
// Make newform 6864.2.a.y 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_y();".
// 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_y();".
// 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], [-4], [1], [-1], [2], [4], [-4], [-2], [0], [-2], [10], [4], [4], [2], [4], [10], [4], [-12], [2], [4], [12], [2], [2], [6], [16], [4], [-14], [-14], [20], [-12], [10], [12], [-10], [20], [-10], [-20], [-16], [-2], [12], [-2], [12], [18], [14], [16], [12], [16], [-12], [6], [26], [24], [2], [-12], [18], [-24], [-6], [-28], [2], [-6], [-12], [-10], [-12], [-20], [-6], [2], [20], [-14], [12], [34], [-6], [32], [0], [26], [20], [28], [18], [-18], [18], [-6], [-20], [-34], [16], [-30], [-36], [-28], [18], [-22], [-2], [-24], [20], [0], [-16], [36], [-4], [0], [-6], [2], [-28], [18], [-44], [30], [-36], [10], [-4], [34], [36], [18], [-36], [26], [-44], [34], [-30], [28], [24], [42], [-44], [12], [-46], [-4], [22], [10], [6], [4], [-12], [-10], [-26], [20], [-16], [2], [44], [48], [0], [-42], [-22], [18], [-30], [-36], [26], [42], [-12], [30], [-40], [4], [14], [-20], [34], [-38], [-4], [36], [50], [-30], [44], [-48], [-28], [-28], [-28], [18], [-6], [-42], [36], [-6], [20], [4], [18], [-44], [16], [-6], [-46], [-22], [36], [-34], [-48], [18], [-16], [26], [-20], [-54], [20], [50], [-16], [-12], [-10], [10], [-52], [58], [-54], [20], [-6], [-48], [-46], [-60], [20], [42], [-36], [-6], [-38], [-26], [66], [-24], [62], [12], [22], [50], [-12], [18], [40], [4], [-6], [-60], [42], [-6], [-56], [12], [-16], [-22], [4], [-14], [68], [2], [2], [-28], [-30], [-64], [-36], [42], [-54], [12], [-68], [12], [14], [60], [52], [10], [-20], [0], [66], [-74], [20], [4], [36], [52], [-56], [30], [-46], [-16], [-8], [36], [-4], [40], [-54], [-14], [-52], [58], [-74], [-20], [38], [-28], [42], [-38], [-68], [-28], [-70], [34], [66], [12], [-30], [-30], [44], [38], [-10], [4], [10], [4], [34], [-16], [-36], [-54], [-30], [28], [16], [64], [32], [10], [76], [-60], [-30], [-82], [32], [-30], [10], [-4], [-38], [-20], [-14], [-10], [24], [-46], [60], [-68], [-78], [-18], [-68], [-76], [20], [-78], [-4], [-34], [28], [26], [-8], [58], [18], [36], [88], [10], [4], [0], [34], [-30], [52], [26], [-42], [-88], [58], [-14], [68], [20], [72], [-26], [26], [42], [-4], [36], [-92], [28], [14], [2], [58], [68], [70], [2], [-26], [-8], [26], [-44], [-66], [52], [24], [34], [-36], [-70], [-54], [-76], [18], [74], [-20], [12], [18], [12], [34], [-54], [-12], [28], [44], [10], [-34], [-12], [-62], [60], [76], [-64], [6], [28], [-2], [-28], [8], [-86], [-46], [18], [42], [42], [36], [18], [12], [44], [0], [86], [-44], [44], [-14], [-6], [-60], [52], [60], [-14], [-28], [-30], [44], [-50], [-70], [-54], [-44], [26], [86], [-28], [-42], [50], [-76], [36], [-94], [46], [-84], [-60], [90], [66], [104], [96], [50], [64], [2], [-30], [-36], [-12], [-54], [42], [84], [-22], [-28], [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_y();".
// 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_y(: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_y();".
// 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_y( : 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![4, 1]>,<17,R![-2, 1]>,<19,R![-4, 1]>],Snew);
    return Vf;
end function;