
// Make newform 588.4.i.h in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_588_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_588_i_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_588_4_i_h();".
// 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_588_4_i_h();".
// 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_588_i();"
function MakeCharacter_588_i()
    N := 588;
    order := 3;
    char_gens := [295, 197, 493];
    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_588_i_Hecke();"
function MakeCharacter_588_i_Hecke(Kf)
    N := 588;
    order := 3;
    char_gens := [295, 197, 493];
    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], [3, -3], [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]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_588_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_588_4_i_h();".
// 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_588_4_i_h(:prec:=2)
    chi := MakeCharacter_588_i();
    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_588_4_i_h();".
// 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_588_4_i_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_588_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![196, -14, 1]>],Snew);
    return Vf;
end function;