
// Make newform 6034.2.a.f in Magma, downloaded from the LMFDB on 25 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_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_6034_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_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], [3], [2], [-1], [0], [1], [-2], [-7], [6], [5], [2], [8], [0], [-9], [6], [10], [4], [8], [1], [-15], [9], [1], [8], [-10], [16], [-15], [-4], [-12], [-5], [10], [12], [-10], [-12], [4], [-15], [-10], [-4], [-18], [-24], [0], [-12], [1], [3], [4], [-9], [-10], [-29], [12], [7], [14], [-18], [17], [1], [-20], [-17], [6], [-11], [-20], [2], [18], [4], [23], [7], [12], [6], [4], [28], [19], [28], [-10], [8], [-21], [-18], [12], [-30], [-8], [-7], [34], [-2], [-35], [20], [-10], [-1], [6], [8], [-38], [27], [2], [-24], [-26], [20], [-25], [11], [-36], [16], [9], [1], [12], [-1], [21], [22], [26], [21], [15], [3], [21], [44], [36], [35], [-33], [1], [9], [-28], [20], [-24], [-30], [19], [33], [18], [-4], [8], [-23], [40], [0], [-8], [-18], [25], [43], [-14], [-12], [1], [41], [46], [47], [-30], [26], [-34], [-38], [-12], [-16], [-11], [-18], [-26], [-50], [34], [50], [58], [3], [40], [-44], [-7], [36], [33], [-56], [12], [22], [-14], [5], [56], [-5], [57], [19], [23], [23], [21], [3], [-7], [-28], [27], [14], [38], [-50], [-6], [41], [-51], [33], [31], [-34], [-20], [4], [36], [-19], [-44], [-33], [-16], [-31], [-46], [13], [24], [49], [-32], [-12], [17], [6], [28], [-1], [10], [20], [-25], [25], [58], [27], [60], [-19], [56], [-14], [8], [-29], [66], [-52], [9], [-38], [-22], [-34], [-14], [37], [40], [-63], [-57], [-31], [7], [-48], [54], [-25], [8], [-58], [63], [-48], [-6], [41], [-67], [6], [55], [-12], [7], [-11], [4], [-29], [-45], [-15], [-40], [-33], [11], [-2], [15], [38], [6], [60], [-9], [36], [-62], [18], [18], [58], [-16], [-38], [54], [16], [6], [22], [-31], [-71], [-31], [-2], [45], [42], [12], [-49], [-21], [-13], [-10], [-50], [-62], [-24], [-34], [-5], [-76], [-26], [30], [-24], [57], [-53], [32], [23], [-8], [-56], [-74], [34], [-22], [-26], [-9], [77], [34], [25], [77], [-54], [-56], [3], [-34], [61], [-33], [63], [0], [-29], [69], [8], [-20], [-6], [-88], [-34], [46], [47], [-30], [-32], [44], [-37], [17], [-88], [70], [-50], [-44], [-72], [-66], [-43], [54], [16], [-4], [71], [75], [23], [-3], [53], [-40], [49], [-58], [80], [-74], [0], [-21], [-65], [5], [82], [30], [17], [-30], [-8], [-78], [-72], [44], [51], [-55], [68], [72], [-1], [-14], [68], [-29], [-55], [8], [-60], [-58], [90], [-13], [18], [30], [-11], [-23], [-8], [6], [-52], [20], [54], [-75], [57], [72], [9], [52], [-70], [-54], [-87], [-62], [44], [-88], [-14], [26], [-22], [41], [46], [-34], [-54], [-36], [31], [-48], [-19], [70], [-30], [-36], [15], [-62], [52], [76], [-72], [32], [-9], [38], [95], [4], [16], [-48], [54], [62], [92], [-77], [72], [70], [8], [72], [8], [46], [-78], [-54], [-27], [-69], [77], [63], [-23], [9], [30], [-18], [-30]];
    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_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_6034_2_a_f(: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_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_6034_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6034_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-3, 1]>,<5,R![-2, 1]>,<11,R![0, 1]>],Snew);
    return Vf;
end function;