
// Make newform 6013.2.a.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6013_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_6013_2_a_b();".
// 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_6013_2_a_b();".
// 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_6013_a();"
function MakeCharacter_6013_a()
    N := 6013;
    order := 1;
    char_gens := [5155, 3438];
    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_6013_a_Hecke(Kf)
    return MakeCharacter_6013_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 := [[2], [-1], [0], [-1], [0], [6], [0], [4], [-2], [8], [-10], [0], [2], [1], [12], [3], [-8], [10], [-2], [8], [-9], [15], [15], [-7], [1], [-15], [16], [-6], [10], [18], [16], [12], [-5], [5], [6], [16], [13], [-8], [-8], [-8], [21], [6], [3], [12], [-12], [27], [7], [-12], [-8], [12], [-18], [-12], [-21], [-5], [-6], [4], [-10], [-28], [-23], [9], [11], [6], [-28], [21], [-1], [17], [22], [22], [-12], [-34], [-24], [18], [30], [-18], [-28], [24], [19], [6], [-22], [10], [1], [34], [24], [-37], [-38], [-35], [15], [33], [3], [-19], [-32], [14], [-13], [-33], [-4], [0], [44], [-26], [6], [14], [-22], [37], [20], [-10], [31], [-19], [-31], [11], [-24], [18], [24], [12], [33], [10], [0], [-36], [-12], [41], [-21], [21], [-33], [4], [-18], [21], [-27], [29], [-32], [39], [8], [42], [10], [-8], [-25], [-18], [18], [12], [-24], [18], [-28], [-22], [-40], [20], [38], [21], [-22], [-51], [-29], [-16], [1], [39], [20], [48], [53], [24], [40], [-5], [-22], [-19], [-28], [-13], [3], [-6], [-62], [-60], [-2], [36], [43], [-1], [11], [-48], [12], [51], [9], [-26], [-44], [18], [-34], [9], [16], [-4], [19], [42], [-16], [-15], [-3], [-18], [-30], [7], [-14], [53], [32], [-6], [-13], [38], [-58], [46], [39], [-49], [22], [50], [-6], [35], [10], [-26], [-18], [2], [-55], [-36], [-36], [-55], [-36], [-51], [-65], [-72], [19], [-43], [-23], [8], [63], [-42], [-41], [56], [18], [16], [33], [-54], [-61], [14], [55], [68], [-20], [-10], [-49], [-25], [-39], [39], [-55], [18], [44], [14], [31], [50], [41], [14], [69], [-40], [-49], [-18], [52], [48], [28], [71], [-2], [-56], [-8], [1], [74], [-5], [29], [-34], [-11], [36], [42], [-37], [38], [38], [10], [-15], [27], [-44], [-7], [-48], [-46], [5], [18], [-52], [20], [27], [48], [79], [-23], [58], [51], [30], [-22], [9], [-13], [18], [31], [-31], [42], [-81], [10], [57], [-9], [-52], [-56], [29], [-71], [-40], [50], [-43], [75], [11], [44], [70], [-26], [-17], [-24], [52], [4], [-26], [-54], [-4], [72], [56], [36], [72], [58], [-86], [-42], [2], [66], [-45], [38], [-28], [-14], [-11], [50], [40], [-31], [-18], [-20], [20], [5], [24], [55], [-70], [-66], [17], [58], [77], [12], [80], [90], [-1], [2], [90], [64], [-12], [10], [-6], [26], [-47], [-40], [25], [-29], [45], [71], [-78], [12], [72], [-63], [-36], [-59], [-85], [67], [34], [-3], [-48], [69], [-41], [-20], [32], [-58], [-9], [-22], [-16], [57], [47], [-79], [-88], [50], [43], [-75], [1], [-44], [-62], [-34], [72], [29], [-39], [24], [-43], [-15], [-16], [-91], [-88], [92], [-2], [-78], [-102], [-92], [-23], [36], [-48], [58], [6], [14], [-20], [-22], [67], [-36], [-30], [8], [40], [80], [-27], [-28], [21], [26], [-50], [50], [-72], [-74], [37], [-55], [52], [10], [-48]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6013_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6013_2_a_b();".
// 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_6013_2_a_b(:prec:=1)
    chi := MakeCharacter_6013_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_6013_2_a_b();".
// 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_6013_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6013_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-2, 1]>],Snew);
    return Vf;
end function;