
// Make newform 6864.2.a.i 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_i();".
// 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_i();".
// 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], [0], [0], [-1], [-1], [-4], [0], [8], [0], [10], [-2], [-6], [2], [-4], [14], [0], [6], [-6], [-8], [-6], [-2], [12], [-8], [-10], [0], [-4], [-4], [6], [2], [-6], [0], [-8], [10], [-14], [4], [-22], [-2], [0], [0], [-4], [-22], [8], [2], [18], [-4], [14], [-2], [-4], [-22], [12], [-16], [14], [-12], [-2], [12], [2], [-28], [-22], [18], [-2], [26], [8], [-8], [-26], [-32], [-10], [-10], [-32], [26], [-4], [-32], [0], [-34], [2], [0], [18], [-2], [-12], [-6], [-36], [18], [-16], [-30], [-14], [20], [-16], [6], [-2], [-2], [36], [-24], [-30], [8], [-6], [12], [28], [-30], [14], [-14], [30], [-14], [0], [-32], [-10], [-14], [-16], [6], [32], [-38], [10], [6], [24], [-38], [-10], [-6], [-34], [-8], [2], [-20], [-26], [-38], [36], [24], [14], [48], [-14], [16], [28], [-2], [4], [-16], [32], [10], [-26], [30], [32], [12], [18], [48], [-16], [30], [-12], [44], [-50], [-36], [-14], [24], [-24], [0], [-14], [-6], [8], [-12], [-12], [48], [-54], [0], [26], [-50], [-20], [16], [56], [20], [-12], [4], [-44], [14], [50], [26], [44], [-18], [4], [-10], [12], [-10], [-20], [-24], [-58], [14], [-22], [24], [26], [2], [12], [-18], [2], [34], [38], [-48], [46], [-36], [48], [36], [32], [-50], [-58], [14], [-22], [12], [54], [-46], [-22], [-42], [60], [-6], [46], [-24], [-14], [40], [34], [-2], [24], [36], [32], [-14], [50], [48], [-24], [-8], [50], [8], [60], [46], [60], [46], [-58], [32], [-42], [36], [-26], [-30], [-28], [-36], [0], [64], [62], [74], [-52], [72], [-56], [14], [-40], [-50], [-2], [-60], [34], [4], [-76], [4], [-62], [18], [48], [54], [-60], [52], [26], [-8], [10], [22], [62], [-12], [-54], [-34], [16], [28], [-36], [-56], [56], [-40], [-18], [-10], [-50], [74], [-26], [14], [8], [-34], [-6], [60], [64], [-38], [12], [-58], [-76], [-24], [26], [-2], [-42], [76], [-22], [36], [-2], [-12], [38], [32], [-36], [30], [40], [-16], [-46], [-66], [-10], [20], [72], [62], [-20], [-10], [4], [78], [12], [48], [-20], [44], [72], [6], [-36], [48], [54], [-68], [46], [34], [16], [-2], [-74], [-34], [-14], [-46], [-52], [72], [-50], [-42], [-66], [20], [52], [72], [-74], [-46], [70], [62], [-18], [6], [42], [-44], [-12], [-60], [14], [-34], [-24], [-82], [-20], [-42], [-40], [26], [-74], [-80], [-36], [-84], [64], [68], [90], [-56], [-8], [-52], [20], [30], [-82], [-76], [18], [52], [32], [-32], [28], [42], [54], [36], [20], [82], [76], [22], [36], [-20], [-32], [18], [-88], [24], [-82], [-10], [52], [-80], [70], [0], [-96], [-44], [-16], [14], [44], [66], [-16], [-78], [-30], [-34], [72], [-104], [30], [-42], [2], [-26], [70], [-64], [66], [0], [-84], [82], [62], [-30], [64], [68], [-88], [-64], [10], [-14], [24], [-60], [-38], [16], [108], [-98], [-106], [40]];
    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_i();".
// 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_i(: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_i();".
// 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_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6864_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![0, 1]>,<7,R![0, 1]>,<17,R![4, 1]>,<19,R![0, 1]>],Snew);
    return Vf;
end function;