
// Make newform 7225.2.a.bv in Magma, downloaded from the LMFDB on 16 June 2024.
// 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_bv();".
// 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_bv();".
// 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, -81, 18, 555, -183, -1284, 301, 1350, -153, -678, 30, 171, -2, -21, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -5, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-5844, 186918, 48264, -1309860, -196404, 2399697, 555554, -1553161, -379936, 451319, 104414, -60751, -12608, 3088, 556], [70773, -831477, -658146, 4595418, 1797825, -7079976, -2583595, 4119887, 1561406, -1097170, -437155, 136148, 57055, -6395, -2816], [-29130, -83328, 228720, 552888, 101535, -1141902, -1076639, 677980, 869425, -162854, -272357, 16504, 37820, -556, -1948], [9939, 931332, -740445, -3244554, 3771429, 3619683, -5656775, -2024630, 3369175, 603334, -940355, -88895, 123887, 5024, -6223], [22524, -612804, -407775, 3420513, 1731081, -5553720, -3008854, 3294164, 1870736, -880180, -516409, 108896, 65542, -5090, -3131], [-186330, -200361, 266379, 1547280, 2483580, -3256641, -5447992, 1956437, 3588869, -475642, -1035334, 48917, 137464, -1685, -6881], [-205626, 584133, 1456332, -4165542, -5077590, 7880103, 8399249, -4802128, -5224045, 1254341, 1464941, -147082, -190238, 6349, 9334], [-24474, 459591, 975147, -2552157, -3985017, 4126860, 5764997, -2341480, -3340819, 578795, 902615, -64054, -114758, 2575, 5560], [162312, -878988, -2213877, 4043052, 8673420, -6276564, -12443263, 3438563, 7221212, -805225, -1961851, 81593, 251254, -2816, -12269], [-79194, 1715991, 1452138, -10380078, -7667232, 17501121, 13935983, -10285768, -8821141, 2664863, 2480147, -314854, -321482, 13867, 15718], [-266667, 3010233, 3627786, -16540758, -13263987, 26441136, 20131297, -15391175, -11984516, 4015120, 3275203, -482048, -417655, 21725, 20198]];
    Rf_basisdens := [1, 1, 1, 1, 16983, 33966, 16983, 33966, 16983, 33966, 33966, 16983, 33966, 33966, 33966];
    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)
    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, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 0, 0, 0, -1, -1, 1, 0, -1, 0, 0, 0, -1, 0, -1], [-1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0], [0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 2, 0, -1, 0, 0, 1, -1, 0, -1, 0], [2, 1, 0, 0, 0, 1, 1, 1, 0, -1, 0, 0, -1, 1, -1], [2, -1, 1, -1, 1, 0, -1, 1, 0, 1, 1, 1, 0, -1, 0], [0, -2, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, -1, 1], [0, -1, 0, 0, 0, 2, -2, 0, 0, -1, -1, 2, 0, 0, 2], [2, 1, 1, 0, -1, -1, 0, 2, -2, -2, 1, 0, 0, 1, 0], [0, 1, 3, 0, 0, 0, -2, 1, 2, 1, 0, 3, 1, 0, 0], [-1, 1, 2, 1, 1, 2, 1, 0, 1, -1, 0, -2, 1, -1, 0], [2, -1, -1, -1, -2, 0, -3, -3, -1, 2, -2, 2, -2, -1, 2], [1, -1, 1, 0, -2, 0, 0, 1, -2, -1, -1, 0, -1, -1, 0], [2, -2, 2, -1, -1, 0, -2, -2, 2, 2, -3, 1, -1, 0, 0], [1, 0, 0, 0, 1, 1, 0, 2, 0, 0, 0, -2, -1, 1, -1], [1, -2, 1, 0, -3, -2, 1, 0, -1, 1, -1, -1, 0, 1, 0], [-1, 2, 3, 0, 0, 1, -2, 2, 1, 0, 0, 2, 3, -2, 1], [0, -1, -2, -1, 1, 0, -2, -4, 2, 1, -1, -1, 0, 0, 2], [0, -1, 0, 1, 0, -1, 1, 1, 1, 1, 1, 0, -1, -2, -1], [1, 0, 1, -1, -2, 0, -1, 0, -2, 1, 1, 4, -2, -1, -2], [2, -2, -2, 0, 2, 0, 2, -1, 0, -1, 1, -5, 2, -1, 1], [0, 1, -2, -1, 1, -1, 3, -2, -1, -1, 0, 1, -2, 2, 1], [3, 3, -1, 0, -1, -4, 3, 2, -3, 0, 2, 0, -2, 1, -3], [2, -3, 0, 0, 2, 1, -4, -2, 2, 1, 0, 0, -1, 1, 3], [-2, 2, -2, 2, 0, -1, 5, 2, 0, -1, -1, 0, -1, 0, -2], [4, 0, 1, 0, 1, 1, 2, -1, -1, -1, 2, -1, 0, -2, 0], [2, 2, -1, 0, -3, -3, -4, 1, -2, 0, 2, 2, -2, 2, -1], [-5, 0, -2, -1, 4, 1, -3, -2, 1, 2, -1, 4, 0, 1, 4], [1, 0, -1, 0, 1, 0, 3, -1, 4, 1, -1, 0, 3, -2, 3], [3, -1, -3, 1, -4, -1, 0, 1, -3, -1, 1, 2, -1, 1, 0], [4, 0, 2, 0, -4, 0, 1, -2, 0, 3, 0, 1, -1, -3, -1], [1, 1, -1, -1, 1, -2, 0, 5, 1, -4, 3, 1, -1, 1, -1], [-5, 0, 1, 2, 1, 0, -1, -2, 3, 3, -2, 2, 1, 0, 0], [1, 2, -3, 1, -2, -2, 5, 1, -2, -1, 1, 1, -1, 3, 1], [8, 0, 2, 2, 2, -2, 3, 5, 0, -1, 2, -3, -2, 1, -5], [3, 4, -1, 0, 1, 1, -4, -2, 0, 0, 2, 1, 1, 0, -1], [-1, -1, -1, -2, -2, -2, -5, 1, -2, -1, 1, 7, -2, 2, -1], [-2, -5, 4, -2, 0, 4, 0, -1, 1, 1, 1, -4, 4, -5, 2], [1, 1, 3, -1, 1, 2, -6, 0, 1, 2, 2, -1, 4, -2, 4], [3, 0, 1, -3, 0, 2, -4, -2, -2, 3, -1, 3, 1, -2, 2], [6, -7, 1, -1, 0, 3, 1, -5, 1, 0, 1, -4, 1, -4, 2], [0, 3, -3, 0, 1, 2, -1, -1, 0, 3, -1, 1, 0, 2, 1], [1, -1, 2, -1, -1, 0, -5, 1, -2, -1, -1, -1, -1, 0, 0], [0, 0, 0, 0, -2, -2, -5, -1, -1, 0, 2, 3, 2, 1, -2], [11, -3, -1, -4, -2, 1, -5, -4, -3, 1, -2, -2, 4, -3, 4], [-4, -3, -1, 1, 1, 1, 3, 4, -2, -5, 6, -8, -1, 0, 0], [3, 1, 3, -1, 1, 2, 0, 1, 3, 0, 2, 2, 5, -3, 0], [1, -4, 2, 3, 3, 1, 3, 1, -2, 1, 1, -6, 2, -6, -1], [-4, -3, 3, 2, -2, 3, 4, 0, -4, -1, 2, -6, 5, -4, -2], [-3, 1, -1, -2, 0, 3, -9, -5, 2, 3, -5, 0, 3, -1, 3], [5, 1, -6, -2, 1, -1, -5, -3, 3, -1, -3, 5, -2, 4, 6], [8, -1, 1, -1, -2, -5, -3, 0, -1, 1, 1, 0, 1, -1, -1], [-6, 3, 3, 3, 2, 3, 0, 4, 5, 0, 1, -4, 2, 2, 2], [-5, 1, 1, 1, 5, 1, -2, -2, 2, 0, -1, 3, -1, -1, 0], [-6, -3, -3, -1, 3, 0, -3, 1, 0, -3, 3, 1, 1, 0, -1], [6, 3, -2, 1, 1, -2, 8, 1, 0, -1, -4, -5, -3, 1, 1], [0, -1, -3, 2, 5, 0, 4, -1, 3, 1, -3, -2, -1, 2, 0], [2, 10, 1, 3, 2, -3, 4, 6, 1, -3, 2, 3, 2, 4, -6], [3, 0, 2, 0, 2, 3, 3, -3, 3, 3, 0, -9, 5, -3, 3], [-3, 4, -2, -1, -1, -2, -2, -1, 0, 4, -2, 4, 0, 4, -2], [-4, 3, -5, 2, 5, 2, -3, -4, 4, 1, 0, -3, 3, -1, 6], [-3, -2, 1, 3, 0, 4, -1, -4, 1, 4, 0, 2, 5, -3, 2], [3, 3, -1, 1, 2, 0, -2, -4, 2, 3, -2, 3, 0, 1, 7], [-5, -1, -2, -2, 3, 0, 0, 2, 3, 3, 1, -4, 1, 3, 1], [6, -3, 1, 3, 1, 2, 7, 0, -2, 1, -3, -5, 2, -5, 2], [-3, -4, 4, 2, 1, 4, -6, 0, 6, 1, -3, 6, 2, 1, 1], [0, -4, 3, -4, -4, -2, -5, -1, -3, 2, 2, -2, 3, -7, 3], [5, 0, -4, -3, -3, 0, 2, -4, -1, 1, -4, -2, 4, -4, 5], [3, 7, -3, 2, 2, 1, 5, -1, 2, 0, -3, 1, 1, 0, -1], [-4, 3, -1, 2, 1, -1, -1, 4, 2, 0, 2, 4, -1, 5, 0], [-3, -4, 5, -5, -2, 4, -10, -5, 2, 5, 4, 5, 10, -6, 2], [-7, 1, -7, -3, 0, -1, 2, -3, -2, 2, 0, 1, 3, 2, 5], [0, -4, 1, 1, 4, 1, 6, -4, 2, 2, -3, -6, -3, -1, -1], [-4, -3, 3, -1, 2, 2, -5, 5, 6, 0, 0, 1, 6, -2, -2], [1, 3, -4, -2, -2, -1, 1, 0, -4, -2, 4, 4, 0, 1, -4], [1, 2, -1, -2, 2, 2, 4, -2, -3, -4, 0, 0, 5, 3, 0], [9, -4, 3, -5, -2, -4, -1, 4, 0, 3, 1, 5, 4, -2, 2], [4, -7, 2, -3, -3, -3, 0, -5, -5, 3, 0, -4, 3, -5, 2], [-5, 2, 2, 3, 0, 3, 4, -3, -2, 0, 2, -6, 7, -5, -2], [2, -4, 2, -3, -2, 3, -9, -5, 0, 5, -2, 8, 6, -5, 6], [6, -1, 3, -2, -6, -2, -1, 4, -1, -1, 1, 2, -1, -5, -4], [3, -4, -1, 2, 1, 2, 1, -6, 0, 0, -3, -6, 2, -2, 0], [-7, 5, 1, -2, 0, 3, 0, -3, 2, 3, -2, 6, 2, -3, 2], [-8, -2, -2, 3, -4, -3, 9, 5, -3, -3, 0, -3, -2, 2, -3], [-4, 0, 3, 2, -3, -1, 8, 11, -3, -6, 6, 4, 1, -1, -5], [5, 1, 2, 3, -4, 1, 16, 6, -7, -7, 3, -9, 1, -4, -2], [1, -1, 2, -6, 3, 1, -11, 2, 1, 2, -1, 9, 0, -1, 0], [0, 5, 5, 0, 1, 2, 7, 5, 2, -6, 0, -2, -3, 5, -5], [3, 5, -3, -2, -4, 3, -1, -6, -4, 2, -4, 0, 3, -1, 3], [0, 5, -3, 0, -3, -5, 2, 1, 1, -1, -7, 4, -4, 3, -1], [-2, 4, -2, 1, 6, 5, 2, 4, -3, -2, 2, -3, 5, 0, 2], [14, 5, -4, 2, 5, -1, -2, 2, 4, -2, 0, 7, -4, 6, 0], [3, 0, -1, -1, 0, -4, -7, 6, 3, 4, 4, 7, 0, 3, -2], [-6, 6, -7, 6, 3, -3, 7, 6, 4, -6, 3, -1, -6, 8, -4], [2, 1, -1, -2, 1, -2, -1, 6, 1, -1, -3, 3, 2, -2, 0], [11, 10, 1, 1, -2, -5, -2, 10, 1, -3, 2, 8, -2, 7, -6], [-6, -5, -1, -6, 2, 4, -8, -10, -1, 5, 2, -4, 7, -9, 6], [4, -6, 7, -1, -2, -1, -4, -1, -7, 5, 3, 0, 3, -1, 0], [-9, 9, -3, 4, -1, 0, -1, -5, 2, 1, -4, 6, -1, 4, -1], [3, -5, -3, -4, -8, -1, 3, -5, -2, 6, -6, 4, -6, 5, 1], [8, 3, 0, 0, -5, -4, 0, -1, -8, 0, -2, 5, -10, 2, -1], [3, -8, 7, -2, -6, 3, 0, -4, -3, -3, 4, -7, 3, -6, -1], [0, 1, 1, 2, 1, 2, 5, 10, 4, -3, 1, 3, -3, -1, -7], [1, 0, 2, -1, 4, -3, -1, -3, -1, 4, -1, -6, 1, 0, -3], [-14, 8, -7, 4, 6, 4, 7, 3, 3, -6, 2, -1, 1, 4, 2], [-4, 9, -3, -2, 8, 1, 0, -6, 3, -1, -5, -3, 2, 3, 2], [-1, -5, -1, -2, 1, 3, -8, 0, 9, 1, 2, 2, 1, -3, 4], [-10, 0, -3, 3, 6, 2, 8, 0, 7, -3, 0, -6, -2, 3, 4], [-2, -4, 6, 0, 3, 5, -4, -8, 2, 1, -2, -8, 7, -10, 6], [-1, -1, 3, -2, 6, 3, -4, -4, 6, 3, -2, 6, 2, -1, 3], [-3, 7, -2, 7, -3, -6, -3, 4, -2, -4, 2, -4, 1, 2, -5], [7, -5, 3, -3, -4, 5, -6, -11, -2, 6, -7, 0, 1, -9, 5], [3, -1, 1, -1, -4, 2, -2, -4, 4, 3, 0, -2, 5, -4, -1], [3, 3, -6, 1, -1, 0, 0, 0, -4, 0, 0, -4, -7, 0, -3], [3, -4, 5, -4, -6, 2, -1, -1, -2, 2, 6, -14, 8, -5, 4], [-7, -2, 5, 3, -1, 3, 9, 0, 4, 0, 3, -4, 10, -6, 6], [2, -5, 4, -4, 4, 8, -10, -4, 5, 0, -6, -4, 5, -5, 11], [8, -3, 4, 1, -5, -1, 0, -2, -1, 6, -1, 1, 4, -7, 2], [11, 0, 6, -4, -1, 5, 9, -1, -2, -1, -1, -7, 1, -5, 1], [2, -1, 1, -1, -1, 5, -8, 3, 2, -5, 3, -3, 3, 1, 7], [7, 2, -4, 0, -1, -5, 6, 1, -2, 4, -4, 5, -8, 8, -7], [-7, -2, -7, 4, 6, 1, 6, -7, 2, 2, 2, -17, -3, -1, -2], [-5, 7, -1, 3, -4, 0, 12, 4, -8, -3, 3, -5, 2, -2, -8], [1, 12, 0, 0, -6, -3, 5, -2, -3, 0, -4, 2, 0, 4, 2], [0, 0, -4, 1, 4, 2, 6, 2, 2, -5, 1, -5, -6, 0, 0], [0, -3, -1, -4, -4, -4, -7, 1, 5, 1, 2, -1, -4, -2, -5], [-12, 1, 2, -3, 6, 4, -5, -2, 2, 6, 0, 0, 10, -4, 5], [11, -4, -1, -4, 7, 6, 1, -7, -3, 1, -1, -3, -1, -6, 11], [-6, 6, -1, -3, 6, 6, -9, -4, 0, 2, -6, 8, 5, -1, 10], [-5, 9, 3, 1, 1, -2, 3, 2, -1, 3, 2, 3, 3, 0, -2], [-2, 3, -6, -1, -3, 2, -1, -6, -8, -4, -2, -1, -2, 6, 1], [-6, 2, 0, 3, -6, -3, -8, 5, -8, 0, 5, 3, -2, 5, -5], [3, 3, -2, 0, -4, 0, -6, 5, -1, -1, 1, 4, 0, 11, -3], [8, 2, 4, -4, -1, 1, -4, -1, -4, 6, 1, 1, 0, 3, -6], [-3, -3, 4, 0, -3, 6, 6, 1, -2, -7, 2, -19, 0, -4, -2], [-5, 10, -3, 6, 7, -5, 10, 12, -3, -11, 7, -9, -4, 5, -5], [9, 7, 3, -4, -5, -3, -14, -2, 2, 6, -6, 9, -3, 2, 7], [-3, 4, -1, 3, -2, 2, -7, 3, 0, 5, -1, 2, -2, 1, -2], [8, -1, 2, 2, 4, 3, -4, 0, 2, 3, 4, 3, 2, 3, -3], [9, -15, 5, -6, -7, -2, -2, 2, -5, 3, 7, -4, -1, -5, -4], [-5, 2, -9, 8, 5, 2, 0, -1, 2, -4, 0, 2, -6, 6, 2], [-5, 0, 2, -1, 0, -1, -7, 4, -2, -2, -5, 9, -3, -4, 6], [3, -2, 5, 0, -10, -4, -6, -4, -3, 5, -5, 13, 1, 2, 0], [12, -9, 10, 4, -1, -3, -2, 5, 0, -1, 6, -8, -3, -9, -6], [7, -7, 1, -2, 5, 4, -12, -9, 7, 5, -5, -3, 5, -5, 5], [1, -2, 5, -1, -5, 7, -3, 4, 1, -2, 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-6, 6, -6, -5], [3, 0, 4, -4, 12, 21, 7, -16, 2, 5, -5, -17, 6, -15, 13], [1, -1, 4, -2, 15, 11, -10, -1, 12, -3, 1, -6, 11, -7, 7], [-4, -1, -3, 3, -6, 1, -5, 1, 7, -4, -1, 4, 2, 4, 1], [-16, -7, 14, 3, 4, 16, 8, 7, -2, -12, 5, -17, 4, -5, 4], [23, 1, 1, -7, -7, 4, -2, -9, -4, 6, -3, -2, 9, -16, 16], [9, -2, -6, -5, -13, -5, -16, -1, -7, 0, -5, 24, -5, 9, -2], [8, 0, 10, 10, -2, 13, 18, -7, 2, 2, -1, -18, -4, -7, 0], [7, 2, -3, 2, 5, 10, -10, -6, -3, -1, -2, 0, 5, -1, -2], [4, 2, 2, -6, -9, -4, 3, -10, -8, 11, -4, 2, 13, -9, 7], [14, -6, 9, 1, -7, 1, 8, 5, 4, 7, -2, 5, 0, 2, -4], [6, 5, -2, 8, -4, -5, -4, 2, 3, 5, -2, 20, -11, 2, -3], [21, -8, -3, -2, 6, 6, 5, -5, -9, 0, 2, -1, 2, -5, 3], [-23, 1, -2, 13, 8, 1, 6, 15, 11, -10, 7, -5, -9, 7, -10], [-21, -8, 6, 2, -7, -4, -6, -9, -1, 10, 2, 0, -1, -7, -12], [17, 12, 6, 2, -5, 4, -6, 4, 2, 0, -6, 6, -19, 7, -2], [-6, -3, 4, 9, 15, -2, 11, -2, 12, 6, -5, -12, -2, 3, 1], [15, -16, 1, -17, -9, 0, -23, -4, -6, 4, -1, 16, 7, -6, 8], [-2, -6, 2, 6, -9, -4, 14, 2, -17, -3, -1, -15, 1, -1, -15], [27, 13, 3, 4, 3, -3, 14, 11, 5, -2, 3, -4, 9, -1, -8], [-17, -1, -8, 7, -1, 13, 3, -12, 8, -1, -9, -14, 10, 6, 9], [1, 13, -7, 8, 0, -11, 0, 8, -2, 0, -4, 6, -3, 8, -5], [0, 7, 0, 8, -4, 1, 6, 3, -8, -2, 4, -8, 0, 0, -11], [-18, 2, -2, -2, 5, 8, 2, -3, 9, 3, 5, 6, 7, 7, -1], [-10, -6, -5, -1, -12, 6, 13, -19, -3, 9, -12, -16, -6, 0, -3], [-34, -3, -4, -4, -1, 9, -16, -19, 10, 12, -15, 4, 4, 1, 18], [25, -3, 17, -1, -14, -4, -12, 10, -6, 1, -2, 15, 0, 3, -13], [5, -4, -1, 7, -5, -10, 3, 20, -2, -1, 10, 10, -9, -4, -19], [20, -17, 8, -5, -4, 3, 9, -1, 0, 2, 10, -2, 16, -14, 7], [-10, 15, -16, 9, 8, -3, 0, 8, 3, -13, 3, -1, -8, 8, -2], [12, 12, 2, 2, 2, -5, 11, 9, 5, 1, 0, 5, -24, 7, -11], [-5, 6, -3, 8, 10, 8, -3, -6, 9, 15, -4, 6, 11, -1, 4], [-17, -2, -4, -4, 2, 3, 3, -11, -1, 2, -12, -21, 5, -3, 6], [-6, 13, -2, 3, 2, -2, 14, 15, -7, -15, 1, 3, -6, 10, -9], [-6, 16, -3, -1, 3, 3, -5, 0, -6, -5, 5, -5, 9, 12, 5], [-1, -12, 18, -4, 6, 6, 5, 12, 2, -4, 11, -11, 10, -8, -4], [11, -13, 3, -7, -14, 4, 8, -20, -4, 19, -14, -1, 2, -12, 7], [17, 4, -7, 4, -3, -11, -7, -2, 3, 4, -8, 23, 0, 6, 9], [15, 13, -3, -1, 6, -2, -4, -7, 10, -1, -10, 1, 2, 5, 5], [10, 17, 0, 4, -5, -2, 14, 10, -9, -9, 5, -6, -2, -4, -4], [0, 4, 3, 1, -4, 2, -28, 12, -2, -4, 3, 27, -4, -5, -7], [10, -3, 10, -2, -9, -3, -7, -7, -4, 8, -8, 20, 12, -14, 4], [23, 7, 4, 4, -12, -20, 30, 14, -15, -11, 1, 3, 0, 5, -11], [-17, 8, 5, 8, -11, -8, -6, 4, 1, 2, -4, 17, 10, 6, -10], [25, -7, 6, -11, -8, 0, -10, 2, -16, 2, 3, 16, 9, -3, 4], [7, -2, -5, 7, 13, 2, 9, 1, -8, -3, 5, -25, 3, -4, -8], [28, 3, -13, 10, -7, -19, 30, 1, -12, -4, -6, -8, -14, 4, -16], [2, -10, 1, -7, -8, -5, 7, -11, -8, 5, -8, -5, -5, 1, 9], [-1, -15, 6, -6, -5, 1, -6, 7, -7, -3, -3, -2, -8, 3, 7], [-35, 13, -13, 6, 23, 3, -2, -2, 14, -11, 0, -13, 10, 6, 11], [26, 3, 7, -10, -11, -9, -24, -5, -2, 18, 1, 21, 4, -5, 4], [-27, -7, 4, 1, 11, 12, -1, -5, 7, 4, -3, -18, 11, -14, 1], [-25, 6, -3, 11, 1, -12, 7, 8, 7, -3, 9, -9, -6, 5, -17], [12, -4, -7, 3, -4, 0, -12, 4, 3, 7, -13, 11, -5, 5, 4], [11, 1, 9, -5, -6, -16, 1, 5, 1, 11, 9, -2, 2, -5, -13], [8, 5, -5, 2, -3, -5, -18, -8, 4, 9, -15, 4, -6, 0, 7], [14, -2, 0, -5, -13, -11, 4, 9, -7, 3, 1, -4, -11, 10, -4], [-17, -21, -4, 1, -2, -3, 0, -7, 3, 3, 1, -13, 6, -5, 8], [-5, 0, -6, -2, 4, -1, -2, -16, -3, 5, -18, 11, -2, 9, -6], [-6, -16, -2, -9, 8, -7, -3, -5, -5, -1, 8, -8, 6, -3, 5], [-13, 15, 10, 1, 4, 2, 11, 4, 3, 2, 2, -16, 9, -5, -10], [2, -7, -1, -5, 8, 12, -11, -18, 3, 6, -8, -11, -1, -3, -1], [16, 3, 12, 8, -3, -1, 6, 22, 8, -8, -2, 13, -3, 9, -10], [0, 9, -11, -5, 4, -3, -6, -2, 6, 5, -9, 16, -1, 10, 3], [21, -1, -3, -13, 11, 1, -22, -9, 3, 9, -9, 15, -20, 5, 2], [-16, -2, 0, 3, 10, 15, 16, 2, 11, -7, 0, -5, 24, 1, 5], [1, 3, 0, 6, -1, -6, 19, 9, -10, -12, -1, -1, -12, 13, -11], [12, 4, -5, 3, -19, -12, 8, 2, -3, 0, 7, 12, 0, 3, -8], [7, -9, 13, 3, -2, 7, -1, -4, -8, 3, 3, -3, 13, -16, 4], [3, 5, -7, -11, -5, -4, -4, -3, -12, 3, -8, -6, -8, 5, 5], [24, -16, 3, 1, -7, -10, 5, 5, -4, 8, 6, -2, -9, -7, -11], [15, -3, -6, -16, 0, -8, 7, -12, -4, 7, 3, -10, -10, -2, -1], [11, 3, -10, -6, -6, -11, 10, 6, -1, 1, 1, -3, -5, 3, -1], [-21, 15, -16, 5, 4, 1, 1, -2, 6, 2, -5, 12, -11, 15, 2], [16, 10, -1, 4, 9, -1, 18, 16, -5, -14, 14, -3, -12, 13, -10], [-5, 7, -19, 3, 4, 1, 13, -7, -4, 1, -5, -19, -8, 15, 6], [10, -1, -7, 3, 3, -11, 1, -12, -6, 2, -7, -18, -4, 1, -9], [32, 6, 4, 1, -1, -3, 0, 10, -3, -9, 9, -1, -7, 2, -10], [1, 1, -1, 3, -1, -6, 7, 9, -4, 8, 2, 12, -10, 10, -5], [11, -10, -5, -10, -6, -5, -9, -14, -3, 8, -2, 2, 6, -8, 10], [24, 3, 5, -1, -8, 8, -18, -3, -1, 2, -12, 9, -6, -6, 5], [-4, 12, -3, -2, -3, 0, -20, 5, 1, -4, 0, -4, -1, 11, 4], [-26, 1, 6, 4, -3, 8, 0, -2, 5, -2, 11, -1, 5, -4, 2], [-17, 6, 22, 5, -4, 2, -14, -4, 5, 4, 2, 0, 14, -15, 5], [-1, -20, 9, -4, 0, 14, -27, 0, -1, 1, 5, -8, 3, -7, -1], [12, -15, 12, -14, -4, 6, -33, -15, -3, 16, -5, 14, 5, -8, 10], [11, -17, 12, -8, -14, 3, -18, -11, 1, 16, -13, 3, -8, -9, 4], [-2, 8, 3, 5, 6, -2, 6, 3, 14, 0, 2, 10, 2, 8, 7], [-26, -9, -6, -8, 3, 8, -12, -11, 9, 4, -3, -9, 13, -7, 11], [7, 0, -2, -4, 12, 8, -17, 6, 3, 4, -3, -1, -18, 0, 1], [18, -11, 3, 2, -14, -7, 17, 5, -16, 3, 1, -20, -14, -11, -1], [13, 1, 8, -3, -2, 3, -14, 4, 10, 2, -5, 17, 8, -16, 1], [4, 19, -6, 9, -20, -14, 7, 7, -7, -5, -6, 21, -18, 8, -21], [12, -6, -18, 5, 4, -3, 22, 1, 5, -1, 1, -6, -7, 3, 4], [31, 9, -2, 1, -6, -14, -3, 7, 0, 4, -4, 9, -14, 20, -4], [-1, -7, 6, 1, -7, 4, -17, -21, -8, 13, -9, -5, 6, -12, -2], [7, 8, 8, 2, 9, 7, 19, 5, -1, -9, 9, -24, 2, 3, 2], [-1, -16, 21, 4, -9, -8, -3, 6, -6, 9, 5, 2, 7, 0, -9], [13, -2, -14, -3, 8, -4, 5, -1, 5, 1, 7, -9, -7, 0, -7], [21, -3, -7, 7, 13, 0, 4, -8, -3, -7, -3, -15, -6, 1, -7], [8, 3, 9, -6, -12, -10, 7, 17, -9, -4, 15, 4, 1, 3, -20], [-6, 2, -15, -2, 5, -12, 15, -5, 0, 6, -8, 11, -6, 9, 0], [1, -9, -4, -4, -5, -12, 22, -8, -16, 3, -2, -7, -13, -3, -5], [3, -4, 6, -1, 4, 5, 5, 0, -4, 3, 5, -14, 6, -7, -9], [-10, -12, 6, 0, 7, 7, -5, -7, 15, 2, -4, -7, -5, 1, 5], [22, -10, 16, -6, -7, 2, -12, -6, -8, 13, 2, 11, 23, -15, 7], [-16, 2, -10, 4, -7, -7, 15, 7, -2, -14, -3, 7, -11, 8, -8], [-15, 8, 8, 10, 12, 6, 5, 14, 0, -15, 12, -17, -1, 3, -4], [1, -10, 4, -15, -2, -3, -16, 2, 8, -3, 1, 4, -4, 3, 12], [-26, -15, 7, -6, -1, -5, 5, 5, -1, 3, 11, -9, 14, -4, -7], [-6, 14, -3, 11, 1, 5, 20, -5, 9, 7, 3, -21, -2, 1, -5], [4, -5, 17, -2, 8, 11, 1, -11, 6, 14, -8, -3, 8, -10, 4], [-3, 10, 12, -5, 10, 7, -27, 7, 8, 4, 11, 3, 16, -3, 0], [-22, 8, 5, -5, -3, 10, 3, -4, -4, 6, -1, 5, 13, -6, 7], [9, 1, -2, -2, 15, 12, 10, -6, 0, 1, -5, -5, 7, -8, 13], [11, -13, 6, -4, 4, 12, -2, -5, -8, -3, 15, -3, -1, -11, -2], [-3, -3, 4, 15, 6, 0, -7, 4, 3, -10, 0, 10, 2, -2, -7], [16, 5, 11, 1, -6, 0, -2, 5, -8, 3, 3, 5, 9, -12, -3], [10, -9, -1, -11, 1, -12, -17, -2, -8, -1, 5, -6, -4, 3, -4], [0, 0, -4, 9, 12, 2, 6, 9, -1, -12, 17, 7, 8, 0, -15], [-36, 2, -10, 10, 7, -2, 18, 8, 5, 2, -2, -8, 1, -4, -1], [-12, 5, 2, -5, -4, 2, -17, -12, 15, 8, -9, 16, 4, 7, 4], [-5, 0, -4, -3, -9, 0, 11, 3, -9, -9, 7, -7, -2, -7, -5], [-19, 19, -5, 15, 8, -5, 12, -5, 0, 1, 0, -6, 8, 3, -4], [21, 14, 7, 2, -9, 1, -15, 9, 4, 3, -1, 13, 4, -1, 0], [2, 5, 13, 5, -1, 11, -14, 7, 13, 10, 0, 13, 5, -8, 1], [11, 1, 3, -9, -7, 3, -23, -10, 1, 6, -2, 0, -1, -3, 11], [15, -15, -9, -6, -1, -7, 8, -1, -4, 3, 8, -27, -11, 2, 2], [-12, 0, -11, 8, -2, -3, -5, 3, 5, -7, 9, 11, 11, 9, 0], [9, -11, -9, 2, 1, 6, -2, 3, -10, -9, 14, -11, 0, -3, -7], [5, 8, -4, 1, 8, 2, -1, 2, 4, 0, -9, 3, -23, 10, 7], [2, 14, -14, -2, 5, 3, -15, -19, 0, 8, -15, -3, 7, 8, 14], [-2, -8, 2, 12, 0, 0, -1, 3, 11, -4, -2, 2, -1, 0, 3], [10, -30, -1, -8, -8, 12, -1, -17, -6, 10, -2, -10, 4, -17, 15], [0, 2, 22, 1, -5, 7, -5, 5, -4, 4, -1, -4, 14, -4, 6], [-8, 9, 3, 3, 15, 12, 25, 6, 13, -3, 1, -5, 20, -8, 2], [4, 19, -13, 13, -1, -5, 38, -2, -6, -6, 0, -23, -5, 11, -9], [13, 17, -8, 8, -4, -23, 24, 15, -7, -5, 0, 8, -2, 9, -18], [2, -3, 9, 6, 3, -2, 16, -3, -2, -1, 0, -14, 19, -3, -2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7225_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7225_2_a_bv();".
// 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_bv(:prec:=15)
    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_bv();".
// 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_bv( : 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, -81, -18, 555, 183, -1284, -301, 1350, 153, -678, -30, 171, 2, -21, 0, 1]>,<3,R![1, 0, -171, -883, -501, 2463, 1340, -3093, -369, 1623, -258, -312, 105, 12, -9, 1]>,<7,R![-3043, -8847, 39129, 59781, -167592, -11424, 158103, -65919, -25560, 21566, -2058, -1539, 385, 12, -12, 1]>,<11,R![2049, 15345, 1170, -64425, -44547, 72996, 78989, -10377, -30498, -2597, 4698, 666, -297, -48, 6, 1]>],Snew);
    return Vf;
end function;