
// Make newform 49.4.c.c in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_49_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_49_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_49_4_c_c();".
// 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_49_4_c_c();".
// 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 := [1, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_49_c();"
function MakeCharacter_49_c()
    N := 49;
    order := 3;
    char_gens := [3];
    v := [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_49_c_Hecke();"
function MakeCharacter_49_c_Hecke(Kf)
    N := 49;
    order := 3;
    char_gens := [3];
    char_values := [[0, -1]];
    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 := 4;
    raw_aps := [[0, 1], [2, -2], [0, -16], [0, 0], [8, -8], [28, 0], [-54, 54], [0, 110], [0, -48], [-110, 0], [-12, 12], [0, 246], [182, 0], [128, 0], [0, -324], [162, -162], [-810, 810], [0, 488], [-244, 244], [-768, 0], [702, -702], [0, -440], [-1302, 0], [0, -730], [294, 0], [688, -688], [0, -1388], [0, -244], [-90, 90], [1318, 0], [-1776, 0], [0, 1118], [-2274, 2274], [-210, 0], [0, 2010], [-1112, 1112], [-124, 124], [0, -2008], [2884, 0], [0, -2228], [820, -820], [3892, 0], [0, 5048], [2962, -2962], [3334, 0], [-1860, 1860], [-4268, 0], [-5432, 0], [2046, -2046], [0, 2980], [0, -4458], [4440, 0], [-3302, 3302], [1582, 0], [0, -2354], [3872, -3872], [-180, 180], [0, -2032], [5426, -5426], [842, 0], [3782, -3782], [-4312, 0], [2674, 0], [3768, -3768], [0, -2438], [0, 3186], [0, -8672], [814, 0], [-9344, 9344], [-5180, 0], [-12178, 12178], [0, -440], [9816, -9816], [0, 442], [-3960, 0], [0, -6708], [13350, -13350], [0, 1356], [0, -6222], [-5150, 5150], [2310, 0], [1262, 0], [4488, -4488], [17038, 0], [0, -16200], [0, 8772], [2130, 0], [0, -10534], [-9268, 0], [-9392, 0], [0, 10806], [-4940, 4940], [5216, -5216], [4412, 0], [0, -19060], [12768, 0], [0, 5500], [7338, -7338], [0, 17582], [0, 1618], [16144, 0], [-4654, 4654], [-10078, 10078], [0, 5930], [19048, -19048], [14366, -14366], [-3626, 0], [0, 1062], [10200, -10200], [-25158, 0], [0, -25664], [-19018, 19018], [17334, 0], [-18730, 18730], [-6928, 0], [-16302, 16302], [4718, 0], [21436, -21436], [0, -4458], [-26640, 0], [-7432, 7432], [58, 0], [0, 21516], [-18108, 18108], [0, 10078], [18762, 0], [0, -6810], [0, -4860], [-13636, 0], [0, -2088], [5160, -5160], [-28152, 0], [0, 16808], [21674, 0], [0, -7422], [13790, 0], [6232, -6232], [1766, -1766], [1204, 0], [7050, -7050], [23282, 0], [0, -10142], [9192, -9192], [-46716, 0], [-11240, 11240], [700, 0], [-37492, 0], [-28894, 28894], [0, 2770], [0, -17688], [0, 33566], [-16758, 0], [11468, 0], [0, 50356], [8716, -8716], [7632, 0], [0, 23080], [0, -45110], [16674, 0], [-43832, 43832], [0, 736], [38138, 0], [26224, 0], [0, -18762], [-38394, 38394], [-5388, 5388], [-25472, 25472], [17096, -17096]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_49_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_49_4_c_c();".
// 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_49_4_c_c(:prec:=2)
    chi := MakeCharacter_49_c();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_49_4_c_c();".
// 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_49_4_c_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_49_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![1, -1, 1]>,<3,R![4, -2, 1]>],Snew);
    return Vf;
end function;