
// Make newform 960.3.j.e in Magma, downloaded from the LMFDB on 01 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_960_j();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_960_j_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_960_3_j_e();".
// 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_960_3_j_e();".
// 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 := [256, 0, 80, 0, 24, 0, 5, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 4, 0, 1, 0, 1, 0, 0], [-64, -16, 0, -24, -12, -5, 4, -1], [192, 144, 32, 24, 20, 5, 4, 1], [0, -8, 0, -22, 0, -3, 0, -1], [-192, 144, -32, 24, -20, 5, -4, 1], [80, 0, 40, 0, 5, 0, 1, 0], [0, -16, 0, 24, 0, -9, 0, -5]];
    Rf_basisdens := [1, 16, 64, 64, 32, 64, 8, 32];
    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_960_j();"
function MakeCharacter_960_j()
    N := 960;
    order := 2;
    char_gens := [511, 901, 641, 577];
    v := [1, 2, 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_960_j_Hecke();"
function MakeCharacter_960_j_Hecke(Kf)
    N := 960;
    order := 2;
    char_gens := [511, 901, 641, 577];
    char_values := [[-1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 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]];
    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, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 1, 0, 0, 0, 0], [0, -3, 0, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, -1, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1], [0, 2, 4, 0, -2, 0, 0, 0], [0, 6, 0, 0, -2, 0, 0, 0], [26, 0, 0, -3, 0, 3, 0, 0], [0, 2, 4, 0, -2, 0, 2, 0], [0, 0, 0, 3, 0, 3, 0, 0], [-26, 0, 0, -6, 0, 6, 0, 0], [0, -4, 0, 0, -8, 0, 0, 0], [0, 12, 0, 0, -8, 0, 0, 0], [0, 0, 0, 3, 0, 3, 0, 0], [0, 8, 16, 0, -8, 0, -1, 0], [-38, 0, 0, 0, 0, 0, 0, 0], [0, 18, 0, 0, -14, 0, 0, 0], [0, 4, 8, 0, -4, 0, -6, 0], [0, 0, 0, 0, 0, 0, 0, 2], [0, -2, -4, 0, 2, 0, -2, 0], [0, 24, 0, 0, -4, 0, 0, 0], [66, 0, 0, 4, 0, -4, 0, 0], [0, 0, 0, -22, 0, -22, 0, -2], [-38, 0, 0, 1, 0, -1, 0, 0], [0, -5, 0, 0, -31, 0, 0, 0], [0, 20, 0, 0, -16, 0, 0, 0], [66, 0, 0, 10, 0, -10, 0, 0], [0, 0, 0, 2, 0, 2, 0, -1], [0, 53, 0, 0, 23, 0, 0, 0], [0, 4, 8, 0, -4, 0, -1, 0], [0, 0, 0, 16, 0, 16, 0, -5], [0, -2, -4, 0, 2, 0, -2, 0], [14, 0, 0, -3, 0, 3, 0, 0], [0, 2, 4, 0, -2, 0, 10, 0], [0, 0, 0, 17, 0, 17, 0, -4], [0, -66, 0, 0, 22, 0, 0, 0], [0, 42, 0, 0, -38, 0, 0, 0], [0, 0, 0, 11, 0, 11, 0, -2], [0, -20, -40, 0, 20, 0, 3, 0], [30, 0, 0, 22, 0, -22, 0, 0], [0, -12, -24, 0, 12, 0, 2, 0], [0, 0, 0, -14, 0, -14, 0, -4], [0, 0, 0, 3, 0, 3, 0, -10], [0, 18, 36, 0, -18, 0, -6, 0], [0, -6, -12, 0, 6, 0, 10, 0], [0, -129, 0, 0, 29, 0, 0, 0], [0, 64, 0, 0, 68, 0, 0, 0], [-146, 0, 0, 18, 0, -18, 0, 0], [0, 0, 0, 6, 0, 6, 0, 1], [0, -12, -24, 0, 12, 0, -14, 0], [158, 0, 0, 36, 0, -36, 0, 0], [0, -4, -8, 0, 4, 0, -9, 0], [0, 0, 0, 26, 0, 26, 0, 9], [0, 62, 0, 0, 22, 0, 0, 0], [150, 0, 0, 11, 0, -11, 0, 0], [0, -22, -44, 0, 22, 0, -18, 0], [0, 0, 0, -29, 0, -29, 0, 16], [-146, 0, 0, 12, 0, -12, 0, 0], [0, -114, 0, 0, -26, 0, 0, 0], [0, 0, 0, 37, 0, 37, 0, 2], [0, 64, 0, 0, 52, 0, 0, 0], [0, 16, 32, 0, -16, 0, 22, 0], [0, 0, 0, 2, 0, 2, 0, -4], [0, 0, 0, 17, 0, 17, 0, 14], [0, 2, 4, 0, -2, 0, -16, 0], [0, 0, 0, -20, 0, -20, 0, -10], [0, 136, 0, 0, -52, 0, 0, 0], [-122, 0, 0, 12, 0, -12, 0, 0], [0, 0, 0, 22, 0, 22, 0, 5], [0, -24, -48, 0, 24, 0, 0, 0], [0, 157, 0, 0, 47, 0, 0, 0], [0, 0, 0, -25, 0, -25, 0, -16], [0, 26, 52, 0, -26, 0, 30, 0], [0, -256, 0, 0, 52, 0, 0, 0], [62, 0, 0, -63, 0, 63, 0, 0], [0, 0, 0, -33, 0, -33, 0, -12], [122, 0, 0, -34, 0, 34, 0, 0], [-122, 0, 0, 58, 0, -58, 0, 0], [0, 12, 24, 0, -12, 0, -7, 0], [-190, 0, 0, 14, 0, -14, 0, 0], [0, -28, -56, 0, 28, 0, 12, 0], [0, 0, 0, -66, 0, -66, 0, 8], [0, -30, -60, 0, 30, 0, -38, 0], [0, -132, 0, 0, -88, 0, 0, 0], [22, 0, 0, 18, 0, -18, 0, 0], [0, 0, 0, 62, 0, 62, 0, 6], [-334, 0, 0, 3, 0, -3, 0, 0], [0, -95, 0, 0, 59, 0, 0, 0], [0, 360, 0, 0, -4, 0, 0, 0], [0, 0, 0, 0, 0, 0, 14, 0], [0, 315, 0, 0, -79, 0, 0, 0], [0, 0, 0, 0, 0, 0, 35, 0], [0, -18, -36, 0, 18, 0, -44, 0], [0, 40, 0, 0, -116, 0, 0, 0], [-70, 0, 0, 21, 0, -21, 0, 0], [-726, 0, 0, -2, 0, 2, 0, 0], [0, -146, 0, 0, -26, 0, 0, 0], [462, 0, 0, -74, 0, 74, 0, 0], [0, 132, 0, 0, 64, 0, 0, 0], [0, 0, 0, 27, 0, 27, 0, -34], [0, -168, 0, 0, -148, 0, 0, 0], [-10, 0, 0, -10, 0, 10, 0, 0], [0, 74, 148, 0, -74, 0, -8, 0], [0, 0, 0, 60, 0, 60, 0, -4], [0, 156, 0, 0, -40, 0, 0, 0], [0, 0, 0, 114, 0, 114, 0, -1], [0, 80, 160, 0, -80, 0, -8, 0], [502, 0, 0, -26, 0, 26, 0, 0], [0, 39, 0, 0, -187, 0, 0, 0], [0, 0, 0, -15, 0, -15, 0, 20], [0, 0, 0, -58, 0, -58, 0, 19], [0, -10, -20, 0, 10, 0, 50, 0], [0, -18, -36, 0, 18, 0, 30, 0], [-650, 0, 0, -54, 0, 54, 0, 0], [0, 306, 0, 0, -14, 0, 0, 0], [0, 76, 0, 0, -8, 0, 0, 0], [0, 0, 0, 49, 0, 49, 0, -32], [0, 8, 16, 0, -8, 0, 49, 0], [-70, 0, 0, 44, 0, -44, 0, 0], [0, 0, 0, -10, 0, -10, 0, 28], [0, 0, 0, 35, 0, 35, 0, 12], [0, -468, 0, 0, 104, 0, 0, 0], [0, 50, 100, 0, -50, 0, -4, 0], [-234, 0, 0, 131, 0, -131, 0, 0], [94, 0, 0, 78, 0, -78, 0, 0], [0, 44, 88, 0, -44, 0, 0, 0], [0, 267, 0, 0, -31, 0, 0, 0], [0, 0, 0, 89, 0, 89, 0, -16], [0, 14, 28, 0, -14, 0, 0, 0], [0, 96, 0, 0, 4, 0, 0, 0], [0, 34, 68, 0, -34, 0, -10, 0], [0, 0, 0, -9, 0, -9, 0, 28], [-518, 0, 0, 90, 0, -90, 0, 0], [122, 0, 0, 54, 0, -54, 0, 0], [0, 0, 0, -153, 0, -153, 0, 10], [0, -394, 0, 0, -146, 0, 0, 0], [0, 0, 0, -119, 0, -119, 0, 22], [-38, 0, 0, 42, 0, -42, 0, 0], [0, -86, -172, 0, 86, 0, 18, 0], [-682, 0, 0, -57, 0, 57, 0, 0], [0, 393, 0, 0, 131, 0, 0, 0], [0, -52, 0, 0, -184, 0, 0, 0], [-366, 0, 0, -74, 0, 74, 0, 0], [0, -80, -160, 0, 80, 0, 44, 0], [0, 0, 0, 1, 0, 1, 0, -48], [0, 0, 0, 28, 0, 28, 0, 21], [0, 26, 52, 0, -26, 0, -74, 0], [0, -386, 0, 0, -170, 0, 0, 0], [0, 0, 0, 161, 0, 161, 0, 20], [-162, 0, 0, 148, 0, -148, 0, 0], [0, 466, 0, 0, -62, 0, 0, 0], [0, -478, 0, 0, -14, 0, 0, 0], [0, 168, 0, 0, 316, 0, 0, 0], [0, 72, 144, 0, -72, 0, 56, 0], [0, -70, -140, 0, 70, 0, -18, 0], [1334, 0, 0, -22, 0, 22, 0, 0], [0, 0, 0, 132, 0, 132, 0, -16], [-1246, 0, 0, 63, 0, -63, 0, 0], [0, 84, 0, 0, 32, 0, 0, 0], [0, 0, 0, 36, 0, 36, 0, 33], [0, -591, 0, 0, 155, 0, 0, 0], [0, -36, -72, 0, 36, 0, -33, 0], [0, 0, 0, 116, 0, 116, 0, 9], [0, 468, 0, 0, 208, 0, 0, 0], [0, -58, -116, 0, 58, 0, 42, 0], [0, 0, 0, 37, 0, 37, 0, -76]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_960_j_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_960_3_j_e();".
// 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_960_3_j_e(:prec:=8)
    chi := MakeCharacter_960_j();
    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_960_3_j_e();".
// 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_960_3_j_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_960_j();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<7,R![64, 0, -92, 0, 1]>,<11,R![24576, 0, 348, 0, 1]>],Snew);
    return Vf;
end function;