
// Make newform 6864.2.a.v 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_v();".
// 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_v();".
// 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], [-4], [1], [-1], [8], [-4], [0], [4], [-2], [-2], [-2], [2], [-4], [6], [0], [-10], [-2], [0], [-14], [-10], [-4], [16], [-2], [4], [4], [-4], [-18], [10], [-6], [16], [-8], [2], [6], [-8], [-6], [-14], [0], [-12], [-12], [-6], [-24], [2], [-2], [-20], [-2], [-14], [-12], [-6], [-16], [-24], [14], [-20], [-2], [-4], [-22], [0], [26], [22], [-26], [-2], [-28], [0], [22], [0], [26], [-18], [16], [-30], [4], [-32], [16], [-26], [-2], [24], [18], [-2], [-28], [2], [36], [-6], [32], [-14], [26], [36], [8], [-26], [2], [2], [20], [0], [22], [-8], [-2], [-4], [36], [-6], [-26], [2], [-10], [-10], [-32], [-4], [30], [2], [-16], [26], [24], [-38], [26], [22], [-8], [-10], [34], [42], [18], [-16], [-6], [28], [-2], [-46], [-24], [24], [2], [12], [18], [-48], [-20], [22], [32], [8], [-24], [42], [-30], [-42], [-24], [40], [18], [-4], [-4], [-6], [44], [36], [-18], [-4], [26], [-28], [32], [-8], [26], [-30], [40], [-12], [-28], [-16], [-46], [32], [2], [42], [-28], [44], [-44], [20], [-12], [36], [4], [38], [-54], [18], [36], [-2], [-12], [-58], [20], [14], [-24], [-48], [-34], [-10], [62], [40], [-38], [30], [-52], [-26], [26], [-2], [-10], [-16], [46], [44], [-32], [-60], [-48], [14], [30], [62], [-26], [44], [-22], [-14], [50], [22], [-12], [58], [-30], [24], [-66], [24], [-38], [54], [0], [44], [48], [10], [-22], [-28], [-32], [-32], [-14], [20], [12], [18], [-68], [-58], [46], [24], [-10], [-44], [-26], [-70], [56], [40], [-32], [-8], [-50], [14], [52], [32], [56], [-10], [-16], [-10], [66], [-76], [54], [60], [64], [36], [34], [-10], [40], [22], [16], [12], [58], [-28], [10], [62], [-50], [36], [-38], [-26], [-24], [20], [-44], [-24], [52], [12], [-18], [-6], [14], [-62], [-66], [-6], [8], [-66], [-38], [28], [32], [46], [28], [-26], [-48], [48], [10], [-54], [-46], [0], [26], [4], [-30], [-60], [86], [60], [-36], [54], [-24], [44], [-14], [-62], [-50], [84], [24], [-58], [-60], [-58], [36], [86], [76], [32], [-16], [68], [-72], [-34], [-60], [80], [-42], [40], [-50], [50], [12], [-54], [26], [-34], [70], [70], [-84], [-36], [54], [-26], [-26], [-4], [80], [-8], [-58], [54], [-26], [-74], [46], [-90], [-50], [-68], [60], [20], [46], [-62], [72], [38], [16], [-26], [32], [-70], [54], [-4], [-4], [-28], [60], [-76], [34], [-84], [-16], [-44], [60], [46], [-30], [88], [58], [68], [-24], [16], [-96], [2], [-90], [60], [-44], [50], [56], [86], [20], [-12], [-20], [46], [-100], [32], [-54], [-10], [-48], [80], [-50], [24], [-32], [-68], [24], [-82], [32], [82], [8], [-66], [-38], [6], [100], [-96], [66], [-2], [98], [10], [-14], [-32], [-14], [60], [20], [-30], [-26], [-30], [72], [28], [-32], [0], [18], [74], [-96], [68], [-70], [-72], [-44], [-94], [-54], [-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_v();".
// 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_v(: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_v();".
// 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_v( : 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![4, 1]>,<17,R![-8, 1]>,<19,R![4, 1]>],Snew);
    return Vf;
end function;