
// Make newform 464.2.u.b in Magma, downloaded from the LMFDB on 25 June 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_464_u();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_464_u_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_464_2_u_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_464_2_u_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, -1, 1, -1, 1, -1, 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_464_u();"
function MakeCharacter_464_u()
    N := 464;
    order := 7;
    char_gens := [175, 117, 321];
    v := [7, 7, 6];
    // 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_464_u_Hecke();"
function MakeCharacter_464_u_Hecke( : Kf := false)
    N := 464;
    order := 7;
    char_gens := [175, 117, 321];
    char_values := [[1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [0, -1, 0, 0, 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 := [[0, 0, 0, 0, 0, 0], [-1, 1, 0, 0, -1, 1], [0, 0, 0, -1, 2, -1], [1, -1, -1, 1, 1, -1], [3, -3, 3, -3, 3, 0], [3, -1, 2, -1, 3, 0], [-3, 0, -1, 3, -3, 1], [3, -5, 3, 0, 2, -2], [0, 4, 1, 1, 1, 4], [3, -3, -1, -3, -1, 1], [-3, 3, 0, 5, -1, 5], [-1, -4, 4, 1, 0, 0], [1, 0, 4, -5, 5, -4], [0, 3, -3, -5, -3, 3], [0, -1, 1, -1, 0, 0], [-3, 3, 0, 0, 7, 0], [5, 0, -1, 3, -3, 1], [5, 2, -1, 1, -2, -5], [0, 0, 0, 0, 0, -11], [-3, -4, 0, -4, -3, 0], [0, -3, -4, 3, -4, -3], [0, -2, 2, 0, 0, 4], [-5, -4, -5, 0, 3, -3], [2, -2, 0, -2, -7, -2], [6, -2, 6, 0, 4, -4], [0, 5, -4, -5, -4, 5], [4, 3, 4, 3, 4, 0], [11, -5, 5, -11, 0, -5], [-4, 6, -7, 7, -6, 4], [-13, 8, -5, 5, -8, 13], [-11, 13, -7, 13, -11, 0], [0, 0, -7, -2, -7, 0], [3, -3, 3, -3, 0, 4], [0, -7, 10, -8, 10, -7], [1, 0, 1, 0, 11, -11], [0, -2, 0, -7, 0, -2], [2, 0, 7, -3, 3, -7], [-3, 5, 11, 5, -3, 0], [-11, 7, -6, 6, -7, 11], [3, 0, -5, 7, -7, 5], [-6, 6, 0, 5, 1, 5], [4, -14, 7, -7, 14, -4], [-4, 0, 1, -1, 1, -1], [3, 11, 3, 0, -3, 3], [0, 11, 1, -1, 1, 11], [17, -9, 17, 0, 11, -11], [10, -6, 6, -10, 0, -7], [4, 1, -1, -4, 0, 4], [0, 0, 0, 8, -1, 8], [-1, -6, -5, 5, 6, 1], [-1, 0, -3, 9, -9, 3], [0, -1, 3, -3, 1, 0], [-4, -5, 5, 4, 0, -1], [3, 4, 3, 0, 13, -13], [-8, 12, -8, 0, -7, 7], [0, 0, 0, 6, -13, 6], [14, 0, 0, -14, 0, -13], [13, 1, 13, 0, 16, -16], [-9, 13, 2, 13, -9, 0], [4, -5, 5, -4, 0, -18], [-21, 25, -24, 24, -25, 21], [-21, 22, -16, 16, -22, 21], [-8, 0, -24, 17, -17, 24], [14, -6, 6, -14, 0, -9], [0, -3, 8, 18, 8, -3], [12, -12, 0, -9, 14, -9], [7, 0, 3, 1, -1, -3], [-17, 3, -5, 3, -17, 0], [8, 0, -10, 10, -10, 10], [26, 0, 6, -10, 10, -6], [-13, 13, 0, 16, -14, 16], [-2, 2, -22, 2, -2, 0], [4, -3, 4, 0, -1, 1], [-10, -1, -10, 0, 9, -9], [-1, 1, 0, -5, -15, -5], [0, 0, -11, -7, -11, 0], [-7, 0, -12, 8, -8, 12], [19, -13, 13, -19, 0, -11], [2, -2, 0, 14, -9, 14], [17, 4, -2, 2, -4, -17], [-6, 23, -12, 23, -6, 0], [0, 0, -18, -8, -18, 0], [-13, 8, -13, 0, -15, 15], [-11, 11, 0, 9, -25, 9], [6, -10, 6, 0, 21, -21], [-21, 2, -2, 21, 0, -6], [0, -27, 15, -15, 15, -27], [7, -15, 8, -8, 15, -7], [-8, 4, -3, 3, -4, 8], [3, 0, 10, -21, 21, -10], [20, -8, 20, -20, 8, -20], [0, -14, -8, 1, -8, -14], [0, 4, 2, -9, 2, 4], [12, -12, 0, -11, 5, -11], [0, 5, -7, -4, -7, 5], [13, -8, 13, 0, 17, -17], [-2, -1, 30, -1, -2, 0], [20, 0, 2, -12, 12, -2], [-22, 0, -5, 13, -13, 5], [9, -6, 9, 0, 5, -5], [1, -6, 1, 0, -17, 17], [0, -11, -10, -1, -10, -11], [9, 0, 18, -3, 3, -18], [12, 7, -12, 7, 12, 0], [15, -10, 10, -15, 0, 21], [13, -12, 23, -23, 12, -13], [-2, -8, -6, 6, 8, 2], [12, 4, 9, 4, 12, 0], [7, 1, 7, 0, -13, 13], [14, 7, -7, -14, 0, 3], [-3, 3, 0, 5, 29, 5], [-1, -26, -1, 0, 6, -6], [-12, 12, -12, 12, 0, 3], [-2, -15, -2, 0, -7, 7], [-8, -25, 14, -14, 25, 8], [10, -8, -7, 7, 8, -10], [-18, 18, 0, 0, -1, 0], [7, 0, 0, -7, 0, -6], [0, 5, -9, 35, -9, 5], [-23, 36, -36, 23, 0, 36], [0, 12, 11, -6, 11, 12], [0, 10, -2, -32, -2, 10], [11, -12, 11, 0, 23, -23], [2, 17, 11, 17, 2, 0], [23, -23, 0, -7, -1, -7], [6, -6, 0, -4, 11, -4], [11, -17, 21, -17, 11, 0], [0, 29, -20, 29, -20, 29], [-6, 6, 0, -6, 24, -6], [-8, 17, -17, 8, 0, 19], [0, -27, 15, -27, 15, -27], [21, -12, -10, -12, 21, 0], [-16, 20, -29, 29, -20, 16], [-1, 3, -10, 10, -3, 1], [9, 31, -3, 3, -31, -9], [0, -9, -10, 20, -10, -9], [9, 23, 9, 0, -4, 4], [29, -37, 29, 0, 24, -24], [0, -13, 10, -22, 10, -13], [-12, -1, -26, 26, 1, 12], [-12, 0, -17, 1, -1, 17], [1, -20, 20, -1, 0, -12], [9, -17, 6, -17, 9, 0], [0, -7, -7, -5, -7, -7], [12, 0, -24, 17, -17, 24], [-14, 14, 0, 23, -21, 23], [-10, 0, -6, -5, 5, 6], [4, -16, 26, -16, 4, 0], [-27, 9, -17, 9, -27, 0], [-16, 19, -10, 10, -19, 16], [-2, -5, 24, -24, 5, 2], [-18, 9, -2, 9, -18, 0], [12, -34, -8, -34, 12, 0], [-2, 0, 16, -10, 10, -16], [15, -5, 5, -15, 0, -30], [-16, 0, -7, 8, -8, 7], [15, 19, -19, -15, 0, 7], [10, 0, 5, 8, -8, -5], [-40, 22, -22, 40, 0, 16], [-18, 20, -16, 20, -18, 0], [-6, 23, -6, 0, -22, 22], [-3, 42, -3, 0, -7, 7], [7, 19, 7, 0, 8, -8], [0, -3, 4, -31, 4, -3], [8, -3, 3, -8, 0, 20], [-3, -11, -31, 31, 11, 3], [5, -5, 0, -29, 32, -29], [10, 3, -40, 3, 10, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_464_u_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_464_2_u_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_464_2_u_b(:prec:=6)
    chi := MakeCharacter_464_u();
    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_464_2_u_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_464_2_u_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_464_u();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, 5, 11, 13, 9, 3, 1]>],Snew);
    return Vf;
end function;