
// Make newform 8034.2.a.c 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_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_8034_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_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], [1], [3], [2], [-1], [0], [4], [0], [7], [-4], [2], [-6], [2], [13], [2], [-5], [4], [-1], [0], [3], [-4], [-1], [-8], [14], [14], [-1], [18], [2], [19], [-20], [2], [13], [-4], [-18], [2], [-13], [-2], [-18], [22], [-4], [-6], [22], [17], [-1], [9], [10], [-27], [24], [10], [6], [-6], [2], [1], [13], [-10], [9], [30], [-31], [-10], [-6], [-30], [-28], [-35], [-17], [18], [17], [-14], [-28], [28], [-24], [-18], [16], [-4], [1], [21], [-14], [-4], [33], [-34], [10], [37], [-10], [26], [13], [33], [42], [9], [24], [-34], [-14], [-15], [12], [12], [-16], [-37], [18], [21], [9], [7], [-1], [18], [-1], [-33], [21], [-15], [41], [-30], [-12], [-44], [28], [26], [-8], [14], [-40], [-24], [-50], [31], [-12], [10], [-42], [-37], [13], [14], [-11], [9], [-11], [-28], [-7], [22], [10], [15], [-24], [4], [48], [13], [14], [14], [-7], [39], [19], [8], [-5], [20], [23], [14], [15], [-18], [40], [20], [4], [46], [20], [-29], [-27], [38], [-43], [-51], [-58], [18], [-36], [-2], [18], [-51], [-54], [-18], [56], [-23], [22], [-58], [5], [10], [45], [-42], [-16], [0], [16], [10], [12], [-1], [44], [62], [40], [52], [47], [27], [-27], [4], [29], [-46], [-30], [-63], [-4], [-18], [47], [30], [52], [3], [-10], [34], [27], [-16], [-17], [55], [45], [-51], [42], [21], [-10], [-37], [-67], [-60], [-31], [66], [-64], [-10], [40], [-26], [-42], [12], [69], [-5], [-64], [-32], [-19], [-29], [-13], [40], [-8], [-9], [-54], [-31], [59], [-45], [-65], [48], [-14], [30], [-63], [-30], [-31], [-70], [49], [65], [-1], [-16], [-44], [-31], [-36], [76], [38], [1], [-43], [42], [-49], [24], [-30], [70], [-59], [-34], [-26], [-72], [10], [-66], [73], [-62], [-8], [-20], [-4], [-38], [-22], [51], [-10], [-6], [7], [-44], [-52], [-69], [-20], [-80], [1], [-42], [30], [27], [14], [57], [-39], [-72], [-36], [75], [-72], [69], [-27], [-42], [-46], [66], [48], [-54], [-15], [66], [-63], [-3], [-9], [-2], [-32], [-56], [-52], [-56], [12], [-38], [-86], [28], [-3], [-63], [27], [53], [24], [-21], [-22], [-49], [-46], [-26], [-34], [-5], [57], [6], [-30], [24], [-42], [-29], [25], [12], [-80], [2], [-39], [9], [54], [27], [42], [-8], [-22], [54], [-39], [-1], [-74], [64], [-17], [-8], [27], [-47], [-50], [-24], [-55], [52], [35], [-18], [-6], [-56], [-87], [-35], [-7], [-56], [66], [-74], [-40], [90], [-37], [92], [71], [-86], [-34], [-33], [20], [-26], [13], [59], [47], [-54], [-54], [-8], [70], [-93], [-1], [11], [-37], [21], [34], [10], [7], [33], [-34], [-20], [72], [-77], [80], [36], [63], [-31], [40], [-64], [16], [-46], [28], [20], [-48], [95], [18], [-40], [68], [47], [-82], [-37], [-1], [-30], [-13], [-14], [72], [-32], [-9], [57], [-62], [-27], [-45], [-32], [-23], [6], [4], [50], [-1], [24]];
    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_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_8034_2_a_c(: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_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_8034_2_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8034_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-1, 1]>,<7,R![-3, 1]>],Snew);
    return Vf;
end function;