
// Make newform 1560.2.a.k in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1560_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_1560_2_a_k();".
// 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_1560_2_a_k();".
// 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_1560_a();"
function MakeCharacter_1560_a()
    N := 1560;
    order := 1;
    char_gens := [391, 781, 521, 937, 1081];
    v := [1, 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_1560_a_Hecke(Kf)
    return MakeCharacter_1560_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 := [[0], [1], [1], [-2], [0], [1], [-4], [6], [6], [4], [8], [-6], [6], [4], [8], [2], [0], [-2], [-4], [-8], [0], [16], [-4], [-6], [16], [0], [-8], [-12], [8], [-16], [8], [-14], [-18], [8], [-6], [8], [-14], [24], [12], [-6], [-2], [26], [-16], [4], [2], [0], [4], [10], [12], [-8], [-4], [0], [-22], [-26], [8], [-14], [24], [-8], [-22], [18], [-4], [-22], [-20], [-8], [-34], [-18], [-10], [-10], [16], [32], [-2], [32], [-16], [-22], [26], [-20], [0], [2], [-18], [6], [-30], [20], [-24], [-26], [8], [-24], [-30], [8], [6], [10], [-12], [-32], [-34], [18], [10], [34], [22], [30], [4], [12], [-12], [30], [16], [14], [-16], [-20], [-36], [-38], [-24], [-26], [0], [2], [6], [18], [-20], [26], [28], [-6], [14], [26], [20], [-14], [-10], [-44], [30], [0], [16], [-12], [28], [42], [50], [-12], [0], [14], [-10], [26], [-6], [-8], [-6], [-38], [30], [18], [8], [12], [46], [16], [34], [56], [-52], [52], [38], [30], [20], [46], [-20], [-20], [56], [-6], [-14], [18], [44], [-60], [-50], [6], [-6], [4], [-16], [-62], [-38], [26], [-36], [-56], [12], [12], [-56], [-34], [-10], [18], [44], [-46], [-50], [-30], [14], [26], [12], [-60], [-2], [28], [-10], [-24], [50], [4], [20], [50], [0], [-24], [-18], [-30], [58], [-22], [-66], [-48], [-66], [62], [-8], [-66], [-32], [56], [-26], [68], [30], [-28], [-44], [-12], [0], [-70], [-52], [30], [-12], [46], [26], [36], [22], [6], [24], [-6], [28], [60], [-44], [-24], [-14], [-68], [24], [-74], [52], [-4], [-18], [46], [30], [52], [36], [-12], [28], [20], [26], [-64], [-62], [-68], [-58], [26], [10], [-30], [-16], [-38], [2], [36], [-10], [-32], [-62], [-8], [-12], [76], [48], [14], [-62], [32], [-42], [58], [-52], [62], [-62], [56], [-4], [-24], [-22], [-26], [-20], [20], [30], [-42], [34], [4], [-54], [80], [12], [52], [46], [-70], [-36], [-38], [-52], [-36], [58], [-36], [-42], [-16], [48], [-42], [42], [32], [-18], [-38], [-16], [-24], [20], [-4], [36], [2], [40], [-26], [6], [-2], [-2], [-44], [40], [-22], [-40], [-8], [72], [-6], [32], [-32], [48], [10], [26], [14], [-38], [-4], [18], [42], [60], [30], [24], [-12], [-26], [52], [68], [-34], [30], [8], [30], [48], [2], [-72], [18], [66], [-26], [36], [24], [-42], [-30], [52], [-78], [44], [-78], [-28], [-48], [-72], [44], [32], [-18], [-38], [82], [-68], [20], [38], [-62], [58], [10], [50], [16], [-44], [60], [-32], [54], [60], [-96], [60], [46], [78], [10], [102], [42], [-54], [18], [-56], [-44], [22], [36], [-54], [-86], [38], [16], [68], [72], [70], [-40], [34], [12], [6], [-4], [0], [-78], [-70], [-18], [16], [-86], [50], [-64], [68], [-66], [98], [0], [24], [90], [-60], [48], [-8], [-86], [26], [-60], [-82], [-88], [-50], [64], [-2], [-68], [-54], [18], [64]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1560_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1560_2_a_k();".
// 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_1560_2_a_k(:prec:=1)
    chi := MakeCharacter_1560_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_1560_2_a_k();".
// 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_1560_2_a_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1560_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![2, 1]>,<11,R![0, 1]>,<17,R![4, 1]>],Snew);
    return Vf;
end function;