
// Make newform 8034.2.a.h 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_h();".
// 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_h();".
// 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], [0], [-1], [3], [-1], [-7], [0], [-6], [-8], [0], [8], [-4], [-8], [-4], [-1], [-6], [2], [-1], [10], [9], [-10], [2], [-6], [8], [-13], [-1], [9], [14], [4], [-19], [-12], [-14], [17], [11], [-4], [7], [-14], [15], [11], [11], [-13], [-15], [26], [-2], [-7], [0], [5], [7], [18], [18], [9], [-7], [4], [20], [-23], [24], [-6], [2], [-26], [16], [8], [-13], [-20], [-22], [-15], [-17], [-25], [-28], [-22], [-9], [3], [-26], [-30], [19], [-34], [-33], [18], [-18], [0], [29], [23], [24], [-16], [-5], [24], [-25], [38], [30], [30], [-11], [-6], [-8], [-25], [20], [-28], [-15], [0], [-3], [-19], [24], [20], [-14], [6], [16], [-39], [18], [-45], [11], [-24], [14], [5], [-34], [16], [19], [-11], [4], [-12], [50], [-15], [30], [23], [24], [39], [12], [-44], [18], [15], [-23], [34], [34], [-22], [16], [-44], [-54], [29], [49], [20], [-24], [-16], [-20], [-11], [48], [-20], [38], [-12], [51], [33], [-42], [-4], [-32], [0], [-1], [42], [5], [-36], [-33], [-56], [0], [35], [48], [-19], [-32], [48], [-38], [47], [46], [-2], [-44], [1], [-54], [-50], [-24], [-11], [-38], [-3], [28], [0], [44], [-9], [-8], [33], [52], [6], [-54], [14], [39], [-30], [-41], [39], [46], [-46], [-12], [18], [-54], [-15], [-60], [41], [-30], [-21], [-44], [12], [41], [-23], [0], [-54], [-22], [-30], [-54], [60], [-43], [26], [40], [-16], [45], [42], [-42], [24], [-8], [54], [-49], [40], [13], [-20], [6], [-65], [62], [-16], [-25], [22], [70], [-51], [59], [-10], [25], [12], [5], [72], [-50], [-19], [36], [-28], [-59], [14], [44], [22], [22], [-38], [14], [54], [31], [-64], [72], [6], [6], [-27], [-72], [20], [-38], [-10], [46], [-45], [26], [41], [-68], [70], [11], [21], [4], [-39], [50], [49], [14], [50], [77], [11], [-58], [-41], [-60], [36], [36], [46], [-17], [-62], [44], [14], [47], [41], [-45], [24], [-10], [-62], [-12], [-13], [8], [-14], [10], [-14], [-66], [-88], [-11], [28], [-19], [-36], [36], [-8], [42], [32], [27], [26], [56], [-15], [-40], [40], [-8], [-61], [41], [-30], [7], [-12], [-46], [-17], [-60], [-15], [-8], [2], [84], [-51], [-67], [-84], [31], [-10], [48], [12], [-36], [14], [66], [54], [-14], [-83], [2], [48], [-58], [45], [-72], [-74], [22], [-22], [-36], [-60], [4], [-58], [0], [10], [-70], [-40], [0], [-39], [-18], [58], [-10], [-29], [72], [93], [72], [-31], [12], [-6], [-34], [-17], [-92], [-80], [6], [0], [2], [10], [-78], [-56], [-3], [82], [21], [16], [-47], [-74], [28], [2], [-16], [10], [35], [6], [1], [41], [-37], [63], [-48], [24], [8], [-4], [-58], [-6], [94], [-26], [-14], [-63], [-81], [-72], [-37], [5], [32], [-68], [-77], [-54], [-16], [0], [-51], [-69], [105], [-63], [-40], [-4], [78], [-23], [48], [107], [66], [-24], [-65], [47], [-44], [-31]];
    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_h();".
// 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_h(: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_h();".
// 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_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8034_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![0, 1]>,<7,R![1, 1]>],Snew);
    return Vf;
end function;