
// Make newform 7225.2.a.bd in Magma, downloaded from the LMFDB on 29 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_7225_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_7225_2_a_bd();".
// 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_7225_2_a_bd();".
// 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 := [-3, -3, 21, 5, -9, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [-3, -1, 1, 0, 0, 0], [2, -4, -1, 1, 0, 0], [0, 7, -4, -2, 1, 0], [-4, 8, 9, -6, -2, 1]];
    Rf_basisdens := [1, 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_7225_a();"
function MakeCharacter_7225_a()
    N := 7225;
    order := 1;
    char_gens := [2602, 2026];
    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_7225_a_Hecke( : Kf := Kf)
    return MakeCharacter_7225_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, -1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 1, 0], [-1, 0, 0, 1, 1, 1], [0, 1, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0], [-1, 0, -1, 0, 0, 0], [2, 2, 2, 0, -1, -1], [3, 1, 1, 0, -1, 0], [2, -1, -2, -2, 1, 0], [1, -1, -2, 0, 0, 0], [2, -3, 2, 2, 0, 0], [0, -1, 1, -1, -1, -1], [3, 0, -2, -1, -1, 1], [0, -1, -1, 3, 3, 0], [2, -5, -3, 2, 2, 0], [3, -3, 0, 2, 0, 0], [3, -3, 2, 4, 0, 0], [2, 2, 0, -2, -2, 1], [5, 1, 2, 1, -1, 0], [2, 0, 0, -1, -2, -1], [0, -3, 3, 3, 1, 0], [4, 3, 1, -2, 2, 0], [-2, 2, 3, -3, -1, -3], [0, 1, -4, -2, -1, -1], [1, 0, -1, 2, 2, 3], [6, 3, 0, 0, 1, 0], [-6, -2, 1, 1, -1, 1], [2, -2, 0, -3, -1, 2], [-6, 0, 1, 1, -1, -3], [-8, -3, 1, 4, -1, 0], [4, -5, -5, -2, 1, 0], [-1, 3, 2, 2, 2, -1], [-4, -3, 3, 2, -1, -3], [-3, -6, 1, 1, -3, 0], [-2, 0, 5, 2, 2, 0], [-2, -3, -1, -2, -1, 2], [14, -2, -3, -4, 1, 0], [7, -2, 0, -1, 4, 2], [-3, 3, -2, -2, 3, 2], [-3, -2, 2, 1, -3, -2], [-2, 6, -1, 1, 2, 0], [1, -1, -4, -2, 2, 2], [3, -2, -5, 1, 2, -1], [4, -7, -3, 3, 1, 3], [6, -7, -3, 3, 0, 2], [3, 1, 4, 5, 1, 2], [5, 5, 6, 0, -1, -1], [9, 3, -2, -3, -4, 1], [2, -4, -2, 5, 1, 0], [5, 0, 1, -6, 0, -1], [-1, -3, -6, -6, 4, 1], [5, -4, -1, 6, 3, 2], [7, 1, 1, -4, -3, -1], [1, 8, 7, 0, -3, -2], [-13, 3, -1, -1, -3, -3], [-3, 1, -2, -7, -4, 2], [14, -9, -1, 7, 5, 0], [6, 2, -1, -1, 5, 4], [0, -5, -2, 1, 1, -2], [-5, 0, 3, 5, 3, -1], [10, 9, 1, -4, 0, 3], [-1, 0, 1, -9, -1, -1], [1, -2, -2, -4, 3, 1], [3, 4, 0, -6, -3, -3], [-8, -4, -1, -3, 4, -1], [4, -5, -7, 4, 1, 2], [12, -4, -3, -6, 1, 3], [-4, 10, 5, 1, -2, -4], [5, -3, -6, 3, 2, 2], [5, 1, 5, 2, -2, -3], [10, 2, 1, -5, -4, -1], [-5, 1, -9, 0, 4, 2], [6, -7, 0, 1, 0, 1], [-8, -8, 0, 7, -1, 0], [5, -4, -6, 2, 3, 0], [-2, -3, -7, -2, 2, 2], [-4, 4, -2, -4, -2, 3], [6, 10, 4, 2, -2, -1], [7, 4, -7, -3, 2, -2], [7, 0, 0, 3, -2, -1], [-3, 6, 1, -4, -5, 4], [-4, 3, -1, -4, -3, 4], [-9, -1, -1, 3, -2, -4], [10, -3, -6, 2, -6, 2], [3, -5, -4, 11, 4, 4], [-19, -5, 1, 0, -3, 0], [9, -7, -6, 3, 6, 3], [8, -7, -2, -5, 3, 2], [3, 4, 7, -9, -9, -3], [-9, -3, 3, 4, -1, -4], [-3, 0, 5, 6, 4, -1], [-2, 11, 0, -5, -4, 0], [8, -2, 6, 2, 2, -1], [8, -10, -8, -5, 3, -2], [-7, -7, -2, 0, 9, 2], [-2, 6, 3, -7, -3, -7], [2, 5, -3, -2, 1, -1], [0, -6, -6, 3, 5, 2], [16, 5, -5, 0, 0, -3], [7, -2, 0, -6, -4, -2], [5, -3, 2, -5, 0, -2], [0, -9, -4, 2, 4, 4], [9, 4, -7, -8, 2, 1], [-8, -2, 2, 4, -2, 1], [-6, 1, -9, -7, -4, -2], [7, 4, -4, -11, 1, -3], [3, 0, 7, 1, 4, 2], [-4, -4, 7, 14, 1, -1], [13, -3, 4, 3, 0, 0], [16, -1, 0, 0, -1, -1], [6, -8, -7, 3, -3, 3], [-9, 0, 2, -3, -2, -7], [-8, 1, 0, 1, -2, 1], [-20, 9, -1, 4, 3, 0], [-3, 11, 2, 4, 2, 1], [3, -10, -12, -7, 4, -2], [5, 7, -5, -6, -4, -1], [-4, -3, -3, 9, -1, 2], [9, 1, -3, 0, 8, -2], [-11, 4, 1, 8, 4, 1], [0, 17, 3, -2, 1, -1], [6, 11, -2, -1, -2, 4], [-7, -3, -9, 4, 4, 0], [-4, -7, -3, 11, 4, 3], [7, -9, 0, -9, -1, 0], [13, 8, -2, -3, -3, 4], [3, -4, 8, 5, -7, -4], [3, -9, -5, 12, -1, 4], [-1, 9, 10, 2, 1, -5], [27, -8, -3, -5, -6, -1], [1, 13, 3, -4, -6, -2], [-9, -4, -1, 9, -3, 4], [-12, 2, 5, -2, 3, 2], [11, -14, 1, 3, 0, -3], [1, 0, -5, -15, 3, 1], [-1, -12, 2, -5, -4, 0], [11, -10, 4, 5, -6, -3], [15, 8, 6, -3, -3, 0], [6, -12, -7, 2, -1, -3], [-15, 11, 1, 4, 3, 5], [-15, 3, -1, 5, 1, -3], [23, 4, 7, -5, -1, 1], [-1, 3, 8, -5, -6, 2], [-5, 10, 0, 3, -2, 1], [-14, -7, -12, 5, 7, 7], [2, -1, 9, 2, -3, 0], [5, 3, 6, -15, -5, -6], [5, 7, 7, -2, 10, 1], [-8, 2, 7, -1, -4, -5], [1, -3, 8, 0, -8, -5], [-3, 15, 6, 3, -8, 1], [20, -2, 4, 3, -5, -4], [-2, -2, -2, -4, -4, -8], [2, 1, -8, 1, 1, -2], [3, 7, -11, -10, -5, 5], [3, 2, -3, 10, 9, -1], [5, 8, -13, -1, 1, 4], [13, 0, -1, -1, -2, -7], [17, 1, -8, -13, 5, 6], [9, 5, -7, -8, -3, -1], [15, 3, 6, 7, 4, -3], [3, 3, 9, 14, -4, -2], [8, -16, 1, 4, 7, 4], [-2, -9, 13, 5, -2, -7], [-17, 1, 13, 5, 3, -8], [1, 5, -5, 5, 5, 4], [23, -5, 3, -6, -13, -5], [-6, -8, 2, 3, -5, -3], [-17, 3, -1, -6, -5, 1], [-16, 7, 5, -4, 3, 1], [5, 10, 0, 0, 2, -1], [3, 17, 7, -4, -4, -1], [-8, 10, -6, 6, 1, -1], [21, 7, 1, -9, 4, 3], [6, 17, 9, -6, -1, 2], [-5, 8, 11, 1, 2, -3], [-30, 2, 1, 1, 0, 0], [-15, 12, 8, 5, 2, 5], [4, -6, -8, -6, 0, 8], [-3, -8, 0, 3, 4, 9], [10, 2, 1, 1, 7, 8], [16, -11, -2, -3, 1, -2], [9, -5, -2, 7, -2, -1], [-4, 13, 4, -4, -7, 3], [-1, -7, 0, -1, -7, -1], [13, -6, 0, 12, 5, 8], [17, 5, -4, -8, -4, -3], [-5, 6, 6, 10, -1, -3], [-9, 4, 11, 16, -1, -2], [17, -7, -14, 6, -1, 5], [13, -13, -7, -5, 1, 7], [-12, -4, 1, -1, -4, -2], [-7, 13, 14, -2, 4, -5], [1, 2, -3, -1, -8, -4], [5, -1, 1, -7, -14, -1], [12, 7, -1, -5, -6, -5], [-19, -6, -1, 14, -2, -3], [-16, 6, -1, 2, 4, 6], [21, 5, -7, -11, 0, 5], [1, -1, 1, 7, -1, -4], [-8, -8, -1, 2, -4, 0], [-17, 5, -5, 6, 6, 5], [7, -1, 0, -12, -9, -2], [10, 13, 14, -3, -7, 4], [1, 2, 2, -18, -11, -10], [14, 13, 2, -1, 3, 3], [-13, 3, -9, 3, 6, 8], [26, 10, 14, 1, -5, -7], [2, 6, 9, 10, 1, 3], [24, 6, 12, 0, -7, -3], [17, -1, -1, 1, -2, -3], [3, -8, -14, -5, 4, 14], [-4, -18, -7, 9, 3, -1], [17, -13, 1, 4, 8, 3], [0, -3, 8, -8, -5, 3], [-33, 1, 1, 10, 0, -1], [13, 13, 9, -10, -2, -10], [7, -8, -3, -9, 4, 1], [-12, 12, 10, 4, -9, -6], [12, -3, -9, -6, 2, 5], [-9, 5, -3, -14, -5, -1], [19, 0, 0, -3, 9, 6], [-18, 5, 4, -4, -14, -11], [1, 0, -7, 4, 1, -2], [33, -5, -7, -6, 4, -6], [-6, -17, -18, 1, 5, 2], [-12, 14, 11, 1, -5, -3], [17, 10, 14, -11, -5, -5], [13, -3, 9, -5, 0, -2], [3, 4, 1, -3, 5, 8], [-5, 3, 8, 11, -7, 0], [15, -1, -6, -3, 6, -3], [13, 11, -14, -5, -6, -5], [20, -2, 1, 4, -8, -2], [-19, 4, 1, 7, -2, 6], [-28, 1, 1, -3, 3, -4], [0, 6, -6, -2, -2, 8], [5, 3, 11, 0, 9, 5], [19, -1, -10, 1, 4, 0], [-6, -2, -14, -13, 1, 5], [-12, 0, -5, -5, -10, -6], [-4, 6, -5, -10, -2, 4], [-5, 6, -2, -1, 3, -4], [-3, -4, -17, -3, -3, 0], [10, -2, -4, -13, -3, 7], [-17, -7, 6, -3, -8, 4], [-31, -14, 8, 1, -1, -10], [-12, 14, 5, -7, 6, -2], [-3, -21, -6, 1, 7, 3], [-29, -9, 2, -8, 5, -3], [-9, -1, -4, 2, -4, -5], [3, 1, 5, 10, 0, 10], [14, -16, -9, 16, 8, 7], [-34, 16, 3, -6, 1, -1], [3, 15, 5, -17, -4, 1], [10, -9, -4, 1, -4, 5], [9, 13, -4, -4, -8, -5], [23, 9, -9, -1, -3, -5], [-12, 1, -14, -4, 14, 6], [5, 9, 10, -3, -6, 5], [30, 14, 3, -3, 1, 5], [-18, 26, 10, -6, -9, -1], [-32, 4, 18, 20, 2, 2], [-13, -14, -8, 3, 6, -3], [-16, 2, 14, 13, -1, 4], [-1, 2, -10, -18, 7, -1], [3, -8, -21, -10, 1, 0], [-32, -8, 5, 18, 4, -8], [-13, 10, 16, 13, 1, 3], [-22, 8, 6, -5, -4, -4], [30, 17, 12, -8, -1, 3], [-8, 13, 1, -8, 5, 1], [6, 16, -5, -6, 7, -2], [-3, -14, 8, 3, -2, 1], [3, -10, 17, 10, -1, -10], [12, -7, -7, 15, 5, 7], [-27, 6, -8, -4, -2, 2], [-10, 2, -3, 2, -3, 6], [2, 16, 8, -7, 0, 3], [12, -23, 8, 12, -9, -4], [26, 2, -2, -14, -9, -11], [6, -2, -5, -4, 7, 8], [9, -11, 4, 3, 11, -1], [-22, -5, 6, -7, -9, -6], [2, 15, 15, 3, -5, -5], [-27, 7, 5, 23, 0, -2], [10, -18, -11, -6, 17, 5], [-22, 18, 13, -13, -6, -9], [-3, -15, 1, -1, -13, -5], [22, 2, -6, -6, 3, 7], [5, 12, -8, -15, -12, 8], [-9, -18, 7, -7, 3, 4], [-3, -15, -11, 0, 15, -4], [10, 16, 19, 13, -16, -6], [-21, -11, -2, 5, 13, -1], [11, 5, -3, -6, -1, 14], [42, 10, 7, 4, -7, 2], [5, 22, -4, -9, 3, 9], [-20, -6, -4, -10, -8, 3], [0, 8, 0, 1, -2, -4], [7, -2, 5, -13, -4, 1], [-22, -2, 2, -1, -7, 6], [17, 5, 4, 8, -2, 8], [-6, 3, 8, 11, 4, 6], [28, -17, 10, 17, 6, 2], [12, -7, -5, 6, 14, 5], [-13, 32, 14, -7, 1, 0], [-3, 6, -3, 4, 11, -3], [11, 4, -7, 8, 7, 12], [12, 7, 7, -19, -3, 1], [-33, -16, 11, -2, 4, -4], [-14, -20, 7, 21, -1, -7], [-25, 13, 2, 10, 4, 4], [5, -7, 10, -7, -11, -5], [15, 1, 2, -9, -6, 0], [18, -1, -9, 2, 2, -2], [7, 9, 4, -6, -7, 6], [-10, -1, -12, -5, -3, 1], [-9, 12, 3, -5, -6, -5], [6, -14, -3, -12, 4, -1], [3, 0, 8, 1, -2, -13], [-2, -7, 13, 21, -1, -1], [23, 3, -6, -13, -4, -6], [23, -6, -4, -16, -7, -2], [-16, 14, 8, 5, 13, 2], [-29, -5, -4, -4, -14, -5], [36, -12, -2, 12, 2, 0], [-4, -9, 6, 33, 4, 2], [-13, -15, -3, 16, 7, 9], [-11, 2, -11, -29, 4, 0], [-17, -22, 6, 3, 4, 9], [-33, 3, 6, 6, 12, 3], [13, 2, -20, -17, -3, -1], [-13, -14, 1, 5, 1, -12], [-12, 6, -3, -5, -1, -3], [3, -5, 5, 8, -7, 4], [-27, 3, 4, 1, -1, 4], [-41, -14, 3, 5, 4, -6], [-36, 15, 15, 13, 1, 0], [10, -14, -7, 14, 6, 2], [5, -5, 9, -9, -17, -1], [1, 5, 1, -13, 1, 3], [-16, 18, -8, -18, -4, -1], [-15, 16, 12, 10, -11, -10], [-32, 3, 10, -6, -5, -6], [-7, 0, -22, -5, 4, -1], [-35, -2, 5, 21, 10, -5], [-3, -2, -18, -10, -12, -2], [-3, -7, 15, 5, -1, -10], [-7, 5, 10, -13, 5, -4], [12, 9, 11, -2, -13, 2], [-8, 7, -14, 19, 7, -4], [-34, -16, 2, 12, -3, 9], [21, -21, 3, -5, -16, -7], [7, -22, -7, -3, 6, 4], [11, -23, -16, 0, 2, 11], [-4, -16, -2, 1, 6, -14], [20, 14, 3, -8, 2, -11], [15, -7, 0, 1, 15, 13], [-21, 15, 12, -5, 5, -4], [-18, 12, -4, -20, -10, 5], [-2, 4, 15, -14, -7, 3], [5, 3, 2, -12, -13, -9], [-21, -16, 0, -1, -4, -7], [1, -19, 12, 5, -2, -4], [34, 0, 2, -2, 1, 1], [10, -7, -13, 4, 6, 11], [-12, -10, -11, -6, 12, 3], [-7, -17, -10, -12, 4, 10], [46, -6, 3, -8, 1, 0], [23, 18, 17, -1, 1, -12], [-9, 10, -16, -19, -7, 8], [-15, 2, -4, -2, -8, 4], [5, -17, -9, 14, 12, 0], [-21, 17, 11, 1, -9, -10], [-11, 3, -11, 11, -3, -6], [38, -1, -9, 14, 10, -3], [12, -1, -1, -6, -15, -11], [-11, -21, 1, 8, -4, -1], [46, 2, 9, -3, -4, -11], [-15, 37, 12, -11, -6, -2], [15, 14, -16, -21, 4, 3], [-30, 0, -8, -1, 1, 0], [13, 19, -8, -11, -4, 9], [7, -5, -14, 22, 12, 7], [-24, 10, 12, -15, -8, -8], [6, 2, 8, -9, -1, -1], [26, 20, 22, -6, -15, -4], [2, 0, -22, 1, 4, 0], [19, -34, -18, 6, 10, -2], [-27, -4, -17, -9, -5, 0], [14, 4, 2, 6, -2, -6], [29, -3, 14, 17, -2, 0], [-7, 10, -20, -9, 3, 7], [22, 6, 6, -4, 8, -3], [-25, -7, -15, -4, 6, -6], [-7, 5, 14, 8, 0, 5], [-15, -3, 20, 32, -1, 4], [10, 14, 14, -5, -21, -4], [-24, -7, 9, -8, -16, -1], [23, -22, -5, 9, 14, 0], [14, -29, -8, 5, -3, 3], [31, -7, -3, 5, 12, 11], [30, -5, -12, -1, 7, 0], [7, -6, 6, -8, -3, 7], [1, -1, -1, 2, 19, -1], [-23, -11, -10, -4, 18, 6], [-20, -9, -3, -14, -7, -3], [-8, 32, 1, -7, 8, 7], [-19, 8, 4, 0, 10, 5], [-5, -2, 24, 23, 2, -4], [27, 0, -16, -13, 10, 3], [-18, 5, -5, -16, -16, -4], [27, -12, -15, -4, 1, -2], [1, -5, 16, 4, -6, 0], [37, -35, -8, -8, -4, -1], [-25, -11, -19, -4, 6, 9], [12, -6, -6, -4, 1, -8], [-5, 4, 16, 2, -9, -16], [-29, 5, -7, -5, -1, 14], [36, -11, -10, 3, 11, 6], [23, 6, -3, 1, 8, 7], [-27, 4, 7, -8, -4, -8], [-13, -8, 9, 5, 3, -6], [-16, 11, -5, 4, -2, 7], [19, -3, -17, -16, 1, 11], [-13, 14, -8, 16, 9, 8], [8, -12, -4, -2, 11, -5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7225_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7225_2_a_bd();".
// 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_7225_2_a_bd(:prec:=6)
    chi := MakeCharacter_7225_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_7225_2_a_bd();".
// 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_7225_2_a_bd( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7225_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-3, 3, 21, -5, -9, 1, 1]>,<3,R![1, -15, 34, 17, -12, -2, 1]>,<7,R![-251, -280, 54, 82, -11, -6, 1]>,<11,R![-1317, 747, 348, -161, -32, 6, 1]>],Snew);
    return Vf;
end function;