
// Make newform 126.3.n.b in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_126_n();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_126_n_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_126_3_n_b();".
// 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_126_3_n_b();".
// 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, 2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
    Rf_basisdens := [1, 1, 2, 2];
    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_126_n();"
function MakeCharacter_126_n()
    N := 126;
    order := 6;
    char_gens := [29, 73];
    v := [6, 5];
    // 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_126_n_Hecke();"
function MakeCharacter_126_n_Hecke(Kf)
    N := 126;
    order := 6;
    char_gens := [29, 73];
    char_values := [[1, 0, 0, 0], [1, 0, 1, 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, 1, 0, 0], [0, 0, 0, 0], [0, 2, 0, 4], [7, 0, 7, 0], [0, 12, 0, 12], [-1, 0, -2, 0], [0, 2, 0, -2], [17, 0, -17, 0], [0, -6, 0, 0], [0, 0, 0, -24], [-14, 0, -7, 0], [47, 0, 47, 0], [0, -56, 0, -28], [31, 0, 0, 0], [0, -34, 0, -68], [0, 54, 0, 54], [0, 34, 0, -34], [-48, 0, 48, 0], [0, 0, -31, 0], [0, 0, 0, 42], [-94, 0, -47, 0], [-41, 0, -41, 0], [0, 4, 0, 2], [0, 24, 0, 48], [24, 0, 48, 0], [0, 72, 0, -72], [-17, 0, 17, 0], [0, -102, 0, 0], [0, 0, 169, 0], [0, 0, 0, -42], [209, 0, 0, 0], [0, -24, 0, -48], [0, -108, 0, -108], [-113, 0, -226, 0], [0, 36, 0, 0], [0, 0, 10, 0], [48, 0, 24, 0], [-86, 0, -86, 0], [0, -148, 0, -74], [0, -18, 0, -36], [0, 6, 0, 6], [25, 0, 50, 0], [0, 54, 0, 0], [0, 0, -287, 0], [0, 0, 0, 90], [240, 0, 120, 0], [82, 0, 0, 0], [-24, 0, -48, 0], [0, -156, 0, 156], [-47, 0, 47, 0], [0, 162, 0, 0], [0, 0, 0, -48], [-528, 0, -264, 0], [0, 284, 0, 142], [0, 66, 0, 132], [0, -180, 0, -180], [0, -8, 0, 8], [24, 0, -24, 0], [0, 0, -337, 0], [0, 0, 0, 174], [-226, 0, -113, 0], [0, -80, 0, -40], [-41, 0, -82, 0], [0, -88, 0, 88], [169, 0, -169, 0], [0, -168, 0, 0], [185, 0, 185, 0], [-359, 0, 0, 0], [0, -330, 0, -330], [336, 0, 672, 0], [0, -118, 0, 118], [0, 240, 0, 0], [178, 0, 89, 0], [289, 0, 289, 0], [7, 0, 0, 0], [0, 202, 0, 404], [0, 162, 0, 162], [47, 0, -47, 0], [0, -66, 0, 0], [-482, 0, -241, 0], [0, -16, 0, -8], [407, 0, 0, 0], [0, 114, 0, 114], [-97, 0, -194, 0], [-312, 0, 312, 0], [0, 90, 0, 0], [0, 0, 0, 78], [-25, 0, -25, 0], [0, 64, 0, 32], [521, 0, 0, 0], [0, -90, 0, -180], [0, 358, 0, -358], [0, 0, 127, 0], [0, 0, 0, -444], [233, 0, 233, 0], [0, -440, 0, -220], [0, 130, 0, 260], [0, 232, 0, -232], [305, 0, -305, 0], [-335, 0, -335, 0], [-658, 0, 0, 0], [0, 192, 0, 192], [0, 6, 0, -6], [0, -600, 0, 0], [0, 0, 449, 0], [-338, 0, -169, 0], [0, 432, 0, 216], [0, -428, 0, -856], [0, 456, 0, 456], [265, 0, 530, 0], [607, 0, -607, 0], [0, 0, 290, 0], [0, 0, 0, -516], [-946, 0, -473, 0], [-58, 0, 0, 0], [0, 678, 0, 678], [-439, 0, -878, 0], [0, -144, 0, 144], [0, 312, 0, 0], [0, 0, 0, -114], [-962, 0, -481, 0], [-263, 0, 0, 0], [0, -204, 0, -408], [0, 678, 0, 678], [713, 0, -713, 0], [0, 0, 0, 690], [1106, 0, 1106, 0], [0, 280, 0, 560], [-247, 0, -494, 0], [23, 0, -23, 0], [0, 0, -487, 0], [0, 0, 0, -360], [545, 0, 545, 0], [-770, 0, 0, 0], [0, 70, 0, 140], [407, 0, 814, 0], [0, -376, 0, 376], [528, 0, 264, 0], [0, 12, 0, 6], [0, -666, 0, -666], [-288, 0, -576, 0], [0, 426, 0, 0], [0, 0, 38, 0], [0, 0, 0, -486], [-962, 0, -481, 0], [0, 200, 0, 100], [-719, 0, -1438, 0], [0, 500, 0, -500], [144, 0, -144, 0], [0, -468, 0, 0], [574, 0, 574, 0], [0, 132, 0, 66], [1735, 0, 0, 0], [0, 80, 0, 160], [0, 0, 751, 0], [0, 0, 0, 882], [-1015, 0, -1015, 0], [0, 446, 0, 892], [1009, 0, 2018, 0], [0, -506, 0, 506], [0, 852, 0, 0], [0, 0, 0, 726], [-895, 0, 0, 0], [0, -86, 0, -172], [0, 888, 0, 888], [0, -482, 0, 482], [0, 0, 655, 0], [530, 0, 265, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_126_n_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_126_3_n_b();".
// 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_126_3_n_b(:prec:=4)
    chi := MakeCharacter_126_n();
    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_126_3_n_b();".
// 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_126_3_n_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_126_n();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![576, 0, -24, 0, 1]>],Snew);
    return Vf;
end function;