
// Make newform 1764.4.k.c in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1764_k();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1764_k_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_1764_4_k_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_1764_4_k_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_1764_k();"
function MakeCharacter_1764_k()
    N := 1764;
    order := 3;
    char_gens := [883, 785, 1081];
    v := [3, 3, 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_1764_k_Hecke();"
function MakeCharacter_1764_k_Hecke(Kf)
    N := 1764;
    order := 3;
    char_gens := [883, 785, 1081];
    char_values := [[1, 0], [1, 0], [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, 0], [0, 0], [0, -14], [0, 0], [4, -4], [-54, 0], [14, -14], [0, 92], [0, -152], [106, 0], [-144, 144], [0, -158], [-390, 0], [-508, 0], [0, 528], [606, -606], [364, -364], [0, 678], [-844, 844], [8, 0], [-422, 422], [0, -384], [-548, 0], [0, -1194], [1502, 0], [-398, 398], [0, 1160], [0, 324], [938, -938], [622, 0], [1200, 0], [0, 1396], [2810, -2810], [-4, 0], [0, 1374], [-2104, 2104], [3206, -3206], [0, -332], [1496, 0], [0, 3322], [-900, 900], [-1902, 0], [0, 4128], [1342, -1342], [3506, 0], [680, -680], [5372, 0], [1072, 0], [2868, -2868], [0, 4798], [0, -5126], [528, 0], [-814, 814], [-1932, 0], [0, 3294], [-7080, 7080], [-7814, 7814], [0, 3168], [7858, -7858], [-6730, 0], [-3020, 3020], [-6834, 0], [-2332, 0], [-8840, 8840], [0, -1046], [0, 7542], [0, -2756], [3954, 0], [6900, -6900], [2426, 0], [1470, -1470], [0, 6872], [-7072, 7072], [0, 818], [-5132, 0], [0, 8576], [-3730, 3730], [0, 6678], [0, -3054], [266, -266], [8844, 0], [-4482, 0], [9936, -9936], [11758, 0], [0, -4104], [0, 9748], [478, 0], [0, 11174], [-11674, 0], [10528, 0], [0, -16604], [8576, -8576], [-9704, 9704], [4092, 0], [0, -17884], [-7704, 0], [0, -14358], [-5082, 5082], [0, -1756], [0, -16230], [17676, 0], [-12250, 12250], [10052, -10052], [0, 25674], [-3732, 3732], [-1214, 1214], [7108, 0], [0, -6162], [2472, -2472], [13750, 0], [0, -11376], [-20382, 20382], [-21178, 0], [-4700, 4700], [-21736, 0], [-13022, 13022], [-3308, 0], [13800, -13800], [0, -2682], [-23836, 0], [-11282, 11282], [-13726, 0], [0, -4974], [-8988, 8988], [0, 10172], [-27446, 0], [0, -3998], [0, -25872], [-12088, 0], [0, 7974], [31764, -31764], [-888, 0], [0, 34656], [-22866, 0], [0, -22570], [1790, 0], [-2990, 2990], [-30756, 30756], [15126, 0], [-6502, 6502], [8252, 0], [0, -21986], [-3736, 3736], [23820, 0], [7942, -7942], [21016, 0], [-24878, 0], [6390, -6390], [0, -46444], [0, -25408], [0, -1078], [-45006, 0], [4028, 0], [0, 29304], [-50916, 50916], [24432, 0], [0, 20360], [0, -23202], [1990, 0], [51130, -51130], [0, -47044], [-46858, 0], [30632, 0], [0, 3804], [-49326, 49326], [-11112, 11112], [-13616, 13616], [-56674, 56674], [42770, 0], [0, -19038], [8756, -8756], [14042, 0], [0, -37416], [-41946, 41946], [-22848, 22848], [11882, 0], [12276, 0], [-19634, 19634], [36664, 0], [0, 13622], [0, -17808], [-19604, 0], [16414, 0], [0, 39046], [45392, -45392], [29826, -29826], [11034, -11034], [-7252, 7252], [0, 24774], [-15360, 15360], [0, -33470], [44316, 0], [0, 28100], [0, -33814], [32844, -32844], [-6298, 6298], [-65810, 65810], [0, -23302], [-58014, 0], [0, 33288], [65862, -65862], [37536, 0], [0, -7474], [-45726, 45726], [30084, 0], [22314, -22314], [0, 69104], [0, -3060], [-51354, 0], [-86812, 86812], [0, -66098], [24062, 0], [17688, 0], [0, 66316], [-54840, 54840], [0, 57914], [-22240, 22240], [-27442, 27442], [0, 93832], [59978, 0], [0, 66242], [2312, 0], [0, 3234], [0, 9440], [27420, 0], [-74386, 0], [0, 75542], [-71776, 71776], [0, 37896], [0, -76092], [9450, -9450], [-16964, 16964], [75376, 0], [-41958, 41958], [14044, 0], [-48720, 48720], [0, -50702], [0, 22686], [-2484, 0], [20984, 0], [38620, -38620], [0, 3668], [-84664, 84664], [0, -84438], [-65582, 0], [0, 77496], [-60528, 0], [-74860, 74860], [-73508, 73508], [65232, 0], [-23354, 0], [0, 113694], [-36360, 36360], [10790, 0], [-123910, 123910], [0, 81244], [-36494, 36494], [88884, -88884], [30158, 0], [0, 32042], [39632, -39632], [-47660, 0], [32734, -32734], [86138, 0], [24606, -24606], [0, -97364], [-69414, 0], [-7798, 0], [80308, 0], [-63954, 63954], [0, 97878], [141124, -141124], [-95478, 95478], [0, 59344], [15470, 0], [0, -112392], [0, 94804], [108954, -108954], [0, 145190], [0, -89340], [-8832, 8832], [-51912, 51912], [-6936, 0], [34114, 0], [0, 10788], [0, 40528], [-29874, 29874], [1058, 0], [-126280, 126280], [19842, 0], [-72090, 72090], [105444, -105444], [0, -1494], [-110140, 0], [-146602, 0], [-26070, 26070], [0, 13520], [46398, 0], [0, 98700], [-102444, 0], [0, 37018], [0, 63558], [67904, -67904]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1764_k_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1764_4_k_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_1764_4_k_c(:prec:=2)
    chi := MakeCharacter_1764_k();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_1764_4_k_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_1764_4_k_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1764_k();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![196, 14, 1]>,<11,R![16, -4, 1]>,<13,R![54, 1]>],Snew);
    return Vf;
end function;