
// Make newform 1045.2.a.b in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1045_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_1045_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_1045_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_1045_a();"
function MakeCharacter_1045_a()
    N := 1045;
    order := 1;
    char_gens := [837, 761, 496];
    v := [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_1045_a_Hecke(Kf)
    return MakeCharacter_1045_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], [0], [1], [0], [-1], [2], [-6], [1], [-8], [-6], [4], [-2], [-10], [4], [0], [-2], [-8], [14], [8], [-4], [2], [-16], [-4], [10], [10], [14], [-4], [-12], [-6], [-6], [-8], [12], [-6], [12], [14], [-16], [-2], [4], [16], [18], [16], [14], [-16], [-2], [-10], [0], [4], [-4], [4], [-10], [-6], [-16], [22], [12], [-30], [-24], [6], [0], [-2], [30], [-20], [-14], [-4], [0], [-22], [30], [-32], [-18], [-4], [-10], [18], [-8], [-32], [-6], [32], [-36], [6], [-18], [-22], [30], [-28], [6], [0], [-22], [-8], [-28], [2], [10], [-10], [-24], [-12], [24], [4], [-28], [36], [-16], [-10], [-30], [-20], [-34], [-36], [22], [36], [-10], [12], [-46], [36], [34], [12], [14], [-32], [46], [-22], [4], [16], [-14], [20], [-16], [-34], [12], [-10], [22], [34], [-40], [20], [-26], [-10], [-24], [-32], [30], [-20], [-8], [4], [-42], [34], [2], [14], [4], [-2], [42], [12], [30], [48], [36], [46], [4], [38], [30], [-44], [4], [2], [-30], [-52], [-40], [-40], [20], [24], [34], [-30], [-14], [-36], [6], [32], [-24], [58], [-4], [-4], [14], [38], [6], [-20], [62], [-8], [26], [44], [58], [12], [-42], [8], [14], [-48], [-20], [-26], [6], [48], [38], [18], [-24], [-26], [64], [-6], [60], [-8], [-18], [4], [10], [-22], [62], [-62], [64], [50], [8], [14], [-2], [-36], [-2], [-40], [4], [66], [16], [-30], [-10], [0], [-16], [24], [-54], [64], [-50], [-36], [30], [-6], [40], [-70], [-8], [44], [-58], [-14], [36], [-40], [-60], [14], [-4], [-16], [30], [-52], [-48], [-14], [-42], [-4], [60], [16], [-20], [-16], [-18], [54], [0], [8], [-48], [44], [-48], [14], [-62], [-48], [10], [-66], [-60], [22], [-12], [22], [-46], [56], [36], [-26], [-38], [-14], [-56], [6], [58], [52], [6], [14], [-32], [-70], [40], [-58], [-56], [68], [34], [-66], [-4], [0], [-32], [-56], [34], [44], [72], [66], [-22], [-40], [-42], [-18], [-4], [78], [12], [10], [-2], [-12], [22], [-52], [-28], [34], [66], [8], [-8], [36], [18], [80], [-2], [64], [-58], [32], [54], [14], [-64], [-72], [78], [-84], [-56], [2], [26], [4], [26], [-30], [-36], [18], [-62], [56], [-8], [80], [-18], [-2], [-2], [56], [-60], [-20], [-44], [30], [-50], [90], [8], [-10], [-38], [-22], [-36], [54], [-92], [-90], [72], [24], [70], [-20], [-38], [78], [-56], [50], [-18], [96], [-20], [-38], [76], [-34], [-78], [4], [-4], [28], [-94], [54], [-96], [-90], [44], [-16], [-56], [34], [0], [-66], [96], [-56], [50], [-30], [62], [-2], [-54], [40], [18], [44], [100], [64], [-42], [-20], [68], [-6], [-66], [12], [36], [36], [-58], [0], [10], [48], [-90], [-14], [-78], [-104], [42], [-30], [-16], [-50], [-42], [16], [92], [-34], [102], [-16], [-20], [2], [-50], [-48], [-28], [18], [64], [-58], [2], [-56], [12], [-22], [22], [-96], [90], [12], [24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1045_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1045_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_1045_2_a_b(:prec:=1)
    chi := MakeCharacter_1045_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_1045_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_1045_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1045_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 1]>],Snew);
    return Vf;
end function;