
// Make newform 1352.2.i.f in Magma, downloaded from the LMFDB on 14 March 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_1352_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1352_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_1352_2_i_f();".
// 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_1352_2_i_f();".
// 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 := [16, 4, 5, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [16, -5, 5, -1], [4, 0, 0, 1]];
    Rf_basisdens := [1, 1, 20, 5];
    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_1352_i();"
function MakeCharacter_1352_i()
    N := 1352;
    order := 3;
    char_gens := [1015, 677, 1185];
    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_1352_i_Hecke();"
function MakeCharacter_1352_i_Hecke( : Kf := false)
    N := 1352;
    order := 3;
    char_gens := [1015, 677, 1185];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [-1, 0, 1, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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]];
    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], [2, 0, 0, 1], [0, -1, 0, -1], [0, 2, 0, 0], [0, 0, 0, 0], [2, 3, -2, 3], [0, 2, 0, 2], [0, 0, 8, 0], [0, 0, 2, 0], [4, 0, 0, 0], [0, -3, -2, 0], [0, 2, -2, 0], [-8, -1, 8, -1], [-8, 0, 0, -3], [-2, 0, 0, -2], [0, 2, 0, 2], [-6, 2, 6, 2], [0, 2, 0, 0], [0, -3, 0, -3], [-6, 0, 0, 0], [8, 0, 0, 0], [-8, 0, 0, -4], [0, 0, -10, 0], [-2, -4, 2, -4], [0, 2, 2, 0], [-8, 0, 0, -4], [0, 0, -4, 0], [10, 0, 0, 5], [-2, -8, 2, -8], [0, 4, 0, 0], [8, 0, 0, 5], [-2, -2, 2, -2], [0, 3, 0, 3], [-6, 4, 6, 4], [8, 0, 0, -1], [-2, 0, 0, 8], [0, -8, 0, -8], [0, -4, -4, 0], [10, 2, -10, 2], [0, -3, 0, 0], [-18, 0, 0, 2], [0, 6, 0, 6], [0, 6, 6, 0], [0, 3, -10, 0], [0, 2, 0, 2], [0, -3, -8, 0], [0, 3, 8, 0], [8, 0, -8, 0], [-6, 0, 0, -5], [6, 0, 0, 5], [0, 0, 0, 1], [14, 4, -14, 4], [-12, 4, 12, 4], [0, 1, -14, 0], [0, 4, 16, 0], [10, 2, -10, 2], [0, 5, 8, 0], [-14, 6, 14, 6], [10, 0, 0, 0], [0, 8, -12, 0], [-18, -1, 18, -1], [8, 0, 0, -2], [-16, 0, 0, -2], [14, 0, 0, -5], [-2, 0, 0, 4], [-16, -8, 16, -8], [-26, 0, 0, -5], [0, 3, 0, 3], [0, -7, 6, 0], [0, -6, 6, 0], [4, 0, 0, -4], [0, -10, 16, 0], [-14, -10, 14, -10], [0, 4, -16, 0], [24, 5, -24, 5], [-10, 0, 0, -12], [2, 0, -2, 0], [0, -10, 22, 0], [-18, -2, 18, -2], [0, 7, -16, 0], [-14, 0, 0, -3], [0, 1, 16, 0], [-22, -11, 22, -11], [0, -2, -16, 0], [8, 0, 0, -5], [-2, 12, 2, 12], [0, -12, 6, 0], [22, -5, -22, -5], [28, 0, 0, -4], [20, 0, 0, -8], [0, 5, -24, 0], [-12, -4, 12, -4], [0, 9, -24, 0], [8, 0, 0, -8], [-8, -6, 8, -6], [0, 0, 18, 0], [-2, 0, 0, 3], [0, 8, -20, 0], [2, 0, 0, 3], [-32, 0, 0, -1], [0, 11, 14, 0], [-16, -9, 16, -9], [0, 3, 34, 0], [16, 0, 0, -5], [-14, 0, 0, 4], [0, 16, -16, 0], [18, 0, 0, -8], [8, 0, 0, -6], [0, -11, -6, 0], [0, 6, 0, 6], [0, 0, -6, 0], [14, 14, -14, 14], [-40, 0, 0, -4], [40, -3, -40, -3], [-2, -4, 2, -4], [0, 14, 0, 14], [0, -2, 0, 0], [0, -2, 10, 0], [4, 8, -4, 8], [0, 12, 10, 0], [0, 13, -14, 0], [14, 0, 0, 6], [24, 0, -24, 0], [0, -16, 24, 0], [38, 0, 0, 6], [-6, -12, 6, -12], [8, -2, -8, -2], [0, 0, 0, 18], [-30, 0, 0, -1], [0, 4, 0, 0], [0, -23, 8, 0], [0, -16, 16, 0], [0, -2, 2, 0], [-26, 8, 26, 8], [0, -14, 22, 0], [-18, 11, 18, 11], [16, 0, -16, 0], [34, 8, -34, 8], [0, 11, 2, 0], [-16, 0, 0, 12], [0, 1, -18, 0], [-16, -10, 16, -10], [0, 0, 0, 14], [0, -8, 2, 0], [-20, -12, 20, -12], [-22, 0, 0, 3], [10, 0, 0, -12], [-28, 0, 0, -16], [-16, 0, 0, 13], [30, -3, -30, -3], [0, 5, 18, 0], [8, 0, 0, 3], [0, 24, -16, 0], [0, 5, 8, 0], [0, 0, 0, -6], [16, 16, -16, 16], [-26, -14, 26, -14], [-6, 0, 0, 12], [2, 0, 0, -21], [0, -6, 8, 0], [-30, -19, 30, -19], [0, 0, 0, -1], [24, -3, -24, -3], [0, -8, -34, 0], [0, 0, 0, -7], [0, 14, 0, 0], [10, 4, -10, 4], [18, 0, 0, 20], [54, 0, 0, 0], [-8, 0, 0, -6], [14, 9, -14, 9], [24, 8, -24, 8], [-18, -18, 18, -18], [-8, 0, 0, 10], [34, -11, -34, -11], [0, -4, 24, 0], [38, 0, 0, -4], [0, -12, 24, 0], [0, -2, -54, 0], [-20, 0, 0, -4], [4, 0, 0, 0], [-10, 0, 0, -12], [-22, 0, 0, 4], [0, 8, 4, 0], [10, -8, -10, -8], [-34, 0, 0, -16], [-16, 0, 0, -14], [0, -2, -18, 0], [-28, -12, 28, -12], [18, 27, -18, 27], [48, 4, -48, 4], [32, 0, 0, 15], [0, -4, -14, 0], [-8, 13, 8, 13], [0, -16, -10, 0], [10, 0, 0, 18], [10, -2, -10, -2], [58, 0, 0, 2], [-8, 0, 0, 6], [22, -17, -22, -17], [40, 0, -40, 0], [0, 3, 6, 0], [-34, 0, 0, 1], [0, 0, -24, 0], [0, -4, 18, 0], [24, 0, 0, 23], [20, 8, -20, 8], [0, -8, 6, 0], [-16, 1, 16, 1], [0, -4, -2, 0], [-10, 0, 0, -8], [0, -16, 0, 0], [-32, -10, 32, -10], [-32, 9, 32, 9], [18, 0, 0, -10], [8, 0, 0, -12], [-46, -13, 46, -13], [0, 24, 4, 0], [34, 0, 0, 19], [0, 10, 34, 0], [-8, 0, 0, 7], [26, 0, 0, 6], [56, -7, -56, -7], [0, -9, -24, 0], [6, 0, 0, -4], [0, -21, 34, 0], [-56, 4, 56, 4], [-32, 6, 32, 6], [16, 0, 0, 18], [0, 8, -46, 0], [0, 8, -28, 0], [0, -20, 12, 0], [-18, 0, 0, 15], [8, 0, 0, 11], [-32, 0, 0, -15], [-2, -16, 2, -16], [0, -17, 38, 0], [20, -4, -20, -4], [0, -14, 24, 0], [0, 2, -16, 0], [0, -7, -8, 0], [32, -8, -32, -8], [0, 3, -34, 0], [30, -12, -30, -12], [40, 0, 0, 12], [32, 5, -32, 5], [0, 4, 64, 0], [-24, -6, 24, -6], [0, -2, 72, 0], [0, -19, -10, 0], [0, 0, 30, 0], [-24, 0, 0, 3], [0, 19, -30, 0], [22, 0, 0, 14], [24, 26, -24, 26], [34, 10, -34, 10], [0, -2, 40, 0], [-2, 0, 0, -6], [-10, -20, 10, -20], [48, 0, 0, 4], [0, 5, -8, 0], [26, 0, 0, 7], [0, 12, 18, 0], [22, -10, -22, -10], [40, -13, -40, -13], [-10, 11, 10, 11], [2, 0, 0, -22], [-24, 4, 24, 4], [2, 14, -2, 14], [38, 0, 0, -14], [32, 0, 0, 6], [0, -8, 22, 0], [0, 0, 0, 0], [-18, -20, 18, -20], [0, 36, -12, 0], [-24, 2, 24, 2], [-2, 0, 0, -32], [-34, 2, 34, 2], [-8, 17, 8, 17], [0, 4, -24, 0], [0, -29, 24, 0], [40, 0, 0, -14], [0, -20, 10, 0], [-8, 0, 0, 16], [40, 0, 0, 22], [30, 0, 0, 17], [10, 0, 0, 31], [-16, 23, 16, 23], [-2, 24, 2, 24], [0, 26, -38, 0], [24, -17, -24, -17], [0, 10, -2, 0], [48, -2, -48, -2], [-30, -4, 30, -4], [-10, 0, 0, -14], [16, 0, 0, 6], [0, 2, -46, 0], [0, 31, -40, 0], [0, 10, -8, 0], [-14, 21, 14, 21], [58, 0, 0, -3], [0, 2, 24, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1352_i_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1352_2_i_f();".
// 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_1352_2_i_f(:prec:=4)
    chi := MakeCharacter_1352_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(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_1352_2_i_f();".
// 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_1352_2_i_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1352_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![16, -4, 5, 1, 1]>,<5,R![-2, -3, 1]>,<7,R![16, 4, 5, -1, 1]>],Snew);
    return Vf;
end function;