
// Make newform 108.3.d.d in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_108_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_108_d_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_108_3_d_d();".
// 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_108_3_d_d();".
// 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 := [16, 0, 24, 0, 32, 0, 6, 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, 72, 0, 128, 0, 16, 0, 3], [0, -88, 0, -256, 0, -48, 0, -9], [88, 0, 96, 0, 24, 0, 3, 0], [-120, 0, -288, 0, -48, 0, -9, 0], [56, 0, -160, 0, -32, 0, -7, 0], [0, 104, 0, -896, 0, -176, 0, -33], [0, 792, 0, 1088, 0, 208, 0, 33]];
    Rf_basisdens := [1, 64, 64, 16, 32, 32, 64, 64];
    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_108_d();"
function MakeCharacter_108_d()
    N := 108;
    order := 2;
    char_gens := [55, 29];
    v := [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_108_d_Hecke();"
function MakeCharacter_108_d_Hecke(Kf)
    N := 108;
    order := 2;
    char_gens := [55, 29];
    char_values := [[-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+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, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -1, -1, 0, 0, 0, 0, 0], [-1, 0, 0, -1, 0, 1, 0, 0], [0, 1, 0, 0, 0, 0, -1, -2], [-2, 0, 0, 1, -1, 3, 0, 0], [0, -3, 3, 0, 0, 0, 2, 0], [0, 0, 0, 0, 1, 0, 0, 0], [0, 1, 10, 0, 0, 0, -3, -2], [0, -4, -4, 0, 0, 0, 0, 0], [4, 0, 0, 4, -4, -4, 0, 0], [-6, 0, 0, -1, 1, -3, 0, 0], [0, 0, -12, 0, 0, 0, -4, 0], [2, 0, 0, 2, 6, -2, 0, 0], [0, -3, 10, 0, 0, 0, 1, 6], [0, 10, 22, 0, 0, 0, 4, 0], [0, -3, 20, 0, 0, 0, -1, 6], [4, 0, 0, 1, -1, 3, 0, 0], [-8, 0, 0, -8, 9, 8, 0, 0], [0, -2, -40, 0, 0, 0, 10, 4], [27, 0, 0, 2, -2, 6, 0, 0], [-7, 0, 0, -7, -16, 7, 0, 0], [0, -2, 0, 0, 0, 0, 2, 4], [0, -3, 3, 0, 0, 0, 2, 0], [-39, 0, 0, 8, -8, 24, 0, 0], [0, -12, -24, 0, 0, 0, -4, 0], [1, 0, 0, 1, -10, -1, 0, 0], [0, -5, 20, 0, 0, 0, 1, 10], [-72, 0, 0, 2, -2, 6, 0, 0], [0, -7, 11, 0, 0, 0, 6, 0], [6, 0, 0, 6, -30, -6, 0, 0], [0, 8, 0, 0, 0, 0, -8, -16], [0, -1, 17, 0, 0, 0, 6, 0], [-2, 0, 0, -2, 33, 2, 0, 0], [0, -22, 14, 0, 0, 0, 12, 0], [17, 0, 0, 17, -16, -17, 0, 0], [52, 0, 0, 10, -10, 30, 0, 0], [-12, 0, 0, -12, 11, 12, 0, 0], [0, -3, -70, 0, 0, 0, 17, 6], [0, -16, 20, 0, 0, 0, 12, 0], [0, 10, 20, 0, 0, 0, -14, -20], [-178, 0, 0, 9, -9, 27, 0, 0], [0, -9, 10, 0, 0, 0, 7, 18], [-119, 0, 0, 10, -10, 30, 0, 0], [0, 27, -57, 0, 0, 0, -28, 0], [-15, 0, 0, -15, 6, 15, 0, 0], [34, 0, 0, 34, -11, -34, 0, 0], [-16, 0, 0, -16, -32, 16, 0, 0], [0, -10, -100, 0, 0, 0, 30, 20], [202, 0, 0, 4, -4, 12, 0, 0], [0, 48, 36, 0, 0, 0, -4, 0], [0, -6, 20, 0, 0, 0, 2, 12], [-41, 0, 0, -26, 26, -78, 0, 0], [0, -12, 60, 0, 0, 0, 0, 24], [0, 62, 62, 0, 0, 0, 0, 0], [0, -2, 60, 0, 0, 0, -10, 4], [0, -39, -87, 0, 0, 0, -16, 0], [33, 0, 0, 33, -40, -33, 0, 0], [-24, 0, 0, 2, -2, 6, 0, 0], [0, -4, -4, 0, 0, 0, 0, 0], [22, 0, 0, 22, 34, -22, 0, 0], [0, -31, -103, 0, 0, 0, -24, 0], [-42, 0, 0, -42, -30, 42, 0, 0], [0, 7, 10, 0, 0, 0, -9, -14], [-35, 0, 0, 22, -22, 66, 0, 0], [0, 74, 2, 0, 0, 0, -24, 0], [2, 0, 0, 2, 13, -2, 0, 0], [-27, 0, 0, -28, 28, -84, 0, 0], [0, -6, 200, 0, 0, 0, -34, 12], [216, 0, 0, 5, -5, 15, 0, 0], [0, -90, -30, 0, 0, 0, 20, 0], [0, 19, -10, 0, 0, 0, -17, -38], [-7, 0, 0, -7, -42, 7, 0, 0], [438, 0, 0, 11, -11, 33, 0, 0], [18, 0, 0, 18, 15, -18, 0, 0], [0, 44, -60, 0, 0, 0, -32, -88], [0, -69, 63, 0, 0, 0, 44, 0], [-348, 0, 0, 26, -26, 78, 0, 0], [0, 26, -82, 0, 0, 0, -36, 0], [-395, 0, 0, -20, 20, -60, 0, 0], [0, 5, 180, 0, 0, 0, -41, -10], [-206, 0, 0, 13, -13, 39, 0, 0], [0, 29, -170, 0, 0, 0, 5, -58], [492, 0, 0, -22, 22, -66, 0, 0], [-28, 0, 0, -28, 60, 28, 0, 0], [0, -30, -20, 0, 0, 0, 34, 60], [0, -53, 121, 0, 0, 0, 58, 0], [64, 0, 0, -14, 14, -42, 0, 0], [0, 45, -3, 0, 0, 0, -16, 0], [67, 0, 0, 67, -28, -67, 0, 0], [0, -39, 60, 0, 0, 0, 27, 78], [0, 38, 80, 0, 0, 0, -54, -76], [21, 0, 0, 21, 130, -21, 0, 0], [0, 11, -180, 0, 0, 0, 25, -22], [4, 0, 0, 4, -116, -4, 0, 0], [0, -27, -210, 0, 0, 0, 69, 54], [0, -21, -105, 0, 0, 0, -28, 0], [0, 55, 121, 0, 0, 0, 22, 0], [-48, 0, 0, -48, 25, 48, 0, 0], [-528, 0, 0, 35, -35, 105, 0, 0], [38, 0, 0, 38, -95, -38, 0, 0], [0, 57, 45, 0, 0, 0, -4, 0], [0, 8, 180, 0, 0, 0, -44, -16], [0, 65, 47, 0, 0, 0, -6, 0], [-62, 0, 0, -62, -17, 62, 0, 0], [117, 0, 0, 38, -38, 114, 0, 0], [0, -33, -140, 0, 0, 0, 61, 66], [0, -126, -210, 0, 0, 0, -28, 0], [0, 4, -80, 0, 0, 0, 12, -8], [388, 0, 0, -14, 14, -42, 0, 0], [1, 0, 0, 1, -18, -1, 0, 0], [-52, 0, 0, -63, 63, -189, 0, 0], [0, 83, -7, 0, 0, 0, -30, 0], [-28, 0, 0, -28, 37, 28, 0, 0], [45, 0, 0, 45, 68, -45, 0, 0], [0, -46, -190, 0, 0, 0, -48, 0], [22, 0, 0, 22, -126, -22, 0, 0], [0, 32, 340, 0, 0, 0, -100, -64], [0, 102, 270, 0, 0, 0, 56, 0], [0, 18, -140, 0, 0, 0, 10, -36], [496, 0, 0, -11, 11, -33, 0, 0], [-725, 0, 0, 28, -28, 84, 0, 0], [0, -81, 87, 0, 0, 0, 56, 0], [0, 56, 220, 0, 0, 0, -100, -112], [-58, 0, 0, -58, -110, 58, 0, 0], [0, 57, 45, 0, 0, 0, -4, 0], [-596, 0, 0, 7, -7, 21, 0, 0], [0, 42, 180, 0, 0, 0, -78, -84], [56, 0, 0, 56, -40, -56, 0, 0], [-348, 0, 0, 14, -14, 42, 0, 0], [138, 0, 0, 138, -66, -138, 0, 0], [0, 45, -250, 0, 0, 0, 5, -90], [-149, 0, 0, -149, 6, 149, 0, 0], [-628, 0, 0, -9, 9, -27, 0, 0], [0, -187, 47, 0, 0, 0, 78, 0], [607, 0, 0, -32, 32, -96, 0, 0], [0, -54, -30, 0, 0, 0, 8, 0], [52, 0, 0, 52, -3, -52, 0, 0], [0, 80, -28, 0, 0, 0, -36, 0], [0, -186, -306, 0, 0, 0, -40, 0], [-126, 0, 0, -126, 6, 126, 0, 0], [0, -25, -313, 0, 0, 0, -96, 0], [-89, 0, 0, -89, -78, 89, 0, 0], [0, -43, 340, 0, 0, 0, -25, 86], [38, 0, 0, 9, -9, 27, 0, 0], [0, -10, -400, 0, 0, 0, 90, 20], [-228, 0, 0, 17, -17, 51, 0, 0], [0, 62, 134, 0, 0, 0, 24, 0], [-74, 0, 0, -74, 169, 74, 0, 0], [0, -91, 130, 0, 0, 0, 65, 182], [-854, 0, 0, 7, -7, 21, 0, 0], [0, 127, 301, 0, 0, 0, 58, 0], [-126, 0, 0, -126, 37, 126, 0, 0], [0, 34, -140, 0, 0, 0, -6, -68], [32, 0, 0, 32, 169, -32, 0, 0], [0, -28, 380, 0, 0, 0, -48, 56], [48, 0, 0, 48, -128, -48, 0, 0], [0, 26, -10, 0, 0, 0, -12, 0], [-171, 0, 0, -34, 34, -102, 0, 0], [0, -103, -175, 0, 0, 0, -24, 0], [0, -53, -300, 0, 0, 0, 113, 106], [0, 117, 87, 0, 0, 0, -10, 0], [-53, 0, 0, -53, 62, 53, 0, 0], [0, -61, -260, 0, 0, 0, 113, 122], [0, -222, -378, 0, 0, 0, -52, 0], [0, 39, -170, 0, 0, 0, -5, -78], [-39, 0, 0, -39, 6, 39, 0, 0], [-476, 0, 0, -86, 86, -258, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_108_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_108_3_d_d();".
// 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_108_3_d_d(:prec:=8)
    chi := MakeCharacter_108_d();
    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_108_3_d_d();".
// 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_108_3_d_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_108_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![64, 0, -56, 0, 1]>],Snew);
    return Vf;
end function;