
// Make newform 1045.2.a.a in Magma, downloaded from the LMFDB on 24 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_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_1045_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_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], [-2], [-1], [-2], [-1], [6], [0], [-1], [0], [6], [4], [0], [-6], [-6], [-4], [12], [-8], [-14], [6], [-8], [16], [-8], [6], [-2], [4], [-18], [-10], [-20], [-2], [4], [4], [4], [2], [12], [6], [-8], [-10], [4], [-8], [6], [-12], [14], [-8], [-6], [-24], [16], [28], [-22], [-8], [-6], [-24], [-24], [14], [-4], [20], [10], [-10], [20], [8], [6], [6], [-14], [-12], [0], [26], [-20], [-28], [10], [-26], [-30], [-18], [12], [-28], [14], [4], [-30], [-26], [-18], [-22], [-22], [-20], [-14], [-8], [-4], [-16], [8], [-14], [32], [30], [8], [-36], [28], [2], [-16], [4], [-10], [-6], [-6], [-28], [22], [-8], [28], [-36], [46], [12], [14], [-36], [-4], [40], [-46], [40], [-36], [26], [4], [24], [-26], [-40], [-12], [34], [-44], [18], [14], [-6], [-18], [28], [18], [-38], [0], [-32], [24], [24], [40], [4], [2], [-14], [-10], [24], [-20], [48], [46], [-28], [2], [24], [48], [-50], [-36], [-24], [6], [-4], [10], [38], [14], [-12], [12], [50], [32], [36], [18], [-12], [-6], [12], [10], [14], [40], [32], [-38], [32], [12], [30], [18], [-36], [14], [-28], [16], [-28], [6], [48], [-54], [36], [-50], [-36], [-12], [52], [2], [48], [-54], [30], [-50], [-50], [-12], [48], [-34], [-4], [18], [-2], [-56], [-58], [14], [0], [54], [2], [-32], [0], [2], [-12], [-66], [-40], [-24], [-18], [-8], [-56], [6], [68], [6], [-16], [-42], [22], [34], [42], [6], [-10], [-16], [-10], [36], [4], [46], [-24], [16], [20], [20], [-38], [-12], [48], [-10], [-12], [6], [34], [-60], [-4], [-48], [74], [68], [16], [-62], [-10], [-40], [-12], [48], [20], [14], [-4], [26], [60], [-62], [-72], [-48], [-62], [-20], [8], [44], [-44], [-60], [-14], [-54], [-54], [-8], [66], [-10], [-60], [-44], [70], [-34], [-14], [56], [22], [-4], [40], [-46], [34], [16], [-16], [32], [26], [-58], [-14], [16], [-34], [-50], [-64], [30], [-18], [12], [-18], [68], [-78], [30], [-32], [6], [44], [2], [28], [-18], [40], [-6], [36], [48], [-18], [2], [24], [16], [66], [42], [18], [62], [-2], [70], [-12], [-24], [-58], [-22], [44], [-30], [18], [6], [44], [-50], [52], [58], [-88], [84], [26], [-32], [52], [-66], [-32], [36], [-50], [66], [-58], [-38], [72], [6], [66], [-44], [-76], [92], [-74], [50], [0], [50], [-4], [-20], [-34], [28], [14], [-74], [24], [-60], [84], [18], [-80], [50], [-54], [84], [-64], [36], [-12], [48], [26], [-68], [-84], [-74], [-10], [20], [54], [32], [88], [32], [-30], [62], [46], [-38], [66], [36], [-52], [-42], [-72], [62], [-34], [78], [90], [28], [-28], [76], [60], [-54], [-56], [-26], [12], [58], [10], [-26], [68], [-36], [-26], [40], [86], [-66], [22], [52], [10], [-52], [-66], [-28], [-72], [74], [-80], [34], [-58], [76], [-66], [-38], [-72], [28], [-24], [-28], [10], [50], [-36], [36]];
    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_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_1045_2_a_a(: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_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_1045_2_a_a( : 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;