
// Make newform 12.4.b.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_12_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_12_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_12_4_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_12_4_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 := [4, 0, 1, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 2, 0, 0], [1, 1, 2, 1], [1, -1, 2, -1]];
    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_12_b();"
function MakeCharacter_12_b()
    N := 12;
    order := 2;
    char_gens := [7, 5];
    v := [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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_12_b_Hecke();"
function MakeCharacter_12_b_Hecke(Kf)
    N := 12;
    order := 2;
    char_gens := [7, 5];
    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 := 4;
    raw_aps := [[0, 1, 0, 0], [0, -1, 0, -1], [0, -2, -1, 1], [0, 0, 1, 1], [0, 10, -5, 5], [-10, 0, 0, 0], [0, -8, -4, 4], [0, 0, -9, -9], [0, -28, 14, -14], [0, 34, 17, -17], [0, 0, 29, 29], [-130, 0, 0, 0], [0, 28, 14, -14], [0, 0, -29, -29], [0, -56, 28, -28], [0, -122, -61, 61], [0, 50, -25, 25], [-442, 0, 0, 0], [0, 0, 95, 95], [0, 300, -150, 150], [410, 0, 0, 0], [0, 0, -11, -11], [0, -362, 181, -181], [0, -188, -94, 94], [770, 0, 0, 0], [0, 334, 167, -167], [0, 0, -175, -175], [0, 186, -93, 93], [-1066, 0, 0, 0], [0, 232, 116, -116], [0, 0, 237, 237], [0, -130, 65, -65], [0, -20, -10, 10], [0, 0, -253, -253], [0, -362, -181, 181], [0, 0, 73, 73], [-730, 0, 0, 0], [0, 0, -33, -33], [0, -4, 2, -2], [0, 250, 125, -125], [0, 390, -195, 195], [1262, 0, 0, 0], [0, -800, 400, -400], [-190, 0, 0, 0], [0, 478, 239, -239], [0, 0, 33, 33], [0, 0, 71, 71], [0, 0, 61, 61], [0, -146, 73, -73], [4094, 0, 0, 0], [0, 1180, 590, -590], [0, 1640, -820, 820], [-1198, 0, 0, 0], [0, -1230, 615, -615], [0, -704, -352, 352], [0, 1228, -614, 614], [0, 10, 5, -5], [0, 0, -1131, -1131], [6350, 0, 0, 0], [0, -1244, -622, 622], [0, 0, 875, 875], [0, -146, -73, 73], [0, 0, -201, -201], [0, -940, 470, -470], [-7030, 0, 0, 0], [0, -110, -55, 55], [0, 0, 547, 547], [1490, 0, 0, 0], [0, -574, 287, -287], [-2074, 0, 0, 0], [0, 1936, 968, -968], [0, -2340, 1170, -1170], [0, 0, -1019, -1019], [4910, 0, 0, 0], [0, 0, -405, -405], [0, -1792, 896, -896], [0, -2114, -1057, 1057], [-10570, 0, 0, 0], [0, 376, 188, -188], [-3574, 0, 0, 0], [0, 4430, -2215, 2215], [3518, 0, 0, 0], [0, 3720, -1860, 1860], [14450, 0, 0, 0], [0, 0, 1945, 1945], [0, -878, 439, -439], [0, -3200, -1600, 1600], [-3430, 0, 0, 0], [0, 874, 437, -437], [0, 0, -2347, -2347], [0, 534, -267, 267], [0, -4400, 2200, -2200], [0, 0, -1263, -1263], [0, -1990, 995, -995], [0, 0, -169, -169], [0, 2292, -1146, 1146], [0, 3250, 1625, -1625], [0, 2092, 1046, -1046], [0, 0, 1299, 1299], [-23962, 0, 0, 0], [0, 0, 1951, 1951], [0, -3590, -1795, 1795], [0, -7258, 3629, -3629], [0, 5236, 2618, -2618], [0, 0, 635, 635], [19490, 0, 0, 0], [0, 394, -197, 197], [0, -5768, -2884, 2884], [0, -700, 350, -350], [-8758, 0, 0, 0], [0, 0, 2525, 2525], [-16450, 0, 0, 0], [0, -1892, -946, 946], [0, 0, -2573, -2573], [0, 0, -1575, -1575], [0, -1568, -784, 784], [0, 0, 2063, 2063], [0, 5196, -2598, 2598], [0, 1162, 581, -581], [0, 1750, -875, 875], [9422, 0, 0, 0], [-17470, 0, 0, 0], [0, 4654, 2327, -2327], [0, 3642, -1821, 1821], [0, 0, 391, 391], [0, -3950, -1975, 1975], [14174, 0, 0, 0], [0, -9480, 4740, -4740], [0, 0, 1033, 1033], [11750, 0, 0, 0], [0, 0, -2577, -2577], [0, 7388, -3694, 3694], [0, 0, 677, 677], [-14290, 0, 0, 0], [0, 3796, 1898, -1898], [-29566, 0, 0, 0], [0, 4798, 2399, -2399], [0, 0, 503, 503], [0, -6878, -3439, 3439], [0, 2332, 1166, -1166], [0, 0, -1053, -1053], [0, -7994, -3997, 3997], [0, 0, -2695, -2695], [0, -3726, 1863, -1863], [-14074, 0, 0, 0], [0, -4820, 2410, -2410], [23630, 0, 0, 0], [0, -7004, -3502, 3502], [0, 0, -1765, -1765], [0, 7248, -3624, 3624], [41750, 0, 0, 0], [0, -536, -268, 268], [0, 0, 5607, 5607], [0, 1844, -922, 922], [0, 0, 4819, 4819], [0, 14680, -7340, 7340], [0, 0, 1833, 1833], [0, -2288, -1144, 1144], [37850, 0, 0, 0], [0, 3130, 1565, -1565], [0, -2234, 1117, -1117], [0, 10036, 5018, -5018], [0, 0, -2047, -2047], [0, -11670, 5835, -5835], [0, 1432, 716, -716], [0, 7492, -3746, 3746], [0, 0, -6243, -6243], [-18370, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_12_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_12_4_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_12_4_b_a(:prec:=4)
    chi := MakeCharacter_12_b();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_12_4_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_12_4_b_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_12_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([],Snew);
    return Vf;
end function;