
// Make newform 960.3.l.f in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_960_l();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_960_l_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_l_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_960_3_l_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 := [9, 0, -4, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-2, 0, 1, 0], [0, -2, 0, 1]];
    Rf_basisdens := [1, 1, 1, 1];
    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_l();"
function MakeCharacter_960_l()
    N := 960;
    order := 2;
    char_gens := [511, 901, 641, 577];
    v := [2, 2, 1, 2];
    // 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_l_Hecke();"
function MakeCharacter_960_l_Hecke(Kf)
    N := 960;
    order := 2;
    char_gens := [511, 901, 641, 577];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [-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+1)]];
    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], [1, -1, -1, 0], [0, 0, -1, 0], [-2, -5, 0, 1], [0, 2, 0, 2], [10, 0, 0, 0], [0, -4, 6, -4], [8, -10, 0, 2], [0, 1, 6, 1], [0, 0, -12, 0], [-8, 0, 0, 0], [22, 20, 0, -4], [0, 8, -6, 8], [14, 15, 0, -3], [0, 5, 30, 5], [0, 4, -6, 4], [0, -12, -12, -12], [16, -20, 0, 4], [-82, 15, 0, -3], [0, -10, -12, -10], [50, -20, 0, 4], [28, -10, 0, 2], [0, 3, -6, 3], [0, 8, -12, 8], [74, 20, 0, -4], [0, 24, 24, 24], [-74, 35, 0, -7], [0, -7, 6, -7], [-20, 60, 0, -12], [0, 24, 42, 24], [106, -15, 0, 3], [0, -14, 72, -14], [0, 32, 18, 32], [56, 70, 0, -14], [0, 24, -78, 24], [-56, -20, 0, 4], [10, 60, 0, -12], [74, 45, 0, -9], [0, -31, 54, -31], [0, 32, 66, 32], [0, 28, 84, 28], [-38, -120, 0, 24], [0, 6, 60, 6], [-130, 20, 0, -4], [0, -12, 54, -12], [-92, -30, 0, 6], [-172, -60, 0, 12], [-14, 45, 0, -9], [0, -29, 42, -29], [-98, 120, 0, -24], [0, -28, -30, -28], [0, 84, -24, 84], [104, -180, 0, 36], [0, -18, 72, -18], [0, -72, 6, -72], [0, 37, 54, 37], [0, -96, 54, -96], [400, -40, 0, 8], [-302, -60, 0, 12], [0, 48, 102, 48], [170, -15, 0, 3], [0, -80, 78, -80], [74, 45, 0, -9], [0, -78, 84, -78], [-274, -120, 0, 24], [0, 44, 66, 44], [-28, 220, 0, -44], [98, -120, 0, 24], [0, 47, 66, 47], [-206, -160, 0, 32], [0, 8, 270, 8], [0, -52, 24, -52], [22, 255, 0, -51], [454, -20, 0, 4], [-88, 90, 0, -18], [0, -101, 42, -101], [0, 8, -258, 8], [214, 0, 0, 0], [0, 32, 264, 32], [-460, 60, 0, -12], [0, -80, -108, -80], [-224, 260, 0, -52], [0, 126, 108, 126], [146, -180, 0, 36], [-428, -90, 0, 18], [0, 9, -114, 9], [0, 40, -186, 40], [-130, 360, 0, -72], [0, 80, -96, 80], [-110, -295, 0, 59], [0, 61, 102, 61], [0, -148, 48, -148], [178, 45, 0, -9], [0, 22, -360, 22], [80, -30, 0, 6], [0, -57, 90, -57], [0, -32, -156, -32], [0, -64, -120, -64], [350, -325, 0, 65], [262, 40, 0, -8], [-262, 25, 0, -5], [0, 80, 150, 80], [0, -65, -198, -65], [0, 40, 6, 40], [-220, -40, 0, 8], [-226, -400, 0, 80], [0, 67, 42, 67], [0, 72, -18, 72], [0, 128, -72, 128], [680, 180, 0, -36], [826, 5, 0, -1], [-302, -280, 0, 56], [0, 180, -138, 180], [128, 110, 0, -22], [304, 420, 0, -84], [0, 32, 90, 32], [-514, -45, 0, 9], [0, -49, 114, -49], [0, -116, -30, -116], [0, -100, -108, -100], [520, 20, 0, -4], [602, 220, 0, -44], [0, 52, -378, 52], [0, -31, 294, -31], [932, 0, 0, 0], [0, -88, 438, -88], [-110, -200, 0, 40], [0, 76, -96, 76], [634, -255, 0, 51], [334, 180, 0, -36], [-880, -30, 0, 6], [0, -53, -102, -53], [-920, 140, 0, -28], [370, -260, 0, 52], [0, 8, -240, 8], [-514, 120, 0, -24], [0, 188, 78, 188], [914, 305, 0, -61], [0, 76, -102, 76], [0, -64, -372, -64], [-124, 60, 0, -12], [0, -32, -90, -32], [190, 375, 0, -75], [0, -179, 102, -179], [-68, 140, 0, -28], [0, 96, 240, 96], [58, 600, 0, -120], [0, -232, -126, -232], [-376, -50, 0, 10], [0, 83, -30, 83], [-662, 220, 0, -44], [0, 72, -198, 72], [386, 665, 0, -133], [0, 59, -342, 59], [926, -305, 0, 61], [0, -154, 156, -154], [268, -270, 0, 54], [0, -8, 474, -8], [-370, 20, 0, -4], [0, 272, 216, 272], [0, 47, -630, 47], [0, 224, 114, 224], [1438, 215, 0, -43], [0, -262, 24, -262], [0, 28, 378, 28], [0, 175, -126, 175], [280, -800, 0, 160], [-446, 240, 0, -48]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_960_l_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_960_3_l_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_960_3_l_f(:prec:=4)
    chi := MakeCharacter_960_l();
    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_l_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_960_3_l_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_960_l();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<7,R![-86, 4, 1]>],Snew);
    return Vf;
end function;