
// Make newform 6045.2.a.a in Magma, downloaded from the LMFDB on 24 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_a();".
// 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_a();".
// 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 := [[-2], [-1], [-1], [-5], [2], [-1], [6], [4], [6], [7], [1], [-6], [9], [11], [-8], [-10], [-7], [2], [-3], [0], [2], [6], [11], [-14], [1], [-6], [10], [-15], [2], [-3], [-3], [18], [-22], [16], [9], [13], [-6], [4], [12], [-1], [23], [-8], [6], [-13], [9], [-16], [-11], [-2], [18], [26], [3], [18], [-7], [-7], [18], [-4], [-30], [-13], [5], [3], [-14], [-24], [-20], [-20], [-5], [4], [13], [22], [0], [14], [5], [27], [16], [-26], [-34], [11], [25], [-2], [-36], [13], [-30], [-12], [3], [19], [17], [15], [-30], [20], [12], [-4], [21], [-39], [-4], [-5], [41], [11], [-38], [22], [29], [2], [-6], [30], [-36], [30], [28], [14], [-3], [0], [-48], [20], [2], [-18], [-8], [-1], [-8], [-43], [30], [-12], [47], [36], [26], [19], [24], [38], [30], [-26], [-18], [25], [40], [19], [36], [-15], [31], [49], [6], [26], [-45], [-38], [18], [-21], [40], [-38], [41], [43], [-28], [-21], [-35], [-10], [40], [-25], [-54], [-39], [15], [9], [-44], [-15], [-41], [36], [2], [34], [24], [56], [12], [-42], [-32], [39], [-26], [-20], [-21], [10], [12], [45], [-24], [-54], [13], [-7], [-22], [-57], [-14], [8], [-49], [25], [16], [54], [-16], [-15], [16], [-47], [6], [13], [35], [8], [28], [20], [-3], [-26], [58], [-10], [-42], [32], [-43], [0], [38], [38], [-12], [42], [-38], [-36], [10], [4], [-29], [-27], [-26], [-36], [58], [-56], [62], [-48], [3], [50], [-62], [-14], [-39], [-44], [-44], [-12], [10], [-33], [27], [19], [-46], [-27], [56], [-17], [-9], [-48], [40], [-32], [41], [57], [26], [-22], [-31], [-37], [-11], [-16], [30], [-36], [55], [40], [30], [66], [59], [19], [-31], [-68], [43], [-38], [-6], [-27], [-22], [22], [31], [-6], [54], [32], [21], [33], [68], [-21], [58], [-52], [44], [8], [56], [-39], [-12], [66], [65], [-69], [48], [-6], [-16], [-2], [31], [-17], [-26], [9], [46], [33], [-72], [28], [-42], [-47], [65], [6], [10], [59], [-76], [40], [52], [-17], [-16], [64], [16], [55], [60], [-65], [78], [-24], [56], [-36], [-16], [36], [-66], [10], [12], [-30], [-26], [-78], [40], [50], [-66], [-65], [-86], [-62], [24], [-33], [90], [-12], [-2], [-83], [-16], [57], [24], [10], [49], [38], [-26], [81], [-30], [-54], [-12], [-52], [12], [38], [14], [-14], [48], [-29], [-69], [92], [-3], [56], [-14], [-56], [36], [56], [-30], [90], [57], [38], [70], [-45], [88], [-25], [-58], [18], [45], [-90], [12], [18], [36], [-69], [51], [48], [48], [43], [-34], [31], [-36], [15], [-42], [3], [41], [94], [-40], [-59], [34], [-58], [-45], [-17], [-87], [-92], [8], [74], [-49], [84], [62], [-58], [39], [-97], [-76], [48], [22], [-36], [23], [92], [56], [66], [-34], [54], [-32], [-66], [-16], [-32], [-18], [76], [-2], [27], [50], [46], [-36], [60], [-45], [7], [33], [90], [96], [51]];
    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_a();".
// 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_a(: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_a();".
// 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_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6045_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![2, 1]>,<7,R![5, 1]>,<11,R![-2, 1]>],Snew);
    return Vf;
end function;