
// Make newform 6864.2.a.c 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_c();".
// 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_c();".
// 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], [0], [-2], [2], [2], [2], [12], [6], [0], [0], [4], [-12], [-6], [4], [-4], [4], [8], [12], [10], [2], [-6], [-12], [-10], [-2], [-18], [0], [2], [6], [-4], [-2], [12], [-22], [-8], [24], [6], [12], [2], [18], [-16], [-10], [16], [-12], [-22], [0], [-20], [-4], [0], [-20], [-12], [-30], [16], [-24], [32], [-26], [6], [8], [6], [-30], [-26], [22], [30], [4], [-10], [-18], [10], [18], [0], [-4], [10], [-28], [-24], [20], [8], [-26], [-32], [-36], [-12], [24], [14], [24], [-16], [30], [-16], [2], [6], [16], [-8], [22], [42], [4], [16], [-10], [-22], [-40], [-34], [-4], [18], [10], [12], [-40], [14], [12], [42], [14], [-18], [-8], [38], [18], [-32], [50], [-2], [-4], [-22], [36], [-6], [-36], [34], [-18], [4], [16], [-42], [-44], [-42], [-52], [6], [-30], [40], [32], [-22], [6], [-32], [-46], [22], [12], [20], [-50], [-22], [8], [0], [-10], [-48], [30], [4], [20], [52], [42], [30], [-20], [12], [-28], [10], [-40], [-18], [50], [-30], [28], [-20], [4], [32], [-46], [28], [-24], [-54], [56], [20], [20], [-4], [0], [4], [44], [6], [-26], [-6], [-8], [-10], [30], [-6], [-46], [-38], [-48], [24], [2], [24], [12], [-32], [14], [24], [-44], [2], [30], [-6], [40], [34], [42], [20], [-30], [24], [60], [-62], [-36], [56], [-62], [-42], [-54], [20], [-10], [-32], [-28], [-20], [24], [6], [-16], [52], [36], [10], [-50], [-28], [2], [46], [6], [10], [62], [30], [0], [32], [-22], [68], [-60], [-36], [-44], [-32], [-6], [-66], [-8], [30], [-52], [-20], [-16], [52], [26], [-16], [-38], [12], [-62], [36], [66], [-34], [32], [-26], [22], [-20], [38], [14], [0], [-8], [56], [30], [14], [62], [30], [-20], [-42], [-42], [58], [-34], [42], [-76], [46], [0], [-70], [-38], [-68], [-2], [52], [22], [-24], [-66], [-48], [-62], [-14], [30], [-18], [-74], [70], [-44], [52], [16], [74], [64], [-46], [-30], [-60], [-12], [6], [-46], [-66], [-62], [24], [-12], [-68], [-2], [-4], [58], [-60], [26], [12], [62], [48], [-12], [72], [18], [-36], [-56], [-26], [-52], [-80], [-22], [-2], [-58], [-22], [-66], [68], [76], [36], [54], [10], [48], [-40], [0], [38], [12], [56], [78], [-14], [40], [-80], [-42], [-58], [-48], [-90], [-70], [-18], [8], [0], [20], [-38], [-10], [-46], [-32], [70], [96], [-84], [0], [-4], [-28], [-58], [-32], [-2], [72], [20], [6], [58], [24], [-62], [64], [44], [48], [82], [16], [42], [-84], [-80], [-44], [-56], [22], [18], [-10], [-76], [-62], [30], [-44], [-54], [-2], [-22], [-34], [-26], [-46], [-68], [-60], [-36], [-26], [-96], [38], [4], [-46], [38], [58], [68], [102], [6], [-32], [24], [-78], [-76], [-100], [-10], [-30], [-8], [32], [34], [-32], [40], [-64], [-98], [100], [-20], [-82], [-24], [-34], [62], [-70], [-32], [58], [52], [-96]];
    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_c();".
// 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_c(: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_c();".
// 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_c( : 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![0, 1]>,<19,R![2, 1]>],Snew);
    return Vf;
end function;