
// Make newform 150.2.g.c in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_150_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_150_g_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_150_2_g_c();".
// 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_150_2_g_c();".
// 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 := [361, -171, 23, -35, 36, -5, -3, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [4276691, -1105295, 1281772, -261151, 197116, -50800, 156, -27571], [5880747, -687257, 1275844, -351515, 389890, 87368, -18958, -45697], [-523849, 23050, 24798, -16673, -38495, -3119, 7027, 2894], [-13281190, 939005, 38090, 989612, -736723, -107003, 29865, 60697], [7026371, 3422970, 501044, -1082509, 265983, 188067, -26109, -72436], [-868243, 101760, -19146, 17029, -68083, -8495, 2617, 5808]];
    Rf_basisdens := [1, 1, 4238558, 4238558, 223082, 4238558, 4238558, 223082];
    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_150_g();"
function MakeCharacter_150_g()
    N := 150;
    order := 5;
    char_gens := [101, 127];
    v := [5, 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_150_g_Hecke();"
function MakeCharacter_150_g_Hecke(Kf)
    N := 150;
    order := 5;
    char_gens := [101, 127];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 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 := 2;
    raw_aps := [[0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0], [-1, 1, 0, 1, 0, 0, 0, 0], [0, 0, -1, 1, 0, 0, 1, 1], [-1, 2, 1, 0, -2, 0, 0, 1], [0, -1, -1, 3, 2, -2, 1, -1], [1, 0, 1, -2, -1, 3, -2, 2], [0, 0, 0, 2, 0, -2, 0, 0], [-4, 0, -2, 0, 0, -4, 0, 0], [-3, -1, -1, 3, 0, 0, 2, -2], [1, 1, 1, 1, 0, -1, 1, 0], [1, 0, -5, -2, -2, -3, -1, 2], [0, -1, -3, -3, 2, -4, 1, -1], [-6, 2, -2, 4, 2, -2, 0, 0], [-2, 2, 2, 0, -2, 2, -4, 2], [5, -1, -1, -2, 3, 7, 2, 1], [1, -2, 2, -7, 2, 2, 1, 0], [2, -2, -1, 0, 1, 1, 1, -1], [4, 2, 4, -6, 0, 6, 2, 0], [2, -2, -2, 2, 4, 0, 4, 0], [-4, -2, -1, 0, 3, -5, -1, -1], [2, 1, 1, -2, 0, 9, -2, 2], [-1, 3, -1, 5, -2, -3, -1, 4], [2, -2, -1, 0, 0, 1, 2, -1], [-3, -1, -1, 2, -1, -8, 2, -3], [-4, 2, -5, 7, 2, -2, 2, 2], [10, 2, 2, -13, -3, 8, -4, 1], [-7, -1, 1, -2, -1, 1, -3, -3], [-4, 5, 3, 1, -2, 0, -1, -3], [1, 2, -1, 2, -6, 3, -3, 4], [5, -2, 5, 0, 0, 4, 2, -1], [-12, -2, -12, 10, 0, -10, -2, 0], [0, 3, -3, -3, -6, 0, -3, 3], [-12, 0, -4, 0, 2, -12, -2, 0], [10, 1, 3, -2, 1, -1, 0, 0], [8, 0, 9, -9, 0, 0, -1, -1], [-11, 1, -4, 5, 1, -1, 1, 1], [0, 0, 0, 14, 0, 0, 0, 0], [-6, -6, -6, 10, 4, -14, 2, -8], [-5, 0, -1, 0, -3, -5, 3, 0], [2, 0, 0, -1, 1, 6, 0, 1], [7, 0, 7, 0, -1, 1, -2, 2], [0, 0, -2, 0, 0, -2, 0, 0], [-5, -4, -3, -1, -4, 4, 2, 2], [11, 0, 0, -9, 2, 14, 0, 2], [5, 5, -3, 8, 5, -5, 3, 3], [4, 8, 0, 0, -4, 8, -4, 4], [-5, 2, -1, 0, -2, -4, 0, 1], [3, -2, 3, 0, 2, 2, 0, -1], [3, -4, -4, 0, 3, 2, 8, -5], [-6, -5, -6, 3, 4, -7, 3, -8], [-2, 4, -8, 0, -2, 0, -2, 2], [-1, -1, -3, 9, 4, -6, 2, -3], [2, -2, 9, -11, -2, 2, -1, -1], [-5, 0, 0, 0, 0, 0, -1, -1], [-2, 0, -4, 12, 4, -8, 2, -4], [3, -2, 3, -12, 0, 12, -2, 0], [-2, -2, -2, 3, 1, 4, 4, -3], [2, 6, 1, 0, -4, 5, -2, 3], [-4, -1, -4, 13, 2, -15, 3, -4], [8, -6, 8, -2, 4, -2, 2, -8], [8, 2, 7, -5, 2, -2, 2, 2], [0, -4, 4, -8, -4, 4, -2, -2], [8, -4, 4, 0, 0, 6, 4, -2], [5, -6, -2, -9, 2, 2, 1, 4], [-11, 8, -11, 17, -5, -12, -2, 10], [-6, 4, -6, 14, 0, -14, 4, 0], [-2, -1, -8, -5, 6, -13, 3, -5], [0, -3, -3, 2, 2, -1, 6, -4], [-1, -5, -2, -3, -5, 5, -5, -5], [-8, -2, -2, 8, 0, 12, 4, -4], [2, 0, 0, 4, -4, 4, -2, 4], [3, -3, 3, -15, 0, 15, -3, 0], [-8, 8, -2, 0, -8, -4, 0, 4], [8, 2, 2, -12, -4, 26, -4, 0], [-6, 6, -6, 18, -2, -16, 2, 4], [15, -8, 10, 0, 7, 11, 1, -4], [-4, 2, 2, 8, 4, 0, -4, 8], [-13, 0, -2, 2, 0, 0, -3, -3], [4, -4, 3, 13, 0, 7, 0, 4], [3, 7, 3, 1, -4, 3, -1, 8], [-5, 1, 1, 3, -2, 18, -2, 0], [-10, 2, -10, 8, 0, -8, 2, 0], [-3, 5, -3, -6, -5, 11, -5, 10], [-9, -6, 7, 0, 2, -12, 4, -3], [-8, 0, 13, -13, 0, 0, -7, -7], [13, 3, -4, 7, 3, -3, 5, 5], [17, 3, 3, 0, 3, -3, 3, 3], [-1, 8, -1, 0, -11, 3, 3, 4], [-6, 12, 18, -24, -12, 18, -6, 0], [-23, 0, -23, 15, 3, -18, 6, -6], [0, 0, 0, 2, 2, -2, 0, 2], [1, 10, 0, -7, -22, 12, -11, 12], [4, -3, 20, -3, -2, 25, -1, 5], [-18, -4, -10, 6, -4, 4, 2, 2], [-8, -2, -2, 14, 6, 6, 4, 2], [-6, 8, -1, -7, -4, -5, -2, -4], [5, 2, 2, -8, -3, -10, -4, 1], [0, -4, -2, 0, 6, -2, -2, -2], [5, -2, -9, -4, -6, -1, -3, 8], [-20, 0, 0, 22, 2, -6, 0, 2], [19, -5, 7, -12, -5, 5, 0, 0], [0, 3, 16, -17, -6, 19, -3, 3], [7, -2, 7, -20, -1, 21, -4, 2], [-2, 6, 6, -2, -4, 10, -12, 8], [19, -6, 14, 0, 5, 16, 1, -3], [-3, 4, 22, -7, -2, 20, -1, -2], [-7, -1, -6, 5, -1, 1, -1, -1], [18, 4, 28, -24, 4, -4, 0, 0], [-22, 1, -17, 18, 1, -1, 2, 2], [9, 1, 21, -20, 1, -1, -1, -1], [3, -6, -5, -10, 6, -5, 3, 0], [-20, 3, -20, 17, 2, -19, 7, -4], [4, 6, 4, -4, -6, 10, -6, 12], [-18, 4, -18, 12, -2, -10, 0, 4], [-2, -4, 4, -4, 12, -4, 6, -8], [8, -4, 0, -4, -4, 4, 0, 0], [-12, -4, -4, 16, 4, -18, 8, -4], [-7, 7, 7, -4, -11, 6, -14, 3], [-6, 1, -4, 17, 10, -15, 5, -11], [-12, -8, -24, 0, 6, -16, 2, -4], [3, 0, -19, 0, -3, 3, 3, 0], [-3, -10, 4, 0, 13, -8, -3, -5], [-7, -2, -7, 1, -1, 0, -4, 2], [6, -10, -2, 12, 8, 0, 4, 2], [5, -8, -4, -4, -8, 8, -3, -3], [-4, 5, -25, 17, -2, -28, -1, -3], [14, 6, 14, -2, 2, 0, 10, -4], [4, -4, -10, 0, 2, 2, 2, -2], [2, 16, 2, 2, -8, 6, 0, 16], [2, 8, 16, 0, 4, 6, -12, 4], [0, 2, -22, 24, 2, -2, 8, 8], [-10, 4, -3, 7, 4, -4, -7, -7], [-11, -2, -2, 0, -2, 2, -9, -9], [-4, -10, -17, 0, 9, -9, 1, -5], [27, 7, 27, -31, -6, 37, -5, 12], [5, 20, 20, 0, -9, 15, -11, 10], [2, -6, -22, 42, 8, -24, 4, -2], [-7, 1, 1, 6, -1, 13, -2, 1], [7, -6, -5, -4, -2, 3, -1, 8], [-10, 0, -24, 0, -10, -10, 10, 0], [-4, 0, 0, 3, -1, -20, 0, -1], [3, -6, -5, 0, 2, 0, 4, -3], [-1, 6, -8, 0, -9, 2, 3, 3], [-1, 0, 0, -4, -5, -30, 0, -5], [28, -4, 0, 0, 4, 26, 0, -2], [-9, 6, 6, 0, -9, -12, -12, 3], [20, -2, 2, -4, -2, 2, -2, -2], [2, 4, 0, 6, -12, 8, -6, 8], [4, -8, -10, 12, 8, -10, 4, 0], [-22, 6, -15, 0, 3, -19, -9, 3], [8, 10, 8, 0, -4, 4, 2, 8], [-12, -8, -12, -2, 4, -2, 0, -8], [-2, 0, 2, 24, 4, -2, 2, -4], [18, -2, 44, -46, -2, 2, -2, -2], [-10, -8, -10, 0, 6, -14, 2, -4], [-22, 6, -22, 30, -2, -28, 2, 4], [-5, -3, -3, 7, 2, -24, 6, -4], [-1, 3, 21, -15, -4, 22, -2, 1], [-5, 5, -31, 5, 0, -36, 0, -5], [20, 6, 6, -23, -3, -2, -12, 9], [-9, 3, 3, 7, -2, -18, -6, 4], [17, -1, 17, -25, 2, 23, 3, -4], [-4, 6, 6, -5, -9, -2, -12, 3], [-10, 2, 11, 0, -9, -9, 7, 1], [12, -8, 12, -18, 0, 18, -8, 0], [-2, 7, 10, -23, -10, 13, -5, 3], [-8, -2, -2, 16, 8, 0, 4, 4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_150_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_150_2_g_c();".
// 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_150_2_g_c(:prec:=8)
    chi := MakeCharacter_150_g();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    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_150_2_g_c();".
// 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_150_2_g_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_150_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![76, -4, -19, 1, 1]>],Snew);
    return Vf;
end function;