
// Make newform 1920.2.a.n 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_n();".
// 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_n();".
// 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], [-2], [2], [-2], [-2], [-2], [-2], [-6], [-4], [2], [-10], [8], [-2], [6], [-2], [-10], [-8], [8], [-6], [-16], [12], [-10], [-6], [-6], [-6], [12], [-10], [-2], [2], [14], [-18], [2], [18], [16], [-14], [12], [2], [18], [10], [-18], [12], [26], [14], [-4], [-22], [-18], [0], [6], [6], [16], [18], [-22], [-18], [-26], [-6], [12], [18], [-14], [0], [26], [-32], [8], [26], [-2], [14], [18], [-28], [22], [6], [36], [10], [-6], [-30], [10], [2], [-18], [-18], [6], [18], [-10], [20], [34], [4], [0], [26], [10], [18], [-26], [0], [8], [22], [-22], [-34], [6], [-6], [-30], [40], [38], [8], [-14], [-20], [6], [-34], [-46], [4], [-26], [-28], [-22], [-22], [6], [38], [10], [16], [18], [12], [2], [38], [2], [-42], [-22], [14], [48], [10], [42], [6], [-16], [22], [-22], [-34], [-42], [4], [-14], [-30], [-26], [-42], [-8], [-34], [42], [38], [-46], [34], [-12], [-10], [-4], [22], [14], [-14], [-54], [-46], [-2], [12], [-6], [20], [-36], [-52], [-6], [-14], [-22], [24], [-18], [-42], [-30], [-18], [-50], [-20], [-26], [-26], [6], [-26], [-2], [-32], [2], [-56], [6], [30], [-6], [26], [46], [-6], [38], [-58], [38], [34], [-22], [2], [60], [6], [-60], [-38], [-24], [10], [2], [-24], [-10], [62], [-34], [6], [-66], [-22], [28], [38], [-62], [30], [-30], [48], [-20], [22], [30], [18], [18], [58], [-68], [44], [-66], [2], [30], [42], [-42], [-50], [-68], [-18], [22], [32], [38], [14], [0], [54], [42], [14], [-2], [68], [18], [-16], [-66], [50], [6], [-18], [32], [-52], [22], [-46], [46], [6], [-4], [-6], [70], [-78], [42], [30], [50], [-62], [-70], [-46], [-30], [22], [4], [58], [-14], [54], [-16], [14], [-38], [78], [14], [-22], [6], [-56], [26], [-10], [-8], [-62], [-64], [58], [58], [28], [-10], [42], [14], [42], [-16], [18], [70], [-28], [-52], [34], [-18], [-84], [46], [-54], [16], [-42], [-38], [-38], [18], [12], [74], [62], [32], [-14], [-14], [-32], [-44], [38], [82], [-28], [-34], [-36], [-22], [-6], [26], [30], [4], [82], [-74], [42], [4], [-30], [30], [90], [50], [42], [-18], [-66], [-14], [46], [-8], [-42], [6], [-10], [2], [72], [-60], [22], [52], [14], [-34], [10], [42], [-54], [-42], [-30], [-40], [-30], [-70], [14], [-20], [28], [-2], [-22], [2], [34], [62], [-34], [-66], [-24], [-70], [-58], [94], [6], [6], [-18], [6], [32], [-14], [30], [-62], [62], [94], [2], [98], [82], [-80], [-66], [74], [-92], [-14], [-22], [-14], [-70], [38], [-74], [62], [-2], [-42], [76], [-62], [56], [70], [-42], [-70], [6], [24], [80], [26], [24], [-54], [-74], [66], [-50], [-18], [50], [-34], [-94], [-56], [78], [-50], [76], [90], [26], [-86], [88], [10], [10], [18], [16], [-98], [78], [-74], [26], [-74], [6], [22], [-94], [18], [-12], [-6], [14], [-100]];
    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_n();".
// 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_n(: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_n();".
// 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_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1920_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![2, 1]>,<11,R![-2, 1]>,<13,R![2, 1]>],Snew);
    return Vf;
end function;