
// Make newform 510.2.w.b in Magma, downloaded from the LMFDB on 15 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_510_w();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_510_w_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_510_2_w_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_510_2_w_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 := [1, 0, 0, 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_510_w();"
function MakeCharacter_510_w()
    N := 510;
    order := 8;
    char_gens := [341, 307, 241];
    v := [4, 2, 7];
    // 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_510_w_Hecke();"
function MakeCharacter_510_w_Hecke( : Kf := false)
    N := 510;
    order := 8;
    char_gens := [341, 307, 241];
    char_values := [[-1, 0, 0, 0], [0, 0, -1, 0], [0, 1, 0, 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 := 2;
    raw_aps := [[1, 0, 0, 0], [-1, 1, 1, 0], [-2, 0, -1, 0], [-2, -2, -1, 1], [-3, 0, 0, -3], [-3, 0, 3, 2], [-3, -2, 0, -2], [-1, 6, -1, 0], [-2, 2, 1, 1], [-2, 2, -1, -1], [0, 0, -1, 1], [2, -1, 1, -2], [-1, 4, -4, 1], [0, -4, -6, -4], [-3, 0, 3, 4], [0, -2, 4, -2], [-5, -2, -5, 0], [6, -1, -1, 6], [-3, 0, 3, 2], [4, -4, 5, 5], [-3, -6, 6, 3], [-8, -5, -5, -8], [0, -4, -6, -4], [0, 10, 0, 10], [-5, 0, 0, -5], [0, 2, 8, 2], [-7, 8, -7, 0], [-3, -2, 2, 3], [-6, 5, -5, 6], [5, -5, 4, 4], [4, -4, 0, 4], [-5, -5, 0, 0], [-1, 12, -1, 0], [1, -1, -4, -4], [0, -10, -8, -10], [1, 0, -1, -8], [9, 10, 9, 0], [-3, 3, -6, -6], [-12, 3, 3, -12], [-6, -7, 7, 6], [-3, 6, -3, 0], [-6, 9, -9, 6], [-12, -4, 0, 4], [9, 9, -10, 10], [6, -6, 1, 1], [0, 0, 11, -11], [9, -6, -6, 9], [-12, -8, 0, 8], [1, 1, 4, -4], [-13, 0, 13, -2], [1, 4, 4, 1], [-12, 0, 0, 0], [-5, -5, 14, -14], [12, -2, 0, 2], [6, -2, 0, 2], [-8, 0, 0, 0], [-10, -10, -17, 17], [-12, 8, 0, -8], [-8, 19, 19, -8], [-11, 0, 11, -12], [9, 9, 2, -2], [7, -2, 7, 0], [-9, 0, 9, -18], [16, 16, -3, 3], [7, -7, -8, -8], [-16, -16, 11, -11], [-3, -2, -3, 0], [-1, 12, -12, 1], [7, -7, 14, 14], [-5, 0, 5, -6], [-11, -8, -11, 0], [3, 0, -3, -8], [-2, -2, -1, 1], [-7, -6, -7, 0], [-15, -15, -12, 12], [16, 8, 0, -8], [1, 0, -1, -2], [-12, 12, -11, -11], [-3, 10, -10, 3], [0, 2, 8, 2], [-3, -12, 12, 3], [0, -10, -24, -10], [24, 9, 9, 24], [0, 12, 12, 12], [-2, 2, 13, 13], [15, 0, -15, -10], [-5, 8, 8, -5], [0, 4, 4, 4], [-7, -18, -7, 0], [11, 0, -11, 4], [0, -6, -18, -6], [14, 14, -5, 5], [0, 0, 5, 5], [17, 2, 17, 0], [1, 18, 18, 1], [12, 12, 3, -3], [30, 4, 0, -4], [9, 10, 10, 9], [-7, 0, 7, 2], [-16, -16, -13, 13], [21, -12, 12, -21], [-1, 0, 1, 14], [0, 12, -22, 12], [3, 8, 3, 0], [-5, -4, -4, -5], [5, 0, -5, 24], [0, -2, -34, -2], [0, -10, -8, -10], [0, -12, 2, -12], [7, -18, 18, -7], [-8, 5, 5, -8], [-7, -10, -7, 0], [-27, 8, -8, 27], [-3, -22, 22, 3], [-23, 0, 23, -4], [-1, -1, 8, -8], [13, -20, -20, 13], [-21, 8, -21, 0], [18, 18, -7, 7], [0, -6, 26, -6], [-5, 0, 5, 2], [-7, 7, -2, -2], [-8, 5, 5, -8], [-27, 6, -6, 27], [-15, 14, -14, 15], [-18, 4, 0, -4], [-4, -4, -5, 5], [8, -8, 25, 25], [-15, -8, -15, 0], [2, 32, 0, -32], [21, 0, -21, 10], [-4, -17, -17, -4], [-10, -10, 17, -17], [2, 8, 0, -8], [-10, -12, 0, 12], [-26, -16, 0, 16], [-46, 0, 0, 0], [9, -16, -16, 9], [-14, 4, 0, -4], [11, 12, 12, 11], [11, -11, 2, 2], [8, 8, 25, -25], [10, -10, -21, -21], [11, 11, 2, -2], [-26, 4, 0, -4], [-26, 9, -9, 26], [-18, 13, 13, -18], [5, -5, 28, 28], [-5, 0, 5, 18], [15, -12, 15, 0], [-16, -16, -19, 19], [19, -19, 4, 4], [-9, -14, -9, 0], [-20, 11, 11, -20], [-35, 35, 10, 10], [-8, 11, -11, 8], [0, 8, -30, 8], [-11, -6, 6, 11], [0, -4, -12, -4], [4, 3, 3, 4], [5, -6, 6, -5], [1, 0, -1, -40], [0, -24, 10, -24], [-5, 18, -5, 0], [0, 18, -32, 18], [20, 5, -5, -20], [2, -2, 9, 9], [4, -4, -31, -31]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_510_w_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_510_2_w_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_510_2_w_b(:prec:=4)
    chi := MakeCharacter_510_w();
    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_510_2_w_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_510_2_w_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_510_w();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![2, -4, 18, 8, 1]>,<11,R![162, 108, 54, 12, 1]>],Snew);
    return Vf;
end function;