
// Make newform 171.3.c.b in Magma, downloaded from the LMFDB on 28 July 2026.
// 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_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_171_3_c_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 := [13, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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 := false)
    N := 171;
    order := 2;
    char_gens := [20, 154];
    char_values := [[1, 0], [-1, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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], [-4, 0], [-5, 0], [10, 0], [0, -1], [-15, 0], [-6, -5], [-35, 0], [0, 5], [0, 10], [0, 6], [0, 10], [-20, 0], [-10, 0], [0, -21], [0, 5], [-40, 0], [0, -11], [0, 30], [105, 0], [0, 10], [40, 0], [0, 0], [0, -34], [50, 0], [0, -16], [0, 21], [0, 55], [0, -34], [0, 36], [-112, 0], [-125, 0], [50, 0], [-70, 0], [0, -10], [10, 0], [-270, 0], [0, 34], [0, -34], [0, 10], [0, -30], [-193, 0], [0, -74], [-90, 0], [123, 0], [0, 65], [0, 56], [0, 71], [-160, 0], [270, 0], [-197, 0], [0, -110], [402, 0], [0, -116], [-310, 0], [0, 30], [105, 0], [-50, 0], [0, -80], [-320, 0], [0, -61], [0, -66], [-395, 0], [125, 0], [0, 1], [0, 55], [0, -16], [40, 0], [98, 0], [185, 0], [225, 0], [50, 0], [0, 121], [0, 135], [0, 56], [478, 0], [750, 0], [0, -80], [0, 10], [-112, 0], [0, -175], [0, -120], [0, 204], [0, -220], [-670, 0], [0, -10], [-755, 0], [-772, 0], [-350, 0], [-70, 0], [370, 0], [0, 144], [632, 0], [380, 0], [45, 0], [0, 230], [0, 170], [0, -129], [-600, 0], [0, 166], [-380, 0], [0, 34], [0, 10], [-790, 0], [675, 0], [280, 0], [-750, 0], [0, 140], [0, -170], [0, 274], [-1200, 0], [350, 0], [560, 0], [-1050, 0], [0, 340], [-1030, 0], [-555, 0], [-50, 0], [0, 55], [0, -55], [0, -166], [0, 19], [0, -66], [-820, 0], [540, 0], [268, 0], [-105, 0], [695, 0], [160, 0], [1028, 0], [0, -146], [0, -10], [60, 0], [655, 0], [-185, 0], [0, -89], [0, -19], [0, 349], [7, 0], [0, -285], [-842, 0], [775, 0], [0, -351], [0, 435], [0, -320], [400, 0], [0, 44], [1432, 0], [0, -36], [0, -109], [-750, 0], [120, 0], [0, 236], [0, 179], [0, -340], [-513, 0], [1217, 0], [-1095, 0], [0, 305], [-640, 0], [0, 324], [-1010, 0], [0, -210], [0, 166], [0, 16], [0, 0], [-170, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_171_c_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_171_3_c_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_171_3_c_b(:prec:=2)
    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_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_171_3_c_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_171_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<2,R![13, 0, 1]>,<5,R![4, 1]>],Snew);
    return Vf;
end function;