
// Make newform 196.3.g.g in Magma, downloaded from the LMFDB on 05 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_196_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_196_g_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_196_3_g_g();".
// 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_196_3_g_g();".
// 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 := [25, 0, -5, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 2, 0, 0], [0, 0, 1, 0], [0, 0, 0, 2]];
    Rf_basisdens := [1, 1, 5, 5];
    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_196_g();"
function MakeCharacter_196_g()
    N := 196;
    order := 6;
    char_gens := [99, 101];
    v := [3, 4];
    // 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_196_g_Hecke();"
function MakeCharacter_196_g_Hecke(Kf)
    N := 196;
    order := 6;
    char_gens := [99, 101];
    char_values := [[-1, 0, 0, 0], [0, 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]];
    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 := [[2, 0, 0, 0], [0, 1, 0, 0], [0, -1, 0, -1], [0, 0, 0, 0], [4, 0, 4, 0], [0, -2, 0, 1], [0, -2, 0, 4], [0, -3, 0, 3], [-24, 0, 12, 0], [-26, 0, 0, 0], [0, -6, 0, 0], [0, 0, 2, 0], [0, 12, 0, -6], [28, 0, -56, 0], [0, 18, 0, -18], [46, 0, -46, 0], [0, 15, 0, 0], [0, -1, 0, -1], [-52, 0, -52, 0], [-56, 0, 112, 0], [0, -16, 0, 32], [32, 0, -16, 0], [0, 0, 0, 21], [0, -8, 0, -8], [0, 12, 0, -6], [0, -9, 0, 18], [0, 18, 0, -18], [200, 0, -100, 0], [-94, 0, 94, 0], [-82, 0, 0, 0], [28, 0, -56, 0], [0, -3, 0, 3], [-94, 0, 94, 0], [0, 0, 0, -21], [0, 0, -26, 0], [4, 0, 4, 0], [0, 5, 0, -10], [200, 0, -100, 0], [0, 0, 0, -42], [0, 41, 0, 41], [60, 0, 60, 0], [0, 54, 0, -27], [-80, 0, 40, 0], [-38, 0, 38, 0], [2, 0, 0, 0], [0, -6, 0, 0], [140, 0, -280, 0], [0, 0, 0, 84], [0, 15, 0, 0], [0, -43, 0, -43], [0, 0, 338, 0], [-140, 0, 280, 0], [0, 26, 0, -52], [0, 0, 0, -105], [0, -8, 0, -8], [88, 0, 88, 0], [0, 5, 0, -10], [0, -24, 0, 24], [214, 0, -214, 0], [-194, 0, 0, 0], [0, -69, 0, 0], [0, -86, 0, 43], [0, 0, 0, 21], [0, -48, 0, 0], [0, 34, 0, 34], [0, 0, 478, 0], [648, 0, -324, 0], [-446, 0, 0, 0], [172, 0, 172, 0], [0, 54, 0, -27], [0, 12, 0, -24], [-472, 0, 236, 0], [0, 36, 0, 0], [0, 0, -698, 0], [-196, 0, 392, 0], [0, 18, 0, -18], [-94, 0, 94, 0], [0, -71, 0, -71], [0, 0, -586, 0], [0, -86, 0, 172], [0, 0, 0, 63], [-166, 0, 0, 0], [60, 0, 60, 0], [0, -212, 0, 106], [0, -192, 0, 192], [-360, 0, 180, 0], [-334, 0, 0, 0], [0, 0, 58, 0], [0, -114, 0, 57], [-280, 0, 560, 0], [0, 81, 0, -81], [0, -6, 0, 0], [-388, 0, -388, 0], [84, 0, -168, 0], [-360, 0, 180, 0], [0, 0, 0, 84], [0, -15, 0, -15], [0, 54, 0, -108], [0, -87, 0, 87], [0, 0, 646, 0], [28, 0, -56, 0], [718, 0, -718, 0], [0, 141, 0, 0], [0, 0, 86, 0], [396, 0, 396, 0], [0, 68, 0, -136], [0, 0, 0, -105], [0, 20, 0, 20], [-416, 0, -416, 0], [0, -128, 0, 64], [0, 144, 0, -144], [-262, 0, 262, 0], [898, 0, 0, 0], [0, 183, 0, 0], [336, 0, -672, 0], [-346, 0, 346, 0], [0, 0, 0, -21], [0, -6, 0, 0], [0, 0, -166, 0], [140, 0, -280, 0], [0, -65, 0, 130], [-194, 0, 0, 0], [0, 69, 0, 69], [508, 0, 508, 0], [0, -171, 0, 171], [-362, 0, 0, 0], [0, 0, 338, 0], [0, -66, 0, 66], [0, 0, 0, 210], [0, -141, 0, -141], [-164, 0, -164, 0], [700, 0, -1400, 0], [-920, 0, 460, 0], [-362, 0, 0, 0], [0, 118, 0, 118], [0, -44, 0, 22], [0, -37, 0, 74], [0, 183, 0, 0], [0, -58, 0, 29], [-1046, 0, 1046, 0], [0, 0, 0, -63], [0, 0, 1318, 0], [312, 0, 312, 0], [-308, 0, 616, 0], [0, -9, 0, 18], [0, 0, 0, -294], [0, 390, 0, -195], [0, 82, 0, -164], [0, 291, 0, -291], [1264, 0, -632, 0], [0, 0, -334, 0], [0, -240, 0, 120], [532, 0, -1064, 0], [0, 18, 0, -18], [340, 0, 340, 0], [-924, 0, 1848, 0], [592, 0, -296, 0], [0, -106, 0, -106], [0, 320, 0, -160], [0, -149, 0, 298], [1320, 0, -660, 0], [338, 0, 0, 0], [-420, 0, 840, 0], [0, 249, 0, -249], [242, 0, -242, 0], [0, -258, 0, 0], [256, 0, 256, 0], [0, -37, 0, 74]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_196_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_196_3_g_g();".
// 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_196_3_g_g(:prec:=4)
    chi := MakeCharacter_196_g();
    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_196_3_g_g();".
// 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_196_3_g_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_196_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![400, 0, -20, 0, 1]>,<5,R![3600, 0, 60, 0, 1]>,<11,R![48, -12, 1]>],Snew);
    return Vf;
end function;