
// Make newform 6045.2.a.f in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6045_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_6045_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_6045_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_6045_a();"
function MakeCharacter_6045_a()
    N := 6045;
    order := 1;
    char_gens := [4031, 4837, 1861, 2731];
    v := [1, 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_6045_a_Hecke(Kf)
    return MakeCharacter_6045_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 := [[0], [1], [-1], [-1], [-3], [1], [-3], [2], [3], [0], [1], [-7], [-9], [-10], [12], [-3], [12], [-1], [-4], [-9], [2], [-1], [-18], [15], [17], [18], [-4], [9], [2], [6], [20], [18], [12], [5], [21], [8], [-4], [11], [-12], [-6], [0], [11], [-6], [23], [12], [-16], [-4], [8], [-6], [14], [9], [-15], [-10], [-12], [-18], [0], [0], [-16], [8], [30], [-4], [12], [11], [24], [-10], [24], [8], [-4], [3], [-10], [12], [24], [-10], [14], [2], [-12], [-12], [11], [30], [-22], [-6], [8], [0], [-16], [-1], [9], [-15], [-37], [-3], [5], [9], [-15], [11], [0], [32], [-12], [27], [30], [-34], [-16], [44], [-12], [-3], [6], [-13], [-7], [36], [-36], [48], [-19], [14], [11], [12], [26], [20], [-42], [5], [33], [30], [6], [32], [-28], [9], [-24], [-10], [-24], [-46], [6], [44], [-49], [-52], [48], [23], [38], [-6], [-40], [-24], [-4], [-21], [36], [38], [-27], [-40], [-12], [38], [15], [-19], [9], [5], [-48], [14], [-24], [2], [15], [-46], [-30], [-7], [15], [2], [39], [-6], [-51], [-40], [36], [48], [-18], [-7], [-10], [38], [-15], [48], [32], [36], [17], [-13], [24], [26], [21], [-34], [-46], [35], [42], [-22], [-6], [-36], [54], [2], [53], [50], [21], [38], [-12], [-25], [-51], [-36], [21], [50], [-28], [48], [-15], [-30], [-37], [5], [53], [45], [-9], [-16], [12], [-3], [23], [20], [-30], [26], [-24], [48], [-64], [-4], [0], [18], [48], [5], [68], [18], [-16], [21], [-1], [-51], [72], [-28], [-21], [-58], [11], [14], [-36], [-10], [-24], [-58], [-24], [60], [-66], [18], [-7], [-58], [38], [30], [-36], [-16], [45], [14], [-12], [-58], [-6], [-24], [-19], [-15], [-3], [2], [-55], [-42], [-79], [62], [57], [-4], [-34], [0], [-55], [-15], [-51], [-31], [-6], [-58], [11], [-1], [68], [-58], [29], [6], [80], [2], [12], [-3], [32], [0], [-46], [-4], [24], [74], [-36], [8], [-24], [-66], [-60], [0], [60], [-46], [60], [8], [-39], [12], [17], [86], [-48], [-61], [63], [32], [-25], [-39], [-13], [-12], [-88], [-45], [-30], [12], [68], [-72], [-61], [39], [-63], [-46], [90], [-28], [14], [66], [-61], [24], [-7], [74], [-73], [39], [54], [8], [57], [80], [48], [-52], [18], [56], [90], [50], [-34], [-1], [-9], [-21], [44], [-84], [-24], [-43], [89], [60], [-15], [86], [-43], [87], [26], [59], [-6], [-15], [24], [42], [-72], [-82], [54], [-48], [60], [20], [44], [-18], [-79], [-7], [30], [20], [-66], [-66], [95], [-52], [27], [18], [-79], [-54], [-10], [-102], [-36], [-88], [54], [-40], [-18], [44], [-22], [20], [-57], [8], [-36], [63], [8], [63], [-4], [56], [66], [5], [54], [-4], [9], [-67], [30], [-51], [-61], [98], [-78], [95], [63], [8], [45], [-81], [-19], [-94], [60], [3], [62], [90], [-21], [-30], [65], [96], [54], [74], [12], [-99], [-15], [56], [-66]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6045_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6045_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_6045_2_a_f(:prec:=1)
    chi := MakeCharacter_6045_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_6045_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_6045_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6045_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![0, 1]>,<7,R![1, 1]>,<11,R![3, 1]>],Snew);
    return Vf;
end function;