
// Make newform 20.3.b.a in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_20_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_20_b_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_20_3_b_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_20_3_b_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
        poly := [1, -1, 1, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 0, 2, 0], [0, 0, 0, 2], [-2, 4, -2, 2]];
    Rf_basisdens := [1, 1, 1, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    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_20_b();"
function MakeCharacter_20_b()
    N := 20;
    order := 2;
    char_gens := [11, 17];
    v := [1, 2];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_20_b_Hecke();"
function MakeCharacter_20_b_Hecke(Kf)
    N := 20;
    order := 2;
    char_gens := [11, 17];
    char_values := [[-1, 0, 0, 0], [1, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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 := 3;
    raw_aps := [[0, 0, -1, 0], [0, 0, 0, 1], [-1, -1, 1, 0], [0, 2, 2, -1], [0, -4, -4, -2], [-6, -2, 2, 0], [2, 8, -8, 0], [0, 4, 4, 0], [0, 2, 2, -3], [-6, -4, 4, 0], [0, -4, -4, 10], [-6, -10, 10, 0], [-22, 6, -6, 0], [0, -4, -4, -3], [0, -6, -6, -13], [-54, -10, 10, 0], [0, -12, -12, 12], [58, 26, -26, 0], [0, -20, -20, 11], [0, 20, 20, 2], [34, -32, 32, 0], [0, 16, 16, -20], [0, 12, 12, 13], [-62, -40, 40, 0], [-78, -12, 12, 0], [74, -8, 8, 0], [0, -22, -22, -23], [0, 24, 24, -17], [42, 74, -74, 0], [146, 20, -20, 0], [0, 38, 38, 23], [0, -28, -28, 42], [-14, 12, -12, 0], [0, 60, 60, 4], [10, -38, 38, 0], [0, -36, -36, -62], [138, 54, -54, 0], [0, -56, -56, 45], [0, -18, -18, 15], [-214, -10, 10, 0], [0, -12, -12, -76], [106, -16, 16, 0], [0, 44, 44, -58], [130, -16, 16, 0], [-86, -70, 70, 0], [0, 32, 32, 64], [0, 28, 28, 42], [0, -70, -70, 13], [0, 16, 16, -1], [138, 20, -20, 0], [146, 140, -140, 0], [0, -40, -40, -36], [-38, 30, -30, 0], [0, 52, 52, -26], [-270, -4, 4, 0], [0, -58, -58, -79], [-262, -70, 70, 0], [0, 12, 12, 86], [-166, 90, -90, 0], [-38, -10, 10, 0], [0, 48, 48, -15], [122, 46, -46, 0], [0, 68, 68, 11], [0, -68, -68, -10], [-350, -176, 176, 0], [218, -46, 46, 0], [0, -44, -44, -70], [-190, -64, 64, 0], [0, 76, 76, 91], [-134, 108, -108, 0], [-158, -144, 144, 0], [0, -16, -16, -16], [0, -34, -34, -157], [410, 154, -154, 0], [0, -60, -60, -32], [0, 114, 114, -43], [202, 154, -154, 0], [218, 94, -94, 0], [210, -72, 72, 0], [-422, -150, 150, 0], [0, 140, 140, -28], [-214, 114, -114, 0], [0, 44, 44, 126], [-174, 60, -60, 0], [0, -128, -128, 48], [0, 88, 88, -19], [426, -102, 102, 0], [674, 120, -120, 0], [-358, -80, 80, 0], [0, -2, -2, -167], [0, -232, -232, -13], [0, 32, 32, 172], [0, -158, -158, 67], [0, 52, 52, 38], [0, -228, -228, 24], [0, -214, -214, 73], [-742, -140, 140, 0], [562, 360, -360, 0], [0, 104, 104, 161], [-358, 80, -80, 0], [0, 40, 40, -1], [618, -86, 86, 0], [0, -184, -184, 89], [-166, -54, 54, 0], [0, -20, -20, 122], [50, 236, -236, 0], [0, -68, -68, -153], [2, 96, -96, 0], [0, 216, 216, 12], [90, -82, 82, 0], [0, 138, 138, 251], [586, -70, 70, 0], [274, 220, -220, 0], [0, 4, 4, -200], [0, 52, 52, -198], [-70, -282, 282, 0], [0, 52, 52, -271], [0, -38, -38, -117], [-422, 42, -42, 0], [0, 28, 28, 92], [-390, -342, 342, 0], [146, 260, -260, 0], [170, 306, -306, 0], [0, -44, -44, 61], [0, -4, -4, 86], [-630, -62, 62, 0], [202, -316, 316, 0], [0, -88, -88, 76], [0, 154, 154, 55], [682, 446, -446, 0], [0, 20, 20, -36], [0, 134, 134, 213], [0, 116, 116, 282], [-710, -134, 134, 0], [-222, -344, 344, 0], [226, 328, -328, 0], [-246, -142, 142, 0], [0, -116, -116, -169], [954, 210, -210, 0], [18, -200, 200, 0], [0, -44, -44, -122], [634, 402, -402, 0], [0, -134, -134, -115], [0, -220, -220, 267], [-422, -70, 70, 0], [0, 208, 208, -220], [-854, -590, 590, 0], [-686, -260, 260, 0], [0, -12, -12, 52], [0, 158, 158, -147], [426, 142, -142, 0], [-86, 262, -262, 0], [0, 64, 64, -103], [0, 18, 18, 199], [0, -104, -104, 27], [0, 108, 108, 442], [0, -88, -88, -204], [714, -254, 254, 0], [274, -20, 20, 0], [922, -160, 160, 0], [0, 140, 140, 259], [1058, 192, -192, 0], [0, 182, 182, -125], [0, 396, 396, -246], [-1294, -68, 68, 0], [0, -122, -122, 65], [0, -244, -244, -14], [-662, -346, 346, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_20_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_20_3_b_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_20_3_b_a(:prec:=4)
    chi := MakeCharacter_20_b();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_20_3_b_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_20_3_b_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_20_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([],Snew);
    return Vf;
end function;