
// Make newform 6045.2.a.b in Magma, downloaded from the LMFDB on 25 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_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_6045_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_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], [2], [-5], [-1], [4], [-2], [-2], [0], [1], [1], [10], [-10], [4], [12], [-10], [14], [-12], [-4], [-7], [-8], [-4], [15], [10], [-3], [-5], [3], [-6], [-3], [-20], [-21], [-18], [12], [4], [13], [-19], [16], [-3], [3], [8], [-12], [-3], [-16], [21], [14], [3], [-13], [28], [-13], [5], [8], [10], [22], [-3], [-10], [0], [13], [2], [-28], [-5], [-18], [-32], [31], [-6], [-6], [0], [-10], [-2], [-10], [3], [18], [28], [-31], [-34], [-27], [-36], [14], [-5], [7], [0], [-22], [-10], [-4], [8], [-31], [30], [-5], [13], [-31], [40], [24], [8], [-30], [4], [-24], [-34], [3], [-10], [-10], [-28], [-35], [-31], [10], [10], [26], [37], [18], [40], [28], [-23], [-18], [-38], [11], [1], [24], [23], [-30], [-15], [-44], [2], [28], [-16], [6], [-10], [-19], [13], [-16], [-16], [14], [-47], [19], [-27], [38], [-42], [18], [-11], [-5], [30], [-22], [12], [-42], [-54], [24], [-44], [-4], [-14], [-39], [26], [32], [-2], [-12], [-56], [0], [-37], [-56], [31], [3], [37], [45], [45], [-16], [-21], [-27], [-54], [36], [8], [23], [3], [8], [33], [-3], [15], [-34], [3], [-16], [-50], [-60], [-33], [-28], [40], [54], [-13], [-5], [-30], [6], [-47], [6], [50], [-60], [4], [30], [-18], [-3], [68], [-24], [-45], [17], [-34], [9], [44], [-28], [32], [11], [28], [12], [28], [-64], [36], [47], [-20], [-60], [32], [-6], [-25], [44], [37], [25], [51], [48], [-14], [40], [4], [26], [-72], [-56], [41], [68], [-32], [-56], [-26], [35], [-48], [-20], [14], [35], [-32], [-4], [-62], [12], [-58], [22], [-22], [46], [47], [64], [-5], [60], [28], [-45], [-54], [-64], [16], [61], [-54], [20], [-10], [21], [-66], [34], [47], [-76], [31], [-10], [-14], [36], [50], [30], [20], [-18], [-45], [63], [-34], [10], [35], [42], [18], [25], [27], [-36], [39], [58], [-56], [-32], [-20], [16], [-26], [35], [-14], [-62], [-34], [-51], [33], [22], [-44], [-84], [2], [-74], [60], [-12], [-31], [2], [-74], [-28], [-24], [40], [-72], [-73], [15], [-34], [60], [-14], [-51], [38], [30], [2], [65], [59], [-42], [20], [38], [-13], [-6], [-74], [-40], [60], [-60], [-48], [-90], [-30], [-20], [-2], [-9], [-33], [2], [-58], [-55], [-6], [68], [60], [61], [-36], [26], [18], [-12], [-20], [22], [-7], [36], [-69], [41], [17], [10], [26], [73], [64], [-59], [41], [44], [-54], [46], [-18], [18], [52], [87], [52], [-48], [-11], [71], [-63], [-25], [66], [19], [25], [-20], [24], [-33], [46], [41], [-10], [62], [-57], [-8], [33], [17], [-66], [-58], [74], [53], [-82], [58], [-18], [-42], [-8], [-44], [31], [-4], [-17], [34], [-11], [71], [35], [46], [14], [45], [17], [0], [-15], [-65], [-60], [34], [-42], [10], [-6], [-103], [57], [77], [-1], [-28], [24], [-24], [85], [-13], [91], [82], [47], [79], [33], [-47], [30]];
    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_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_6045_2_a_b(: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_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_6045_2_a_b( : 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![-2, 1]>,<11,R![5, 1]>],Snew);
    return Vf;
end function;