
// Make newform 1380.2.f.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1380_f();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1380_f_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_1380_2_f_a();".
// 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_1380_2_f_a();".
// 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_1380_f();"
function MakeCharacter_1380_f()
    N := 1380;
    order := 2;
    char_gens := [691, 461, 277, 1201];
    v := [2, 2, 1, 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_1380_f_Hecke();"
function MakeCharacter_1380_f_Hecke(Kf)
    N := 1380;
    order := 2;
    char_gens := [691, 461, 277, 1201];
    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, 0], [0, 0, 0, -1, 0, 0], [0, -1, 0, 0, 1, 0], [0, 0, 0, -1, 0, 0], [0, -1, 0, 0, 0, 0], [0, 0, 0, 1, -1, 2], [0, 0, 0, 0, -1, 2], [1, -2, 1, 0, 0, 0], [0, 0, 0, -1, 0, 0], [2, 0, -3, 0, 0, 0], [-4, -1, 1, 0, 0, 0], [0, 0, 0, 0, -3, 1], [-3, -1, -4, 0, 0, 0], [0, 0, 0, 1, -3, -1], [0, 0, 0, -4, -2, 1], [0, 0, 0, -1, -2, -1], [-2, 0, -1, 0, 0, 0], [-7, 4, -1, 0, 0, 0], [0, 0, 0, 2, -5, -1], [-1, -3, 4, 0, 0, 0], [0, 0, 0, -7, -1, -2], [2, -6, 2, 0, 0, 0], [0, 0, 0, 8, -3, 2], [-2, 0, -4, 0, 0, 0], [0, 0, 0, -3, 7, -5], [-7, 1, -2, 0, 0, 0], [0, 0, 0, -3, 1, -1], [0, 0, 0, -1, -4, 5], [3, 0, 7, 0, 0, 0], [0, 0, 0, -1, 2, 3], [0, 0, 0, -3, 5, -8], [-2, -6, -6, 0, 0, 0], [0, 0, 0, -5, -1, 5], [-3, -2, 0, 0, 0, 0], [8, 1, 0, 0, 0, 0], [-1, 1, 1, 0, 0, 0], [0, 0, 0, -7, -4, 2], [0, 0, 0, -2, -2, -2], [0, 0, 0, -6, 10, -7], [0, 0, 0, 2, 8, -2], [1, -3, 1, 0, 0, 0], [-4, -6, 4, 0, 0, 0], [-6, 1, -2, 0, 0, 0], [0, 0, 0, 2, 6, 4], [0, 0, 0, 2, 6, 6], [-2, 2, -12, 0, 0, 0], [-15, 4, -8, 0, 0, 0], [0, 0, 0, -12, 4, 0], [0, 0, 0, 0, -8, 4], [4, 10, 0, 0, 0, 0], [0, 0, 0, 0, -4, 12], [1, 9, -10, 0, 0, 0], [-11, 6, 5, 0, 0, 0], [-2, 4, 2, 0, 0, 0], [0, 0, 0, 8, 4, 7], [0, 0, 0, 15, -4, 3], [-2, 8, -13, 0, 0, 0], [-7, 0, -2, 0, 0, 0], [0, 0, 0, -6, 2, -2], [-8, 13, -14, 0, 0, 0], [0, 0, 0, -1, 0, -6], [0, 0, 0, 6, 1, 4], [0, 0, 0, 7, -1, 8], [-18, 4, -2, 0, 0, 0], [0, 0, 0, -12, 5, -9], [0, 0, 0, -12, 14, -1], [4, -1, 1, 0, 0, 0], [0, 0, 0, 6, 4, 4], [0, 0, 0, 0, -4, -6], [13, 2, 4, 0, 0, 0], [0, 0, 0, -6, 8, -5], [16, -9, 18, 0, 0, 0], [0, 0, 0, 6, 5, -7], [0, 0, 0, -21, 9, -5], [-4, 0, 8, 0, 0, 0], [0, 0, 0, -6, 17, -2], [12, -8, 12, 0, 0, 0], [0, 0, 0, 12, 2, -6], [-16, 4, -2, 0, 0, 0], [7, -6, -12, 0, 0, 0], [-22, 5, -10, 0, 0, 0], [-1, 0, 13, 0, 0, 0], [-6, 4, 4, 0, 0, 0], [0, 0, 0, -1, -10, 10], [-5, 11, 9, 0, 0, 0], [0, 0, 0, -4, 6, -15], [22, 2, 1, 0, 0, 0], [0, 0, 0, 8, -1, -5], [-12, -2, -2, 0, 0, 0], [0, 0, 0, -9, -7, 18], [0, 0, 0, -5, 0, -13], [-10, -1, 0, 0, 0, 0], [0, 0, 0, -7, -1, -2], [5, -19, 10, 0, 0, 0], [-4, -5, -15, 0, 0, 0], [0, 0, 0, 8, -5, 6], [5, 9, -9, 0, 0, 0], [8, -5, -8, 0, 0, 0], [0, 0, 0, -18, -10, -4], [-19, 3, 5, 0, 0, 0], [0, 0, 0, 12, -18, 10], [0, 0, 0, 21, -2, -7], [0, 0, 0, -1, -12, -13], [14, 6, -12, 0, 0, 0], [11, -12, -9, 0, 0, 0], [0, 0, 0, 10, 8, -16], [0, 0, 0, -8, -2, -2], [0, 0, 0, 0, 4, 19], [-4, 14, -4, 0, 0, 0], [-16, 7, -17, 0, 0, 0], [0, 0, 0, 1, 11, 8], [0, 0, 0, -18, -8, 10], [0, 0, 0, 16, -19, 12], [-20, -14, 6, 0, 0, 0], [-11, 2, -15, 0, 0, 0], [-4, -9, -8, 0, 0, 0], [0, 0, 0, -16, -1, 13], [0, 0, 0, -4, 0, 1], [0, 0, 0, 22, -16, -3], [-8, -11, 2, 0, 0, 0], [16, -12, -10, 0, 0, 0], [0, 0, 0, -27, -3, -4], [0, 0, 0, -11, -8, -1], [0, 0, 0, 10, -20, 9], [21, -1, 3, 0, 0, 0], [-12, 9, 6, 0, 0, 0], [3, -22, 3, 0, 0, 0], [-10, 8, 15, 0, 0, 0], [0, 0, 0, 11, 2, -14], [0, 0, 0, -22, 9, 1], [0, -5, -5, 0, 0, 0], [0, 0, 0, 18, 6, -22], [-1, 10, -21, 0, 0, 0], [0, 0, 0, -1, -6, -10], [6, -12, 17, 0, 0, 0], [-3, -16, 5, 0, 0, 0], [0, 0, 0, 4, -2, -16], [0, 0, 0, 2, -5, 21], [0, 0, 0, -9, 6, 9], [3, -13, -6, 0, 0, 0], [-3, -8, 20, 0, 0, 0], [9, -7, -5, 0, 0, 0], [0, 0, 0, -12, -10, 10], [0, 0, 0, -1, 6, -9], [-6, 3, -19, 0, 0, 0], [2, 8, 6, 0, 0, 0], [0, 0, 0, -8, 2, -16], [0, 0, 0, 14, -18, 8], [-13, 12, -20, 0, 0, 0], [0, 0, 0, 12, 0, -20], [0, 0, 0, -16, 8, 18], [18, -11, 18, 0, 0, 0], [0, 0, 0, 3, 5, 14], [0, 0, 0, -20, 6, -4], [0, 0, 0, 0, -7, -7], [-2, 14, 22, 0, 0, 0], [-20, 10, -14, 0, 0, 0], [20, -4, -1, 0, 0, 0], [0, 0, 0, -1, -7, 9], [6, -1, 14, 0, 0, 0], [0, 0, 0, -28, -6, -2], [0, 0, 0, 2, 8, -12], [0, 0, 0, 13, 5, 12], [42, 12, -6, 0, 0, 0], [0, 0, 0, -6, 3, -20], [0, 0, 0, 8, 9, -12], [19, 22, -2, 0, 0, 0], [0, 0, 0, -10, 4, -6], [0, 6, -10, 0, 0, 0], [0, 0, 0, 38, 2, 7], [-24, 0, 18, 0, 0, 0], [1, 20, -12, 0, 0, 0], [12, 10, -10, 0, 0, 0], [0, 0, 0, 3, -26, -8], [2, 16, -10, 0, 0, 0], [-10, 7, -8, 0, 0, 0], [29, 3, -1, 0, 0, 0], [-18, -12, -17, 0, 0, 0], [0, 0, 0, -4, 9, 17], [13, -4, 17, 0, 0, 0], [0, 0, 0, -3, -3, -22], [10, 3, -8, 0, 0, 0], [0, 0, 0, 12, -26, 24], [0, 0, 0, 20, 0, 19], [0, 0, 0, -17, 8, -17], [-14, 5, 2, 0, 0, 0], [0, 0, 0, -21, -13, -4], [0, 0, 0, -34, 3, -7], [0, -7, 15, 0, 0, 0], [23, 15, -12, 0, 0, 0], [0, 0, 0, -17, 37, -8], [0, 0, 0, 32, 12, 6], [17, -4, -5, 0, 0, 0], [-9, 15, 14, 0, 0, 0], [0, 0, 0, 5, 7, -29], [0, 0, 0, 9, -10, 17], [35, -10, 21, 0, 0, 0], [0, 0, 0, 7, 13, -1], [0, 0, 0, -4, 18, -6], [0, 0, 0, 4, 20, -14], [16, 17, -10, 0, 0, 0], [21, -1, 31, 0, 0, 0], [0, 0, 0, -17, 3, 26], [19, 8, 19, 0, 0, 0], [-14, -1, -4, 0, 0, 0], [0, 0, 0, 8, -16, 4], [-5, 2, 29, 0, 0, 0], [0, 0, 0, 0, 2, 15], [-1, -19, 4, 0, 0, 0], [3, 7, -19, 0, 0, 0], [0, 0, 0, -36, 2, -8], [28, -6, -2, 0, 0, 0], [0, 0, 0, -25, 3, 17], [0, 0, 0, -21, 21, -3], [-8, -18, 17, 0, 0, 0], [5, -14, 13, 0, 0, 0], [0, 0, 0, 2, 6, 28], [-25, 17, -26, 0, 0, 0], [0, 0, 0, 11, 14, 25], [0, 0, 0, -14, 6, 7], [-12, -11, -11, 0, 0, 0], [-22, 18, -32, 0, 0, 0], [6, 14, 9, 0, 0, 0], [0, 0, 0, -46, -1, -7], [0, 0, 0, -36, 14, 9], [13, 4, -24, 0, 0, 0], [0, 0, 0, 24, -6, -14], [-19, 3, -10, 0, 0, 0], [0, 0, 0, 26, 6, -16], [47, -15, 24, 0, 0, 0], [0, 0, 0, 5, -11, 24], [-31, -4, -19, 0, 0, 0], [14, 22, -16, 0, 0, 0], [18, -10, -5, 0, 0, 0], [0, 0, 0, -44, 11, -11], [0, 0, 0, -15, 7, 15], [-19, 24, -21, 0, 0, 0], [0, 0, 0, -34, 20, -20], [-16, 0, -3, 0, 0, 0], [6, 18, -13, 0, 0, 0], [0, 0, 0, 10, -21, 12], [8, -7, -11, 0, 0, 0], [0, 0, 0, -32, 40, -26], [3, 1, 35, 0, 0, 0], [0, 0, 0, 30, -4, 6], [30, 18, -10, 0, 0, 0], [0, 0, 0, 49, 1, 8], [58, -5, 14, 0, 0, 0], [-14, 2, -12, 0, 0, 0], [0, 0, 0, -4, 25, -18], [0, 0, 0, -5, -19, 19], [34, -4, -18, 0, 0, 0], [0, 0, 0, 38, -1, 8], [2, -8, 26, 0, 0, 0], [0, 0, 0, -2, -16, 23], [-46, 0, -13, 0, 0, 0], [-7, 14, 9, 0, 0, 0], [0, 0, 0, -8, -12, -12], [0, 0, 0, 8, -12, 3], [0, 0, 0, -28, 0, -16], [0, 0, 0, 18, 7, -23], [0, 0, 0, 23, -18, -11], [-8, -15, -27, 0, 0, 0], [0, 0, 0, 33, -6, -10], [0, 0, 0, 14, -2, 16], [11, 20, 13, 0, 0, 0], [-32, 3, 14, 0, 0, 0], [12, 26, -14, 0, 0, 0], [0, 0, 0, -31, -2, 8], [0, 0, 0, 34, -22, -6], [-8, 15, -23, 0, 0, 0], [0, 0, 0, 18, 23, -11], [0, 0, 0, 4, -25, -5], [0, -4, 0, 0, 0, 0], [0, 0, 0, -5, -23, -16], [0, 0, 0, 21, 23, -18], [0, 0, 0, 10, -24, 13], [6, -7, -19, 0, 0, 0], [27, 14, 15, 0, 0, 0], [18, 7, -22, 0, 0, 0], [0, 0, 0, -40, 2, -2], [7, -22, 19, 0, 0, 0], [0, 0, 0, -40, -4, -12], [-6, -28, 20, 0, 0, 0], [0, 0, 0, 19, 21, -16], [-16, -22, -19, 0, 0, 0], [0, 0, 0, 53, -12, -8], [0, 0, 0, -24, -6, -14], [-29, 7, 5, 0, 0, 0], [-24, 0, -30, 0, 0, 0], [30, -18, 14, 0, 0, 0], [0, 0, 0, 30, -1, 10], [0, 0, 0, -34, -18, 8], [-26, 23, -40, 0, 0, 0], [0, 0, 0, 21, 3, 8], [26, 2, 10, 0, 0, 0], [5, -14, 13, 0, 0, 0], [0, 0, 0, 18, 12, 21], [-14, 4, -23, 0, 0, 0], [0, 0, 0, -8, 30, -18], [0, 0, 0, -17, 16, 12], [0, 0, 0, 26, 21, -6], [36, -16, 10, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1380_f_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1380_2_f_a();".
// 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_1380_2_f_a(:prec:=6)
    chi := MakeCharacter_1380_f();
    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_1380_2_f_a();".
// 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_1380_2_f_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1380_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![1, 0, 1]>],Snew);
    return Vf;
end function;