
// Make newform 5625.2.a.q in Magma, downloaded from the LMFDB on 10 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_5625_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_5625_2_a_q();".
// 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_5625_2_a_q();".
// 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, 3, 29, -1, -11, 0, 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], [-1, 0, 5, -6, -1, 1], [-1, 24, 5, -10, -1, 1], [-13, 24, 9, -10, -1, 1], [3, 34, -7, -12, 1, 1]];
    Rf_basisdens := [1, 1, 4, 4, 4, 2];
    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_5625_a();"
function MakeCharacter_5625_a()
    N := 5625;
    order := 1;
    char_gens := [4376, 1252];
    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_5625_a_Hecke( : Kf := Kf)
    return MakeCharacter_5625_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, 0, 0, 0], [0, 0, 0, 0, 0, 0], [1, -1, 0, 0, 0, 0], [0, -1, 0, 0, 0, 1], [0, -1, 0, -1, 0, -1], [-3, 0, 1, -1, -1, -1], [3, 0, 1, 2, -1, 0], [-2, -1, 0, -1, -1, 1], [0, -1, -1, 0, 0, -1], [-1, -2, -2, 1, 1, 1], [4, -1, 1, 0, 0, 1], [1, 1, 3, 1, 1, 1], [1, -1, 2, 1, -1, 0], [-4, -2, 0, -3, 1, 0], [-2, 0, 2, 4, 1, 0], [-4, 1, -1, -3, 0, -2], [3, -1, -3, 1, -1, 1], [1, -1, -2, 1, -1, -2], [3, 1, -1, 3, 2, 1], [2, 0, 1, -2, -2, 1], [-1, 2, 1, 1, -2, -2], [2, 0, 1, -1, 2, -1], [5, 1, -3, 8, -2, 0], [0, -3, -2, 0, -1, 2], [1, 2, 2, -4, -1, 0], [-1, 2, 2, 3, -1, -1], [7, 3, 4, 5, -1, 0], [0, 1, 3, 1, -1, 0], [-5, 1, -2, -5, 2, -1], [4, 4, 0, -5, -1, 0], [10, -1, -1, 0, -1, 0], [-9, -4, -3, -1, 1, 0], [0, 2, 0, -8, 0, 0], [10, -3, -1, -1, -1, 0], [9, -1, 3, 4, -1, 3], [9, -1, -4, -2, -1, 3], [3, 3, 6, -2, 0, 0], [-7, 3, -3, -5, 2, -3], [0, 1, 4, -1, -4, -2], [6, 5, 3, -1, -4, 0], [-3, -3, -1, -6, 0, 2], [3, 4, 0, 1, -1, -1], [0, 1, 2, -1, 4, -1], [-6, 2, 0, 5, -2, -2], [6, -3, 1, 1, 0, 0], [4, -5, -5, 3, 2, -4], [-1, 2, 0, 3, -3, -3], [8, -2, -2, 9, -3, -2], [5, -4, -1, 1, 3, 2], [-5, 0, 0, -3, -2, 3], [-4, 0, -1, -4, 3, -1], [-4, 0, 5, 0, -2, 2], [-2, 4, 1, 2, -1, 1], [7, 2, -2, 4, 1, 5], [-1, -1, 1, -2, -5, 3], [3, -9, 2, 4, -1, 2], [5, 4, 7, -1, 0, 0], [5, 3, 0, 3, 2, -3], [5, 0, -3, 11, -1, 2], [1, -4, -2, -3, 5, 3], [-4, 3, 3, 3, 3, 3], [3, -6, -3, 3, 2, 4], [-3, -3, -2, -3, -1, -4], [7, -3, -5, 0, 3, 7], [-4, -4, -2, -5, 3, 6], [1, -4, -6, -4, -4, -1], [-5, 2, 2, -6, -2, 3], [2, 4, 1, 12, -1, 1], [-2, 1, -1, 8, -4, 1], [5, -2, 1, 9, 2, 2], [3, -6, 3, 8, -3, 0], [-10, 7, 3, -7, -4, 0], [13, 8, 4, 6, 0, 1], [-9, 4, -1, 1, -4, -2], [7, 3, 0, 3, -7, 0], [-5, 3, 4, -1, -6, 2], [7, 8, -1, 2, -1, 2], [1, -6, 0, -4, 1, 5], [-12, -1, -1, -12, 2, -1], [13, 1, -9, 5, -6, 1], [-14, 10, 3, -4, -2, 2], [15, -2, 2, -1, 3, 5], [7, 4, 5, 14, -4, 4], [21, 1, -4, 3, -1, 2], [-17, 3, -1, -18, -1, -1], [17, 2, -2, 4, 1, -2], [1, 6, 2, -3, -1, -9], [5, 2, 8, 5, -4, -4], [-10, 4, -1, 7, -6, -3], [5, 1, 6, 0, 0, -2], [17, 1, -5, -4, -1, 1], [-8, -5, -7, -5, -2, -6], [7, 2, -4, 8, -2, -1], [10, 6, 2, 12, 2, 0], [5, 5, -4, -7, -1, 2], [16, 0, 0, 4, 3, -3], [-1, 0, -4, -23, 0, -1], [-5, 0, -2, 10, 4, -1], [3, -4, -8, 11, 2, 6], [-17, 7, -1, -4, -1, 1], [-21, -3, -3, -4, -2, -6], [-11, 9, 0, -3, -1, -6], [3, 2, -1, -6, 4, -3], [-1, -6, 4, -15, 7, 1], [9, -4, 1, 18, -3, -2], [12, 4, -1, 17, 2, -1], [-10, 0, 7, 5, -7, 0], [-1, -1, 10, 0, -2, 0], [-6, 6, -2, -15, -2, -2], [-16, 6, 7, -3, 0, 5], [0, 12, 2, -4, 3, -5], [-3, 6, -3, -14, -2, -6], [-1, 1, -7, 10, -5, -5], [-2, 9, 4, 6, 2, 5], [6, -8, -6, -3, 1, 0], [2, 2, -1, 12, -5, 3], [-15, 8, 4, -7, 1, -5], [-3, 11, -2, 6, -3, -5], [-1, -8, 3, -6, -1, 2], [14, -4, 6, -10, 4, 2], [-7, 2, 1, -21, -1, -1], [-21, 3, -4, -4, -4, -2], [10, 1, -3, -12, 3, -4], [7, 2, 0, -14, -2, 3], [-3, 1, -2, -6, 1, -6], [4, 0, 7, 8, -6, 1], [2, 0, 13, 5, 4, 3], [-18, -6, -3, -12, 7, 5], [23, 0, -5, -1, 6, 0], [-1, -5, -1, -3, 4, -9], [-9, 7, 0, -7, -3, 2], [-5, 6, 6, -6, 4, 3], [-6, 14, -6, -5, 0, -2], [-21, -3, -3, -22, 6, 7], [14, -5, -5, 14, 7, 2], [0, 1, 2, -2, 1, 7], [9, -1, 0, 7, -1, 6], [-2, 3, -12, 8, 5, 1], [-2, 3, 5, 7, 1, -9], [-13, -2, -2, 3, 3, 3], [-7, 5, -6, -10, 4, 4], [15, -1, 4, 3, 7, 6], [-4, -3, 1, -24, 1, 4], [2, 5, -8, 7, -2, 3], [25, -12, -5, 21, -2, 0], [-4, -3, 10, -12, 1, -7], [-16, -5, 1, -19, -2, -4], [18, -13, 4, 21, 1, 7], [-17, -6, 3, -7, 2, -4], [3, -4, -7, -5, -3, -2], [-28, 6, 6, -12, -6, -6], [-7, 13, 1, -1, 0, -7], [-4, 9, 0, 9, 3, -3], [17, 4, -1, 4, -5, -8], [9, -5, -10, -8, 4, -2], [-11, 11, 12, -7, -1, -2], [-9, 7, 4, 0, 1, 11], [3, 3, -10, 4, -3, -3], [-6, 3, 5, -16, 2, 11], [7, -8, -9, 4, 1, 2], [10, -13, 1, -3, 1, -3], [24, 9, 1, 14, -3, 2], [6, -7, -3, 6, 5, -6], [16, -1, 0, -9, 0, -3], [2, 3, -1, -4, 7, -2], [-3, 1, 10, -13, -3, -8], [25, -3, 11, 16, -9, 1], [8, 4, -13, -18, 6, 1], [3, -7, 3, 10, 4, 9], [3, -8, -5, -7, 8, -4], [-14, 6, 7, -4, -6, -3], [-9, -6, -3, -29, -6, 4], [-18, -13, -2, -18, -3, -2], [-9, 8, -6, -2, 6, -3], [19, -13, -9, 24, 0, 6], [-20, 1, 3, 4, -5, -2], [-8, -19, -7, -5, 3, 1], [-23, -6, -6, 6, 2, 3], [33, -7, 0, 7, 8, 5], [-29, 3, 7, 1, 4, -3], [-11, -1, 10, 8, 6, -2], [-6, -4, 9, 5, -3, 2], [16, -15, 1, 15, -5, 5], [28, 8, 6, 8, -6, 0], [13, 3, 7, 10, -7, 7], [27, 2, 7, 20, 2, 7], [-22, -4, 3, 6, 7, -3], [4, -9, -7, 11, 1, -5], [14, -10, 7, -11, 2, 3], [10, 1, 1, -1, -4, 4], [-2, 2, 5, 1, 6, 7], [20, 5, -12, -4, 4, 5], [-29, -10, -13, -8, 6, -2], [3, -1, -12, 2, -10, 0], [2, -9, -5, 12, 2, -4], [-18, 6, 3, -19, 6, -1], [36, 1, 1, -3, -1, -2], [0, -5, -10, -16, 9, -3], [25, -7, -9, 28, 9, 3], [-23, -5, -5, -2, -6, 5], [15, -11, -3, 10, -1, 1], [-1, -3, -3, 2, -5, -3], [8, 6, 5, 0, -6, -3], [4, -3, 4, -8, 10, 7], [-18, -4, 9, 2, 0, -6], [-15, 0, 0, -33, 7, 5], [17, 3, 0, -12, 2, 8], [-18, -28, -3, 0, 12, -1], [-19, 4, 3, -11, -8, 2], [14, 6, -3, 21, 7, 1], [-18, 16, 0, -10, 2, -8], [-28, 9, -6, -22, 0, -5], [4, 4, -1, -12, -1, -1], [32, 2, -2, 22, 6, 2], [6, -6, -20, 4, 1, -2], [-33, -9, -14, -12, 12, 0], [-28, 11, 7, -10, -6, -6], [3, -8, 9, 4, -5, 6], [-8, 8, 4, -17, 6, -3], [17, 6, -1, 9, -9, -3], [26, -8, -3, 13, 6, 7], [-11, 6, -2, 22, -6, 9], [8, -11, 8, 20, -2, 7], [3, 3, -5, -19, -6, 1], [24, -10, -3, 6, 2, 6], [15, -10, 2, 6, -13, 1], [21, 11, 0, 2, 0, 12], [-12, 3, 0, -12, 12, -1], [30, 2, 8, 4, -8, 0], [5, -11, 1, 12, 10, 2], [-8, 6, 9, -1, 12, -3], [8, -6, -15, 7, 2, -3], [-11, 5, 11, 8, 0, 2], [1, 7, -1, 16, 3, 7], [32, -4, 1, 16, -3, 3], [6, 9, 16, 13, -2, 6], [-25, -13, -4, 1, 2, 6], [-27, -17, -4, -31, -1, 2], [10, 8, 8, -21, -3, 2], [4, 0, 6, 13, -1, -2], [13, 8, 13, 6, 1, 2], [-26, -8, 6, -27, 1, -6], [-13, -1, 9, -21, 1, 1], [24, 4, -2, 21, 2, 4], [8, 10, -9, -8, -3, -9], [-20, -10, -4, -19, -5, 0], [-13, -1, 9, -2, 9, 7], [-13, -2, 3, -10, 5, -4], [11, -6, -4, 17, -9, 5], [-7, -3, -8, -7, -8, -5], [2, 6, 10, 30, 5, -7], [34, 6, 10, 10, -2, 6], [8, 12, 1, 7, 3, -3], [-24, 1, -5, -13, -6, -4], [28, 6, 11, 13, -12, -7], [-11, -5, 0, 4, 11, -3], [-15, -4, 2, 4, 8, 1], [-20, -16, -3, -6, 4, 8], [-25, -10, 0, 12, 1, 6], [-11, 2, 6, 19, -13, 9], [25, 4, -4, -13, -1, 3], [-6, -2, 4, -6, -11, -2], [13, 3, 3, 8, -8, 7], [25, 21, 2, 1, -4, -3], [35, -5, 9, -3, 0, 7], [13, -7, -6, -4, 15, 2], [1, 2, 4, -17, -4, -2], [-34, 2, -15, -9, 0, -5], [-21, -2, -5, 10, 6, -10], [-1, -2, 3, 8, 8, -4], [-26, -6, 4, 8, -4, -6], [-34, -10, -16, -20, -5, -6], [5, -9, -9, -5, -2, -13], [10, 1, 18, 5, -10, 2], [-20, -12, 6, -3, -7, -4], [35, -12, 5, 14, 9, 8], [24, -4, -1, 31, 3, 8], [-32, -8, -4, -28, -3, -3], [1, 11, 5, -1, 14, -11], [9, 1, -5, 0, -9, 1], [6, -5, -6, 10, -10, 7], [6, -5, -4, 6, 4, 11], [-16, -7, -10, -28, -1, -7], [-5, -17, 11, 11, 8, 7], [-4, 16, -7, -4, 3, -5], [18, 3, 5, 19, 3, 10], [7, 3, -5, 12, -4, 14], [19, 5, -5, -8, 1, -11], [-15, 11, -3, -10, -2, -9], [-3, -9, 6, -8, 5, 13], [-1, -8, 2, 4, 10, -3], [25, -6, -16, -8, 1, -2], [4, -11, 8, -14, 10, 7], [28, -13, -20, 12, 7, 5], [-9, -5, -9, 31, 9, -11], [26, -17, 0, -6, -6, -3], [24, 5, 12, 10, -11, 8], [12, 6, -4, -15, 3, 0], [6, -9, -2, 4, 10, 13], [-22, -7, -2, -12, -1, -9], [2, -3, -1, 6, 12, -3], [-7, 1, -11, -12, -3, -17], [-29, -23, -12, -6, 6, 0], [-33, -1, 8, -17, 11, 6], [-36, 1, -3, -8, 10, 0], [-2, -3, 0, -6, 6, -13], [22, 11, 20, 8, -7, 0], [-18, 18, 12, -14, -2, -4], [25, 8, -8, 12, -6, -3], [1, -7, -14, -21, 7, 6], [-4, -8, 3, 26, 1, 5], [-31, 12, -2, -16, 3, -7], [19, -10, -6, -9, 1, -9], [29, -2, 7, 36, 7, 4], [19, 7, 2, -13, 0, -11], [20, 18, 5, 2, -5, -3], [17, 12, -7, 9, 4, 12], [38, -11, -17, 26, -2, 5], [-10, -13, 9, -4, -6, -14], [-6, -8, -1, 8, 1, -11], [21, 5, 7, 10, 4, -7], [-20, -2, -3, -9, -1, -7], [-10, -9, 11, -16, 9, 0], [-1, -5, 1, 19, -7, 4], [34, 16, 10, 23, 8, -5], [-18, -10, 0, -11, -5, 0], [-34, 3, 13, 12, 5, -8], [37, 3, 7, 45, -18, -7], [-7, 19, 0, -15, 2, 0], [-28, -18, -1, -20, -2, -9], [-44, 5, -18, -9, 8, -7], [4, -17, 6, -2, 0, 5], [19, -4, 16, 8, 12, 11], [1, 5, 8, 28, 0, 12], [8, -3, -2, 9, -7, -7], [-14, 24, -4, 0, -1, -11], [44, -15, -1, 0, -2, 4], [3, 0, 3, -8, -2, 18], [-19, -5, 14, -13, 5, -6], [-7, -20, -14, 3, 10, 4], [27, 11, 17, 4, -11, -5], [15, 1, -4, 3, -2, -4], [-16, 28, -2, -8, 2, -14], [-11, -5, -6, 5, 6, 5], [15, -11, 4, -16, 0, 6], [2, 11, 3, 22, 2, -1], [-36, 3, -14, -20, -6, -13], [36, -1, -6, 21, -1, 7], [-5, 9, 13, -22, -10, 11], [-22, -12, 1, -21, 0, 1], [18, 14, -4, 20, -9, -6], [-21, -13, 12, -5, 0, -9], [-1, -11, 2, -20, 8, 2], [-27, 3, 1, -2, -13, -15], [-10, -8, 7, 33, -5, 7], [-13, -6, -9, -16, 13, -2], [8, -18, -3, 26, -3, 9], [-2, 7, 18, 15, -8, 14], [4, -6, 15, 17, 0, 1], [11, -1, -13, 2, 14, 12], [32, -6, 0, 20, 13, 5], [-17, 4, 0, -53, 3, -1], [-17, 0, 6, -23, 5, -3], [-7, 5, -6, 13, 1, -8], [-12, 1, -15, -32, 6, -11], [19, -20, -16, 15, 6, 11], [36, -6, -18, 7, -11, -2], [24, -9, 2, 31, 2, 0], [41, 11, -3, 5, -6, 5], [-54, 11, 7, -56, 1, -2], [0, -18, -5, -6, 19, -5], [8, 14, 17, 30, -5, 13], [15, -11, -15, -5, 8, 21], [39, 3, 16, 18, -6, 6], [14, -16, -14, -17, -7, -2], [-4, -2, -9, 13, 12, 5], [-13, 6, -16, 3, 6, -4], [-8, -10, -24, 2, 11, 5], [26, -10, 5, 38, 2, 0], [23, -25, 2, 33, -3, 6], [-13, -12, -21, 18, 14, -7], [64, -10, -8, 8, 0, -4], [-14, -13, -14, -9, 6, 9], [8, 15, -2, -18, -8, -1], [-13, 5, -2, 23, -3, 4], [2, -10, 7, 14, -17, -9], [20, -3, -5, -10, 0, -3], [7, -22, -23, 21, 6, 16], [24, 9, -1, -4, -7, -14], [13, -9, 3, -12, 12, -1], [-11, 11, 15, 21, 6, -11], [12, 2, 1, -2, -13, 9], [12, 10, 10, -21, -11, 16], [25, 8, 24, -2, -14, 1], [0, 22, 11, -20, -14, -1], [14, 8, 3, -19, -2, 13], [6, 4, -2, -11, -6, 9], [-14, 27, -3, -1, 10, 0], [-35, -10, -19, -23, -2, -6], [-8, 9, 12, -4, -11, 0], [-5, 10, 0, 14, 13, -1], [-5, -5, -2, 12, -12, 12], [10, 22, 18, 6, -3, 10], [-13, -20, -7, -21, -12, -2], [13, -15, -15, 16, 11, -11], [14, 1, -5, -11, 17, 2], [16, -2, 8, -8, 7, 10], [-31, -27, -12, -2, 6, 4], [-28, 8, 12, -30, 12, -4], [-3, 19, 0, -17, 5, -6], [-25, -2, -4, 19, -1, 7], [-3, 6, -24, -15, 11, -3], [5, 2, -7, 4, 17, -6], [-3, 3, 12, 29, 6, -6], [-38, 21, 8, -4, 2, -3], [-31, 16, 7, -44, -9, -4], [21, 10, 0, 43, -11, -1], [1, -7, -1, -16, 7, 9], [12, 15, -10, 3, 13, 9], [-15, -19, 1, -35, -11, -1], [-29, 18, 5, 0, 0, -7], [-37, -10, -17, 6, 3, 0], [4, 16, 2, 20, 0, -4], [27, -6, 15, -2, 2, 13], [15, -12, -2, 22, 6, -7], [0, -19, 15, 32, -1, 2], [10, -7, -13, 19, -11, 4], [12, -13, -21, 27, 0, 16], [5, -19, -3, -8, -1, -11]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5625_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5625_2_a_q();".
// 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_5625_2_a_q(:prec:=6)
    chi := MakeCharacter_5625_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_5625_2_a_q();".
// 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_5625_2_a_q( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5625_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, -3, 29, 1, -11, 0, 1]>,<7,R![20, -20, -25, 25, 4, -6, 1]>],Snew);
    return Vf;
end function;