
// Make newform 572.2.i.c in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_572_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_572_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_572_2_i_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_572_2_i_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 := [4, -12, 38, -26, 47, 2, 59, 6, 9, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-300404, 5900, -18408, 1085227, -21476, 295, -57824, 767, -413, -2483], [-1548188, 2051300, -6400056, -1739951, -7466732, 102565, -984578, 266669, -143591, 19399], [774739, -397900, 1241448, 95293, 1448356, -19895, 198199, -51727, 27853, -3302], [-1427971, 1341100, -4184232, -3035092, -4881604, 67055, -494521, 174343, -93877, 18803], [34658, 2835430, -867399, 3508271, 150497, 4373135, 451373, 675496, -77584, 75101], [-75212, -1202335, 626586, -3695354, -3918833, -4919210, -1042322, -607339, 5116, -75074], [-5874, -255794, 83447, -396279, -210913, -509807, -79679, -71072, 4886, -8279], [2593264, -8307730, 11803173, -10724477, -876659, -21851195, -2588951, -3339832, 367648, -387047]];
    Rf_basisdens := [1, 1, 929970, 929970, 154995, 464985, 929970, 464985, 20666, 929970];
    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_572_i();"
function MakeCharacter_572_i()
    N := 572;
    order := 3;
    char_gens := [287, 365, 353];
    v := [3, 3, 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_572_i_Hecke();"
function MakeCharacter_572_i_Hecke(Kf)
    N := 572;
    order := 3;
    char_gens := [287, 365, 353];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 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 := 2;
    raw_aps := [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, -1, 0, -1, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [0, -1, 0, 0, 1, 1, 0, 0, 1, 0], [-1, -1, -1, 0, 1, 0, 1, 0, 1, -1], [2, -1, -1, 0, -1, 0, -2, 0, -1, -1], [0, 0, 0, 0, 0, -2, -1, 0, -1, -2], [0, -2, 0, 0, 0, -2, -1, 1, 0, -2], [-2, 0, 0, -2, 1, -1, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 2, -1, 1, 0], [0, 1, 0, 0, 0, -1, 1, 0, 1, -1], [-3, -1, -1, -1, 0, 0, 3, 1, 0, 1], [-3, 0, -2, 1, 1, 2, 0, 0, 0, 0], [0, 0, 3, 1, 0, -3, 0, 0, 0, 0], [2, 0, 0, -1, -1, 0, -2, 1, -1, 3], [4, 2, 2, 1, 0, 0, -4, -1, 0, -1], [0, 2, 0, 0, 0, -1, -1, -3, -2, -1], [0, 1, 1, -1, -1, 0, 0, 1, -1, -3], [-8, 0, -1, 0, 2, 1, 0, 0, 0, 0], [1, 0, 3, 0, -1, 4, 0, 0, 0, 0], [-4, 0, -2, -1, 1, 3, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 5, -3, 1, 0], [4, 0, 0, 2, -3, 0, -4, -2, -3, -3], [0, 2, 0, 0, 0, -2, 2, -3, 1, -2], [6, 0, 3, 3, -2, 2, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 0, -3, 2, 0], [-1, 0, 2, -2, -4, -3, 0, 0, 0, 0], [6, 0, 0, 0, 1, 0, -6, 0, 1, -4], [0, 1, 0, 0, 0, 3, -1, -1, -4, 3], [2, 0, 0, 1, -5, 1, 0, 0, 0, 0], [1, 1, 1, -3, 1, 0, -1, 3, 1, -1], [2, 6, 6, -2, -2, 0, -2, 2, -2, 4], [-7, 0, 0, -1, 0, 0, 7, 1, 0, 2], [-10, 0, 1, 3, -1, 1, 0, 0, 0, 0], [1, 0, 1, -5, 1, -1, 0, 0, 0, 0], [4, 4, 4, 3, 0, 0, -4, -3, 0, -2], [0, 2, 0, 0, 0, -1, 5, 4, 0, -1], [0, -5, -5, 1, 1, 0, 0, -1, 1, 5], [0, -2, 0, 0, 0, 1, 10, 1, 1, 1], [1, 0, -6, -2, 3, 3, 0, 0, 0, 0], [0, -3, -3, -4, 3, 0, 0, 4, 3, 7], [0, -1, 0, 0, 0, -4, 6, 6, -5, -4], [0, -10, 0, 0, 0, 5, 6, 4, 5, 5], [-5, -1, -1, -2, -2, 0, 5, 2, -2, -7], [0, -1, 0, 0, 0, -5, -6, -4, -1, -5], [0, 0, 0, 0, 0, -4, 2, 3, -2, -4], [5, 2, 2, 4, -1, 0, -5, -4, -1, -2], [-15, 0, 2, 0, -1, 4, 0, 0, 0, 0], [4, 0, -2, -4, 2, -8, 0, 0, 0, 0], [-2, 0, -3, 5, -2, 4, 0, 0, 0, 0], [7, 3, 3, 0, 0, 0, -7, 0, 0, -2], [3, -3, -3, -6, 2, 0, -3, 6, 2, 1], [0, 1, 0, 0, 0, 2, 5, 1, 4, 2], [0, 0, 0, 0, 0, -2, -5, -4, 5, -2], [21, 1, 1, -1, -3, 0, -21, 1, -3, 0], [0, 3, 0, 0, 0, 5, 10, -1, 3, 5], [-16, 0, 0, 1, 1, 0, 16, -1, 1, 0], [-7, 0, -3, -6, 6, -2, 0, 0, 0, 0], [0, -7, 0, 0, 0, -3, 9, 4, -2, -3], [-2, -7, -7, 6, 4, 0, 2, -6, 4, -5], [-6, 0, 4, 3, -1, 1, 0, 0, 0, 0], [5, 0, -8, 3, 6, -3, 0, 0, 0, 0], [1, 0, -5, -1, 6, 5, 0, 0, 0, 0], [9, 0, 3, 1, -4, 2, 0, 0, 0, 0], [2, -6, -6, 0, 4, 0, -2, 0, 4, -2], [2, 0, 4, -5, 5, -8, 0, 0, 0, 0], [-2, -1, -1, -4, 6, 0, 2, 4, 6, 6], [0, -1, 0, 0, 0, 4, 9, 3, -1, 4], [0, 0, 0, 0, 0, -1, 11, 6, -3, -1], [-3, 0, 6, -4, -5, 0, 0, 0, 0, 0], [0, -7, 0, 0, 0, 6, -5, 7, -1, 6], [5, 5, 5, 6, 5, 0, -5, -6, 5, 3], [0, 7, 0, 0, 0, 4, 2, -3, 0, 4], [1, 4, 4, 2, 4, 0, -1, -2, 4, -3], [4, 0, 12, 8, -5, 4, 0, 0, 0, 0], [5, 0, 0, 0, -1, 0, -5, 0, -1, -4], [0, 7, 0, 0, 0, -5, 8, -7, -2, -5], [-9, 8, 8, 3, -4, 0, 9, -3, -4, -6], [0, -2, 0, 0, 0, 3, -10, 6, -5, 3], [3, 0, 2, 2, 1, 7, 0, 0, 0, 0], [0, -4, 0, 0, 0, 4, 5, 11, 1, 4], [1, -1, -1, 3, -4, 0, -1, -3, -4, -10], [0, -9, 0, 0, 0, 10, 2, 4, 4, 10], [0, 0, -12, 5, 4, 6, 0, 0, 0, 0], [-9, -1, -1, 3, -4, 0, 9, -3, -4, 7], [0, 4, 0, 0, 0, -5, -2, -13, -2, -5], [-10, -6, -6, 1, -5, 0, 10, -1, -5, -8], [-10, 0, 2, 1, 2, -2, 0, 0, 0, 0], [11, 0, 16, 5, -4, 3, 0, 0, 0, 0], [0, 7, 0, 0, 0, -3, 2, -6, -3, -3], [3, -7, -7, 4, 1, 0, -3, -4, 1, 2], [0, 3, 0, 0, 0, 1, 7, -6, 0, 1], [-8, 0, -3, -4, -7, -1, 0, 0, 0, 0], [1, -8, -8, 1, 3, 0, -1, -1, 3, 4], [0, -7, 0, 0, 0, 7, 4, -1, 4, 7], [0, 0, 5, -9, -1, -4, 0, 0, 0, 0], [0, 4, 0, 0, 0, 7, 5, 11, 4, 7], [13, 0, 7, 6, -7, 7, 0, 0, 0, 0], [-6, 0, 9, 1, 1, 3, 0, 0, 0, 0], [0, -5, 0, 0, 0, -2, -4, -6, 5, -2], [-12, -5, -5, -8, 6, 0, 12, 8, 6, 2], [0, 2, 0, 0, 0, 7, 2, 4, -5, 7], [-3, 0, 2, -1, 0, -13, 0, 0, 0, 0], [10, 0, -2, -2, 3, 0, 0, 0, 0, 0], [0, 15, 0, 0, 0, -11, -3, -4, -4, -11], [24, 0, 2, 1, -3, -4, 0, 0, 0, 0], [-5, 0, 13, -5, -4, -5, 0, 0, 0, 0], [0, 15, 0, 0, 0, -3, 5, 2, -5, -3], [-1, 3, 3, 12, -2, 0, 1, -12, -2, -5], [0, -2, 0, 0, 0, 0, -14, 1, 3, 0], [-10, 3, 3, 5, -5, 0, 10, -5, -5, 4], [-11, 0, 2, 7, -2, 7, 0, 0, 0, 0], [3, -18, -18, -3, 9, 0, -3, 3, 9, -2], [2, 4, 4, -2, 4, 0, -2, 2, 4, -3], [7, 6, 6, -3, 1, 0, -7, 3, 1, 6], [0, -5, 0, 0, 0, -4, 23, -3, -3, -4], [0, 7, 0, 0, 0, 12, 0, -1, 4, 12], [-7, 1, 1, -4, -5, 0, 7, 4, -5, -4], [0, -2, 0, 0, 0, -2, -8, 6, -3, -2], [0, 4, 0, 0, 0, 3, -26, 1, -2, 3], [2, 0, -11, 1, 0, -4, 0, 0, 0, 0], [0, -9, -9, -1, 1, 0, 0, 1, 1, 11], [0, 0, 0, 0, 0, -9, -4, -2, 11, -9], [-6, 0, 9, 13, 0, 10, 0, 0, 0, 0], [-8, -9, -9, -5, 4, 0, 8, 5, 4, 5], [8, 1, 1, 5, -1, 0, -8, -5, -1, 7], [0, 0, -5, 11, 6, 2, 0, 0, 0, 0], [-3, 0, -17, -6, 5, -15, 0, 0, 0, 0], [0, -10, 0, 0, 0, 11, -1, 8, -2, 11], [0, 0, 0, 0, 0, -14, -6, -3, -8, -14], [0, -14, 0, 0, 0, -10, 4, -4, 6, -10], [0, 0, 0, 0, 0, -14, 20, -2, -2, -14], [-11, 1, 1, -1, 8, 0, 11, 1, 8, -1], [0, -3, 0, 0, 0, -11, 8, -1, -1, -11], [-1, 0, 0, 2, 6, 0, 1, -2, 6, -7], [15, -1, -1, 4, 3, 0, -15, -4, 3, 12], [4, -12, -12, -4, 2, 0, -4, 4, 2, 2], [0, -10, 0, 0, 0, 8, 7, 7, 10, 8], [-6, 0, -11, -1, 6, -10, 0, 0, 0, 0], [0, 16, 0, 0, 0, 5, -3, 6, -4, 5], [22, -5, -5, -1, 2, 0, -22, 1, 2, 0], [18, 0, -16, -1, 4, 2, 0, 0, 0, 0], [0, 18, 0, 0, 0, -4, -23, -6, -4, -4], [-3, 0, 0, -13, 2, 0, 3, 13, 2, -1], [6, 0, -6, -5, 2, -5, 0, 0, 0, 0], [-4, 0, -14, 1, 8, 5, 0, 0, 0, 0], [-8, 0, -1, -6, 7, -7, 0, 0, 0, 0], [-11, 0, 7, 5, -3, 2, 0, 0, 0, 0], [4, -5, -5, 4, 2, 0, -4, -4, 2, -5], [0, -3, 0, 0, 0, 7, -6, -7, 8, 7], [-7, 0, -15, -7, -1, 2, 0, 0, 0, 0], [0, 3, 0, 0, 0, 6, -4, 3, -2, 6], [0, -14, 0, 0, 0, 10, 8, 7, 0, 10], [2, 0, 8, -1, 7, 9, 0, 0, 0, 0], [-2, -10, -10, -6, -7, 0, 2, 6, -7, 9], [6, -8, -8, 4, -3, 0, -6, -4, -3, -2], [-5, 0, 5, 2, -4, 6, 0, 0, 0, 0], [12, 0, -8, -4, -2, -8, 0, 0, 0, 0], [0, 7, 0, 0, 0, 4, -2, -3, 4, 4], [13, -11, -11, -5, 3, 0, -13, 5, 3, -2], [-8, 0, -3, 8, 7, 7, 0, 0, 0, 0], [-3, -2, -2, -3, -6, 0, 3, 3, -6, 3], [0, -8, 0, 0, 0, 9, -21, -4, 5, 9], [20, 0, -9, -5, 8, 10, 0, 0, 0, 0], [0, -12, 0, 0, 0, 0, 20, 1, 12, 0], [-2, 0, 0, -3, 10, 0, 2, 3, 10, -11]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_572_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_572_2_i_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_572_2_i_c(:prec:=10)
    chi := MakeCharacter_572_i();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    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_572_2_i_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_572_2_i_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_572_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![4, -12, 38, -26, 47, 2, 59, 6, 9, -1, 1]>],Snew);
    return Vf;
end function;