
// Make newform 176.2.m.d in Magma, downloaded from the LMFDB on 21 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_176_m();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_176_m_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_176_2_m_d();".
// 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_176_2_m_d();".
// 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 := [121, 88, 100, 28, 19, 2, 5, -3, 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], [440220, 319181, -332708, 71401, 33439, -15725, 2098, 528], [528473, 501961, 104068, -61888, 195125, -30517, -9973, 5794], [-701074, 18601, -77439, -58164, -171974, 183537, -59487, 7409], [2809884, 2179908, 380907, 521745, 200474, -106264, 16927, 8817], [-2305776, -1085964, -979996, -102852, -228404, 58156, 3536, -11971], [665808, -742824, 6060, -266949, 65418, -113608, 57068, -13790]];
    Rf_basisdens := [1, 1, 1168519, 1168519, 1168519, 1168519, 1168519, 1168519];
    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_176_m();"
function MakeCharacter_176_m()
    N := 176;
    order := 5;
    char_gens := [111, 133, 145];
    v := [5, 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_176_m_Hecke();"
function MakeCharacter_176_m_Hecke(Kf)
    N := 176;
    order := 5;
    char_gens := [111, 133, 145];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 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 := 2;
    raw_aps := [[0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, -1, 0, 1, 0], [0, 0, 0, 1, -1, -1, -1, 0], [-2, 0, 1, -1, -1, -1, 0, 2], [1, 0, -3, 2, 1, 1, -1, -1], [1, 0, 0, 1, 0, -1, 1, 0], [0, -3, -2, 0, -1, 2, 2, 3], [3, 0, -3, 4, 0, 0, 0, -4], [0, 0, 2, 0, 2, 0, 0, 2], [2, 0, 1, -1, -1, -1, 0, -2], [1, 0, 2, 1, 0, 1, 1, 0], [2, 2, -1, 5, 1, 1, -2, -2], [4, -1, -4, 2, 1, 0, -1, -2], [-1, 1, 1, 0, 1, -1, 0, 0], [3, 1, -3, 3, 2, 0, -2, -3], [-5, 0, 0, -5, 0, -1, -1, 0], [5, 0, 0, 0, 0, 0, 0, -5], [0, 4, -2, 5, 1, -3, -3, 2], [0, 2, 1, 0, -3, -2, 0, -1], [0, -4, 4, -7, -1, 3, 3, 2], [-2, -3, 0, -4, 0, 0, 3, 2], [1, 0, 2, 1, 0, -3, -3, 0], [0, 0, -1, 3, -2, -2, -2, 5], [3, -1, -1, 0, -1, 1, 0, 0], [1, 0, 0, 1, 0, -2, 2, 0], [-1, 0, 8, -1, 0, -3, 1, 0], [-2, -2, -1, -11, 1, 1, 2, 2], [8, -1, -8, 4, -1, 0, 1, -4], [-2, -4, -8, 0, 0, 4, 0, -4], [5, 4, -5, 4, 4, 0, -4, -4], [0, 8, -2, 5, 5, -3, -3, 0], [-12, -2, -1, 0, -1, 2, 0, 1], [0, 4, 3, -1, 2, -2, -2, 3], [4, 2, -3, -5, 3, 3, -2, -4], [0, 2, -2, -1, 5, 3, 3, 0], [-9, 1, 9, -1, -2, 0, 2, 1], [-9, 7, 9, -9, 0, 0, 0, 9], [-6, 0, 12, -6, 0, -3, 6, 0], [13, 0, -14, 13, 0, 3, -1, 0], [5, -1, -5, 13, 2, 0, -2, -13], [-5, -2, 5, -6, 0, 0, 0, 6], [0, -4, 8, -11, -1, 3, 3, 4], [4, 0, 3, 9, -3, -3, 0, -4], [0, -2, -8, 3, 3, 5, 5, -18], [-4, -2, 12, 0, 0, 2, 0, 14], [-6, 2, 4, 0, 0, -2, 0, 2], [-9, 0, -2, -9, 0, 0, -4, 0], [-3, 5, 3, 5, 0, 0, 0, -5], [-1, 0, 6, 2, -6, -6, 0, 1], [11, 0, -4, 11, 0, 1, -3, 0], [-3, 0, 8, -3, 0, 4, 4, 0], [-3, -11, 3, -3, -4, 0, 4, 3], [4, -4, -15, 0, 1, 4, 0, -11], [7, 0, -14, 7, 0, 3, 1, 0], [2, 9, 4, 4, -4, -4, -9, -2], [2, 6, 10, 0, -2, -6, 0, 4], [0, 2, 2, -1, 1, -1, -1, 0], [14, 0, 1, 3, -1, -1, 0, -14], [9, 0, -8, 9, 0, -1, -3, 0], [0, 0, 5, -1, -4, -4, -4, 13], [-5, -11, 5, -5, -2, 0, 2, 5], [8, -6, 5, -5, -5, -5, 6, -8], [3, 1, 5, 0, -7, -1, 0, 4], [1, 5, -1, 1, 4, 0, -4, -1], [0, -3, -2, 0, -1, 2, 2, -13], [5, 0, -8, 5, 0, -5, 5, 0], [11, -3, -23, 0, 5, 3, 0, -20], [-3, 0, -4, 8, 4, 4, 0, 3], [0, -1, 4, -10, 5, 6, 6, 9], [-5, -1, 5, -13, 4, 0, -4, 13], [0, 0, -5, 0, 7, 0, 0, -5], [12, -4, -13, 9, 13, 13, 4, -12], [-10, -2, -5, -27, 5, 5, 2, 10], [-12, 6, 6, 0, -6, -6, 0, 0], [0, 8, -9, 11, 6, -2, -2, 5], [3, 0, -10, 3, 0, 3, 1, 0], [-12, 0, 7, 1, -7, -7, 0, 12], [14, -4, -6, 0, -2, 4, 0, -2], [0, -12, 3, -13, -2, 10, 10, 11], [9, 0, 0, 9, 0, -7, 3, 0], [-15, 3, 3, 0, 7, -3, 0, 0], [-7, 11, 7, -3, 10, 0, -10, 3], [3, 0, -14, 3, 0, -1, 1, 0], [7, 0, -6, 18, 6, 6, 0, -7], [14, -6, -8, 0, -8, 6, 0, -2], [8, 3, -8, -16, -1, 0, 1, 16], [4, 0, -22, 4, 0, 5, -6, 0], [0, -4, 15, -17, -2, 2, 2, -1], [6, 4, -2, 0, 2, -4, 0, -6], [-16, -8, 4, 0, 0, 8, 0, 12], [0, -2, 16, -17, -1, 1, 1, 24], [0, 6, -16, 15, 7, 1, 1, -16], [25, -11, -25, 13, -4, 0, 4, -13], [-8, 1, 6, -14, -6, -6, -1, 8], [-12, 7, 8, -12, -8, -8, -7, 12], [9, 9, -9, 9, 8, 0, -8, -9], [-7, 5, 7, -17, 2, 0, -2, 17], [-29, -4, 2, -6, -2, -2, 4, 29], [0, 7, 8, -10, 9, 2, 2, -7], [-7, 0, -8, -7, 0, 11, 1, 0], [11, 2, -11, 2, 8, 0, -8, -2], [0, -4, -9, -15, 9, 9, 4, 0], [11, 0, 14, 11, 0, 0, -4, 0], [13, 2, -13, 22, -4, 0, 4, -22], [0, 0, 12, 0, 4, 0, 0, 12], [0, 3, 2, 6, -5, -8, -8, -7], [-5, -8, 2, 2, -2, -2, 8, 5], [12, -8, -9, 0, 11, 8, 0, -1], [0, 0, 18, -23, 5, 5, 5, 8], [-2, 9, 4, 4, -4, -4, -9, 2], [5, 0, 6, 5, 0, -5, 7, 0], [1, -3, -1, -21, -2, 0, 2, 21], [16, 4, -13, 0, 7, -4, 0, -17], [3, 4, -3, 8, -4, 0, 4, -8], [-2, 6, -9, -3, 9, 9, -6, 2], [20, -5, -20, 22, -3, 0, 3, -22], [0, -4, -11, 11, -4, 0, 0, -5], [-27, 0, 18, -27, 0, 5, -3, 0], [14, -6, -1, 5, 1, 1, 6, -14], [0, -2, -15, 0, 1, 2, 0, -13], [-6, -8, 12, 0, 4, 8, 0, 20], [-16, 0, 6, -16, 0, 5, 10, 0], [0, 10, 20, -17, 7, -3, -3, -2], [-16, -4, 12, 0, -12, 4, 0, 16], [8, 0, -4, 8, 0, 1, -4, 0], [-11, 11, 11, 5, 10, 0, -10, -5], [0, -6, -20, 11, 3, 9, 9, -18], [8, -2, 9, 7, -9, -9, 2, -8], [10, -2, -6, 0, 10, 2, 0, -4], [-6, -8, 5, 15, -5, -5, 8, 6], [-10, 0, 8, -10, 0, 5, 2, 0], [0, -2, 2, 3, -7, -5, -5, 18], [-23, 7, 23, -9, 8, 0, -8, 9], [-5, 0, 16, -5, 0, -3, 9, 0], [12, 0, 14, 12, 0, -7, -10, 0], [-14, 12, 20, 0, -16, -12, 0, 8], [-5, 9, 5, -19, 0, 0, 0, 19], [0, -3, -20, 24, -7, -4, -4, -13], [0, -8, 8, -15, -1, 7, 7, 12], [0, -15, -18, 8, -5, 10, 10, 11], [12, 7, -12, 0, 11, 0, -11, 0], [2, -10, 3, -7, -3, -3, 10, -2], [-9, 0, 2, -9, 0, -3, 7, 0], [-6, 0, -16, -6, 0, 9, 6, 0], [8, -2, 5, -5, -5, -5, 2, -8], [1, -17, -1, -5, -4, 0, 4, 5], [0, -4, 6, 5, -15, -11, -11, 14], [-23, 5, 25, 0, 1, -5, 0, 20], [-8, 10, 29, 0, -3, -10, 0, 19], [0, 0, 8, -15, 7, 7, 7, 26], [-9, 11, 9, 7, 8, 0, -8, -7], [-21, 3, 3, 0, -13, -3, 0, 0], [-22, -9, 22, -6, -11, 0, 11, 6], [-14, -4, 13, 7, -13, -13, 4, 14], [0, 13, 20, -16, 9, -4, -4, 7], [-7, 0, -26, -7, 0, -1, 3, 0], [0, -14, -20, 15, -9, 5, 5, 16], [0, 9, 22, -8, -5, -14, -14, 35], [6, 0, 2, 6, 0, 9, 8, 0], [0, -12, -28, 21, -5, 7, 7, -12], [4, 6, -11, 0, 13, -6, 0, -17], [-6, -9, -14, -34, 14, 14, 9, 6], [30, 14, -12, 0, 8, -14, 0, -26], [-25, -11, 25, -1, -10, 0, 10, 1], [-3, 0, -20, -3, 0, 13, -1, 0], [18, 12, -3, 19, 3, 3, -12, -18], [28, 0, -2, 0, 6, 0, 0, -2], [-22, -2, 7, -15, -7, -7, 2, 22]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_176_m_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_176_2_m_d();".
// 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_176_2_m_d(:prec:=8)
    chi := MakeCharacter_176_m();
    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_176_2_m_d();".
// 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_176_2_m_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_176_m();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![121, 99, 20, -29, 49, -19, 10, -1, 1]>],Snew);
    return Vf;
end function;