
// Make newform 28.3.b.a in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_28_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_28_b_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_28_3_b_a();".
// 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_28_3_b_a();".
// 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, 2]];
    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_28_b();"
function MakeCharacter_28_b()
    N := 28;
    order := 2;
    char_gens := [15, 17];
    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_28_b_Hecke();"
function MakeCharacter_28_b_Hecke(Kf)
    N := 28;
    order := 2;
    char_gens := [15, 17];
    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 := 3;
    raw_aps := [[0, 0], [0, 1], [0, -1], [5, -1], [-6, 0], [0, 1], [0, -4], [0, 5], [-30, 0], [-6, 0], [0, 0], [10, 0], [0, 10], [10, 0], [0, -4], [90, 0], [0, -5], [0, 5], [-70, 0], [42, 0], [0, -22], [74, 0], [0, 13], [0, 30], [0, 16], [0, 15], [0, -34], [-150, 0], [-86, 0], [90, 0], [50, 0], [0, -15], [210, 0], [0, -25], [-198, 0], [34, 0], [0, -27], [170, 0], [0, -34], [0, -39], [42, 0], [0, -5], [-246, 0], [-230, 0], [-150, 0], [0, 70], [-326, 0], [0, -24], [0, 81], [0, 15], [210, 0], [114, 0], [0, 20], [0, 35], [0, 48], [330, 0], [0, -95], [0, -20], [170, 0], [114, 0], [0, -69], [0, 31], [0, 21], [0, 50], [0, 46], [90, 0], [-38, 0], [-230, 0], [-150, 0], [0, 125], [0, 56], [-318, 0], [0, -44], [-310, 0], [442, 0], [0, 88], [-726, 0], [0, -39], [474, 0], [0, -10], [0, -55], [314, 0], [-366, 0], [0, -12], [0, -30], [90, 0], [-606, 0], [-230, 0], [0, 25], [-310, 0], [0, 21], [0, -80], [130, 0], [234, 0], [394, 0], [0, -14], [0, -155], [0, 130], [0, 103], [-406, 0], [-790, 0], [-150, 0], [0, 93], [762, 0], [842, 0], [0, -64], [0, -57], [0, 196], [-726, 0], [0, -210], [0, 136], [-1030, 0], [90, 0], [0, -185], [1034, 0], [282, 0], [0, -39], [0, -42], [1290, 0], [474, 0], [0, 235], [-350, 0], [0, -9], [-630, 0], [0, -75], [234, 0], [-326, 0], [0, -140], [0, -222], [0, -99], [74, 0], [-510, 0], [754, 0], [730, 0], [0, -50], [0, 200], [0, 191], [0, -59], [0, 21], [-486, 0], [0, 135], [234, 0], [1130, 0], [90, 0], [0, -155], [0, -250], [0, 283], [0, -202], [0, 235], [330, 0], [410, 0], [0, -180], [730, 0], [0, 218], [250, 0], [-78, 0], [1354, 0], [0, 80], [0, 146], [0, 265], [-390, 0], [-270, 0], [-1150, 0], [0, -185], [-30, 0], [0, -254], [-1718, 0], [0, -249]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_28_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_28_3_b_a();".
// 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_28_3_b_a(:prec:=2)
    chi := MakeCharacter_28_b();
    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_28_3_b_a();".
// 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_28_3_b_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_28_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([],Snew);
    return Vf;
end function;