
// Make newform 1920.2.a.l in Magma, downloaded from the LMFDB on 26 December 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1920_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_1920_2_a_l();".
// 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_1920_2_a_l();".
// 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_1920_a();"
function MakeCharacter_1920_a()
    N := 1920;
    order := 1;
    char_gens := [511, 901, 641, 1537];
    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_1920_a_Hecke(Kf)
    return MakeCharacter_1920_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]];
    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], [4], [2], [0], [0], [0], [8], [2], [2], [-8], [-2], [-4], [0], [-6], [14], [-14], [4], [-8], [-10], [-6], [12], [14], [18], [-10], [16], [0], [14], [16], [16], [-6], [12], [-16], [2], [-10], [-4], [20], [-16], [22], [6], [-2], [-12], [-26], [-6], [18], [0], [-16], [-16], [-14], [-20], [-4], [14], [22], [16], [-16], [-14], [-22], [-28], [-2], [20], [26], [12], [-12], [-2], [26], [-28], [-26], [0], [26], [4], [-4], [-20], [-4], [-20], [32], [6], [8], [18], [14], [-10], [2], [-12], [-26], [-6], [24], [-38], [26], [-6], [-4], [28], [-16], [0], [-30], [12], [0], [-10], [2], [4], [-26], [20], [-2], [24], [18], [-20], [-38], [-8], [-24], [-44], [10], [-28], [16], [12], [-16], [10], [6], [4], [0], [-34], [2], [2], [-42], [-42], [-36], [40], [-42], [18], [-44], [0], [-44], [20], [0], [34], [28], [-14], [-6], [-6], [-20], [-18], [30], [-28], [10], [40], [20], [-2], [4], [-8], [48], [4], [0], [-4], [30], [36], [32], [12], [8], [2], [-30], [2], [-18], [12], [-20], [-32], [-26], [-48], [16], [-2], [-12], [34], [-2], [14], [-50], [0], [-2], [50], [-58], [40], [-30], [-36], [22], [8], [30], [-36], [32], [56], [-58], [8], [44], [30], [60], [6], [-60], [56], [-30], [-32], [64], [-18], [48], [8], [-64], [30], [-10], [-48], [-2], [-6], [-30], [54], [24], [-42], [40], [22], [42], [16], [20], [48], [34], [28], [-14], [24], [-42], [-22], [-6], [46], [-44], [-20], [6], [32], [12], [-28], [54], [-8], [-68], [26], [-58], [28], [24], [6], [38], [-22], [64], [0], [-4], [-56], [34], [4], [-48], [-12], [-50], [-68], [-64], [12], [18], [-48], [-10], [-70], [66], [-58], [76], [-38], [62], [-72], [-76], [26], [16], [-28], [68], [14], [-70], [-68], [-66], [62], [4], [-34], [-22], [46], [8], [-48], [82], [-74], [-54], [48], [10], [-8], [62], [20], [-36], [-6], [70], [-34], [66], [14], [48], [76], [-42], [-24], [42], [46], [-62], [-38], [-28], [-66], [-10], [-10], [-8], [-40], [-62], [-36], [2], [-72], [-84], [64], [-54], [-22], [-84], [72], [50], [54], [24], [14], [-22], [68], [30], [70], [-80], [-16], [50], [36], [-12], [-48], [78], [-70], [30], [62], [48], [24], [4], [-2], [12], [-94], [48], [12], [16], [-26], [-94], [-14], [18], [-78], [44], [0], [18], [28], [-6], [70], [0], [-14], [-12], [-8], [46], [60], [-24], [44], [-30], [-96], [18], [84], [58], [42], [-36], [6], [82], [68], [48], [26], [78], [-60], [10], [-16], [66], [-14], [-22], [-38], [60], [-20], [52], [-76], [-24], [42], [-76], [84], [-8], [82], [78], [14], [-4], [24], [10], [34], [6], [-40], [22], [-34], [76], [64], [36], [-22], [58], [36], [-62], [-52], [-50], [10], [102], [-28], [-16], [-34], [-14], [-24], [-12], [60], [24], [-86], [76], [80], [-18], [-2], [-38], [52], [18], [-24], [60]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1920_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1920_2_a_l();".
// 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_1920_2_a_l(:prec:=1)
    chi := MakeCharacter_1920_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_1920_2_a_l();".
// 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_1920_2_a_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1920_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-4, 1]>,<11,R![-2, 1]>,<13,R![0, 1]>],Snew);
    return Vf;
end function;