
// Make newform 6025.2.a.e in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6025_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_6025_2_a_e();".
// 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_6025_2_a_e();".
// 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 := [-1, 4, 0, -5, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [1, 0, -4, 0, 1], [-3, 0, 5, 0, -1], [3, -4, -5, 1, 1]];
    Rf_basisdens := [1, 1, 1, 1, 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_6025_a();"
function MakeCharacter_6025_a()
    N := 6025;
    order := 1;
    char_gens := [2652, 2176];
    v := [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_6025_a_Hecke(Kf)
    return MakeCharacter_6025_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+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, 1, 0, 1, 0], [1, -1, 0, -1, -1], [0, 0, 0, 0, 0], [2, 1, -1, -1, 0], [-1, 2, -2, 1, 1], [0, 2, -1, 1, 1], [1, 1, -3, -1, 2], [0, 2, -1, 1, -1], [2, -1, 3, 1, 0], [2, -1, 1, -2, -2], [-3, -2, 0, 0, 1], [-1, -1, 4, 2, 0], [2, 0, 1, 0, 0], [7, -1, 4, 2, 1], [1, 0, -3, -2, -1], [6, -1, -1, 0, -1], [-2, 1, -1, -1, -2], [-3, 5, -1, 0, -2], [0, -1, 2, 2, -5], [-1, -2, 4, -2, 0], [6, 1, -1, 0, 2], [3, -1, -1, 2, 2], [3, 1, 1, -4, 0], [4, -7, 6, -2, 3], [8, -3, 0, 2, -1], [4, 4, 3, 6, 6], [1, -4, -3, -3, 7], [6, -2, 3, 3, -2], [0, 0, -4, -8, -4], [5, 1, 4, 3, 5], [1, -2, 2, -2, -4], [5, -5, 1, -1, 4], [8, -3, 5, 5, 3], [6, -7, 1, -6, -3], [5, -1, 0, 3, -4], [6, -1, 4, 2, 6], [6, 4, -4, 3, 1], [-5, 10, -4, 3, 2], [-5, -2, -3, -9, 1], [5, 6, -6, -3, -1], [-2, 2, -10, -3, 0], [0, 6, -4, -1, 5], [-2, 6, 0, 0, -9], [-1, -11, 11, 0, 2], [1, -2, -1, -6, -6], [-4, -1, -4, 5, 0], [-7, 0, -4, -1, 5], [14, -3, 1, -1, 6], [-3, 2, -9, -10, -4], [10, -1, -3, -4, 0], [-3, -3, 0, 2, 0], [5, 7, -6, -4, 4], [1, 0, 0, 0, 0], [4, 5, 0, -3, -1], [-11, 4, -4, 4, 9], [7, 10, 1, 2, -1], [2, 5, 4, 12, 3], [-2, -6, 13, -2, 3], [-15, 0, -2, -1, -11], [7, 8, -5, 3, 7], [8, 5, 1, 9, -3], [12, -9, 14, -5, -1], [4, -6, -9, -8, -8], [-6, -5, -11, 2, -3], [10, -12, 0, -12, -6], [-9, -4, 3, 5, -3], [-10, -13, 3, 6, -1], [-18, 13, -9, 0, -9], [-5, 3, 7, 2, -3], [-14, 10, -6, 10, 10], [12, -11, -7, -8, -6], [5, -5, 1, -10, -5], [14, 12, -8, -1, 14], [-3, -13, 8, -1, -12], [-3, 3, 12, 15, 7], [11, -10, 3, 2, 4], [-3, -6, 1, 15, 6], [12, -14, 2, -1, 8], [-13, -4, 8, -5, 1], [-13, -8, 3, 3, -1], [5, 9, 7, 7, 17], [1, 11, -6, 3, -7], [-11, -9, -3, 4, -2], [-2, -2, -1, -9, 6], [-4, 6, -2, 1, 5], [6, 10, 4, -1, -8], [5, -8, 2, -11, -3], [-9, 7, -11, 2, -12], [6, 2, 1, 9, -12], [4, -3, 6, 11, 3], [3, 1, 6, 1, 7], [-2, 10, -2, 5, 5], [-11, -14, 3, -4, 1], [2, -5, 1, -8, -5], [8, -5, 16, 6, -10], [-13, -2, 0, 14, 5], [-12, -3, -2, 1, -6], [4, -12, 0, 1, -7], [12, -10, 3, -8, 7], [21, -1, 10, 6, 16], [8, -5, -5, -7, 16], [4, -6, 0, -2, 6], [-7, 3, -5, -3, -17], [-7, -12, 11, -6, 3], [6, 5, -12, -5, -6], [-9, -16, 0, -7, 0], [-1, 7, -8, 7, 8], [-6, 11, -25, -6, 3], [5, -11, 1, 6, 3], [3, -8, 18, -5, 11], [-10, 9, -22, 2, -1], [-2, -17, 5, 6, -4], [2, 10, -7, -2, 14], [-18, -8, -5, 7, 3], [4, -5, 16, -1, -8], [-11, -3, 23, 15, -2], [3, -8, 0, -9, 3], [2, -22, 3, -6, -9], [0, 21, -9, -4, -5], [-11, -1, 10, -10, 9], [-16, 0, -18, -2, 14], [18, 9, -10, 1, 1], [7, -1, -13, 1, 1], [5, -18, 8, 9, 7], [1, 9, 12, -4, 1], [-7, 8, -8, 8, 3], [6, -1, -1, -9, -11], [7, -21, -5, -9, 15], [-24, -5, -1, 3, -7], [6, -1, -6, -15, 2], [8, 15, -4, 5, 2], [6, 12, -1, 17, 14], [4, -6, -5, -1, 1], [19, -2, 13, 0, 11], [21, -6, 18, 3, 6], [5, -10, -10, -7, -8], [-9, -6, -5, -4, 12], [0, -3, 13, -1, 10], [-6, 2, -8, -4, -18], [4, 1, -7, -19, -8], [7, -16, 1, -6, -15], [25, -16, 9, -8, 2], [9, -6, 9, 21, 12], [-9, -12, 22, 6, -7], [8, 6, -2, 0, 10], [20, -9, -11, -9, 2], [-18, 9, -9, 4, 7], [1, -4, 6, 5, 10], [-13, -19, 9, -15, 5], [1, -6, 0, 5, -1], [-22, -1, -8, -10, -8], [30, 9, -6, 3, 6], [18, 7, 5, -4, 3], [-8, -1, 23, 19, 8], [3, 5, -10, 7, 10], [14, 14, 4, 17, 6], [-2, -2, -10, -24, -13], [19, -22, -3, -12, -7], [-26, 4, -9, -7, -1], [-3, 7, -16, 4, 15], [8, 8, 6, -7, -12], [-29, -17, 7, -7, -11], [11, 4, 7, 5, -11], [14, 9, -10, 4, 16], [4, 12, 14, -4, 9], [-7, 6, 11, 11, -22], [-21, -22, 9, 1, -11], [-1, -21, -3, -17, -19], [-5, -5, 20, -2, -5], [-22, 4, -11, -7, -13], [-12, 6, -7, 5, -15], [7, -20, 14, -7, -2], [1, 1, -17, 2, -16], [13, 17, 10, 3, 26], [-15, 1, 14, 5, 13], [-3, -14, 22, -2, 6], [0, -11, 1, 11, 15], [-13, 6, -4, 6, -1], [-10, 14, 4, 4, 1], [-24, -16, 15, -2, -5], [18, -2, -5, -3, 8], [-2, 10, -7, -12, -9], [21, -6, 1, 15, 1], [-6, 9, 5, 8, -27], [-31, 5, -5, 13, 9], [12, -3, 0, -5, 4], [-6, 1, -7, -10, 10], [0, -2, -9, -8, -9], [22, 9, 7, 4, -13], [5, 21, 5, -5, -11], [0, -8, 20, 26, 5], [10, 1, -3, 8, 4], [-7, -23, 12, 1, -4], [-30, 2, 10, 13, -7], [-5, 8, 28, 21, 1], [-5, -16, -2, 0, 1], [-1, 22, -14, 9, 1], [22, -8, 22, 6, 3], [19, -9, 4, -11, -6], [7, 16, 9, 9, -9], [5, 15, -12, -12, 3], [-8, -24, 2, -6, -1], [-12, 3, 5, -2, 4], [3, 18, 2, -3, -18], [-1, -3, 3, -4, 2], [-3, 35, -15, 8, -10], [-28, 7, -23, -1, 12], [-6, -1, 7, 19, 1], [-9, 6, -16, -29, -14], [-24, -19, 14, 7, 2], [-17, 20, -12, 4, -12], [1, -23, -8, -13, 3], [0, 25, 4, 10, 24], [-1, 23, -19, -5, -5], [4, -26, 3, -10, -16], [7, 10, -4, 17, 19], [9, 14, 9, -5, -16], [-17, 8, -5, 8, 10], [-10, 3, -8, -14, 19], [21, -21, 2, -3, -5], [-24, 19, -18, 3, 5], [-15, 9, -1, 23, 1], [17, -32, -8, -22, -7], [15, -15, 10, 5, -2], [-6, 23, 5, 23, 28], [-21, 3, -10, -18, 0], [0, -4, -13, 0, 10], [6, 1, 24, 19, -13], [29, -3, -15, -7, -8], [-17, -16, 12, 8, -10], [-1, -17, -15, -17, -20], [-22, 7, -13, -7, -6], [19, -21, 3, -22, -10], [10, -22, -2, -6, -28], [3, 30, -5, 4, 19], [13, -15, 25, 3, 4], [-20, -1, -16, -33, 0], [35, 4, 10, 14, 1], [21, -14, 12, 2, -4], [42, -4, 12, -8, 1], [7, -25, 19, -6, 1], [-31, 19, 1, 12, 3], [-26, 2, 3, -19, 6], [18, -3, 9, 26, 4], [18, 0, 6, -5, 15], [4, 6, 2, 1, 23], [38, -1, -7, -12, -2], [-6, 22, -9, 23, 0], [-21, 23, -25, 12, 4], [25, -1, 18, -8, -1], [2, 2, 6, 7, 3], [-24, 21, 0, -9, -5], [-1, -4, -14, -2, -7], [3, -3, 2, 11, -25], [14, -30, -15, 0, -10], [-6, 2, -3, -15, 9], [-16, 26, 5, 25, 5], [1, 18, 16, 25, 11], [-31, -16, -1, 1, -27], [-6, -15, -9, 3, 10], [28, -3, 2, -6, 22], [-16, 6, -1, -25, 3], [-13, -13, 21, 14, -5], [-17, 25, 17, 19, 14], [-15, 1, -4, -10, 20], [1, 16, 4, 1, 25], [-13, 8, 28, 28, 5], [6, 15, -27, 13, -8], [-20, 28, -39, 5, -9], [20, 5, 15, -9, 14], [-3, -13, -16, 12, -17], [-8, -6, 16, 2, -18], [34, 8, 1, -3, -12], [30, 12, -5, 19, -8], [2, 20, -3, -15, -10], [-21, -7, 5, 10, -4], [12, 18, -18, 9, 19], [-8, 2, -10, 10, 22], [-9, -1, 23, 1, 11], [23, -8, 7, 2, -2], [-22, 12, 5, 10, -13], [2, -22, -8, 1, 1], [13, -18, -27, -5, -16], [34, -4, 13, -24, -16], [-3, 17, -6, 18, 9], [-1, 17, -45, -16, 8], [8, 4, 2, -16, -27], [30, 14, -8, 8, -4], [22, 7, 14, -14, 6], [-1, 15, 11, 7, 14], [8, 3, 15, -10, -14], [-11, -13, -11, -22, -35], [-2, -8, 24, -1, 2], [32, 1, 12, 14, 1], [-21, -5, -21, 4, -6], [-27, -35, 30, 1, -13], [-27, -2, 38, 18, 3], [-16, -11, -20, -20, 6], [17, -10, 31, 18, 9], [3, 21, -1, 33, -1], [-5, 32, -15, -6, -19], [-9, -25, 5, 8, -23], [-20, 19, -36, 2, -4], [-10, -27, 16, -5, 3], [7, 5, -2, -24, -17], [-8, 14, 16, -2, -12], [-60, -15, 1, -2, -6], [-29, 23, -15, -7, -26], [21, 7, -11, -17, -8], [8, 25, 7, 0, 12], [0, 15, 14, 19, 25], [-11, -19, 16, -7, -38], [3, 22, 8, 21, 13], [5, 24, -6, 4, -14], [29, -14, 31, 9, -3], [-10, -4, 19, 18, 20], [15, 8, 8, 12, 5], [51, 1, 2, 3, -6], [11, 17, -29, -6, -8], [35, 11, -8, 16, -2], [-57, 9, 7, 2, 4], [0, -17, 14, 11, -21], [0, -30, 7, -6, -29], [0, -11, -5, -1, -8], [-2, 26, 0, 17, -28], [14, -15, 12, -25, -27], [-16, 2, -23, 4, 5], [21, 19, 13, 5, 36], [-20, -7, -15, 16, -9], [18, 15, 15, 1, -3], [-22, -12, 4, -32, -8], [-2, -5, 19, -4, 5], [43, 16, -4, 2, 16], [7, 4, -7, 4, -12], [-31, -11, 3, -10, 20], [32, 24, -2, -4, 20], [-11, 15, -18, 16, 3], [34, -4, -9, -24, -13], [-6, 19, -2, 29, 10], [45, -21, 19, -14, -5], [-28, 32, 21, 20, -5], [-6, -37, 3, 9, 10], [-20, 31, 17, 8, 10], [-3, -10, 0, -14, 28], [10, 16, -23, -19, -2], [15, -16, -3, 6, -9], [40, -8, -21, -18, 5], [30, 12, -1, -2, 12], [17, 12, -5, -22, -13], [32, -26, 21, -11, -3], [-35, -8, -11, -5, -17], [9, 9, 8, 31, 6], [-24, -26, -1, -17, 11], [-3, 38, 6, 6, 21], [-48, -9, -7, 0, 2], [9, -4, -18, -2, -3], [22, -15, 4, 8, 17], [15, 9, -27, -6, 23], [1, -5, -8, -1, 14], [12, -32, -10, 4, 11], [2, -4, 21, 21, 3], [-1, 27, -1, 2, -26], [47, 9, -4, 4, 9], [-7, -12, 1, 10, -32], [24, -15, 2, -10, 13], [-24, 8, -16, 30, 17], [-19, 25, -7, 2, -11], [-19, -15, 7, -5, 15], [-5, 38, -7, 26, 29], [21, 22, -12, 17, 23], [33, 12, 3, 7, -21], [-3, -16, 10, -6, -12], [-10, -2, -27, -18, -11], [44, 4, -15, -17, 2], [-3, -31, 6, -10, -7], [45, -12, 14, 21, 22], [-14, 22, -16, 5, 32], [17, 10, 19, 42, 1], [13, -12, -8, -19, -30], [-30, 0, 14, 2, -23], [28, 9, -8, -26, 3], [-21, 15, 26, -3, 13], [-46, 2, -3, -16, -4], [17, 9, -5, 37, 3], [2, 1, 22, -13, -5], [-3, 3, -12, -25, 2], [10, 12, -2, 34, 5], [10, 9, -37, -37, -12], [8, -30, -5, -9, 15], [4, -20, -1, -9, 15], [-4, 36, 7, -5, -9], [6, 11, 9, -19, 3], [-53, 8, -14, 4, 13], [27, -14, -20, -31, -9], [2, -21, 7, -29, -9], [40, -7, 23, -11, -17], [10, 7, 9, 0, 10], [-17, -10, -6, 25, 2], [7, -16, -4, -29, -24], [3, -22, -7, -21, 35], [-28, 58, -5, 17, -19], [36, -27, 0, 9, -1], [31, 13, -17, 14, -6], [-34, 21, -31, 15, 21], [21, -24, 6, -31, -36], [-1, 33, 3, 23, -7], [-4, 16, 9, -18, -16], [-19, -36, 16, -31, 2], [-6, -13, -10, 1, -38], [13, -23, -15, -6, -3], [19, -41, -10, -33, -1], [34, -17, 40, 24, 14], [17, -36, 37, -1, 13], [13, -2, -1, -22, -35], [30, 33, -21, -2, 14], [7, -23, 15, 2, -28], [-9, -13, -8, 6, -34], [-35, 11, -20, -32, -36], [17, -18, 13, -10, -17], [-13, -21, 2, -2, -35], [27, 0, -6, -14, -7], [-36, -21, 21, 15, 8], [56, 26, -13, 11, -4], [-1, 9, -3, 33, 3], [-38, 21, -2, 33, 26], [3, 9, 40, 3, 5], [-7, -14, -26, -26, 0], [6, -8, 14, -7, 0], [-28, -33, 35, 1, -23], [-23, 8, -25, -33, -32]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6025_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6025_2_a_e();".
// 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_6025_2_a_e(:prec:=5)
    chi := MakeCharacter_6025_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_6025_2_a_e();".
// 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_6025_2_a_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6025_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 1, 5, -6, -1, 1]>,<3,R![-17, -9, 17, 2, -5, 1]>],Snew);
    return Vf;
end function;