
// Make newform 225.2.h.e in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_225_h();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_225_h_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_225_2_h_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_225_2_h_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 := [625, 0, -1250, 0, 1250, 0, -700, 0, 205, 0, -10, 0, 5, 0, -5, 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, 136237250, 0, -206804175, 0, 105496925, 0, -4566370, 0, -1261485, 0, -1775155, 0, 529550, 0, -9094], [104474000, 0, -87914875, 0, 88524275, 0, -1879805, 0, -2183265, 0, -1233230, 0, 486170, 0, 18559, 0], [0, -113406125, 0, 105654425, 0, -103007550, 0, 21524610, 0, 734880, 0, 729395, 0, -683920, 0, 85587], [0, 103147500, 0, -84275075, 0, -11269000, 0, 36243345, 0, -1814665, 0, -1605420, 0, -327040, 0, 191309], [0, -120294875, 0, 255540700, 0, -186657050, 0, 45929540, 0, -360830, 0, 1298755, 0, -1248275, 0, 216278], [-85476750, 0, 95202125, 0, -87919850, 0, 45336155, 0, -1956335, 0, -301695, 0, -812710, 0, 226756, 0], [-3771750, 0, -94337250, 0, 87194950, 0, -44976455, 0, 1796010, 0, 357520, 0, 787630, 0, -237426, 0], [-22852125, 0, 30950625, 0, -17331500, 0, 5037260, 0, -210920, 0, 108960, 0, -128005, 0, 26057, 0], [0, 54880500, 0, -199991675, 0, 190202500, 0, -66501065, 0, 1061130, 0, -371875, 0, 1471550, 0, -323013], [0, 81913375, 0, -188253675, 0, 115300575, 0, -59745955, 0, 4065110, 0, 321835, 0, 1021010, 0, -330256], [-87855125, 0, 161761375, 0, -149141900, 0, 76510440, 0, -3682905, 0, -665635, 0, -1338325, 0, 389298, 0], [0, 260525500, 0, -350742825, 0, 274576900, 0, -91265695, 0, 2317165, 0, -997060, 0, 2060985, 0, -443034], [-36685125, 0, 49697625, 0, -27808800, 0, 8130605, 0, -383635, 0, 194030, 0, -196825, 0, 40791, 0], [0, -244363125, 0, 406932950, 0, -286289900, 0, 116417835, 0, -5390820, 0, 218325, 0, -2324530, 0, 597957], [-393095875, 0, 538425125, 0, -378889375, 0, 126219210, 0, -3468945, 0, 1509785, 0, -2843530, 0, 607697, 0]];
    Rf_basisdens := [1, 54753875, 54753875, 54753875, 54753875, 54753875, 54753875, 54753875, 4977625, 54753875, 54753875, 54753875, 54753875, 4977625, 54753875, 54753875];
    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_225_h();"
function MakeCharacter_225_h()
    N := 225;
    order := 5;
    char_gens := [101, 127];
    v := [5, 3];
    // 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_225_h_Hecke();"
function MakeCharacter_225_h_Hecke(Kf)
    N := 225;
    order := 5;
    char_gens := [101, 127];
    char_values := [[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, -1, 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, 1, 0, 0, 0, 0, 0, 0, -1, 0, 1, 0, -1, 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, 0, 0, 0, 0, 1, 0, 0, 0], [-1, 0, -1, 0, 0, 0, -1, -2, -1, 0, 0, 0, 0, 1, 0, 0], [0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, -1, 0, 1, 0], [0, 0, -2, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 1, 1, -1, 0, 0, 0, -1, -2, 0, 0, 0, -1, 0], [3, 0, -1, 0, 0, 0, 2, 2, -1, 0, 0, -3, 0, -1, 0, 1], [0, -2, 0, 0, 0, -2, 0, 0, 0, -2, 0, 0, -1, 0, 1, 0], [0, -1, 0, 0, 3, -3, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0], [4, 0, 3, 0, 0, 0, 3, 5, 3, 0, 0, -2, 0, -1, 0, 1], [0, 0, -1, 0, 0, 0, 0, -1, 2, 0, 0, 2, 0, 0, 0, -2], [0, 4, 0, -3, -3, 4, 0, 0, 0, 3, -1, 0, 1, 0, 0, 0], [-5, 0, -2, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, 2], [0, 3, 0, -2, -1, 1, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0], [0, -2, 0, 5, -1, 1, 0, 0, 0, 0, 0, 0, 5, 0, -3, 0], [0, -3, 0, 5, 1, -3, 0, 0, 0, -1, -6, 0, 6, 0, 0, 0], [2, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -2, 0, 0, 0, -2], [3, 0, 2, 0, 0, 0, 1, 10, 2, 0, 0, -8, 0, -2, 0, 2], [0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -7, 0], [9, 0, 2, 0, 0, 0, 10, 2, 0, 0, 0, -9, 0, -2, 0, -2], [0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 7, 0, 3, 0, 0], [0, 0, 0, -1, -3, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0], [0, 1, 0, -6, 0, -2, 0, 0, 0, 1, 6, 0, -4, 0, 4, 0], [4, 0, 0, 0, 0, 0, 0, 4, -2, 0, 0, 1, 0, -4, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 2, 0, -3, 0, 2, 0], [2, 0, 0, 0, 0, 0, 0, 2, -5, 0, 0, -4, 0, 3, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 3, 0, -3, 0, 3, 0], [0, 0, -3, 0, 0, 0, -3, -6, 1, 0, 0, 4, 0, 0, 0, -1], [0, -6, 0, -1, 4, -6, 0, 0, 0, -4, 3, 0, -3, 0, 0, 0], [-4, 0, 2, 0, 0, 0, -1, 2, 0, 0, 0, 4, 0, -2, 0, -3], [0, 0, 0, -2, 5, 1, 0, 0, 0, 1, -1, 0, 0, 0, 2, 0], [0, -1, 0, 1, -1, -1, 0, 0, 0, 1, -2, 0, 2, 0, 0, 0], [-4, 0, 0, 0, 0, 0, -10, 0, 0, 0, 0, 4, 0, 0, 0, 4], [0, 5, 0, 0, -5, 0, 0, 0, 0, 2, -3, 0, -4, 0, -3, 0], [-8, 0, 4, 0, 0, 0, 5, 5, 0, 0, 0, 0, 0, 0, 0, -4], [7, 0, 3, 0, 0, 0, -5, -3, 2, 0, 0, 0, 0, -2, 0, -1], [0, 0, 5, 0, 0, 0, 1, 3, 0, 0, 0, -1, 0, 0, 0, 0], [0, 0, 0, 2, -4, 8, 0, 0, 0, 8, 3, 0, 0, 0, -2, 0], [0, 0, 0, -3, 0, 5, 0, 0, 0, 0, 3, 0, -2, 0, 2, 0], [0, 3, 0, 4, -4, 4, 0, 0, 0, 0, 0, 0, 4, 0, -3, 0], [8, 0, 1, 0, 0, 0, 7, 5, 1, 0, 0, -4, 0, -1, 0, 1], [0, -6, 0, -2, 9, -6, 0, 0, 0, -9, -6, 0, 6, 0, 0, 0], [-12, 0, 2, 0, 0, 0, -5, -2, 3, 0, 0, 0, 0, -3, 0, 1], [0, -5, 0, -7, 4, -4, 0, 0, 0, 0, 0, 0, -7, 0, 7, 0], [-8, 0, 4, 0, 0, 0, 2, 5, 3, 0, 0, 0, 0, -3, 0, -1], [5, 0, 1, 0, 0, 0, -7, 1, 0, 0, 0, -5, 0, -1, 0, 5], [-4, 0, -5, 0, 0, 0, 11, -5, 0, 0, 0, 4, 0, 5, 0, -1], [0, 5, 0, -6, 0, 3, 0, 0, 0, 5, 6, 0, -6, 0, 6, 0], [5, 0, 0, 0, 0, 0, 0, 5, -5, 0, 0, -1, 0, -3, 0, 0], [0, 0, 0, -5, 4, 2, 0, 0, 0, 2, 10, 0, 0, 0, 5, 0], [0, -4, 0, 3, 0, -3, 0, 0, 0, -4, -3, 0, -6, 0, 6, 0], [0, 0, 3, 0, 0, 0, -2, -5, 5, 0, 0, 7, 0, 0, 0, -5], [0, -6, 0, 0, 6, 0, 0, 0, 0, -7, -5, 0, 6, 0, -5, 0], [0, -3, 0, 0, 3, 0, 0, 0, 0, 6, 9, 0, -3, 0, 9, 0], [0, 7, 0, -3, 0, 7, 0, 0, 0, 0, 2, 0, -2, 0, 0, 0], [0, 0, 0, 0, 0, 2, 0, 0, 0, 2, -4, 0, 0, 0, 0, 0], [-2, 0, 0, 0, 0, 0, 0, -2, -2, 0, 0, 13, 0, 5, 0, 0], [0, 0, -1, 0, 0, 0, 16, -1, 0, 0, 0, 0, 0, 1, 0, 1], [0, 0, 0, 0, -3, -4, 0, 0, 0, -4, 5, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, -7, -1, -1, 0, 0, 0, 0, -7, 0, 7], [0, -6, 0, 0, 6, 0, 0, 0, 0, -2, 2, 0, 6, 0, 2, 0], [7, 0, -7, 0, 0, 0, -9, -11, -2, 0, 0, 0, 0, 2, 0, 5], [0, 4, 0, 4, 0, 5, 0, 0, 0, 4, -4, 0, 5, 0, -5, 0], [0, 0, -4, 0, 0, 0, -19, -11, -2, 0, 0, 17, 0, 0, 0, 2], [0, 0, 0, -4, 4, -3, 0, 0, 0, -3, 2, 0, 0, 0, 4, 0], [4, 0, -3, 0, 0, 0, 6, 11, -3, 0, 0, -14, 0, 2, 0, -2], [0, 0, -5, 0, 0, 0, -1, -15, -7, 0, 0, -6, 0, 0, 0, 7], [0, -9, 0, 1, -2, 2, 0, 0, 0, 0, 0, 0, 1, 0, -2, 0], [-8, 0, 1, 0, 0, 0, -4, -7, -3, 0, 0, 0, 0, 3, 0, -4], [0, -5, 0, -1, -3, 3, 0, 0, 0, 0, 0, 0, -1, 0, -4, 0], [0, -4, 0, -5, 2, -4, 0, 0, 0, -2, 1, 0, -1, 0, 0, 0], [-4, 0, 1, 0, 0, 0, 1, 2, 1, 0, 0, -1, 0, 5, 0, -5], [2, 0, -9, 0, 0, 0, 8, -9, 0, 0, 0, -2, 0, 9, 0, -2], [0, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 1, 0, 3, 0, 0], [0, 0, 0, -5, 5, -8, 0, 0, 0, -8, 2, 0, 0, 0, 5, 0], [0, 7, 0, -4, 0, 3, 0, 0, 0, 7, 4, 0, -10, 0, 10, 0], [4, 0, 0, 0, 0, 0, 0, 4, -1, 0, 0, -1, 0, 1, 0, 0], [0, 5, 0, 0, -5, 0, 0, 0, 0, 8, 3, 0, -1, 0, 3, 0], [0, 0, -5, 0, 0, 0, 2, -3, 2, 0, 0, 0, 0, 0, 0, -2], [0, 0, 0, 8, -8, -6, 0, 0, 0, -6, 5, 0, 0, 0, -8, 0], [-3, 0, 0, 0, 0, 0, 0, -3, -2, 0, 0, 0, 0, -5, 0, 0], [0, 0, 0, -10, 5, -1, 0, 0, 0, -1, 12, 0, 0, 0, 10, 0], [2, 0, 7, 0, 0, 0, 4, 2, 7, 0, 0, 5, 0, 2, 0, -2], [-7, 0, -6, 0, 0, 0, -7, -6, 0, 0, 0, 7, 0, 6, 0, -8], [0, 9, 0, 0, -9, 0, 0, 0, 0, 2, 4, 0, -15, 0, 4, 0], [0, -4, 0, 0, 4, 0, 0, 0, 0, 6, 8, 0, -1, 0, 8, 0], [0, 0, -4, 0, 0, 0, 1, 4, 3, 0, 0, 0, 0, -3, 0, 7], [0, 1, 0, 13, 0, -1, 0, 0, 0, 1, -13, 0, 5, 0, -5, 0], [0, 0, 2, 0, 0, 0, -2, -9, 6, 0, 0, 8, 0, 0, 0, -6], [0, 0, 0, 9, -1, -3, 0, 0, 0, -3, 4, 0, 0, 0, -9, 0], [0, -5, 0, 8, 7, -7, 0, 0, 0, 0, 0, 0, 8, 0, -5, 0], [0, 0, 6, 0, 0, 0, 18, 9, 7, 0, 0, -11, 0, 0, 0, -7], [0, 9, 0, -16, -5, 9, 0, 0, 0, 5, 18, 0, -18, 0, 0, 0], [8, 0, 6, 0, 0, 0, -6, 6, 12, 0, 0, 0, 0, -12, 0, 6], [0, 1, 0, -1, -2, 2, 0, 0, 0, 0, 0, 0, -1, 0, 13, 0], [0, 3, 0, -15, -7, 3, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0], [0, 10, 0, -13, -10, 10, 0, 0, 0, 0, 0, 0, -13, 0, 9, 0], [12, 0, 5, 0, 0, 0, -8, 5, 0, 0, 0, -12, 0, -5, 0, -2], [0, 0, -6, 0, 0, 0, 21, 9, -8, 0, 0, -29, 0, 0, 0, 8], [9, 0, 0, 0, 0, 0, 0, 9, -2, 0, 0, 9, 0, -2, 0, 0], [0, 7, 0, 0, -7, 0, 0, 0, 0, -9, 4, 0, -2, 0, 4, 0], [0, -3, 0, 12, -1, -3, 0, 0, 0, 1, -15, 0, 15, 0, 0, 0], [0, 0, 0, -1, 13, 0, 0, 0, 0, 0, -4, 0, 0, 0, 1, 0], [14, 0, 0, 0, 0, 0, 0, 14, 7, 0, 0, -4, 0, -9, 0, 0], [2, 0, 5, 0, 0, 0, 2, 5, 0, 0, 0, -2, 0, -5, 0, 3], [0, 6, 0, -13, 7, 6, 0, 0, 0, -7, 3, 0, -3, 0, 0, 0], [0, -3, 0, 0, 3, 0, 0, 0, 0, -4, 1, 0, -15, 0, 1, 0], [0, 1, 0, 0, -1, 0, 0, 0, 0, 7, 9, 0, 7, 0, 9, 0], [-5, 0, -8, 0, 0, 0, -1, -7, -6, 0, 0, 0, 0, 6, 0, 2], [7, 0, -2, 0, 0, 0, 2, -4, -6, 0, 0, 0, 0, 6, 0, -4], [0, 0, 6, 0, 0, 0, 0, 9, 1, 0, 0, 1, 0, 0, 0, -1], [0, 0, 0, -12, 3, -7, 0, 0, 0, -7, 5, 0, 0, 0, 12, 0], [-4, 0, -8, 0, 0, 0, -4, 3, -8, 0, 0, -11, 0, 0, 0, 0], [9, 0, -7, 0, 0, 0, 10, -5, -7, 0, 0, -2, 0, 1, 0, -1], [0, -7, 0, 4, -4, -7, 0, 0, 0, 4, 10, 0, -10, 0, 0, 0], [5, 0, -15, 0, 0, 0, 6, -3, -9, 0, 0, 0, 0, 9, 0, 6], [0, 10, 0, -10, -5, 5, 0, 0, 0, 0, 0, 0, -10, 0, 10, 0], [0, -3, 0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 1, 0, 10, 0], [0, -4, 0, -3, 10, -4, 0, 0, 0, -10, -5, 0, 5, 0, 0, 0], [-18, 0, -10, 0, 0, 0, -20, -10, 0, 0, 0, 18, 0, 10, 0, -2], [1, 0, 6, 0, 0, 0, 0, 6, 0, 0, 0, -1, 0, -6, 0, 3], [0, -7, 0, 12, 0, -6, 0, 0, 0, -7, -12, 0, 15, 0, -15, 0], [0, 0, 0, 0, 3, -13, 0, 0, 0, -13, -7, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, -13, -15, -7, 0, 0, 6, 0, 0, 0, 7], [0, 12, 0, 0, -12, 0, 0, 0, 0, 2, 4, 0, -6, 0, 4, 0], [0, 0, 10, 0, 0, 0, -7, 15, 2, 0, 0, 9, 0, 0, 0, -2], [0, 0, 0, -1, -1, 5, 0, 0, 0, 5, -3, 0, 0, 0, 1, 0], [10, 0, -8, 0, 0, 0, 12, -8, 0, 0, 0, -10, 0, 8, 0, -5], [9, 0, 5, 0, 0, 0, -1, -4, 5, 0, 0, 9, 0, -10, 0, 10], [15, 0, 2, 0, 0, 0, 1, 2, 0, 0, 0, -15, 0, -2, 0, -5], [0, 0, 0, 0, 0, 0, 0, 0, 0, 5, -11, 0, 3, 0, -11, 0], [-4, 0, 17, 0, 0, 0, -1, 1, 2, 0, 0, 0, 0, -2, 0, -15], [-7, 0, -6, 0, 0, 0, 21, 9, -12, 0, 0, 0, 0, 12, 0, -6], [0, 2, 0, 7, 0, -5, 0, 0, 0, 2, -7, 0, 0, 0, 0, 0], [16, 0, 3, 0, 0, 0, 6, 26, 3, 0, 0, -23, 0, -10, 0, 10], [0, -2, 0, -2, 0, -1, 0, 0, 0, -2, 2, 0, -4, 0, 4, 0], [0, 0, 0, 0, 0, 0, 17, -4, 0, 0, 0, -17, 0, 0, 0, 0], [0, -7, 0, 12, 10, -10, 0, 0, 0, 0, 0, 0, 12, 0, -2, 0], [0, 18, 0, -10, -14, 18, 0, 0, 0, 14, 18, 0, -18, 0, 0, 0], [10, 0, 13, 0, 0, 0, 3, 13, 0, 0, 0, -10, 0, -13, 0, 1], [0, -8, 0, -1, 1, -1, 0, 0, 0, 0, 0, 0, -1, 0, 13, 0], [-25, 0, 6, 0, 0, 0, -25, 6, 0, 0, 0, 25, 0, -6, 0, -8], [0, -3, 0, 9, 0, -4, 0, 0, 0, -3, -9, 0, 13, 0, -13, 0], [-30, 0, 0, 0, 0, 0, 0, -30, 1, 0, 0, 18, 0, 1, 0, 0], [0, -11, 0, -7, 0, -3, 0, 0, 0, -11, 7, 0, 2, 0, -2, 0], [10, 0, 0, 0, 0, 0, 0, 10, 6, 0, 0, -24, 0, -3, 0, 0], [0, -6, 0, 0, 6, 0, 0, 0, 0, -3, -18, 0, 15, 0, -18, 0], [0, 0, -11, 0, 0, 0, -10, -9, -7, 0, 0, 3, 0, 0, 0, 7], [0, 9, 0, -19, 4, 9, 0, 0, 0, -4, 3, 0, -3, 0, 0, 0], [-9, 0, -1, 0, 0, 0, 1, -1, 0, 0, 0, 9, 0, 1, 0, 7], [0, 0, 0, 4, -14, 2, 0, 0, 0, 2, 10, 0, 0, 0, -4, 0], [-6, 0, -4, 0, 0, 0, -13, -19, -4, 0, 0, 15, 0, -7, 0, 7], [0, 14, 0, 10, -16, 14, 0, 0, 0, 16, 7, 0, -7, 0, 0, 0], [-27, 0, 8, 0, 0, 0, -17, -14, 3, 0, 0, 0, 0, -3, 0, -5], [0, -9, 0, 0, 0, -10, 0, 0, 0, -9, 0, 0, 1, 0, -1, 0], [3, 0, 8, 0, 0, 0, 13, -2, 8, 0, 0, 10, 0, 10, 0, -10], [0, 13, 0, -7, -7, 7, 0, 0, 0, 0, 0, 0, -7, 0, -6, 0], [0, 0, -11, 0, 0, 0, 11, -6, -1, 0, 0, -12, 0, 0, 0, 1], [0, -3, 0, 8, -2, -3, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0], [0, -4, 0, 3, -6, 6, 0, 0, 0, 0, 0, 0, 3, 0, -10, 0], [0, -6, 0, 10, 10, -10, 0, 0, 0, 0, 0, 0, 10, 0, -8, 0], [-20, 0, -3, 0, 0, 0, -20, -24, -3, 0, 0, 21, 0, 0, 0, 0], [0, -22, 0, 6, 7, -7, 0, 0, 0, 0, 0, 0, 6, 0, -17, 0], [0, -11, 0, -8, 0, -16, 0, 0, 0, -11, 8, 0, -10, 0, 10, 0], [0, 0, 0, 6, 11, 3, 0, 0, 0, 3, -11, 0, 0, 0, -6, 0], [0, 0, 0, 0, 0, 0, 17, -16, 6, 0, 0, -11, 0, 0, 0, -6], [25, 0, 0, 0, 0, 0, 0, 25, -4, 0, 0, -31, 0, 4, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_225_h_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_225_2_h_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_225_2_h_e(:prec:=16)
    chi := MakeCharacter_225_h();
    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_225_2_h_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_225_2_h_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_225_h();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![25, 0, 125, 0, 250, 0, 25, 0, 385, 0, 75, 0, 20, 0, 5, 0, 1]>],Snew);
    return Vf;
end function;