
// Make newform 1155.2.c.c in Magma, downloaded from the LMFDB on 23 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_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_1155_2_c_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 := [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, -1, -1, 0], [0, 0, 0, 0, -1, 1], [-1, 0, -2, 0, 0, 0], [0, 0, 0, 0, -1, -1], [2, -2, 1, 0, 0, 0], [-4, 1, -4, 0, 0, 0], [0, 0, 0, -1, -1, 0], [-3, 0, -1, 0, 0, 0], [0, 0, 0, -2, 1, -2], [0, 0, 0, 0, 2, -1], [0, 0, 0, 7, 0, 2], [5, -1, 0, 0, 0, 0], [-8, -3, -1, 0, 0, 0], [0, 0, 0, -2, 0, 0], [-3, -4, 3, 0, 0, 0], [0, 0, 0, -8, 0, -2], [0, -1, 4, 0, 0, 0], [0, 0, 0, 8, 3, -1], [6, -2, -3, 0, 0, 0], [0, 0, 0, 0, -1, 4], [-6, 0, -4, 0, 0, 0], [0, 0, 0, -4, 5, -4], [0, 0, 0, -6, -2, -1], [2, 3, -4, 0, 0, 0], [0, 0, 0, 8, 1, -5], [0, 0, 0, 1, -4, -1], [-1, 2, 1, 0, 0, 0], [0, 0, 0, 1, 3, -7], [-1, 1, 9, 0, 0, 0], [-10, 0, 0, 0, 0, 0], [0, 2, 2, 0, 0, 0], [0, 0, 0, -1, -4, -3], [0, 0, 0, -1, -1, 0], [0, 0, 0, -5, 3, -5], [0, 0, 0, -1, -3, -9], [11, -4, 13, 0, 0, 0], [-2, 8, 6, 0, 0, 0], [2, 2, 10, 0, 0, 0], [0, 0, 0, 8, -10, 2], [0, 0, 0, 8, -10, 10], [6, 2, -8, 0, 0, 0], [-6, -6, 6, 0, 0, 0], [0, 0, 0, -4, 11, -8], [0, 0, 0, 1, 10, -6], [-8, -3, -4, 0, 0, 0], [0, 0, 0, 8, -10, 7], [-19, 3, -2, 0, 0, 0], [-17, -7, -1, 0, 0, 0], [0, 12, 4, 0, 0, 0], [0, 0, 0, -18, 6, -7], [0, 0, 0, 14, 0, 4], [-3, 1, -12, 0, 0, 0], [-4, 9, 5, 0, 0, 0], [0, 0, 0, 25, 1, -5], [-13, -5, 1, 0, 0, 0], [0, 0, 0, -3, 1, -10], [0, 0, 0, -11, 10, -8], [0, 0, 0, 12, 2, -8], [-8, -12, 4, 0, 0, 0], [0, 0, 0, -7, -6, 11], [0, 0, 0, 13, -9, 1], [11, -10, 2, 0, 0, 0], [0, 0, 0, 4, -3, 0], [0, 0, 0, -6, 0, -8], [1, 12, -12, 0, 0, 0], [0, 0, 0, 4, 2, 12], [-2, -4, -1, 0, 0, 0], [0, 0, 0, 9, 6, -13], [0, 0, 0, -7, -6, 3], [0, 7, 7, 0, 0, 0], [0, 0, 0, -14, -2, -1], [5, -6, 9, 0, 0, 0], [0, 0, 0, -15, 1, -1], [-1, -10, -7, 0, 0, 0], [7, 1, 7, 0, 0, 0], [12, 0, -11, 0, 0, 0], [-7, -6, 0, 0, 0, 0], [8, -6, 16, 0, 0, 0], [0, 0, 0, -4, 18, 6], [-9, 2, -10, 0, 0, 0], [0, 0, 0, 13, -7, -3], [11, -6, -3, 0, 0, 0], [0, 0, 0, -14, -1, -6], [14, -8, -6, 0, 0, 0], [0, 0, 0, -26, 10, 0], [0, 0, 0, 14, 4, -4], [21, -6, -1, 0, 0, 0], [0, 0, 0, -2, 18, 4], [1, -5, -14, 0, 0, 0], [22, -9, 15, 0, 0, 0], [0, 0, 0, 1, -2, 4], [12, -16, 13, 0, 0, 0], [7, -13, 6, 0, 0, 0], [0, 0, 0, -14, -4, -2], [-2, 9, 14, 0, 0, 0], [0, 0, 0, -5, 20, 1], [0, 0, 0, -8, -2, -12], [0, 0, 0, -1, 7, 7], [5, -15, -12, 0, 0, 0], [8, 7, -12, 0, 0, 0], [0, 0, 0, 24, 6, 8], [0, 0, 0, -26, -4, 3], [0, 0, 0, 1, -1, 3], [-6, -2, -12, 0, 0, 0], [6, 5, -5, 0, 0, 0], [0, 0, 0, -6, -6, -16], [0, 0, 0, -6, 8, 18], [0, 0, 0, -15, -1, 21], [-2, 4, 12, 0, 0, 0], [16, -1, -11, 0, 0, 0], [4, 2, 14, 0, 0, 0], [0, 0, 0, -15, -8, 19], [0, 0, 0, 12, -20, 6], [0, 0, 0, 3, 2, 10], [-13, 17, -2, 0, 0, 0], [4, -13, -12, 0, 0, 0], [0, 0, 0, -5, -2, 15], [0, 0, 0, 6, -11, -9], [0, 0, 0, 32, -4, -4], [-12, -2, -22, 0, 0, 0], [-2, -12, -11, 0, 0, 0], [34, -3, 1, 0, 0, 0], [19, -13, 22, 0, 0, 0], [0, 0, 0, -5, 10, -11], [0, 0, 0, 22, -22, 4], [-8, -16, -6, 0, 0, 0], [0, 0, 0, -16, 8, -12], [-11, 0, 4, 0, 0, 0], [0, 0, 0, -32, 16, -10], [-30, 6, -18, 0, 0, 0], [-1, -2, -14, 0, 0, 0], [0, 0, 0, 22, -32, 14], [0, 0, 0, 18, -14, -2], [0, 0, 0, -20, -14, 4], [12, -6, 14, 0, 0, 0], [2, 24, -6, 0, 0, 0], [-21, 13, -18, 0, 0, 0], [0, 0, 0, -5, -21, 2], [0, 0, 0, -30, -8, -4], [16, 4, 12, 0, 0, 0], [4, 2, 15, 0, 0, 0], [0, 0, 0, -5, -1, -8], [0, 0, 0, 10, -16, -8], [-20, 13, -22, 0, 0, 0], [0, 0, 0, -3, -16, 24], [0, 0, 0, 12, 15, 14], [-24, -6, -7, 0, 0, 0], [0, 0, 0, -8, 26, -8], [0, 0, 0, 25, -6, -4], [0, 0, 0, -13, 5, -16], [-10, 4, -14, 0, 0, 0], [6, 4, -6, 0, 0, 0], [1, -19, 7, 0, 0, 0], [0, 0, 0, 27, -5, -4], [9, -14, 13, 0, 0, 0], [0, 0, 0, 2, -21, 21], [0, 0, 0, 6, -16, -4], [0, 0, 0, -2, 11, -14], [6, -6, 19, 0, 0, 0], [0, 0, 0, 21, -12, 18], [0, 0, 0, 0, -20, 8], [-3, -10, -14, 0, 0, 0], [0, 0, 0, 8, 10, 6], [14, 7, -14, 0, 0, 0], [0, 0, 0, -8, 2, -5], [-34, 6, -9, 0, 0, 0], [-2, 21, 0, 0, 0, 0], [22, -2, -1, 0, 0, 0], [0, 0, 0, -15, 12, 13], [-16, -4, -6, 0, 0, 0], [16, -12, -18, 0, 0, 0], [-12, -13, 18, 0, 0, 0], [-14, -14, 6, 0, 0, 0], [0, 0, 0, -2, -10, -12], [-23, 5, -31, 0, 0, 0], [0, 0, 0, -10, -20, 8], [-2, -22, 4, 0, 0, 0], [0, 0, 0, 4, -4, 14], [0, 0, 0, -7, 14, 12], [0, 0, 0, 15, -6, -18], [15, 11, 14, 0, 0, 0], [0, 0, 0, 29, 6, 5], [0, 0, 0, 17, -26, 17], [-26, -19, 2, 0, 0, 0], [-9, 2, -11, 0, 0, 0], [0, 0, 0, -32, -2, -14], [0, 0, 0, 24, 2, -22], [1, 9, 3, 0, 0, 0], [15, -1, -6, 0, 0, 0], [0, 0, 0, -14, 26, -9], [0, 0, 0, -34, 20, -2], [12, 20, -14, 0, 0, 0], [0, 0, 0, -3, 11, -26], [0, 0, 0, 5, 4, 20], [0, 0, 0, -7, 13, 9], [-31, 15, -4, 0, 0, 0], [45, 6, 6, 0, 0, 0], [0, 0, 0, 32, 1, -16], [27, 5, 7, 0, 0, 0], [2, 14, -7, 0, 0, 0], [0, 0, 0, 32, -20, -4], [14, 1, -2, 0, 0, 0], [0, 0, 0, -20, 34, -26], [-2, 18, -23, 0, 0, 0], [34, -7, 22, 0, 0, 0], [0, 0, 0, -17, 6, 9], [-24, -4, -5, 0, 0, 0], [0, 0, 0, 18, 10, 8], [0, 0, 0, -12, 6, 1], [-16, 0, -4, 0, 0, 0], [-24, -4, -2, 0, 0, 0], [0, 0, 0, -8, -17, 12], [4, 4, -26, 0, 0, 0], [0, 0, 0, 5, -15, 7], [0, 0, 0, -8, 19, -25], [-10, -10, 16, 0, 0, 0], [-35, -7, 3, 0, 0, 0], [13, 8, 5, 0, 0, 0], [0, 0, 0, 21, -5, -18], [0, 0, 0, -39, 4, 14], [26, -7, 34, 0, 0, 0], [0, 0, 0, 4, -28, -7], [23, 8, 7, 0, 0, 0], [0, 0, 0, 12, -4, 26], [29, 3, -12, 0, 0, 0], [0, 0, 0, 15, -11, 4], [-7, -17, -1, 0, 0, 0], [18, -19, 6, 0, 0, 0], [-29, -15, 0, 0, 0, 0], [0, 0, 0, 1, -28, 15], [0, 0, 0, -13, 20, -20], [-16, 19, -26, 0, 0, 0], [0, 0, 0, 28, -4, 2], [0, 26, 4, 0, 0, 0], [6, 16, -22, 0, 0, 0], [0, 0, 0, -33, 7, -1], [20, 29, -1, 0, 0, 0], [0, 0, 0, -9, -31, -19], [-2, -23, 15, 0, 0, 0], [0, 0, 0, -13, -4, 18], [25, -2, -11, 0, 0, 0], [0, 0, 0, 8, 8, 10], [-15, -13, 0, 0, 0, 0], [18, -25, 20, 0, 0, 0], [0, 0, 0, 42, -10, 0], [0, 0, 0, -22, -12, 16], [-3, -4, 1, 0, 0, 0], [0, 0, 0, 22, -30, 8], [0, 23, -14, 0, 0, 0], [0, 0, 0, 3, -6, 12], [4, 16, -23, 0, 0, 0], [24, 31, -3, 0, 0, 0], [0, 0, 0, 27, -35, 6], [0, 0, 0, 6, -4, 15], [0, 0, 0, 52, 8, -14], [0, 0, 0, -9, -4, 19], [0, 0, 0, -14, 36, 2], [11, -8, 12, 0, 0, 0], [0, 0, 0, 27, -17, 2], [0, 0, 0, -26, -18, -11], [14, 1, 6, 0, 0, 0], [-32, 10, 6, 0, 0, 0], [-30, 0, -23, 0, 0, 0], [0, 0, 0, 43, -4, -9], [0, 0, 0, -2, 12, -33], [0, 6, 6, 0, 0, 0], [0, 0, 0, 38, -4, 12], [0, 0, 0, 15, 22, 7], [-24, -19, 10, 0, 0, 0], [0, 0, 0, 15, -13, 12], [0, 0, 0, -45, -3, 5], [0, 0, 0, -21, 8, 4], [-12, -24, 6, 0, 0, 0], [-8, 23, -34, 0, 0, 0], [2, 28, 20, 0, 0, 0], [0, 0, 0, -9, 18, -10], [-1, 1, -15, 0, 0, 0], [0, 0, 0, 2, 14, -16], [-34, 7, -23, 0, 0, 0], [0, 0, 0, 29, -13, 18], [19, -30, 21, 0, 0, 0], [0, 0, 0, 25, 19, -8], [0, 0, 0, 16, -12, 12], [0, -18, 4, 0, 0, 0], [21, 6, -17, 0, 0, 0], [10, -20, 32, 0, 0, 0], [0, 0, 0, 22, 12, -7], [0, 0, 0, -22, 10, 23], [8, 0, -8, 0, 0, 0], [0, 0, 0, 10, -4, -26], [34, 8, 12, 0, 0, 0], [-18, 4, 0, 0, 0, 0], [0, 0, 0, -28, -18, -15], [14, 24, 12, 0, 0, 0], [0, 0, 0, 22, -40, 0], [0, 0, 0, -50, 16, 2], [0, 0, 0, -34, 32, -5], [-8, -35, 8, 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_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_1155_2_c_c(: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_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_1155_2_c_c( : 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![4, 0, 20, 0, 12, 0, 1]>],Snew);
    return Vf;
end function;