
// Make newform 171.3.c.f in Magma, downloaded from the LMFDB on 12 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_171_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_171_c_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_171_3_c_f();".
// 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_171_3_c_f();".
// 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 := [25, -5, -4, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [5, -1, 1, -1], [5, 9, 1, -1], [-35, -7, 7, 3]];
    Rf_basisdens := [1, 5, 5, 5];
    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_171_c();"
function MakeCharacter_171_c()
    N := 171;
    order := 2;
    char_gens := [20, 154];
    v := [2, 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_171_c_Hecke();"
function MakeCharacter_171_c_Hecke(Kf)
    N := 171;
    order := 2;
    char_gens := [20, 154];
    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]];
    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], [3, 0, -1, 0], [9, 0, -1, 0], [3, 0, 1, 0], [0, 3, 0, 1], [-1, 0, 5, 0], [1, -7, -2, 1], [-14, 0, 2, 0], [0, 6, 0, 2], [0, 9, 0, -1], [0, -13, 0, -7], [0, 5, 0, -3], [1, 0, 15, 0], [-17, 0, -9, 0], [0, 3, 0, -9], [0, 9, 0, -7], [-19, 0, 3, 0], [0, 44, 0, -8], [0, 29, 0, 5], [21, 0, 3, 0], [0, -4, 0, 10], [-2, 0, -28, 0], [0, -47, 0, 1], [0, -3, 0, 13], [-118, 0, 10, 0], [0, 18, 0, -20], [0, 44, 0, 8], [0, -28, 0, 10], [0, -24, 0, 20], [0, -38, 0, -8], [35, 0, -47, 0], [127, 0, 17, 0], [1, 0, -13, 0], [147, 0, 23, 0], [0, -29, 0, 5], [66, 0, -28, 0], [10, 0, 16, 0], [0, 10, 0, -34], [0, -8, 0, 0], [0, 22, 0, 2], [0, 6, 0, -12], [143, 0, -29, 0], [0, -54, 0, -22], [-286, 0, -10, 0], [-143, 0, -29, 0], [0, 85, 0, 5], [0, -49, 0, 21], [0, -77, 0, -45], [-27, 0, 67, 0], [151, 0, -23, 0], [111, 0, 55, 0], [0, -9, 0, 35], [-109, 0, -19, 0], [0, 56, 0, 12], [215, 0, 55, 0], [0, -87, 0, 45], [-14, 0, 40, 0], [349, 0, -45, 0], [0, -110, 0, 38], [9, 0, -13, 0], [0, -6, 0, 26], [0, 86, 0, -46], [-73, 0, -21, 0], [90, 0, 76, 0], [0, -29, 0, -49], [0, 8, 0, 28], [0, -127, 0, 9], [299, 0, -15, 0], [-539, 0, 23, 0], [150, 0, 92, 0], [-265, 0, 71, 0], [-398, 0, -40, 0], [0, -22, 0, 4], [0, 78, 0, 54], [0, 65, 0, 1], [-173, 0, -61, 0], [29, 0, -73, 0], [0, -80, 0, -52], [0, -159, 0, 41], [182, 0, 40, 0], [0, 133, 0, -49], [0, 209, 0, -43], [0, 32, 0, 20], [0, 140, 0, 30], [-341, 0, -63, 0], [0, -250, 0, 10], [85, 0, 39, 0], [75, 0, 35, 0], [-175, 0, -5, 0], [427, 0, -75, 0], [-694, 0, 54, 0], [0, -43, 0, -37], [310, 0, -144, 0], [-103, 0, 87, 0], [738, 0, -54, 0], [0, 103, 0, -33], [0, -136, 0, 52], [0, 84, 0, -12], [541, 0, -137, 0], [0, -309, 0, 23], [-37, 0, 27, 0], [0, -267, 0, 13], [0, -203, 0, 5], [162, 0, 100, 0], [325, 0, -161, 0], [-413, 0, 5, 0], [230, 0, -4, 0], [0, 80, 0, 124], [0, -225, 0, 23], [0, -173, 0, -51], [221, 0, -85, 0], [287, 0, 61, 0], [-126, 0, 220, 0], [-175, 0, 191, 0], [0, 89, 0, 57], [489, 0, -221, 0], [-121, 0, -85, 0], [-533, 0, 59, 0], [0, -32, 0, 92], [0, 134, 0, 120], [0, -293, 0, -37], [0, 414, 0, -74], [0, -373, 0, -13], [377, 0, -57, 0], [498, 0, 102, 0], [-54, 0, 80, 0], [63, 0, -33, 0], [9, 0, -77, 0], [426, 0, -232, 0], [41, 0, -213, 0], [0, -220, 0, -36], [0, -154, 0, -180], [-283, 0, -37, 0], [-857, 0, 89, 0], [-227, 0, -93, 0], [0, 361, 0, 17], [0, 250, 0, 98], [0, 234, 0, 22], [-889, 0, -3, 0], [0, 155, 0, 47], [355, 0, -325, 0], [33, 0, 215, 0], [0, -74, 0, 106], [0, 204, 0, 78], [0, -90, 0, -170], [554, 0, 64, 0], [0, -381, 0, 59], [-1207, 0, -41, 0], [0, -401, 0, -5], [0, 178, 0, -4], [-393, 0, 321, 0], [1009, 0, 127, 0], [0, -251, 0, 45], [0, -113, 0, 131], [0, -268, 0, 88], [250, 0, -140, 0], [566, 0, 220, 0], [-227, 0, 19, 0], [0, -212, 0, -216], [-458, 0, -72, 0], [0, -482, 0, 82], [810, 0, -52, 0], [0, 548, 0, 8], [0, -33, 0, 7], [0, -262, 0, 66], [0, 193, 0, 247], [-387, 0, 299, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_171_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_171_3_c_f();".
// 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_171_3_c_f(:prec:=4)
    chi := MakeCharacter_171_c();
    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_171_3_c_f();".
// 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_171_3_c_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_171_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<2,R![16, 0, 11, 0, 1]>,<5,R![-8, -5, 1]>],Snew);
    return Vf;
end function;