
// Make newform 6034.2.a.d in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6034_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_6034_2_a_d();".
// 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_6034_2_a_d();".
// 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_6034_a();"
function MakeCharacter_6034_a()
    N := 6034;
    order := 1;
    char_gens := [1725, 869];
    v := [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_6034_a_Hecke(Kf)
    return MakeCharacter_6034_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], [0], [-4], [1], [4], [-2], [4], [4], [-8], [-6], [-4], [6], [-6], [4], [0], [-2], [8], [-8], [-12], [8], [16], [16], [14], [0], [-2], [10], [8], [-8], [2], [-6], [8], [6], [-18], [-8], [-18], [4], [-4], [12], [0], [0], [12], [-18], [-16], [10], [-18], [-20], [-8], [6], [0], [-20], [-18], [0], [-20], [-6], [-12], [-28], [-2], [-20], [-18], [-18], [4], [14], [-8], [8], [16], [-6], [28], [-6], [-4], [14], [6], [-8], [-4], [-10], [20], [12], [-30], [16], [-18], [-4], [-18], [-22], [1], [-22], [26], [-20], [30], [-30], [-12], [20], [8], [38], [8], [12], [-20], [-30], [-18], [14], [28], [22], [28], [-18], [36], [10], [-16], [-32], [10], [-6], [40], [-4], [-22], [34], [-2], [-18], [-12], [-26], [40], [18], [-2], [-4], [40], [26], [-12], [-44], [34], [-22], [-46], [-18], [-20], [44], [-12], [-8], [-52], [26], [-38], [-10], [-52], [14], [-36], [-46], [-40], [-10], [-16], [-28], [-14], [40], [34], [-24], [-26], [12], [-22], [38], [32], [46], [4], [-40], [-56], [36], [46], [-18], [4], [-22], [-40], [-44], [-34], [-18], [-32], [28], [-34], [4], [-20], [-16], [12], [26], [38], [16], [20], [38], [-56], [-12], [-28], [-4], [10], [-28], [-16], [54], [-2], [-56], [-58], [18], [14], [12], [32], [-8], [40], [12], [54], [46], [-36], [-18], [30], [10], [36], [-56], [-60], [58], [2], [0], [50], [10], [-34], [-32], [-64], [-30], [-12], [-52], [-24], [60], [32], [2], [58], [-10], [-66], [40], [-18], [10], [36], [-16], [16], [0], [-30], [-6], [0], [-54], [8], [42], [14], [14], [24], [-50], [52], [-32], [30], [38], [44], [-36], [52], [-12], [-40], [76], [-18], [48], [72], [44], [-32], [-12], [2], [74], [38], [-50], [32], [68], [-14], [-56], [-76], [62], [82], [-44], [-40], [-70], [54], [20], [6], [-56], [-70], [50], [0], [18], [-46], [-60], [-78], [64], [42], [-10], [58], [52], [82], [-42], [40], [-2], [-66], [64], [-58], [-4], [-26], [36], [-30], [-6], [54], [-88], [32], [50], [24], [32], [-40], [-18], [32], [76], [-16], [-82], [-48], [50], [78], [68], [24], [-52], [64], [24], [-54], [30], [-18], [-50], [-12], [56], [-46], [82], [44], [36], [-64], [10], [-10], [2], [-60], [-64], [-76], [26], [-62], [-42], [-76], [-14], [30], [-66], [-18], [24], [18], [-44], [-22], [-60], [-70], [78], [52], [-62], [66], [22], [-10], [-14], [-34], [16], [34], [-32], [-62], [-72], [32], [-60], [-70], [-58], [-58], [-64], [6], [20], [0], [-12], [-58], [62], [74], [4], [-36], [-80], [18], [-76], [-18], [26], [24], [42], [-70], [8], [-72], [-28], [-12], [0], [-82], [62], [-4], [4], [76], [-18], [-20], [0], [76], [-22], [-76], [34], [-48], [-64], [18], [22], [14], [66], [-32], [-14], [44], [74], [12], [100], [62], [40], [32], [-70], [44], [36], [-94], [-22], [72], [72], [-16], [66], [-12], [30], [-90], [44]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6034_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6034_2_a_d();".
// 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_6034_2_a_d(:prec:=1)
    chi := MakeCharacter_6034_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_6034_2_a_d();".
// 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_6034_2_a_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6034_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![0, 1]>,<5,R![4, 1]>,<11,R![-4, 1]>],Snew);
    return Vf;
end function;