
// Make newform 108.5.c.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_108_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_108_c_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_5_c_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_108_5_c_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 := [6, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [0, 18]];
    Rf_basisdens := [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_108_c();"
function MakeCharacter_108_c()
    N := 108;
    order := 2;
    char_gens := [55, 29];
    v := [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_108_c_Hecke();"
function MakeCharacter_108_c_Hecke(Kf)
    N := 108;
    order := 2;
    char_gens := [55, 29];
    char_values := [[1, 0], [-1, 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 := 5;
    raw_aps := [[0, 0], [0, 0], [0, 1], [-31, 0], [0, -5], [-241, 0], [0, 5], [-271, 0], [0, 5], [0, 10], [-778, 0], [1079, 0], [0, -50], [-298, 0], [0, 75], [0, 70], [0, 65], [-2641, 0], [5609, 0], [0, -100], [7199, 0], [329, 0], [0, 30], [0, 185], [-15961, 0], [0, 440], [1649, 0], [0, -95], [-3238, 0], [0, -315], [-13858, 0], [0, 610], [0, 115], [25409, 0], [0, 140], [12569, 0], [21002, 0], [-16831, 0], [0, -675], [0, -570], [0, -500], [10079, 0], [0, 845], [6599, 0], [0, -55], [-57871, 0], [-1231, 0], [-60058, 0], [0, 840], [-3118, 0], [0, 820], [0, -1620], [-54121, 0], [0, -1730], [0, 570], [0, -940], [0, 2065], [-67231, 0], [34442, 0], [0, -1550], [-83578, 0], [0, -2035], [98102, 0], [0, -2165], [30839, 0], [0, 170], [-191551, 0], [-155401, 0], [0, -4380], [207239, 0], [0, 2620], [0, -2145], [62249, 0], [34679, 0], [198089, 0], [0, -200], [0, 1415], [202682, 0], [0, -920], [-105001, 0], [0, -435], [-252121, 0], [0, 7595], [-22678, 0], [-124858, 0], [0, 2010], [0, 1665], [33002, 0], [0, 4135], [63929, 0], [0, 7645], [0, -3030], [319769, 0], [0, -5305], [36302, 0], [0, -10365], [0, -3185], [0, 6815], [-195391, 0], [60119, 0], [311369, 0], [0, 5825], [0, -930], [0, 695], [-140911, 0], [-128161, 0], [0, 6995], [0, 9710], [0, 15530], [-373798, 0], [195329, 0], [-322201, 0], [0, 15325], [249449, 0], [-357751, 0], [0, -20], [-527098, 0], [0, -510], [0, -1390], [0, 7490], [-367801, 0], [93479, 0], [0, 9095], [0, 13980], [-19738, 0], [0, -4895], [-688081, 0], [0, -15090], [-355258, 0], [482522, 0], [571142, 0], [0, 4095], [-835711, 0], [406079, 0], [0, 7485], [358319, 0], [0, 11430], [182369, 0], [0, 27180], [0, -11020], [664742, 0], [0, 23585], [-745711, 0], [0, 8705], [-459001, 0], [0, -24570], [1119839, 0], [0, -24360], [-128311, 0], [0, -4945], [817679, 0], [0, -26105], [-757351, 0], [0, -18410], [706289, 0], [0, 6450], [1470422, 0], [0, 30600], [183959, 0], [0, -2485], [0, 34905], [0, -21075], [-561031, 0], [0, -22545], [0, -12430], [0, 33425], [-905671, 0], [-770518, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_108_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_108_5_c_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_108_5_c_b(:prec:=2)
    chi := MakeCharacter_108_c();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 5));
    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_5_c_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_108_5_c_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_108_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,5,sign)));
    Vf := Kernel([<5,R![1944, 0, 1]>],Snew);
    return Vf;
end function;