [12], [-52], [-30], [30], [-35], [-42], [12], [62], [13], [-78], [-46], [-50], [15], [20], [28], [51], [49], [0], [-3], [-20], [-66], [34], [-20], [-1], [32], [34], [-46], [-69], [-68], [37], [36], [-66], [-58], [36], [-14], [-4], [-60], [-19], [48], [-40], [-18], [42], [-36], [3], [60], [4], [-78], [-26], [48], [-60], [5], [-67], [-6], [-28], [-21], [11], [-17], [63], [-20], [60], [-28], [-30], [42], [45], [-11], [-66], [70], [-9], [-54], [-49], [66], [-77], [-43], [-30], [-8], [-18], [46], [91], [31], [42], [66], [-22], [-72], [-28], [-12], [-7], [-60], [-80], [51], [38], [-46], [-44], [3], [-42], [-44], [18], [-15], [56], [-44], [-66], [-54], [97], [19], [42], [46], [-32], [-45], [36], [-21], [15], [54], [62], [81], [-48], [-15], [-88], [10], [-12], [-26], [-53], [45], [28], [36], [6], [-34], [-28], [-36], [-72], [11], [39], [67], [84], [30], [38], [-51], [61], [-60], [32], [50], [101], [-66], [13], [78], [-99], [77], [-36], [-41], [-8], [-87], [61], [54], [-74], [-21], [-68], [18], [-48], [62], [32], [21], [4], [51], [-59], [-36], [63], [41], [23], [-18], [60], [28], [-42], [48], [78], [28], [-30], [21], [-26], [-12], [-15], [42], [44], [66]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6498_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6498_2_a_e();".
// 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_6498_2_a_e(:prec:=1)
    chi := MakeCharacter_6498_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_6498_2_a_e();".
// 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_6498_2_a_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6498_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![0, 1]>,<7,R![4, 1]>,<11,R![3, 1]>,<13,R![2, 1]>,<29,R![0, 1]>],Snew);
    return Vf;
end function;