
// Make newform 1155.2.c.d in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1155_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1155_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_1155_2_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_1155_2_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 := [2, -4, 4, 2, 2, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [38, 2, -1, -4, 8, -1], [29, 10, -5, -20, 17, -5], [-13, 32, 30, 5, -10, 7], [27, -70, -11, -21, 19, -11], [26, -64, -37, -10, 20, -14]];
    Rf_basisdens := [1, 23, 23, 23, 23, 23];
    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_1155_c();"
function MakeCharacter_1155_c()
    N := 1155;
    order := 2;
    char_gens := [386, 232, 661, 211];
    v := [2, 1, 2, 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_1155_c_Hecke();"
function MakeCharacter_1155_c_Hecke(Kf)
    N := 1155;
    order := 2;
    char_gens := [386, 232, 661, 211];
    char_values := [[1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 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, -1], [0, 0, 0, -1, 0, 0], [0, 0, -1, 0, 0, 1], [0, 0, 0, 1, 0, 0], [1, 0, 0, 0, 0, 0], [0, 0, 0, 3, -1, 0], [0, 0, 0, 0, 1, 1], [3, -2, 0, 0, 0, 0], [0, 0, 0, 2, 1, -1], [8, 0, 1, 0, 0, 0], [2, -3, 2, 0, 0, 0], [0, 0, 0, 5, -3, 0], [-5, -2, 1, 0, 0, 0], [0, 0, 0, 0, -3, -2], [0, 0, 0, -4, -2, -3], [0, 0, 0, 1, -2, -2], [1, -1, 4, 0, 0, 0], [-2, 1, -5, 0, 0, 0], [0, 0, 0, 6, -8, 4], [-3, 2, -1, 0, 0, 0], [0, 0, 0, 4, 0, 2], [6, -3, 2, 0, 0, 0], [0, 0, 0, 4, 1, -1], [6, -6, 5, 0, 0, 0], [0, 0, 0, 0, -5, 6], [2, 4, 4, 0, 0, 0], [0, 0, 0, 0, -3, 2], [0, 0, 0, -10, 10, -5], [-8, 5, -2, 0, 0, 0], [0, 0, 0, 2, 3, 3], [0, 0, 0, 9, 2, 5], [-7, -4, 3, 0, 0, 0], [0, 0, 0, 1, -1, 1], [9, 7, -1, 0, 0, 0], [6, -4, -4, 0, 0, 0], [-12, 2, 2, 0, 0, 0], [0, 0, 0, -9, -4, -5], [0, 0, 0, -3, 5, 4], [0, 0, 0, 1, -3, 1], [0, 0, 0, -11, 11, -3], [-5, -6, -7, 0, 0, 0], [-2, -4, -2, 0, 0, 0], [-2, -6, 10, 0, 0, 0], [0, 0, 0, -4, -2, -2], [0, 0, 0, -12, 6, 2], [2, -6, 8, 0, 0, 0], [-2, 2, -2, 0, 0, 0], [0, 0, 0, 0, -5, -2], [0, 0, 0, -17, 2, 4], [-2, 5, 2, 0, 0, 0], [0, 0, 0, 0, -2, -5], [1, 3, 8, 0, 0, 0], [-11, 3, 5, 0, 0, 0], [0, -4, 4, 0, 0, 0], [0, 0, 0, 6, -2, -9], [0, 0, 0, 10, 4, -4], [-11, -3, -4, 0, 0, 0], [-2, 5, 9, 0, 0, 0], [0, 0, 0, 15, -5, 5], [-15, 5, 3, 0, 0, 0], [0, 0, 0, 1, -3, 14], [0, 0, 0, -1, -10, 2], [0, 0, 0, -4, 2, -12], [0, 0, -12, 0, 0, 0], [0, 0, 0, -15, 18, -7], [0, 0, 0, 9, 7, -11], [11, -10, 18, 0, 0, 0], [0, 0, 0, 2, 1, -8], [0, 0, 0, 6, -8, 12], [25, -2, 2, 0, 0, 0], [0, 0, 0, 0, -10, 0], [-12, 2, 3, 0, 0, 0], [0, 0, 0, 1, 6, 5], [0, 0, 0, 1, -4, 13], [-4, -1, 11, 0, 0, 0], [0, 0, 0, 14, -6, -7], [5, 20, -7, 0, 0, 0], [0, 0, 0, -3, 9, 7], [-13, -4, -11, 0, 0, 0], [-7, -5, -7, 0, 0, 0], [4, 4, 13, 0, 0, 0], [9, 6, 0, 0, 0, 0], [4, 10, -4, 0, 0, 0], [0, 0, 0, 8, -18, 2], [3, 12, 8, 0, 0, 0], [0, 0, 0, -3, 13, 1], [-9, 8, -11, 0, 0, 0], [0, 0, 0, 12, 7, -18], [-14, 12, 2, 0, 0, 0], [0, 0, 0, -18, -2, 8], [0, 0, 0, -22, 8, -8], [-5, -4, 1, 0, 0, 0], [0, 0, 0, -2, 10, -4], [17, -1, 8, 0, 0, 0], [-2, 3, -9, 0, 0, 0], [0, 0, 0, -17, -2, 18], [-16, 8, -23, 0, 0, 0], [7, -5, 6, 0, 0, 0], [0, 0, 0, 6, 16, -2], [-8, -9, -12, 0, 0, 0], [0, 0, 0, -5, -2, -9], [0, 0, 0, 4, -18, -8], [0, 0, 0, 9, 13, -7], [5, 1, 14, 0, 0, 0], [-26, 5, -6, 0, 0, 0], [0, 0, 0, 16, -2, -8], [0, 0, 0, -6, 16, -3], [0, 0, 0, -5, -7, 1], [10, -6, 0, 0, 0, 0], [-32, 1, -1, 0, 0, 0], [0, 0, 0, -10, 14, -12], [0, 0, 0, -34, 0, 2], [0, 0, 0, 9, -5, -11], [-2, 4, -16, 0, 0, 0], [24, 11, -3, 0, 0, 0], [-16, 10, -22, 0, 0, 0], [0, 0, 0, 5, -20, -7], [0, 0, 0, -24, -4, -2], [0, 0, 0, -11, 16, -6], [7, -3, 4, 0, 0, 0], [6, 23, -6, 0, 0, 0], [0, 0, 0, 15, -16, -3], [0, 0, 0, 2, -13, -13], [0, 0, 0, 16, -4, 16], [-8, -14, -6, 0, 0, 0], [0, 10, 17, 0, 0, 0], [-2, 13, -11, 0, 0, 0], [-1, 11, -6, 0, 0, 0], [0, 0, 0, 7, 6, -9], [0, 0, 0, -30, 10, 0], [8, 20, -2, 0, 0, 0], [0, 0, 0, -20, -12, 12], [17, 12, 0, 0, 0, 0], [0, 0, 0, 16, 12, -14], [10, -6, -18, 0, 0, 0], [-1, 12, 0, 0, 0, 0], [0, 0, 0, 14, -16, 2], [0, 0, 0, 6, -2, -10], [0, 0, 0, -8, -2, 12], [44, -2, 2, 0, 0, 0], [18, -12, -6, 0, 0, 0], [-1, -7, -12, 0, 0, 0], [0, 0, 0, -31, 9, -2], [0, 0, 0, -18, 0, 12], [8, -8, -4, 0, 0, 0], [16, -2, 3, 0, 0, 0], [0, 0, 0, 11, -9, -8], [0, 0, 0, 10, -28, 4], [2, -15, 16, 0, 0, 0], [0, 0, 0, 27, -22, 12], [0, 0, 0, 38, 3, -6], [-4, 18, -3, 0, 0, 0], [0, 0, 0, -16, 10, 12], [0, 0, 0, -49, 2, 2], [0, 0, 0, 1, -5, 24], [-10, 20, 10, 0, 0, 0], [-22, -12, 2, 0, 0, 0], [33, -11, 15, 0, 0, 0], [0, 0, 0, 3, 7, -16], [11, 4, 3, 0, 0, 0], [0, 0, 0, -4, -3, 17], [0, 0, 0, -26, 16, -4], [0, 0, 0, -4, -9, 14], [14, 6, -5, 0, 0, 0], [0, 0, 0, -9, 2, -14], [0, 0, 0, 16, 4, -28], [-7, 2, 14, 0, 0, 0], [0, 0, 0, 16, 22, 14], [28, -3, 20, 0, 0, 0], [0, 0, 0, 28, -10, 17], [-4, 16, -21, 0, 0, 0], [12, -7, -6, 0, 0, 0], [-4, 20, 11, 0, 0, 0], [0, 0, 0, 17, 10, -17], [0, -4, 14, 0, 0, 0], [24, 8, -10, 0, 0, 0], [30, 17, -8, 0, 0, 0], [-14, -2, -10, 0, 0, 0], [0, 0, 0, -34, 26, -8], [-13, -9, 3, 0, 0, 0], [0, 0, 0, 22, 8, -20], [-6, -18, 16, 0, 0, 0], [0, 0, 0, -16, -12, -14], [0, 0, 0, -9, -2, 2], [0, 0, 0, 9, 16, 6], [-5, -1, 10, 0, 0, 0], [0, 0, 0, 1, 12, 3], [0, 0, 0, 9, -2, 11], [-20, 15, -16, 0, 0, 0], [-31, 8, -9, 0, 0, 0], [0, 0, 0, 16, -2, 14], [0, 0, 0, -28, -10, 14], [13, 5, -17, 0, 0, 0], [-21, -21, 6, 0, 0, 0], [0, 0, 0, 2, 6, -9], [0, 0, 0, -6, 16, -30], [12, 20, 10, 0, 0, 0], [0, 0, 0, -41, -11, -2], [0, 0, 0, -1, 12, 10], [0, 0, 0, 15, 15, -25], [1, 11, 6, 0, 0, 0], [13, -24, 24, 0, 0, 0], [0, 0, 0, -28, 1, 10], [9, -21, -11, 0, 0, 0], [2, 2, 1, 0, 0, 0], [0, 0, 0, -12, 24, -4], [28, -7, 8, 0, 0, 0], [0, 0, 0, 24, 10, -2], [-8, 8, -11, 0, 0, 0], [0, -19, 12, 0, 0, 0], [0, 0, 0, 15, 4, -1], [0, -28, 3, 0, 0, 0], [0, 0, 0, -10, -10, -20], [0, 0, 0, 40, -10, 7], [16, -8, -8, 0, 0, 0], [-16, -12, -10, 0, 0, 0], [0, 0, 0, -14, -21, -12], [-8, -28, 14, 0, 0, 0], [0, 0, 0, 41, 1, -5], [0, 0, 0, 14, 1, -13], [38, 2, -12, 0, 0, 0], [7, 31, 9, 0, 0, 0], [5, 26, 9, 0, 0, 0], [0, 0, 0, -33, -7, -10], [0, 0, 0, 7, 0, -16], [12, -21, 8, 0, 0, 0], [0, 0, 0, 36, -12, -5], [-5, 6, 11, 0, 0, 0], [0, 0, 0, -48, -4, 18], [-15, -13, -14, 0, 0, 0], [0, 0, 0, 9, -13, -4], [19, -3, -11, 0, 0, 0], [20, 31, -12, 0, 0, 0], [-17, -3, -22, 0, 0, 0], [0, 0, 0, -15, 20, -23], [0, 0, 0, 9, 4, 30], [22, -1, -4, 0, 0, 0], [0, 0, 0, -12, -8, 18], [-56, -2, 0, 0, 0, 0], [-6, -16, -26, 0, 0, 0], [0, 0, 0, 19, -1, -17], [-2, 29, -9, 0, 0, 0], [0, 0, 0, -5, -31, 1], [-30, 9, -21, 0, 0, 0], [0, 0, 0, 5, -8, -12], [27, 16, -17, 0, 0, 0], [0, 0, 0, 4, -12, 6], [37, -1, -4, 0, 0, 0], [-12, -7, 14, 0, 0, 0], [0, 0, 0, -2, -26, 20], [0, 0, 0, 10, -8, 28], [51, -10, 7, 0, 0, 0], [0, 0, 0, -22, 30, -16], [18, -17, -12, 0, 0, 0], [0, 0, 0, 25, 6, 2], [46, -6, 9, 0, 0, 0], [-16, 11, 29, 0, 0, 0], [0, 0, 0, 15, -3, 26], [0, 0, 0, -26, 8, -3], [0, 0, 0, -24, 36, -22], [0, 0, 0, 3, 2, 29], [0, 0, 0, 6, -20, 2], [-25, -14, 18, 0, 0, 0], [0, 0, 0, 3, 31, -10], [0, 0, 0, -14, -18, 23], [16, -3, 32, 0, 0, 0], [32, -2, 22, 0, 0, 0], [18, 8, 1, 0, 0, 0], [0, 0, 0, -9, -14, -15], [0, 0, 0, -22, 28, -1], [-28, 6, 10, 0, 0, 0], [0, 0, 0, 10, 20, 0], [0, 0, 0, -21, 18, -27], [34, 3, 4, 0, 0, 0], [0, 0, 0, 31, 23, -4], [0, 0, 0, 15, 13, 1], [0, 0, 0, -35, 30, -12], [0, 12, -2, 0, 0, 0], [-10, 9, 4, 0, 0, 0], [-10, -8, -8, 0, 0, 0], [0, 0, 0, -35, 46, -20], [67, -3, 5, 0, 0, 0], [0, 0, 0, 30, 18, 4], [-32, -17, 9, 0, 0, 0], [0, 0, 0, 29, -41, 6], [-1, -32, 9, 0, 0, 0], [0, 0, 0, -17, -23, -20], [0, 0, 0, 20, -20, 8], [68, -10, 16, 0, 0, 0], [-53, 8, -15, 0, 0, 0], [-26, 8, -44, 0, 0, 0], [0, 0, 0, -30, 32, -13], [0, 0, 0, -2, -14, 3], [52, -8, 8, 0, 0, 0], [0, 0, 0, -34, 4, 14], [-2, -12, 24, 0, 0, 0], [26, 16, -4, 0, 0, 0], [0, 0, 0, 32, 18, -1], [18, 16, -4, 0, 0, 0], [0, 0, 0, -10, 8, 16], [0, 0, 0, 18, 0, -14], [0, 0, 0, -18, 0, 19], [-6, 7, 22, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1155_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1155_2_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_1155_2_c_d(:prec:=6)
    chi := MakeCharacter_1155_c();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    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_1155_2_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_1155_2_c_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1155_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![4, 0, 16, 0, 8, 0, 1]>,<13,R![100, 0, 164, 0, 28, 0, 1]>],Snew);
    return Vf;
end function;