
// Make newform 1860.2.a.i in Magma, downloaded from the LMFDB on 09 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1860_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_1860_2_a_i();".
// 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_1860_2_a_i();".
// 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 := [14, 9, -11, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, -7, 0, 1], [0, 11, 0, -1], [-6, -1, 1, 0]];
    Rf_basisdens := [1, 2, 2, 1];
    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_1860_a();"
function MakeCharacter_1860_a()
    N := 1860;
    order := 1;
    char_gens := [931, 1241, 1117, 1801];
    v := [1, 1, 1, 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;

function MakeCharacter_1860_a_Hecke( : Kf := Kf)
    return MakeCharacter_1860_a();
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], [-1, 0, 0, 0], [1, 0, 0, 0], [-1, 1, 0, 0], [0, 1, 1, 0], [1, 0, -1, 0], [0, 0, 0, -1], [0, 0, 0, 0], [0, 0, 0, 1], [5, 0, 1, 1], [1, 0, 0, 0], [1, 1, 0, 0], [2, -1, -1, 0], [0, -1, -1, 0], [0, -2, 0, -1], [-2, -2, 0, -1], [-1, -2, -1, -3], [2, -2, -2, -2], [3, 2, 1, 2], [1, 1, 0, 1], [3, 1, 2, 0], [6, 1, -1, 1], [-4, 0, 0, -1], [7, 1, 0, 1], [2, 1, 1, -2], [6, 1, 1, 2], [1, -2, -1, 0], [2, 0, 0, 1], [10, 1, 1, 1], [-4, 0, 2, 0], [2, 0, 2, 0], [-5, 2, -1, -1], [4, -2, -2, -3], [10, -4, -2, -2], [-2, -1, -1, -4], [4, -1, -1, -1], [2, 3, -1, 0], [-1, 0, -1, 0], [-2, -4, -2, 0], [0, -2, 0, 0], [4, 2, 2, 2], [10, 2, -2, 2], [-7, -2, -3, -1], [-2, -1, 3, 0], [4, 2, 2, -1], [12, -3, -3, -1], [6, -4, 2, 2], [-6, 1, -1, 0], [2, -4, 0, -1], [8, 4, -2, 2], [-4, 0, 2, 3], [-6, -1, 1, 2], [0, -2, -4, 0], [-12, 4, 4, 2], [4, -2, 0, 1], [-10, -4, 2, -2], [-3, 6, 3, 1], [-2, 2, 4, 2], [3, -3, 2, 0], [-4, 4, 2, 0], [3, -1, -2, 2], [-12, 0, -2, 1], [-7, -1, 0, -4], [-21, -1, 0, 1], [-1, 0, -3, 0], [8, -6, 0, -1], [6, -1, 1, 3], [-3, 3, -2, 0], [2, 2, 4, 4], [2, 3, 3, 3], [-16, -2, -2, -1], [-7, 0, -1, -3], [0, 0, 0, 0], [0, -2, 0, -2], [0, 6, 2, 0], [-12, 4, 0, -1], [-9, 1, 0, -1], [-10, -4, -4, 2], [-7, -1, 0, -1], [6, 4, 0, -2], [-5, -3, -2, 1], [4, -1, 1, 1], [-5, -1, -4, -3], [-1, -1, 0, 4], [0, 0, 0, -4], [2, 0, -4, -3], [5, -5, 0, -1], [-13, 5, 6, 2], [15, 0, 3, 3], [12, 3, 3, 6], [-8, 4, -2, -3], [5, 1, 4, 3], [-10, 2, 0, 2], [6, -6, 0, -2], [16, 0, -4, 0], [-12, -2, 0, 3], [5, 5, 2, 1], [6, -1, 3, 0], [22, 2, 4, 2], [8, 1, -1, 5], [-9, 4, -1, 0], [4, -2, -2, -3], [-6, -2, -2, -1], [3, -4, -1, -3], [14, -6, 0, -2], [-28, -3, -1, -2], [14, 0, 2, 2], [0, -4, -2, 2], [3, -3, 6, 3], [-12, 2, -4, -4], [9, 0, 1, 0], [-1, -2, -1, 4], [14, -2, -6, 0], [18, -1, 5, 5], [0, 4, -4, 0], [19, -6, 1, 1], [6, -6, -4, -6], [8, 4, 0, 2], [0, 4, 10, 3], [-1, -7, -6, -1], [-8, -2, 4, -4], [17, 2, 9, 6], [38, 2, 0, -1], [6, 4, -4, 1], [10, 4, 2, -2], [-4, 1, 3, 0], [6, -4, -8, -4], [12, -5, -5, -4], [-9, -6, -3, -6], [-14, -2, -6, 0], [20, -3, 1, -1], [-26, -4, 2, 2], [-30, -2, -4, -4], [9, -4, 3, 2], [-5, 6, 1, 3], [2, 1, 5, 5], [4, 4, 4, 3], [10, 3, 5, 10], [-20, 2, -2, -3], [1, -1, -4, 5], [-2, 0, 2, 4], [23, -7, -4, -5], [-20, -3, 5, -2], [24, -6, -2, -5], [-20, -8, 2, -2], [-19, -6, -3, 1], [-20, 3, 1, 0], [-4, 8, 2, 0], [16, -4, 0, -4], [12, 2, 6, 0], [-4, -7, -1, 2], [11, 9, 4, -1], [-2, -3, 3, -6], [-8, -6, 4, -3], [9, -5, -4, -2], [-16, -1, 3, 0], [0, 4, 0, 2], [1, 6, -5, 1], [-30, -1, -1, 2], [-1, -3, -4, -3], [20, 2, 6, 1], [20, -8, -8, -5], [2, -7, -9, 0], [3, 1, 6, 3], [-10, 0, -4, -4], [-8, 2, -2, -2], [-14, -2, -4, -2], [-32, -6, 4, -2], [-12, -10, 0, -2], [4, -2, -10, -1], [10, -3, -5, -4], [8, -2, 4, 4], [-23, -3, 0, 1], [18, 5, 5, 8], [-2, 0, -2, -6], [11, -4, -5, 1], [10, -2, -4, -6], [20, -6, -4, 0], [17, 5, -2, 6], [-8, 2, 0, -4], [-15, 3, 8, 0], [-12, 0, 0, 0], [-18, 7, 7, 6], [-14, -2, 4, 3], [18, 4, 4, 3], [5, 0, 1, -5], [8, -5, 5, 2], [-13, 0, -1, 6], [8, 8, 2, -4], [15, -2, -1, -3], [23, 9, -2, 0], [0, -12, -2, -7], [4, 5, 1, -1], [27, -2, 1, -1], [44, 2, 0, 3], [-18, 12, 10, 5], [-12, 0, 2, 2], [-14, -6, -2, -6], [10, 4, 4, 12], [2, -2, -6, -3], [-30, 2, 6, 0], [18, -6, -8, -2], [12, 7, 5, 10], [-6, 4, 8, -4], [-7, -6, -3, 5], [26, -4, 2, -3], [2, -4, -6, 0], [-4, 4, 8, 3], [-4, 13, 3, 8], [-28, 0, 0, 0], [-1, -2, -5, -2], [-5, 5, 4, 1], [3, -7, -4, -4], [32, -8, -2, -3], [-28, -3, 5, 0], [-14, 7, 3, -1], [13, 1, 2, 4], [14, 7, -5, 0], [-4, 4, 4, -5], [8, -4, -6, 2], [-6, -8, -8, -2], [-2, 2, -4, 0], [-8, 9, 11, 2], [9, 10, 3, -2], [-8, 0, -10, -1], [-2, -8, 0, -2], [4, -2, 4, -3], [12, 7, -1, -2], [22, -4, 2, 2], [-3, 5, -8, -5], [1, 2, 1, 10], [-26, 0, -2, 6], [12, -4, 4, 2], [23, -3, -4, 1], [22, 3, 1, 8], [-8, 6, -2, 0], [4, -6, -8, 0], [-10, 0, -4, -2], [-18, -7, -5, 2], [-38, -3, 7, -4], [42, 2, -2, 1], [28, 6, 6, 0], [-12, -7, 1, 8], [-20, -10, -4, -4], [-6, 2, 4, 1], [-29, 2, -1, -3], [-34, -7, 3, -4], [22, -1, -7, 2], [-34, 6, -4, -2], [-12, -4, -2, -1], [-32, 2, -4, 2], [-18, 9, 5, 4], [14, 10, 4, -4], [-12, -5, -7, -7], [-26, 2, 6, -2], [-15, 4, 3, -3], [18, -6, -6, -4], [20, 11, -1, 4], [-8, 6, 0, 7], [28, 6, 8, 6], [-35, 4, 1, -2], [28, -4, -8, -3], [-2, 4, 8, 10], [-28, -5, 9, 0], [-38, 6, 4, 3], [-4, -2, 10, 2], [-14, -2, 2, 12], [-20, 5, -5, -8], [33, 7, 4, -2], [14, 2, 2, 4], [-8, 5, 11, 5], [8, 5, 5, -4], [-9, -11, 2, 2], [-12, -14, -2, 4], [-24, 12, 2, 2], [-5, 9, 8, 10], [-24, 10, 0, -3], [-18, 4, 0, 10], [2, 2, 6, 6], [6, 4, 2, -2], [-16, -2, -4, 5], [14, 2, 0, 0], [-32, -6, 4, -5], [24, 8, 2, -4], [15, 0, -1, -6], [-8, -2, 6, 6], [-7, 1, -8, -8], [0, -2, -10, 1], [-30, 2, -8, -10], [-67, -2, -1, -1], [16, 13, -5, 2], [-12, 0, 6, -7], [18, -12, -6, 1], [-13, 3, 0, 1], [3, -6, -1, -6], [-9, -5, 2, 3], [22, 3, 1, 1], [-16, -8, -10, -7], [22, -8, -10, -4], [14, -2, 8, 10], [60, -9, -7, -4], [-4, -14, -2, 5], [28, -20, -8, -8], [6, 0, -12, -5], [12, 3, -1, 3], [20, 2, -12, -2], [0, -14, -10, -11], [12, 0, -2, -12], [10, -8, 6, 4], [18, 10, -2, 4], [-8, 0, -4, 11], [-19, -11, 2, 3], [48, 4, -2, 0], [18, 2, 8, 2], [14, 14, -2, 7], [-18, 12, 4, 4], [-24, -2, 10, 2], [-22, -7, 3, -2], [-28, -12, -2, 0], [-11, 5, 2, -7], [-8, -3, -11, -3], [-37, 12, 5, 6], [26, 1, -3, 0], [25, 10, 11, -2], [40, 2, -8, 3], [12, 12, 6, -4], [-10, 14, 12, 8], [-19, 2, -5, -10], [16, -4, 8, 2], [-44, -2, -2, 5], [-30, 14, 10, 8], [0, -2, 0, -1], [-40, -6, -2, 0], [32, 0, 8, -1], [24, -2, 2, 8], [22, -6, -6, -5], [20, 6, 8, 0], [0, 0, 6, -1], [22, -1, -1, -1], [16, 1, 5, 12], [-5, 8, -3, 0], [2, -6, -4, 1], [-19, -8, -3, -13], [-32, -8, 0, 4], [-46, 8, 0, 2], [3, -18, -1, -7], [2, 11, 11, -1], [30, -6, -12, -6], [42, 2, 0, 6], [16, 6, 4, -3], [8, 9, 1, -9], [27, 6, 11, 6], [-8, 14, 12, 10], [12, -13, -5, -14], [20, -2, 4, -8], [22, 0, -2, -5], [14, -4, -10, 0], [10, -3, -5, -2], [-10, -2, 0, 5], [16, 2, 4, 7], [-12, -13, -7, 0], [7, -11, -8, -9], [24, -4, 0, -5], [5, 10, 5, 15], [-7, 1, -10, 2], [15, -13, -4, -10], [36, -4, 2, -5], [24, -1, -5, 0], [42, 5, 1, -1], [13, -10, -7, 5], [42, 0, -2, 6], [-26, 4, -8, -9], [6, -13, -5, -4], [12, 8, 0, 2], [23, 12, 9, 0], [-12, -3, 9, -6], [-49, -6, -1, 3], [-8, -5, 9, -2], [-6, 11, 3, 6], [33, -3, -4, 2], [17, -5, 0, 3], [12, 4, 12, 9], [8, -14, -6, 0], [-30, 14, -4, -3], [-28, 0, 8, 0], [44, 2, 0, -5], [12, -6, -18, -6], [-5, 14, -1, 2], [-34, 20, 6, 8], [-24, -2, -10, -5], [-38, -2, -2, 2], [-4, 10, 2, 9], [-31, 5, -4, 3], [11, -7, -8, -10], [-23, 1, -4, -3], [-10, -4, 0, -10], [38, 6, 0, -2], [8, 7, -7, -4], [-2, -3, -9, -9], [5, -2, -1, -1], [32, 8, -10, 4], [4, -4, -2, 9], [21, 8, 17, 8], [8, 2, 4, 13], [31, 4, -9, 7], [-2, -2, 0, -16], [-40, 7, 1, 4], [3, 11, 10, 11], [-10, 11, 9, 6], [-2, -1, -7, 4], [-3, 8, 3, 14], [-4, 12, 10, 16], [12, 18, -6, 0], [-16, -15, -7, -13], [-6, -14, 2, -10], [4, 12, -2, -8], [28, 1, -7, 8], [33, 7, -4, 8], [-24, 6, 16, 5], [8, 4, 10, 9], [3, -1, -6, -15], [41, -13, -6, -4], [12, -10, -2, -7], [-7, 0, -5, 7], [40, -9, -3, -6], [-54, 0, -6, -5], [10, -12, 4, 3], [29, 10, -5, 3], [-24, -8, -12, -20], [-36, 4, 0, -6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1860_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1860_2_a_i();".
// 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_1860_2_a_i(:prec:=4)
    chi := MakeCharacter_1860_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 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_1860_2_a_i();".
// 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_1860_2_a_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1860_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![116, -54, -20, 4, 1]>],Snew);
    return Vf;
end function;