
// Make newform 1045.2.a.c in Magma, downloaded from the LMFDB on 16 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_c();".
// 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_c();".
// 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
        poly := [-2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
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, 1], [2, 0], [1, 0], [0, -2], [-1, 0], [4, 2], [-2, -2], [-1, 0], [4, 0], [6, -2], [4, -2], [-4, 2], [-6, -2], [-4, -2], [0, 0], [0, -2], [4, 6], [2, -4], [6, -4], [12, -2], [-2, -6], [-12, 0], [0, 2], [-2, -8], [-4, 10], [2, 4], [10, 0], [2, 8], [2, -2], [-4, -6], [-2, 12], [4, 4], [-2, 8], [4, 12], [2, 4], [4, 4], [-6, -8], [12, 4], [-14, 0], [0, 2], [4, 6], [6, -8], [0, 12], [-4, 14], [6, -14], [8, -4], [-24, 0], [2, 0], [10, 8], [-6, 16], [2, 6], [-8, 0], [-14, -2], [-4, 8], [16, -10], [-4, -6], [-18, -4], [-24, 0], [-6, 10], [2, 6], [-8, -6], [-8, 2], [-22, -8], [8, 4], [10, 12], [4, -14], [-4, -14], [-20, 6], [0, -22], [-2, -16], [-10, 0], [8, 8], [24, 8], [-16, 2], [-16, -2], [-2, -8], [6, -16], [2, 8], [18, 0], [-10, 2], [4, -12], [-22, 8], [-4, -16], [12, 18], [20, 0], [32, -4], [18, 4], [2, -18], [22, 12], [4, 16], [24, 0], [-8, -12], [2, -8], [-12, -8], [-12, -16], [-12, 14], [2, 0], [-6, 0], [22, 4], [-6, -24], [-6, -20], [10, 18], [-18, 4], [6, 10], [-36, 4], [-6, -8], [32, 0], [34, 2], [-8, -14], [34, 2], [10, -12], [-14, -10], [6, 12], [36, 8], [0, -4], [14, -24], [-16, -4], [4, 12], [-14, 0], [24, -12], [26, 12], [-8, -18], [-16, 10], [6, -8], [-12, 16], [-6, -24], [-42, 0], [-24, 8], [8, 12], [-2, 6], [12, 0], [22, 4], [-32, -14], [-18, 8], [-2, 36], [-18, 12], [-12, -14], [2, -12], [-8, -2], [-38, -4], [4, -8], [22, 16], [0, 4], [30, 4], [-10, -12], [-16, 6], [6, 26], [16, -18], [20, 8], [30, -12], [24, -6], [-42, 0], [20, 20], [14, 8], [2, -4], [-8, 18], [16, -12], [-46, 0], [-26, 6], [10, -10], [8, 8], [-24, -10], [12, -26], [16, -6], [24, -2], [46, 4], [8, -34], [10, -10], [-14, -2], [-30, 16], [4, -8], [-18, -4], [-40, -8], [10, 2], [36, -6], [14, -32], [-4, -12], [2, 0], [14, -8], [-6, 24], [-20, -4], [12, -36], [-20, 10], [-8, 30], [12, 16], [-2, -32], [-24, -22], [-26, -20], [-34, 10], [-16, -12], [0, -10], [16, 10], [-16, -10], [-14, 4], [-20, 10], [20, 10], [2, 8], [-10, -8], [-22, 26], [-16, -2], [22, 30], [-36, -12], [4, -6], [-22, 22], [-52, 12], [-6, -16], [-8, -12], [2, 0], [-22, 32], [0, 38], [-6, -38], [-22, 24], [12, 4], [-18, 12], [8, 32], [30, -4], [48, -2], [38, 14], [30, 12], [-18, 32], [6, -6], [16, 0], [-10, -8], [-36, -12], [-6, -24], [6, 16], [0, 22], [-4, 30], [62, -8], [-4, -44], [-26, 28], [-36, 4], [-44, -16], [34, 22], [40, 8], [0, 6], [22, 8], [46, 14], [-20, 24], [-32, -30], [38, -4], [-4, 32], [16, -4], [2, -32], [-28, -2], [24, -20], [56, 0], [36, -2], [12, 0], [16, -26], [-10, -42], [34, 24], [28, -20], [-6, -32], [38, -6], [-4, 36], [-42, 24], [14, -40], [24, 14], [-18, -6], [34, 28], [30, 12], [-22, -20], [16, 2], [-22, 4], [20, -38], [42, 4], [-42, -24], [-22, 0], [14, -22], [-50, 0], [30, -16], [22, -28], [-16, 32], [-12, -18], [40, -16], [20, -24], [-18, 22], [2, -38], [-4, -12], [-34, 20], [16, -32], [-52, 2], [-38, 22], [16, 38], [40, 16], [18, -40], [20, 10], [-8, 20], [-18, 30], [22, -32], [12, 20], [36, -18], [-40, 32], [-48, 10], [26, -24], [-44, 22], [30, -12], [28, -16], [-36, -10], [6, 34], [24, -6], [8, -40], [-6, 8], [-20, 0], [60, 10], [-26, -12], [-38, 4], [-48, -16], [-18, 50], [24, 18], [2, 0], [54, -26], [-66, 8], [-12, 6], [-14, 14], [4, -32], [-20, 4], [-30, -12], [10, 36], [-28, 8], [-18, 12], [62, -18], [54, -4], [-58, -6], [70, 8], [44, 26], [-18, -44], [-26, -20], [-70, 14], [34, 28], [8, -50], [8, -4], [52, -6], [-4, 0], [28, -28], [-2, 8], [56, -18], [10, 8], [-12, 50], [48, -18], [6, 12], [74, 6], [-68, 6], [-8, 38], [0, 48], [-2, 0], [-34, 28], [40, 20], [-46, 12], [52, -12], [40, -18], [-6, 16], [62, 4], [30, -10], [-40, -2], [16, 8], [-12, -56], [-18, -10], [-10, 36], [74, -6], [14, 36], [10, 20], [12, -24], [0, -28], [12, 2], [-62, 14], [70, -8], [10, 46], [12, -60], [60, -6], [-16, 38], [18, -2], [-16, 40], [-6, -44], [16, 10], [-24, 28], [26, -2], [18, -40], [-12, -42], [-50, -24], [-14, 24], [16, -14], [10, 10], [-24, -20], [-6, -20], [32, 16], [-70, 4], [52, 34], [-38, 8], [2, 0], [0, -66], [-20, 8], [16, -24], [-16, 6], [48, 38], [-12, 32], [-42, -28], [0, 6], [-42, -4], [42, 38], [50, 0], [2, -36], [56, 6], [74, -2], [0, -12], [10, -16], [-34, 22], [54, -24], [-4, -60], [0, -42], [-26, -30], [10, 44], [44, 8], [-50, 2], [6, 64], [-44, -16], [-34, 16], [38, 4], [-42, 28], [-34, 24], [24, -6], [40, -36], [-4, 24], [-28, -14], [24, 22], [-66, 20], [-34, 24], [68, 8], [0, -48]];
    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_c();".
// 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_c(:prec:=2)
    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_c();".
// 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_c( : 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, -2, 1]>],Snew);
    return Vf;
end function;