
// Make newform 8034.2.a.f in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8034_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_8034_2_a_f();".
// 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_8034_2_a_f();".
// 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_8034_a();"
function MakeCharacter_8034_a()
    N := 8034;
    order := 1;
    char_gens := [5357, 1237, 5773];
    v := [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_8034_a_Hecke(Kf)
    return MakeCharacter_8034_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 := [[1], [-1], [4], [-5], [1], [1], [2], [0], [1], [0], [0], [-5], [9], [3], [-3], [-6], [0], [13], [-4], [8], [-10], [-8], [0], [4], [-8], [19], [1], [12], [-1], [1], [7], [4], [15], [-16], [11], [-12], [8], [5], [18], [13], [17], [-2], [16], [5], [-8], [3], [1], [13], [12], [-8], [9], [-20], [7], [28], [-14], [-14], [30], [18], [8], [18], [-13], [-30], [-16], [-15], [-19], [-22], [10], [13], [-3], [-14], [-30], [-22], [18], [-35], [-2], [21], [-3], [-2], [15], [28], [9], [-24], [10], [8], [1], [28], [2], [-3], [9], [-20], [15], [25], [-14], [24], [-14], [-9], [-13], [-3], [-26], [-10], [22], [30], [-24], [3], [0], [23], [28], [-10], [26], [-22], [34], [16], [28], [20], [-49], [12], [-33], [-11], [-5], [-48], [30], [41], [46], [0], [-44], [30], [-16], [38], [16], [-7], [-25], [0], [-30], [-9], [-18], [-31], [-33], [-13], [-42], [22], [-22], [-37], [-3], [39], [-28], [-30], [0], [-28], [19], [3], [51], [50], [-10], [-37], [2], [42], [27], [-29], [16], [-38], [5], [30], [-20], [0], [-1], [-14], [-52], [-22], [16], [-61], [-10], [-5], [-24], [-19], [-20], [-12], [32], [20], [-44], [24], [-8], [-5], [1], [36], [-64], [-36], [-60], [-55], [38], [-44], [-14], [-8], [20], [-50], [-32], [36], [8], [-34], [-44], [-42], [-58], [10], [48], [38], [52], [48], [6], [-54], [-4], [46], [27], [-27], [39], [12], [70], [38], [38], [3], [28], [57], [-68], [39], [-48], [-72], [0], [50], [1], [9], [-48], [-10], [-8], [-26], [48], [-19], [-54], [-39], [67], [2], [-12], [46], [14], [-12], [-43], [-50], [54], [-36], [-54], [-34], [-23], [55], [-64], [-39], [8], [-74], [-46], [25], [25], [12], [-42], [42], [-16], [39], [-39], [34], [15], [-79], [14], [4], [14], [-30], [-33], [-10], [36], [-50], [-33], [7], [-24], [-14], [-38], [77], [29], [-16], [8], [6], [-21], [8], [-58], [-27], [32], [10], [-70], [-70], [45], [44], [31], [63], [-28], [28], [36], [-36], [-73], [-36], [-20], [-27], [-35], [-38], [-58], [-39], [-54], [82], [22], [2], [74], [84], [-13], [17], [9], [0], [-14], [17], [61], [15], [-68], [48], [-26], [8], [41], [0], [-26], [26], [-88], [42], [62], [-60], [68], [72], [22], [46], [-38], [25], [-10], [-6], [-24], [24], [-74], [0], [-52], [51], [35], [-54], [59], [-74], [-28], [26], [50], [93], [-14], [60], [-19], [1], [62], [-12], [32], [-5], [-88], [1], [54], [-64], [29], [-8], [52], [32], [-39], [-42], [62], [-20], [-83], [72], [-12], [46], [-10], [9], [-20], [3], [-84], [-9], [64], [22], [-74], [-60], [-79], [30], [-36], [92], [66], [-58], [63], [22], [-84], [51], [62], [-47], [98], [70], [-55], [-56], [82], [62], [-32], [-6], [94], [-52], [-54], [7], [-42], [-25], [16], [5], [-47], [-9], [-36], [-8], [73], [37], [-103], [-74], [27], [-54], [-74], [38]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8034_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8034_2_a_f();".
// 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_8034_2_a_f(:prec:=1)
    chi := MakeCharacter_8034_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_8034_2_a_f();".
// 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_8034_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8034_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-4, 1]>,<7,R![5, 1]>],Snew);
    return Vf;
end function;