
// Make newform 625.2.d.o in Magma, downloaded from the LMFDB on 31 December 2025.
// 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_o();".
// 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_o();".
// 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 := [625, 0, 1125, 0, 900, 0, 395, 0, 11, 0, -49, 0, -4, 0, 1, 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], [0, -35093875, 0, -210825325, 0, -420964990, 0, 97097069, 0, -1538146, 0, 6573709, 0, -2422021, 0, 22849], [181387875, 0, 27131400, 0, 77053830, 0, 136185777, 0, -35207443, 0, -1547103, 0, -2081693, 0, 948392, 0], [52412250, 0, 77779875, 0, 54634025, 0, 3932417, 0, -5799603, 0, -701513, 0, 97172, 0, 122987, 0], [563878625, 0, 771306350, 0, 424939915, 0, 94590252, 0, -61900293, 0, -4990978, 0, -312743, 0, 1379032, 0], [-167096475, 0, -152170830, 0, -83450271, 0, -18275316, 0, 12381889, 0, 939009, 0, 69219, 0, -281280, 0], [0, 846395125, 0, 222668250, 0, -474206065, 0, -137248467, 0, 43919978, 0, 8678488, 0, 1428203, 0, -1157122], [-215410900, 0, -246968035, 0, -135351515, 0, -29602957, 0, 20140738, 0, 1544473, 0, 108648, 0, -441862, 0], [-860602375, 0, -884417625, 0, -716617430, 0, -262226761, 0, 118381374, 0, 8884104, 0, 2379174, 0, -2817591, 0], [-222107450, 0, -281904075, 0, -210462216, 0, -41997039, 0, 27217946, 0, 2783831, 0, 144621, 0, -626638, 0], [0, -189940875, 0, -254856700, 0, -113117275, 0, -4440499, 0, 13501191, 0, 1184711, 0, -260409, 0, -257689], [0, 1573606875, 0, 2678832450, 0, 1546922530, 0, 137417033, 0, -175485672, 0, -18411087, 0, 2442853, 0, 3711283], [0, 4625990250, 0, 5075395150, 0, 3788808280, 0, 752782897, 0, -491452273, 0, -48798533, 0, -2500148, 0, 11296137], [0, 416073000, 0, 432663800, 0, 285294535, 0, 84515054, 0, -44316636, 0, -3517956, 0, -651386, 0, 1033909], [0, 722601875, 0, 843562400, 0, 466969935, 0, 90901286, 0, -66369199, 0, -5426879, 0, -201274, 0, 1468861], [0, 7417196625, 0, 7862490225, 0, 5454301045, 0, 1385305413, 0, -789938892, 0, -66698807, 0, -8550017, 0, 18299323]];
    Rf_basisdens := [1, 858900625, 171780125, 15616375, 171780125, 34356025, 858900625, 34356025, 171780125, 34356025, 78081875, 858900625, 858900625, 78081875, 78081875, 858900625];
    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 := false)
    N := 625;
    order := 5;
    char_gens := [2];
    char_values := [[-1, 0, 0, 0, -1, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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]];
    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, -1, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 1, -1, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0], [0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, -1, 0, 0, -1], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 1, 0, -1], [-2, 0, 0, 0, 0, -2, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, -1, -2], [-3, 0, 1, 1, -3, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 2, -2, 1, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 1, 0, 0, 0, -1, -1, 1, -1, 1, -2], [-1, 0, -1, -2, 4, 0, 0, -1, -2, 0, 0, 0, 0, 0, 0, 0], [0, -3, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 2, -1, -3], [0, -1, 0, 0, 0, 0, 5, 0, 0, 0, -3, -2, 4, -1, -3, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -3, 2, 4, -1, -3, 0], [-2, 0, 3, 2, 2, 0, 0, -2, 2, 0, 0, 0, 0, 0, 0, 0], [0, 0, -3, 0, -1, -1, 0, -1, -3, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, -2, -2, 0, 2], [-8, 0, -2, -2, -8, -6, 0, -6, -3, -3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 1, 0, -2, 2, 0], [-2, 0, 1, 1, -2, -6, 0, -6, -3, -3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -5, 0, 0, 0, 1, 1, -1, -4, 0, 4], [0, 0, 1, 0, 7, 8, 0, 7, 1, 1, 0, 0, 0, 0, 0, 0], [0, 8, 0, 0, 0, 0, -8, 0, 0, 0, 4, 4, -3, -1, 4, 0], [9, 0, -2, -1, 9, 9, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0], [0, -3, 0, 0, 0, 0, 2, 0, 0, 0, 3, -5, 3, -6, 3, 0], [0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, -3, 3], [-9, 0, 1, -2, -4, 0, 0, -9, -2, 0, 0, 0, 0, 0, 0, 0], [0, 6, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, 6, 0, 0, 0], [0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 4, 4, -2, -4], [2, 0, 0, 1, 0, 2, 0, 8, 3, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2, 0, 0, 0, -2, 7, 0, 7, -7, 0], [0, 0, -5, 0, -6, -6, 0, -6, -5, -2, 0, 0, 0, 0, 0, 0], [0, 0, 1, -2, -5, -5, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [-2, 0, 0, 5, -8, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0], [0, -2, 0, 0, 0, 0, 7, 0, 0, 0, -7, 4, -2, 4, -4, 1], [0, 0, 0, 0, 0, 0, -6, 0, 0, 0, 9, 9, -1, 1, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 5, -1, 1, -5], [-2, 0, -5, -5, -2, -6, 0, -6, -2, -2, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 5, 0, -1, 0, 4, 5, 5, 0, 0, 0, 0, 0, 0], [2, 0, 2, 4, 8, 0, 0, 2, 4, 0, 0, 0, 0, 0, 0, 0], [0, -5, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, -7, -5], [0, 2, 0, 0, 0, 0, 0, 0, 0, 0, -3, 3, -5, 8, -3, 0], [12, 0, -1, 2, 4, 4, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, -3, 0, 2, -2, 0, 2, -3, -4, 0, 0, 0, 0, 0, 0], [0, 6, 0, 0, 0, 0, -12, 0, 0, 0, 0, -4, -12, -6, 12, 12], [0, 4, 0, 0, 0, 0, -6, 0, 0, 0, 0, 6, -10, -2, 6, 10], [0, 0, 5, 5, 0, -3, 0, -3, 3, 3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -8, 0, 0, 0, 7, 7, 3, 8, 0, -8], [0, 0, 1, 0, -4, 0, 0, -4, 1, -4, 0, 0, 0, 0, 0, 0], [9, 0, 5, 0, 9, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, -3, -4, -4, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 0, 0, 0, -10, 0, 0, 1, 2, 2], [0, -7, 0, 0, 0, 0, -8, 0, 0, 0, 8, -1, -7, -1, 1, 6], [5, 0, 0, -4, 0, 5, 0, 1, -3, -4, 0, 0, 0, 0, 0, 0], [-2, 0, -2, -2, -2, -12, 0, -12, 2, 2, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 5, 0, 0, 0, 0, 9, 7, 3, -5, -7], [-3, 0, 0, -6, 0, -3, 0, -7, -8, -6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 7, 0, 0, 0, -8, -8, 0, -1, 0, 1], [0, 3, 0, 0, 0, 0, 0, 0, 0, 0, -5, 0, 0, 7, -4, 3], [0, -6, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, -4, 6, -6], [0, 0, 3, 0, 10, 22, 0, 10, 3, -1, 0, 0, 0, 0, 0, 0], [0, 10, 0, 0, 0, 0, 10, 0, 0, 0, -10, -2, 10, -2, 2, 2], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -7, 8, 0, -8], [2, 0, 0, -2, 0, 2, 0, -2, 4, -2, 0, 0, 0, 0, 0, 0], [0, -7, 0, 0, 0, 0, -4, 0, 0, 0, 4, -11, -7, -11, 11, -6], [0, -1, 0, 0, 0, 0, -12, 0, 0, 0, 5, 7, -11, 6, 5, 0], [-5, 0, 3, 4, -5, -5, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 5, -4, 2, -7, 5, 0], [-4, 0, -8, -5, 16, 0, 0, -4, -5, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -5, 0, 0, 0, 2, 2, 10, 9, 0, -9], [0, -3, 0, 0, 0, 0, 13, 0, 0, 0, 0, 1, 11, 10, -13, -11], [10, 0, 1, 1, 10, 22, 0, 22, 2, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 11, 0, 0, 0, -1, -1, 11, 2, 0, -2], [0, 0, 1, 0, -7, 1, 0, -7, 1, 1, 0, 0, 0, 0, 0, 0], [0, -3, 0, 0, 0, 0, 4, 0, 0, 0, 1, -5, -5, 4, 1, 0], [-12, 0, 9, 2, -3, -3, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0], [3, 0, -3, 0, 7, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0], [-16, 0, 0, 5, 0, -16, 0, -2, 4, 5, 0, 0, 0, 0, 0, 0], [5, 0, -4, -4, 5, 16, 0, 16, -1, -1, 0, 0, 0, 0, 0, 0], [-8, 0, 0, 1, 0, -8, 0, 8, -2, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -10, 0, 0, 0, 6, 6, -14, -3, 0, 3], [0, 0, 1, 0, 6, 0, 0, 6, 1, 11, 0, 0, 0, 0, 0, 0], [0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 5, -3], [3, 0, -7, -11, 11, 11, 0, 0, 0, -7, 0, 0, 0, 0, 0, 0], [0, -12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, -8, -12], [0, 0, 2, 0, 7, 8, 0, 7, 2, 11, 0, 0, 0, 0, 0, 0], [0, -14, 0, 0, 0, 0, -15, 0, 0, 0, 15, 8, -14, 8, -8, -5], [0, 0, 0, 0, 0, 0, 7, 0, 0, 0, -12, -12, 12, -15, 0, 15], [-16, 0, -3, -3, -16, -22, 0, -22, 1, 1, 0, 0, 0, 0, 0, 0], [0, 8, 0, 0, 0, 0, 1, 0, 0, 0, -1, -6, 8, -6, 6, 9], [2, 0, -7, -9, 8, 0, 0, 2, -9, 0, 0, 0, 0, 0, 0, 0], [-14, 0, 5, 0, 12, 12, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0], [0, 10, 0, 0, 0, 0, 0, 0, 0, 0, -4, 4, -6, 10, -4, 0], [-1, 0, 4, 4, -25, 0, 0, -1, 4, 0, 0, 0, 0, 0, 0, 0], [-4, 0, 0, 0, -4, -15, 0, -15, -5, -5, 0, 0, 0, 0, 0, 0], [0, -7, 0, 0, 0, 0, 1, 0, 0, 0, 0, 5, -11, -6, -1, 11], [1, 0, 2, 9, -15, 0, 0, 1, 9, 0, 0, 0, 0, 0, 0, 0], [0, -8, 0, 0, 0, 0, 5, 0, 0, 0, -8, 3, 15, -7, -8, 0], [0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 15, 0, 0, 1, 2, -3], [0, -7, 0, 0, 0, 0, -2, 0, 0, 0, 2, -3, -7, -3, 3, -14], [-11, 0, 0, -10, 0, -11, 0, -12, -11, -10, 0, 0, 0, 0, 0, 0], [18, 0, 5, 5, 18, 18, 0, 18, 0, 0, 0, 0, 0, 0, 0, 0], [0, 8, 0, 0, 0, 0, 4, 0, 0, 0, 0, 6, 2, 12, -4, -2], [0, -2, 0, 0, 0, 0, -5, 0, 0, 0, 5, 6, -2, 6, -6, 9], [0, -7, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, -4, -8, -7], [6, 0, 11, 3, 12, 12, 0, 0, 0, 11, 0, 0, 0, 0, 0, 0], [-4, 0, -3, -9, 23, 23, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0], [0, 4, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 7, 10, 4], [0, 6, 0, 0, 0, 0, -2, 0, 0, 0, 2, -18, 6, -18, 18, 15], [0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 6, 6, 4, 6, 0, -6], [6, 0, 0, 3, 0, 6, 0, 4, -2, 3, 0, 0, 0, 0, 0, 0], [14, 0, 0, -7, 0, 14, 0, -6, -6, -7, 0, 0, 0, 0, 0, 0], [-12, 0, 4, -7, -14, 0, 0, -12, -7, 0, 0, 0, 0, 0, 0, 0], [0, -10, 0, 0, 0, 0, 0, 0, 0, 0, 16, 0, 0, -6, -2, -10], [0, -2, 0, 0, 0, 0, 8, 0, 0, 0, -10, 2, 2, 8, -10, 0], [0, 16, 0, 0, 0, 0, -6, 0, 0, 0, 0, 6, 1, -1, 0, 0], [14, 0, 6, -2, 20, 0, 0, 14, -2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -12, -14, 0, -12, 0, -5, 0, 0, 0, 0, 0, 0], [0, 21, 0, 0, 0, 0, -2, 0, 0, 0, 0, 11, 3, 19, 2, -3], [0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, -5, 3, 2, -2, -3], [0, 0, 0, 0, 0, 0, -11, 0, 0, 0, 3, 3, -11, 6, 0, -6], [-2, 0, -11, -2, -14, 0, 0, -2, -2, 0, 0, 0, 0, 0, 0, 0], [-8, 0, -2, 9, -15, -15, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0], [-15, 0, 4, -1, -12, 0, 0, -15, -1, 0, 0, 0, 0, 0, 0, 0], [2, 0, 0, -3, 0, 2, 0, 12, 10, -3, 0, 0, 0, 0, 0, 0], [0, 7, 0, 0, 0, 0, -5, 0, 0, 0, 0, 9, 9, 2, 5, -9], [0, 0, 0, 0, 0, 0, -12, 0, 0, 0, 15, 15, -13, 5, 0, -5], [0, 0, 10, 0, 6, 10, 0, 6, 10, 13, 0, 0, 0, 0, 0, 0], [0, 12, 0, 0, 0, 0, 0, 0, 0, 0, -12, 0, 0, 14, -8, 12], [6, 0, -5, 5, -2, -2, 0, 0, 0, -5, 0, 0, 0, 0, 0, 0], [0, -8, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, -8, -14, -8], [0, 0, -4, 0, 3, 11, 0, 3, -4, 8, 0, 0, 0, 0, 0, 0], [20, 0, 0, 6, 0, 20, 0, 6, 2, 6, 0, 0, 0, 0, 0, 0], [0, 12, 0, 0, 0, 0, -8, 0, 0, 0, 0, 8, 11, 4, 8, -11], [0, -22, 0, 0, 0, 0, -6, 0, 0, 0, 6, -2, -22, -2, 2, 4], [0, 6, 0, 0, 0, 0, 9, 0, 0, 0, 1, -10, 8, -9, 1, 0], [1, 0, -10, -14, 1, 0, 0, 1, -14, 0, 0, 0, 0, 0, 0, 0], [0, 0, -3, 0, 30, 12, 0, 30, -3, -4, 0, 0, 0, 0, 0, 0], [2, 0, 12, 12, 2, 14, 0, 14, 13, 13, 0, 0, 0, 0, 0, 0], [0, 4, 0, 0, 0, 0, 4, 0, 0, 0, 0, 4, -8, 8, -4, 8], [0, -13, 0, 0, 0, 0, 9, 0, 0, 0, 0, 1, -1, -4, -9, 1], [-20, 0, 1, 1, -20, -13, 0, -13, 2, 2, 0, 0, 0, 0, 0, 0], [0, 0, 2, 0, -2, 26, 0, -2, 2, 9, 0, 0, 0, 0, 0, 0], [0, -10, 0, 0, 0, 0, -2, 0, 0, 0, -1, 3, -1, 2, -1, 0], [0, -5, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 18, 1, -5], [8, 0, -7, 7, 2, 0, 0, 8, 7, 0, 0, 0, 0, 0, 0, 0], [0, 9, 0, 0, 0, 0, 13, 0, 0, 0, -13, -13, 9, -13, 13, 9], [0, 5, 0, 0, 0, 0, -19, 0, 0, 0, 0, -3, -12, -14, 19, 12], [20, 0, 0, -4, 0, 20, 0, 2, -2, -4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 2, 2, 18, -12, 0, 12], [0, 18, 0, 0, 0, 0, 8, 0, 0, 0, -8, 0, 18, 0, 0, -6], [0, 13, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, -23, 23, 13], [0, 0, -7, 0, 2, -12, 0, 2, -7, -6, 0, 0, 0, 0, 0, 0], [-14, 0, 0, 0, 0, -14, 0, 4, -11, 0, 0, 0, 0, 0, 0, 0], [-15, 0, -6, -6, -15, -8, 0, -8, -6, -6, 0, 0, 0, 0, 0, 0], [0, -6, 0, 0, 0, 0, 9, 0, 0, 0, -9, 8, -6, 8, -8, -8], [-11, 0, 3, 6, -11, 0, 0, -11, 6, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 18, -14, -4, -14, 18, 0], [0, -8, 0, 0, 0, 0, 26, 0, 0, 0, -8, -18, 23, -15, -8, 0], [0, 0, 0, 0, 0, 0, 14, 0, 0, 0, -6, -6, 8, 8, 0, -8], [2, 0, -15, -15, 2, -16, 0, -16, -15, -15, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, -21, 4, 2, 0, -4], [0, 0, 0, 0, 0, 0, 4, 0, 0, 0, -7, -7, -5, 9, 0, -9], [26, 0, -5, 5, 20, 0, 0, 26, 5, 0, 0, 0, 0, 0, 0, 0], [0, -10, 0, 0, 0, 0, -5, 0, 0, 0, -8, 13, 11, -3, -8, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_625_d_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_625_2_d_o();".
// 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_o(: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_o();".
// 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_o( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_625_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 0, -17, 0, 863, 0, 1246, 0, 755, 0, 146, 0, 38, 0, 8, 0, 1]>,<3,R![256, 0, -448, 0, 4448, 0, 4704, 0, 2105, 0, 399, 0, 53, 0, 7, 0, 1]>],Snew);
    return Vf;
end function;