
// Make newform 6864.2.a.e in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6864_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_6864_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_6864_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_6864_a();"
function MakeCharacter_6864_a()
    N := 6864;
    order := 1;
    char_gens := [2575, 1717, 4577, 4369, 2641];
    v := [1, 1, 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_6864_a_Hecke(Kf)
    return MakeCharacter_6864_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 := [[0], [-1], [-2], [0], [1], [1], [-6], [4], [8], [-10], [0], [6], [10], [-4], [-8], [-10], [12], [14], [12], [0], [-6], [-8], [-12], [2], [-14], [-2], [-8], [-12], [14], [-14], [8], [12], [18], [12], [-10], [0], [-2], [12], [0], [6], [-20], [-10], [0], [18], [6], [24], [20], [-16], [20], [-10], [18], [-8], [2], [20], [18], [-8], [-2], [8], [-10], [-6], [-4], [6], [20], [8], [-6], [22], [20], [34], [20], [-2], [-6], [16], [16], [38], [20], [8], [-10], [14], [-22], [10], [12], [22], [-24], [-14], [16], [4], [-22], [-6], [-2], [-16], [12], [24], [-8], [-12], [28], [-24], [6], [42], [28], [-34], [4], [-18], [44], [18], [-4], [2], [12], [-30], [8], [-6], [-8], [38], [18], [20], [-24], [-30], [-36], [8], [-34], [-20], [-10], [2], [-18], [-36], [44], [-10], [-10], [-32], [8], [-34], [20], [48], [32], [22], [-54], [18], [30], [-28], [-18], [2], [-20], [6], [8], [-20], [-34], [0], [22], [18], [20], [-24], [-18], [50], [44], [-24], [4], [16], [-32], [10], [10], [-2], [36], [-46], [-16], [36], [42], [-48], [-48], [22], [2], [38], [12], [-2], [-24], [-6], [16], [10], [28], [30], [16], [-2], [48], [28], [38], [42], [-8], [54], [-2], [12], [26], [40], [-62], [-4], [28], [6], [-36], [-38], [-30], [-34], [2], [24], [-34], [8], [-58], [18], [-20], [-18], [-16], [-20], [10], [52], [-14], [-10], [-56], [-36], [32], [-54], [8], [10], [-48], [-42], [-58], [16], [-6], [-32], [28], [-26], [-54], [-64], [16], [12], [-50], [68], [-24], [50], [-28], [40], [34], [-42], [-60], [8], [-60], [60], [-40], [78], [34], [24], [-16], [-12], [-36], [0], [-2], [66], [-48], [10], [-42], [-12], [54], [-4], [22], [-22], [-40], [-36], [6], [14], [-54], [28], [22], [-30], [-20], [-34], [46], [76], [42], [-24], [34], [-8], [-4], [-2], [58], [60], [-16], [-24], [8], [70], [44], [0], [50], [-26], [8], [-22], [30], [-4], [-22], [-68], [-34], [54], [48], [-74], [-12], [-28], [10], [-66], [-56], [-36], [-44], [18], [-44], [-82], [-16], [-10], [80], [-82], [10], [28], [64], [26], [84], [72], [50], [26], [-28], [58], [54], [32], [26], [-14], [-4], [20], [-48], [-34], [-50], [30], [72], [36], [-68], [-4], [-34], [34], [58], [-56], [86], [-22], [70], [-40], [6], [-84], [-26], [-92], [-24], [22], [4], [-38], [6], [-40], [70], [82], [72], [-52], [42], [64], [38], [-30], [48], [28], [12], [-86], [30], [16], [58], [60], [-76], [8], [-66], [64], [-66], [20], [80], [-14], [26], [42], [-26], [66], [16], [66], [-28], [-48], [0], [-42], [-52], [16], [-46], [-2], [60], [12], [-64], [-22], [-24], [42], [68], [54], [-66], [-46], [-72], [-30], [54], [0], [62], [34], [-68], [4], [18], [-18], [36], [4], [-6], [-66], [72], [8], [-38], [8], [-50], [6], [88], [100], [10], [102], [-36], [42], [-12], [24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6864_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6864_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_6864_2_a_e(:prec:=1)
    chi := MakeCharacter_6864_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_6864_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_6864_2_a_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6864_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![2, 1]>,<7,R![0, 1]>,<17,R![6, 1]>,<19,R![-4, 1]>],Snew);
    return Vf;
end function;