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