
// Make newform 48.6.c.d in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_48_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_48_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_48_6_c_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_48_6_c_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 := [196, 0, -14, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [-14, 0, 2, 0], [7, 28, -1, -1], [0, 0, 0, 6]];
    Rf_basisdens := [1, 7, 7, 7];
    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_48_c();"
function MakeCharacter_48_c()
    N := 48;
    order := 2;
    char_gens := [31, 37, 17];
    v := [1, 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_48_c_Hecke();"
function MakeCharacter_48_c_Hecke(Kf)
    N := 48;
    order := 2;
    char_gens := [31, 37, 17];
    char_values := [[-1, 0, 0, 0], [1, 0, 0, 0], [-1, 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 := 6;
    raw_aps := [[0, 0, 0, 0], [0, 2, -1, 0], [0, 0, 0, -1], [0, 11, 0, 0], [0, 21, 42, 0], [814, 0, 0, 0], [0, 0, 0, -44], [0, -11, 0, 0], [0, -54, -108, 0], [0, 0, 0, 121], [0, -3009, 0, 0], [-1594, 0, 0, 0], [0, 0, 0, 198], [0, -2607, 0, 0], [0, -636, -1272, 0], [0, 0, 0, -429], [0, 1761, 3522, 0], [23870, 0, 0, 0], [0, 8773, 0, 0], [0, -1266, -2532, 0], [17578, 0, 0, 0], [0, -2057, 0, 0], [0, 627, 1254, 0], [0, 0, 0, 1970], [-49070, 0, 0, 0], [0, 0, 0, 759], [0, 22187, 0, 0], [0, -6171, -12342, 0], [-140690, 0, 0, 0], [0, 0, 0, -716], [0, 80223, 0, 0], [0, 5775, 11550, 0], [0, 0, 0, -1858], [0, -23287, 0, 0], [0, 0, 0, 2299], [0, -51821, 0, 0], [-324226, 0, 0, 0], [0, -59059, 0, 0], [0, -19866, -39732, 0], [0, 0, 0, 7909], [0, 2067, 4134, 0], [-256202, 0, 0, 0], [0, 21504, 43008, 0], [203522, 0, 0, 0], [0, 0, 0, 10879], [0, -145717, 0, 0], [0, 306845, 0, 0], [0, -153345, 0, 0], [0, 49071, 98142, 0], [-485594, 0, 0, 0], [0, 0, 0, -946], [0, -8844, -17688, 0], [1339954, 0, 0, 0], [0, 16521, 33042, 0], [0, 0, 0, 27632], [0, -71874, -143748, 0], [0, 0, 0, 25421], [0, 473143, 0, 0], [-1111946, 0, 0, 0], [0, 0, 0, -13838], [0, 331617, 0, 0], [0, 0, 0, 33671], [0, 315117, 0, 0], [0, 77394, 154788, 0], [1562474, 0, 0, 0], [0, 0, 0, 62657], [0, -703543, 0, 0], [-2765246, 0, 0, 0], [0, -67023, -134046, 0], [3705086, 0, 0, 0], [0, 0, 0, -52152], [0, -32538, -65076, 0], [0, -1202041, 0, 0], [-1140106, 0, 0, 0], [0, -606743, 0, 0], [0, 133200, 266400, 0], [0, 0, 0, 45759], [-50866, 0, 0, 0], [0, 0, 0, 57860], [-2748614, 0, 0, 0], [0, -162897, -325794, 0], [4092838, 0, 0, 0], [0, -52668, -105336, 0], [-4623838, 0, 0, 0], [0, -1175229, 0, 0], [0, -61335, -122670, 0], [0, 0, 0, 53736], [-1524182, 0, 0, 0], [0, 0, 0, -75603], [0, -224857, 0, 0], [0, 12387, 24774, 0], [0, 64680, 129360, 0], [0, 1443755, 0, 0], [0, 318549, 637098, 0], [0, 1309869, 0, 0], [0, 143682, 287364, 0], [0, 0, 0, -114159], [0, 0, 0, -142242], [0, 1969825, 0, 0], [-7136162, 0, 0, 0], [0, -946099, 0, 0], [0, 0, 0, 25333], [0, -209781, -419562, 0], [0, 0, 0, 300850], [0, 3184753, 0, 0], [-3752398, 0, 0, 0], [0, 5421, 10842, 0], [0, 0, 0, -139348], [0, 205434, 410868, 0], [296186, 0, 0, 0], [0, 1434191, 0, 0], [-769978, 0, 0, 0], [0, 0, 0, -32098], [0, 1476217, 0, 0], [0, 2139571, 0, 0], [0, 0, 0, -197944], [0, -4800483, 0, 0], [0, 611502, 1223004, 0], [0, 0, 0, 187197], [0, -51909, -103818, 0], [-1306570, 0, 0, 0], [2117522, 0, 0, 0], [0, 0, 0, -476025], [0, -236355, -472710, 0], [0, -462171, 0, 0], [0, 0, 0, 205073], [-22330010, 0, 0, 0], [0, -751092, -1502184, 0], [0, 4836771, 0, 0], [16449598, 0, 0, 0], [0, -706827, 0, 0], [0, 14454, 28908, 0], [0, -5209385, 0, 0], [-21140810, 0, 0, 0], [0, 0, 0, 465938], [-10478974, 0, 0, 0], [0, 0, 0, -290609], [0, 4093045, 0, 0], [0, 0, 0, 161305], [0, 0, 0, -766810], [0, 3228753, 0, 0], [0, 0, 0, 582659], [0, -1444877, 0, 0], [0, 119889, 239778, 0], [9085726, 0, 0, 0], [0, -1297746, -2595492, 0], [-7952362, 0, 0, 0], [0, 0, 0, -894454], [0, 3680825, 0, 0], [0, 804120, 1608240, 0], [12774542, 0, 0, 0], [0, 0, 0, 562068], [0, -3763771, 0, 0], [0, 402402, 804804, 0], [0, 11777953, 0, 0], [0, -1471380, -2942760, 0], [0, -10213533, 0, 0], [0, 0, 0, 640288], [4295258, 0, 0, 0], [0, 0, 0, 374869], [0, 619299, 1238598, 0], [0, 0, 0, 376178], [0, -12225125, 0, 0], [0, -277419, -554838, 0], [0, 0, 0, -161820], [0, 1041162, 2082324, 0], [0, 10076495, 0, 0], [34137158, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_48_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_48_6_c_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_48_6_c_d(:prec:=4)
    chi := MakeCharacter_48_c();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 6));
    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_48_6_c_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_48_6_c_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_48_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<5,R![2016, 0, 1]>,<11,R![-296352, 0, 1]>],Snew);
    return Vf;
end function;