
// Make newform 625.2.d.n in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_625_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_625_d_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_625_2_d_n();".
// 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_625_2_d_n();".
// 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, 0, 1205, 0, 3809, 0, 4820, 0, 3166, 0, 1165, 0, 239, 0, 25, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1474, 0, -12623, 0, -4230, 0, 15463, 0, 13949, 0, 4515, 0, 619, 0, 29, 0], [1601, 0, -25961, 0, -63150, 0, -52142, 0, -20887, 0, -4422, 0, -470, 0, -19, 0], [673, 70883, 12623, 237289, 4230, 297480, -15463, 183139, -13949, 59873, -4515, 10389, -619, 886, -29, 29], [431, 10906, -1224, 29676, -4869, 39643, -4282, 32176, -1388, 15210, -142, 3907, 6, 488, 1, 22], [431, -10906, -1224, -29676, -4869, -39643, -4282, -32176, -1388, -15210, -142, -3907, 6, -488, 1, -22], [-1556, 159574, -77159, 474718, -150381, 534120, -110467, 297029, -38668, 88283, -6717, 13968, -541, 1082, -16, 32], [1556, 159574, 77159, 474718, 150381, 534120, 110467, 297029, 38668, 88283, 6717, 13968, 541, 1082, 16, 32], [3891, 98578, -17926, 240239, -90759, 227865, -120384, 100622, -70055, 18841, -19770, 240, -2583, -316, -119, -23], [164, 17682, -1224, 91029, -4869, 161716, -4282, 136696, -1388, 60887, -142, 14429, 6, 1678, 1, 72], [-950, 89311, -138110, 221134, -371667, 217803, -377263, 116507, -185932, 38195, -46914, 7668, -5683, 833, -250, 35], [-21, 11052, -149932, 11761, -375897, -19152, -361800, -41577, -171983, -27682, -42399, -8280, -5064, -1113, -221, -52], [-2335, 0, 95085, 0, 241140, 0, 230851, 0, 108723, 0, 26487, 0, 3124, 0, 135, 0], [950, 89311, 138110, 221134, 371667, 217803, 377263, 116507, 185932, 38195, 46914, 7668, 5683, 833, 250, 35], [1634, -43770, -121014, -100789, -352830, -99777, -379880, -54951, -195717, -17948, -51003, -3441, -6320, -351, -282, -14], [2391, -55321, 245786, -285710, 665577, -489378, 689538, -394625, 346813, -168712, 88980, -38772, 10914, -4415, 484, -187]];
    Rf_basisdens := [1, 801, 801, 1602, 534, 534, 1602, 1602, 1602, 534, 1602, 1602, 801, 1602, 1602, 1602];
    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_625_d();"
function MakeCharacter_625_d()
    N := 625;
    order := 5;
    char_gens := [2];
    v := [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_625_d_Hecke();"
function MakeCharacter_625_d_Hecke(Kf)
    N := 625;
    order := 5;
    char_gens := [2];
    char_values := [[-1, 0, 0, 0, 1, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, 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+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, 0, 0, 0, 0, -1, -1, 1, -1, 0, 0, 0, -1, 0, 0, 0], [-1, -1, 0, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, -1, 0, -1, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [0, 2, 0, 2, 0, 0, -1, -1, 1, 0, 0, -2, -1, 0, -1, 0], [-2, -1, 0, -2, -3, 1, 0, 1, 0, 0, 0, -1, -1, 0, -2, 0], [2, 0, 1, 0, 0, -1, 0, 0, 0, 3, 0, 0, 0, 0, -1, -1], [0, 0, -1, 1, 0, -1, 1, -2, 1, -1, 0, 0, 0, -1, 1, 1], [0, 0, 0, 0, 1, -3, 2, -1, 1, -1, 0, 0, 2, 0, 1, 1], [-1, -2, 0, 0, -1, -1, -2, 0, -1, 1, -1, 0, -1, 1, -2, -1], [0, 0, 2, -2, 0, 2, 0, 0, 0, -1, 0, -1, 0, 1, -2, -2], [0, 1, 3, 1, 1, 3, 0, 1, -1, -1, -1, -3, -1, 3, -2, -1], [4, 2, 3, 4, 3, 1, 0, -1, -2, 0, -2, -1, 1, 3, 1, -2], [-1, 2, 2, 0, 2, 3, -1, 1, 0, 0, 0, -1, -1, 0, -1, 0], [-3, 1, 0, 2, 4, 2, 0, 0, -1, -2, -1, 0, -1, 0, -1, 0], [-3, -5, 0, -2, -2, 2, 2, 0, -1, -2, 1, 0, -1, 1, -3, -1], [2, 3, 2, 2, 2, 3, 0, -1, 0, 0, -1, -3, 1, 2, 0, -1], [0, 0, 1, 0, -1, 1, 1, 1, -1, 1, -1, -1, 1, 0, -1, -3], [0, 0, -1, 0, 0, -1, -1, -2, -1, 2, 0, -3, 0, -1, 1, 1], [3, 0, 0, 0, -3, -3, -3, 0, -1, 3, 1, 0, -1, -2, 0, 2], [0, -1, -1, -1, -5, -3, 0, -3, 3, 5, 1, 2, 0, 0, 0, -1], [4, -4, 0, -3, -8, 1, 1, 0, 0, -1, -1, 0, 0, 1, -1, -1], [3, 0, 2, 4, 0, -1, -1, -1, -1, 7, 0, 0, 0, -1, -2, -2], [0, 1, -4, 1, 1, -2, -1, 0, 0, -1, 4, 3, -1, 0, 3, 3], [-3, -3, 0, -4, 0, -4, 1, 0, 1, 4, -2, 0, 1, -3, 1, 3], [-5, 1, 2, 0, 0, 5, 2, -2, 0, 0, 1, -5, 1, -1, -5, 0], [-3, 3, 0, 5, 6, 4, 0, 0, 0, -4, -2, 0, 0, 2, -2, -2], [2, 1, 1, 0, -2, 3, 0, 0, 0, 0, 1, -5, -1, -1, -5, 0], [-1, -6, -2, -6, -3, -2, 0, 3, -2, 5, 0, 2, -3, -2, -4, 0], [0, -1, 1, 3, 1, -3, 0, -3, 2, -3, 1, 3, 3, 1, 2, 1], [0, -1, -2, -1, -2, 0, 1, 3, -3, 2, 2, 3, 1, 0, 0, 1], [5, 0, -3, -4, 0, -8, -1, -2, -1, 6, 0, 5, 0, -4, 3, 3], [5, -3, -3, -2, 2, -4, 0, -1, 0, 7, 3, 4, 1, -3, 1, 3], [0, -6, 3, -6, -4, -2, 2, -2, 2, 4, -3, 3, 2, 0, 2, 2], [6, -2, 4, 0, 1, -1, -2, 2, 0, 0, -3, 2, 4, 3, 2, 0], [-3, -4, -5, 0, -4, -6, -1, 1, 0, 0, 0, 2, -3, 0, 2, 0], [-6, -3, 2, 0, -5, -1, 6, -6, 0, 0, -1, -4, 2, 1, -4, 0], [5, -8, 3, -1, -1, -4, 0, 1, -5, 6, -2, 4, -1, 3, -4, -2], [3, 0, 0, 6, 0, -3, -4, 2, -4, 10, 0, 3, 0, 2, 0, 0], [0, -4, 5, -4, -4, -2, -2, 5, -5, 4, -5, -1, -2, 0, -2, -4], [7, 13, 0, 10, 6, -3, -1, 0, 1, 3, 2, 0, 1, 0, 3, 0], [-2, 0, 0, 5, 0, 2, 5, -5, 5, -4, 0, 0, 0, 0, 0, 0], [-9, 3, -1, 1, 8, 1, 0, 3, 1, -10, -1, -1, -3, -1, 2, -1], [-6, -1, -2, 0, -1, -3, -3, 3, 0, 0, -2, 2, 3, 2, 2, 0], [1, 11, 0, 12, 10, -4, -7, 0, -5, 4, 3, 0, -5, 2, -1, -2], [-6, 4, -4, 0, 6, 9, 0, 0, 0, 0, 3, -3, 0, -3, -3, 0], [0, 4, -4, 4, 3, 7, -1, 4, -4, -3, 4, 0, -1, 0, 2, 2], [0, -1, -4, -1, -3, -5, -3, 4, -4, 3, 4, 5, -3, 0, -7, -2], [0, 8, -1, 8, -1, -12, -4, -2, 2, 1, 1, -7, -4, 0, 2, 3], [-8, -6, 0, -6, 2, 6, 10, 0, 8, -6, -3, 0, 8, 0, 0, 0], [0, 0, 1, -2, 0, 1, 2, 5, 2, -6, 0, -6, 0, 1, -1, -1], [0, -2, 1, -2, 3, -4, -7, 4, -4, -3, -1, 1, -7, 0, -3, -2], [8, -15, 0, -5, -11, -10, 0, 0, -5, 13, 0, 10, 0, 0, -5, 0], [6, 0, -4, 0, 2, -2, -6, 6, 0, 0, 3, 4, -3, -3, 4, 0], [3, 11, -5, 0, 11, 5, 1, -1, 0, 0, 3, 6, 2, -3, 6, 0], [0, -3, -2, -1, -11, -4, 0, 6, -5, 1, 3, 4, -6, -2, 1, 3], [-8, 0, 1, -2, 0, 9, -2, 7, -2, -7, 0, -8, 0, 5, -1, -1], [3, 3, 0, 2, 0, -6, 0, 0, -5, 6, -4, 0, -5, -2, 1, 2], [0, -5, 3, -5, 3, 1, -3, 4, -4, -3, -3, 2, -3, 0, -3, -4], [-9, 0, -2, -2, 0, 7, -2, 1, -2, -8, 0, 8, 0, -1, 2, 2], [-11, 0, -5, -2, 0, 6, 4, -1, 4, -4, 0, -5, 0, -7, 5, 5], [2, -2, 2, 0, -13, -9, -4, 4, 0, 0, 0, -4, -10, 0, -4, 0], [-1, -7, 0, 0, 1, 6, 0, 0, 0, 0, -2, -5, 4, 2, -5, 0], [0, 1, -1, 1, -7, 2, -4, 6, -6, 7, 1, 0, -4, 0, -12, -2], [-10, 10, 0, 0, 12, 10, 0, 0, 3, -10, 5, -10, 0, 0, 0, 5], [6, 0, -7, -4, 0, -13, 0, 5, 0, -1, 0, 10, 0, -4, 7, 7], [-11, 0, 1, -11, 0, 12, 0, 5, 0, -5, 0, -3, 0, -2, -1, -1], [-3, -4, -1, 1, 10, -6, 0, -3, -4, -4, 5, 6, 3, -1, 2, 5], [4, 0, 0, 3, -4, 5, -2, 0, -5, -5, -3, 0, -5, 4, -3, -4], [3, 9, 2, 0, 3, -3, 4, -4, 0, 0, 1, 6, 7, -1, 6, 0], [1, 9, 0, -1, 8, 7, -1, 0, -2, -7, 1, 0, -2, 1, 10, -1], [3, 2, 0, 5, 3, -3, 0, 4, -5, -2, 2, 3, -4, 0, 5, 2], [-13, 0, 0, -2, 0, 13, 9, -2, 9, -10, 0, -8, 0, -3, 0, 0], [0, 4, -2, 4, -1, -2, 4, -4, 4, 1, 2, -2, 4, 0, 11, 2], [-6, -3, 0, 1, 3, 5, -3, 0, 2, -5, -6, 0, 2, 1, -4, -1], [-1, 0, -1, 1, 0, 0, 3, -3, 3, 8, 0, 2, 0, -5, 1, 1], [0, -6, 6, -6, -11, 5, 0, 6, -6, 11, -6, 0, 0, 0, -11, -7], [2, -2, 0, -5, -4, 0, -3, 0, 3, 0, 7, 0, 3, -6, 3, 6], [1, 3, -4, 0, 0, 5, 1, -1, 0, 0, -2, -5, 3, 2, -5, 0], [3, -5, -5, 1, 1, -11, 0, -7, -2, 2, 2, 11, 7, -5, 6, 2], [0, 0, -6, -5, 0, -6, 3, -8, 3, 15, 0, -2, 0, -1, 6, 6], [9, 3, 0, -8, -6, -5, 2, 0, -1, 5, 2, 0, -1, -9, 11, 9], [0, 0, 2, 11, 0, 2, -1, -2, -1, -9, 0, -10, 0, 11, -2, -2], [-13, 0, 1, -6, 0, 14, 1, -2, 1, -6, 0, -4, 0, 1, -1, -1], [0, 3, 1, 3, 3, -9, 8, -6, 6, -3, -1, -4, 8, 0, -3, 3], [11, -10, -1, 0, -4, -7, -2, 2, 0, 0, 1, 3, 6, -1, 3, 0], [8, 2, -13, 0, 1, -6, 1, -1, 0, 0, 6, 7, 3, -6, 7, 0], [2, -4, 1, 0, -17, -22, -5, 5, 0, 0, -9, 5, -6, 9, 5, 0], [0, 4, -4, 4, -7, -3, -1, -6, 6, 7, 4, 0, -1, 0, 7, 2], [11, 10, 3, 10, 19, 3, 0, -7, 4, 1, -2, -3, 7, 3, 7, -2], [5, 0, 0, -3, 0, -5, 0, -5, 0, 10, 0, 0, 0, 2, 0, 0], [6, -3, 0, -4, -9, -13, 1, 0, -5, 13, -9, 0, -5, 2, 1, -2], [7, 1, -6, 6, 0, -11, 0, -3, 6, 1, 5, 11, 3, -6, 12, 5], [-5, -4, -4, -8, -6, 0, 0, 4, 3, 3, 1, 0, -4, -4, -4, 1], [-5, -4, -5, 0, 6, -1, -6, 6, 0, 0, 5, 7, -3, -5, 7, 0], [-7, -3, 0, 6, 4, 1, -9, 0, -8, -1, 0, 0, -8, -2, -9, 2], [6, -5, 9, 6, -13, -2, 0, -8, 11, 0, -4, 2, 8, 9, -3, -4], [2, 3, 0, -3, 1, 3, -7, 0, 1, -3, 5, 0, 1, 0, 6, 0], [0, 10, -2, 10, 19, 19, 5, 2, -2, -19, 2, -8, 5, 0, 3, -3], [-14, -1, -1, -12, -1, 10, 0, 6, -8, -2, -1, -10, -6, -1, -11, -1], [11, 1, 0, -3, -10, -4, 3, 0, 5, 4, -2, 0, 5, -3, 4, 3], [-12, 1, 5, 0, 16, 10, -4, 4, 0, 0, -2, 6, 2, 2, 6, 0], [26, 3, 0, 11, -8, -8, 0, -3, -3, 15, 0, 8, 3, 0, 11, 0], [-20, 0, 4, -5, 0, 24, 3, -3, 3, -15, 0, -7, 0, -1, -4, -4], [-3, -16, 0, 1, -13, -18, -11, 0, -7, 18, -10, 0, -7, 5, -17, -5], [0, -7, 2, -7, -8, 5, 4, -9, 9, 8, -2, 5, 4, 0, 4, 4], [-4, -15, 7, -6, -8, -2, 0, -4, -1, 2, -7, 2, 4, 7, -13, -7], [14, 1, -1, 0, 6, -9, 10, -10, 0, 0, 1, 15, 12, -1, 15, 0], [1, -16, 14, 0, -23, -7, 2, -2, 0, 0, -8, -16, -4, 8, -16, 0], [-6, 0, -2, 0, -14, -1, -3, 3, 0, 0, -6, -13, -14, 6, -13, 0], [2, -10, -7, 0, -3, -9, 6, -6, 0, 0, 7, 6, 7, -7, 6, 0], [4, -7, -1, -4, -2, -4, 0, -9, 0, 8, 4, 4, 9, -1, -3, 4], [16, 0, 12, 4, 0, -4, -1, -2, -1, 1, 0, 7, 0, 8, -12, -12], [-9, 0, -5, -15, 0, 4, 1, 13, 1, -12, 0, 2, 0, 2, 5, 5], [9, 0, 1, 2, 0, -8, -6, 3, -6, 27, 0, 3, 0, -2, -1, -1], [4, -7, 8, 1, -1, 0, 0, 7, -11, 3, -2, 0, -7, 8, -7, -2], [-6, -6, -7, 0, -16, -13, -3, 3, 0, 0, -2, -3, -7, 2, -3, 0], [-19, 10, 0, -12, 29, 27, 7, 0, 13, -27, -1, 0, 13, -10, 22, 10], [2, 14, 0, 9, 12, 3, 1, 0, -3, -3, 2, 0, -3, 9, 5, -9], [-16, 16, -2, -2, 14, 16, 0, 4, 5, -14, 5, -16, -4, -2, 0, 5], [0, 1, 5, 1, -11, 6, 2, -3, 3, 11, -5, -6, 2, 0, 7, 5], [0, 9, -9, 9, -8, -10, -18, 14, -14, 8, 9, 0, -18, 0, -2, 3], [0, -11, -1, -11, 4, -6, 12, -6, 6, -4, 1, 12, 12, 0, -1, -2], [-9, 0, 4, 17, 0, 13, 2, 0, 2, -12, 0, -1, 0, 10, -4, -4], [-1, -7, 3, 1, 4, -5, 0, -3, -11, -2, -2, 5, 3, 3, -2, -2], [-10, -6, 12, 0, -12, 7, -1, 1, 0, 0, -4, -19, 2, 4, -19, 0], [-4, 4, 9, 9, 11, 4, 0, -1, -5, -13, -9, -4, 1, 9, 0, -9], [9, 0, -2, 5, 0, -11, 5, 1, 5, -13, 0, 4, 0, 6, 2, 2], [0, 5, -4, 5, 11, 3, -15, 12, -12, -11, 4, -1, -15, 0, -1, 2], [-10, 0, -3, -10, 0, 7, -7, 7, -7, 7, 0, -3, 0, -6, 3, 3], [0, -5, 3, -5, 12, 22, 4, 0, 0, -12, -3, 2, 4, 0, 4, -3], [5, -4, 8, 0, 9, 8, 6, -6, 0, 0, -12, 1, 5, 12, 1, 0], [12, -6, 5, 0, 9, 1, 6, -6, 0, 0, 0, 8, -2, 0, 8, 0], [6, -7, -7, 0, -8, -10, -3, 3, 0, 0, 7, 2, -5, -7, 2, 0], [0, -6, 8, -6, -5, -15, -9, 16, -16, 5, -8, -2, -9, 0, 0, 6], [7, 0, 6, 3, 0, -1, -7, 2, -7, -2, 0, -8, 0, 0, -6, -6], [0, -8, 7, -8, 18, 7, 6, -2, 2, -18, -7, 1, 6, 0, 8, 5], [-10, 14, 0, -3, 4, 17, 0, 0, -3, -7, 1, -17, 0, 0, -3, 1], [-1, -12, 0, -5, -11, 7, 6, 0, -9, -7, -6, 0, -9, 0, -7, 0], [10, -4, -4, -7, -4, -1, 0, 2, 1, 17, -7, 1, -2, -4, -3, -7], [0, -7, 6, -7, 7, 8, 7, -3, 3, -7, -6, 1, 7, 0, 0, 2], [-9, -8, 0, -12, 1, -2, 5, 0, 5, 2, 9, 0, 5, -3, 4, 3], [0, -4, 8, -4, -11, -5, 1, -12, 12, 11, -8, -4, 1, 0, 0, -3], [0, -12, 0, -12, -30, -18, -3, 9, -9, 30, 0, 12, -3, 0, -9, 0], [-1, 12, 0, 6, 13, 5, 2, 0, 2, -5, 4, 0, 2, -4, 6, 4], [0, -13, 4, -13, -5, -6, -1, 5, -5, 5, -4, 9, -1, 0, 7, 9], [20, 6, 0, 2, -14, -11, 4, 0, 8, 11, 3, 0, 8, -8, 4, 8], [8, 13, 11, 0, 3, 9, 4, -4, 0, 0, -4, -6, 1, 4, -6, 0], [-3, -2, -8, -8, -18, -2, 0, -1, 10, 5, 4, 2, 1, -8, 0, 4], [-3, -2, 3, 1, -16, 0, 0, 5, 3, -4, -2, 0, -5, 3, -2, -2], [0, -8, 8, -8, -7, -2, 9, 1, -1, 7, -8, 0, 9, 0, -14, -9], [5, 0, -8, -7, 0, -13, 10, 0, 10, -16, 0, -6, 0, -9, 8, 8], [-25, 0, 1, -12, 0, 26, -3, 10, -3, -16, 0, -1, 0, 1, -1, -1], [-11, 8, 8, 10, -4, 6, 0, -11, 15, -21, -6, -6, 11, 8, 2, -6], [-11, 4, 3, 0, 2, 13, 9, -9, 0, 0, -3, -11, -4, 3, -11, 0], [0, -3, -1, -3, -21, -29, -12, 8, -8, 21, 1, 4, -12, 0, -15, -2], [-5, 0, -1, -2, 0, 4, 7, 0, 7, -3, 0, -6, 0, -11, 1, 1], [0, 11, 0, -8, 11, 2, -2, 0, 4, -2, 5, 0, 4, -11, 19, 11], [6, 4, -2, 7, -4, -5, 0, -7, 3, -1, 9, 5, 7, -2, 9, 9], [15, -6, 1, 3, -28, -8, 0, -2, 6, 12, 1, 8, 2, 1, 2, 1], [-1, -1, 0, -16, 0, 1, 2, 0, 8, -1, -7, 0, 8, 7, 15, -7], [-7, 15, 0, -2, 22, -2, -1, 0, 13, 2, -3, 0, 13, -11, 17, 11], [11, 0, 3, 21, 0, -8, -5, -5, -5, 14, 0, 0, 0, 1, -3, -3], [-23, -9, 0, -11, 14, 10, -4, 0, 7, -10, -10, 0, 7, 0, 2, 0], [0, 10, 1, 10, 1, 3, 5, 2, -2, -1, -1, -11, 5, 0, 4, 2], [8, 0, -9, -13, 0, -17, -2, -3, -2, 19, 0, 12, 0, -4, 9, 9], [-12, 23, -5, 6, 17, 12, 0, 4, 3, -18, 0, -12, -4, -5, 11, 0], [-15, 6, 0, -8, 21, 0, 11, 0, 1, 0, -3, 0, 1, -1, 14, 1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_625_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_625_2_d_n();".
// 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_625_2_d_n(:prec:=16)
    chi := MakeCharacter_625_d();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_625_2_d_n();".
// 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_625_2_d_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_625_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![81, -405, 1053, -1080, 1278, 1665, 2406, 60, 1045, -185, 336, -20, 48, 10, 8, 0, 1]>,<3,R![841, 2610, 6327, 10330, 14803, 16150, 15869, 9055, 4220, 1255, 529, 295, 168, 50, 17, 5, 1]>],Snew);
    return Vf;
end function;