
// Make newform 8640.2.a.bu in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8640_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_8640_2_a_bu();".
// 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_8640_2_a_bu();".
// 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_8640_a();"
function MakeCharacter_8640_a()
    N := 8640;
    order := 1;
    char_gens := [2431, 3781, 6401, 3457];
    v := [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_8640_a_Hecke(Kf)
    return MakeCharacter_8640_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], [0], [1], [1], [-6], [1], [0], [-1], [6], [6], [-8], [7], [6], [-4], [12], [-6], [0], [-11], [-7], [-6], [11], [1], [-6], [12], [-13], [0], [13], [18], [10], [18], [16], [-12], [18], [5], [-12], [-11], [-2], [11], [12], [12], [-6], [7], [-12], [-19], [24], [-5], [17], [16], [-6], [10], [-24], [30], [17], [-6], [0], [-12], [24], [25], [-2], [0], [-28], [30], [-16], [0], [17], [12], [-19], [23], [6], [1], [36], [-18], [-23], [7], [11], [24], [0], [-14], [6], [-31], [24], [19], [-18], [26], [-8], [12], [-36], [-10], [-12], [13], [-12], [-18], [-23], [30], [32], [-6], [30], [30], [5], [43], [35], [12], [12], [24], [-13], [-31], [-6], [-6], [-18], [14], [13], [-29], [-6], [11], [25], [0], [-4], [-30], [-12], [-48], [13], [17], [12], [18], [-28], [18], [-5], [6], [-8], [22], [20], [-24], [13], [-23], [-42], [-37], [6], [17], [0], [48], [-4], [24], [-17], [-36], [7], [12], [-41], [-42], [41], [-12], [-5], [18], [-37], [48], [-19], [24], [16], [-18], [-7], [-42], [-24], [0], [31], [36], [18], [-42], [-5], [10], [5], [48], [-30], [22], [-24], [-37], [-17], [30], [20], [-42], [-41], [-50], [-53], [-36], [22], [12], [36], [-54], [1], [35], [-7], [48], [-37], [24], [65], [12], [-24], [6], [23], [-35], [30], [30], [-12], [-35], [-23], [-55], [6], [-54], [-35], [42], [-42], [23], [-7], [0], [-5], [-42], [-18], [35], [52], [42], [-6], [-18], [43], [4], [-60], [16], [-54], [-11], [-12], [54], [37], [36], [-23], [-52], [16], [-30], [-1], [60], [-67], [12], [30], [-12], [-6], [71], [-53], [-71], [-36], [-6], [-53], [-24], [44], [54], [-50], [-24], [12], [41], [18], [-36], [-5], [8], [42], [-70], [13], [42], [-5], [79], [-24], [-28], [-30], [-30], [68], [-48], [49], [47], [26], [25], [-82], [49], [78], [-74], [14], [-54], [18], [4], [-54], [13], [83], [-24], [-85], [0], [55], [6], [18], [66], [-42], [36], [10], [-24], [-5], [-48], [-6], [17], [-31], [-30], [-44], [0], [-67], [62], [-24], [19], [-24], [7], [-84], [-60], [-18], [-61], [30], [26], [-60], [-78], [38], [78], [-73], [53], [-78], [-32], [6], [-43], [-16], [44], [30], [78], [67], [-78], [-35], [-42], [-52], [30], [13], [-90], [2], [-56], [67], [-66], [54], [-71], [24], [36], [70], [-4], [60], [-78], [-13], [-61], [12], [64], [-47], [54], [78], [-30], [-30], [54], [-77], [-72], [18], [-60], [47], [17], [48], [-47], [23], [-36], [-7], [36], [-6], [31], [-71], [6], [12], [-31], [-78], [11], [60], [-30], [-29], [42], [-49], [-12], [64], [-23], [-61], [48], [74], [-48], [42], [-19], [-18], [-37], [61], [-60], [20], [102], [-50], [-6], [-8], [42], [84], [52], [13], [42], [47], [-42], [-34], [-12], [-60], [41], [-13], [-18], [-72], [-77], [-42], [-54], [-66], [-26], [-48], [-96], [74], [12], [-18], [-90], [56], [54], [-43], [36], [89], [66], [-41], [-12], [-19], [70], [-7], [73], [-36], [42], [85], [12], [-31], [24], [20], [60], [47], [-50], [65], [78], [24], [60], [23], [102], [10], [30], [49], [72], [68], [-56], [-54], [-59], [-61], [59], [19], [42], [54], [32], [76], [-48], [-36], [77], [60], [-26], [-108], [-92], [-48], [-96], [-13], [78], [2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8640_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8640_2_a_bu();".
// 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_8640_2_a_bu(:prec:=1)
    chi := MakeCharacter_8640_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(3457) - 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_8640_2_a_bu();".
// 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_8640_2_a_bu( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8640_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-1, 1]>,<11,R![6, 1]>,<13,R![-1, 1]>,<17,R![0, 1]>,<19,R![1, 1]>],Snew);
    return Vf;
end function;