
// Make newform 952.2.q.c in Magma, downloaded from the LMFDB on 03 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_952_q();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_952_q_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_952_2_q_c();".
// 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_952_2_q_c();".
// 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 := [4, -46, 493, -454, 572, -236, 243, -77, 73, -11, 10, -1, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-19343310930, 206109928375, 91009820464, 88169625807, 84408952449, 47679772520, 34405365162, 6721614649, 5580755644, 953976395, 594547151, -14290269], [1913729518, 704011652, -7499345466, -13215682037, -2935777339, -3401431493, -1434631570, -1371961151, -201644976, -190246065, -28890745, -23459088], [63298769751, -834611470, 8861526665, 18149771418, -202916299, 8472122107, -2299126891, 3240981346, -340552273, 459694944, -44344815, 52349833], [-59327961246, 7388017946, -78685590803, -47871365271, -29054323579, -37809665813, -12697228905, -15166525730, -1840052477, -2399231901, -265752455, -257870635], [-420518644, 219426585385, -50958965191, 38303270703, 14183839014, 1051557875, -5484345390, -12788835221, -545641141, -2260600454, -265470224, -360135604], [-70890182357, 10908076206, -116182318133, -92818906755, -43733210274, -54816823278, -19870386755, -22026331485, -2848277357, -3350462226, -410206180, -375166075], [-3162064816, 486733017119, -407985236512, 553198196826, -219017220138, 235387399577, -70934664141, 70254417359, -10145020219, 9626429080, -909946583, 956864759], [9673867152, -1458529828417, 1248494393608, -1695894133314, 657457493012, -723106442945, 217402246205, -217245214769, 31116165203, -29798677128, 2818529379, -2975293943], [64293926046, -689402844873, 721448199671, -842612670003, 381844881219, -350473904134, 126528440196, -108818857755, 18266973422, -14897292195, 1685045763, -1544721230], [-154455560466, 1656641624443, -1872588932236, 2083271752170, -988476089850, 878017604337, -329840085609, 270708900859, -47534303897, 37047961040, -4441338787, 3795706303]];
    Rf_basisdens := [1, 1, 21130868701, 21130868701, 21130868701, 21130868701, 21130868701, 21130868701, 42261737402, 42261737402, 21130868701, 42261737402];
    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_952_q();"
function MakeCharacter_952_q()
    N := 952;
    order := 3;
    char_gens := [239, 477, 409, 785];
    v := [3, 3, 2, 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_952_q_Hecke();"
function MakeCharacter_952_q_Hecke( : Kf := false)
    N := 952;
    order := 3;
    char_gens := [239, 477, 409, 785];
    char_values := [[1, 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, 1, 0, 0, 0], [1, 0, 0, 0, 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, 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, 0, 0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 1, 0, 0, 0, -1, 0, 0, -1, 0, 0], [0, -1, 0, 0, 0, -1, 0, 1, 1, 0, -1, -1], [0, 0, 0, 0, 1, 1, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, -1, -1, 1, 1, 0, 0, 0, 0, 1, -2, 0], [1, -1, -1, 1, 0, 0, 0, 0, -1, 0, -1, 0], [-3, 0, 2, 0, 1, -1, -2, 0, 0, 0, 0, 0], [0, 3, 0, 0, 0, -1, -3, 0, -3, 1, -1, 0], [2, -3, 1, 3, 0, 0, 0, 0, -2, 0, 0, -1], [0, 0, 2, 0, -1, -2, -2, -1, 0, 0, 0, 0], [4, 0, -1, -1, -1, -2, 1, 2, 0, 0, 0, 0], [1, 2, -2, -2, -1, 0, 0, 0, -1, -1, 3, 0], [0, -2, 0, 0, 0, 1, 2, 0, 6, 2, 1, 0], [0, 2, 0, 0, 0, -1, 0, -1, 1, -1, -1, 1], [-3, -1, 3, 1, 2, 0, 0, 0, 3, 2, -1, 3], [0, 2, 0, 0, 0, 0, -4, 1, -6, -2, 0, -1], [-2, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 0], [0, 3, 0, 0, 0, 2, -3, 1, -4, -1, 2, -1], [3, 0, 1, 0, -1, 0, 0, 0, -3, -1, 3, 0], [-2, 0, -2, -4, -1, -2, 2, -1, 0, 0, 0, 0], [6, 2, -2, -2, -1, 0, 0, 0, -6, -1, 1, 2], [-3, 0, 4, 0, 0, -3, -4, 1, 0, 0, 0, 0], [0, -1, 0, 0, 0, -2, 1, 1, -4, -3, -2, -1], [-7, -1, 3, 1, -3, 0, 0, 0, 7, -3, 3, 2], [5, 2, -5, -2, 1, 0, 0, 0, -5, 1, -1, -2], [0, -6, 0, 0, 0, -4, 0, -2, 2, 1, -4, 2], [1, 0, -3, -1, 0, 5, 3, -3, 0, 0, 0, 0], [-3, 0, 3, 1, -3, -3, -3, 0, 0, 0, 0, 0], [1, 2, -4, -2, -3, 0, 0, 0, -1, -3, -3, 0], [0, -5, 0, 0, 0, -3, -1, 2, 8, -3, -3, -2], [2, 0, -3, -2, -2, 4, 3, -2, 0, 0, 0, 0], [1, -5, -1, 5, -2, 0, 0, 0, -1, -2, -2, -1], [0, 1, 0, 0, 0, -5, 1, -4, 3, 5, -5, 4], [0, 4, 0, 0, 0, -2, 2, 2, 1, 5, -2, -2], [0, -5, 5, 5, -1, 0, 0, 0, 0, -1, -2, -1], [-5, 0, 0, -1, -4, -5, 0, 6, 0, 0, 0, 0], [-4, -1, 3, 1, -4, 0, 0, 0, 4, -4, -2, -1], [0, 4, 0, 0, 0, 0, -4, 3, -2, -4, 0, -3], [0, 0, -5, -1, -3, 2, 5, 2, 0, 0, 0, 0], [8, -2, -2, 2, -2, 0, 0, 0, -8, -2, -4, 0], [0, -7, 0, 0, 0, 4, 3, 1, 8, -5, 4, -1], [-2, 0, -4, 0, -6, 2, 4, -1, 0, 0, 0, 0], [0, -3, 0, 0, 0, -5, 0, 0, -5, 2, -5, 0], [-2, 0, 5, 1, -3, -4, -5, 3, 0, 0, 0, 0], [-13, 0, 0, 2, -5, 3, 0, -2, 0, 0, 0, 0], [0, -2, 0, 0, 0, 2, 1, -1, 4, -5, 2, 1], [-4, -3, 5, 3, -4, 0, 0, 0, 4, -4, 2, -2], [6, 2, 2, -2, 2, 0, 0, 0, -6, 2, 6, 6], [-2, 0, -2, 0, 5, -2, 2, -1, 0, 0, 0, 0], [0, -2, 0, 0, 0, 6, 4, -3, -4, -3, 6, 3], [-9, 0, 0, -2, 2, -1, 0, 3, 0, 0, 0, 0], [-1, -6, 4, 6, 0, 0, 0, 0, 1, 0, -2, -6], [0, -5, 0, 0, 0, 0, 5, -6, 0, 2, 0, 6], [0, -2, 0, 0, 0, -3, 6, 0, -3, -2, -3, 0], [0, 10, -4, -10, 1, 0, 0, 0, 0, 1, 4, 1], [0, -3, 0, 0, 0, 3, 3, -6, 11, -1, 3, 6], [-8, 0, 5, -1, 0, -9, -5, 1, 0, 0, 0, 0], [0, -2, 0, 0, 0, -3, 5, -1, 1, 10, -3, 1], [-6, 0, -1, 9, -6, 1, 1, 0, 0, 0, 0, 0], [-4, 0, 4, 2, 4, -6, -4, 1, 0, 0, 0, 0], [0, -6, 0, 0, 0, -2, -1, -3, 4, 2, -2, 3], [-2, 1, -1, -1, 3, 0, 0, 0, 2, 3, -4, -3], [3, -8, 4, 8, -6, 0, 0, 0, -3, -6, -1, 4], [2, 7, 7, -7, 4, 0, 0, 0, -2, 4, 0, -1], [-1, 0, -2, -6, 2, -1, 2, -5, 0, 0, 0, 0], [0, 8, 0, 0, 0, 0, -4, -2, -10, 0, 0, 2], [-3, 0, 0, 0, 2, -5, 0, 5, 0, 0, 0, 0], [0, -7, 0, 0, 0, 3, 1, -5, 0, 2, 3, 5], [7, 4, 2, -4, -5, 0, 0, 0, -7, -5, 5, -4], [0, -4, 0, 0, 0, 4, 7, 2, 0, 5, 4, -2], [7, 0, -4, 0, -1, 0, 0, 0, -7, -1, 2, -6], [4, 0, 3, 0, 9, 4, -3, 5, 0, 0, 0, 0], [-4, -7, -3, 7, 0, 0, 0, 0, 4, 0, -4, -2], [0, -3, 0, 0, 0, 3, -3, -3, 10, -5, 3, 3], [-6, 3, -5, -3, 4, 0, 0, 0, 6, 4, 0, 3], [5, -1, 5, 1, 1, 0, 0, 0, -5, 1, -1, -4], [0, -2, 0, 0, 0, -2, -14, 1, -3, -6, -2, -1], [-6, 0, 6, 1, 5, -4, -6, 1, 0, 0, 0, 0], [-12, 0, 3, 5, 4, 6, -3, -2, 0, 0, 0, 0], [0, 5, 0, 0, 0, 2, -10, 8, 0, -3, 2, -8], [7, 0, -1, 13, 2, 3, 1, 2, 0, 0, 0, 0], [7, 0, 1, 0, 3, 0, 0, 0, -7, 3, -1, 9], [1, 6, -2, -6, -8, 0, 0, 0, -1, -8, 5, -2], [-4, 0, -6, -2, 6, 6, 6, 0, 0, 0, 0, 0], [9, -1, -7, 1, 1, 0, 0, 0, -9, 1, -5, -5], [-7, 0, 4, 4, -2, -2, -4, -5, 0, 0, 0, 0], [17, 0, -2, -2, -3, 5, 2, -10, 0, 0, 0, 0], [-4, 6, 0, -6, 2, 0, 0, 0, 4, 2, 6, 5], [0, -4, 0, 0, 0, -3, -6, -1, 5, -1, -3, 1], [0, -7, 0, 0, 0, 3, 6, -4, 3, 6, 3, 4], [-2, 0, 1, -1, 1, 0, -1, -4, 0, 0, 0, 0], [-3, 4, -7, -4, 4, 0, 0, 0, 3, 4, -1, -1], [4, 0, -6, -1, -7, 6, 6, 0, 0, 0, 0, 0], [12, 4, -10, -4, -7, 0, 0, 0, -12, -7, 6, -1], [0, -6, 0, 0, 0, 3, 6, 1, -3, 6, 3, -1], [-6, -1, 9, 1, -1, 0, 0, 0, 6, -1, 6, 11], [-4, 4, 8, -4, 2, 0, 0, 0, 4, 2, 0, 4], [10, 0, 3, -4, 0, 0, -3, 0, 0, 0, 0, 0], [0, -8, 0, 0, 0, -5, 2, 4, 16, -6, -5, -4], [0, -7, 0, 0, 0, 0, 9, -5, -2, -2, 0, 5], [9, 20, -8, -20, 4, 0, 0, 0, -9, 4, -6, 0], [0, -1, 0, 0, 0, -3, 7, 7, -17, 6, -3, -7], [0, -8, 0, 0, 0, -2, 2, 1, 7, -4, -2, -1], [2, 0, 3, 11, -2, -4, -3, -5, 0, 0, 0, 0], [5, 3, 1, -3, 3, 0, 0, 0, -5, 3, 0, -8], [0, -2, 0, 0, 0, 6, -2, 0, 6, -2, 6, 0], [5, 0, 3, 9, 2, -5, -3, 1, 0, 0, 0, 0], [-20, -2, 7, 2, 8, 0, 0, 0, 20, 8, -8, 5], [0, 4, 0, 0, 0, 1, -14, 7, -8, -7, 1, -7], [-4, 0, -3, -1, -2, 2, 3, 6, 0, 0, 0, 0], [0, 6, 0, 0, 0, 9, -1, -4, -3, 3, 9, 4], [-3, 0, 7, 7, -9, -7, -7, 4, 0, 0, 0, 0], [0, 18, 0, 0, 0, 3, -4, 8, -3, 3, 3, -8], [-15, 0, 5, 14, 0, 3, -5, -1, 0, 0, 0, 0], [0, 3, 0, 0, 0, 11, 3, -11, 3, -6, 11, 11], [8, 0, 2, 0, 1, 0, 0, 0, -8, 1, 6, 2], [3, 0, -2, -4, -1, -7, 2, -9, 0, 0, 0, 0], [0, -4, 0, 0, 0, -9, 2, 0, 13, 5, -9, 0], [-8, 0, 8, 2, -5, 2, -8, 3, 0, 0, 0, 0], [5, -12, 8, 12, 4, 0, 0, 0, -5, 4, -7, 5], [0, -1, 0, 0, 0, -1, -2, 7, -7, 2, -1, -7], [3, 10, 0, -10, -1, 0, 0, 0, -3, -1, -5, 10], [-13, 0, -2, -2, -9, -8, 2, 8, 0, 0, 0, 0], [2, -14, 0, 14, 3, 0, 0, 0, -2, 3, 6, 6], [4, 1, -12, -1, -1, 0, 0, 0, -4, -1, -2, -5], [-12, 0, 1, 5, -9, -4, -1, 5, 0, 0, 0, 0], [11, 2, 8, -2, 3, 0, 0, 0, -11, 3, 10, 0], [0, -4, 0, 0, 0, 12, 2, 2, -22, -5, 12, -2], [0, 0, -9, 2, -11, 2, 9, 1, 0, 0, 0, 0], [17, -3, -4, 3, 2, 0, 0, 0, -17, 2, 5, -2], [-3, 0, -1, 7, 0, 8, 1, -11, 0, 0, 0, 0], [14, 4, 2, -4, 2, 0, 0, 0, -14, 2, -3, 5], [-8, 0, 0, 0, -1, -5, 0, 0, 0, 0, 0, 0], [0, 6, 0, 0, 0, 1, 10, -5, -4, 11, 1, 5], [0, 2, 0, 0, 0, 1, -12, 2, -5, -3, 1, -2], [12, 0, -6, -10, 2, 5, 6, -6, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 8, -3, 10, 11, 0, 3], [20, 0, -11, -10, -9, 4, 11, -3, 0, 0, 0, 0], [-9, -9, 7, 9, -16, 0, 0, 0, 9, -16, 11, 6], [0, 6, 0, 0, 0, -1, 13, -5, -5, 12, -1, 5], [5, 0, 3, -2, -2, -9, -3, 7, 0, 0, 0, 0], [0, 6, 0, 0, 0, -2, -8, -8, 9, -5, -2, 8], [-3, 0, 4, 2, 9, -11, -4, 7, 0, 0, 0, 0], [-8, 0, 4, -10, -5, -2, -4, 3, 0, 0, 0, 0], [0, 1, 0, 0, 0, -6, 3, 15, -10, -1, -6, -15], [-6, -8, 0, 8, -13, 0, 0, 0, 6, -13, 0, 0], [4, 0, -2, 0, -7, 0, 0, 0, -4, -7, 8, 7], [8, -8, 6, 8, -1, 0, 0, 0, -8, -1, 2, -5], [2, 0, -5, 9, -10, 6, 5, 0, 0, 0, 0, 0], [20, 0, -7, 11, 4, 4, 7, -4, 0, 0, 0, 0], [17, 5, -8, -5, 5, 0, 0, 0, -17, 5, -3, 0], [0, -11, 0, 0, 0, -9, -3, -1, -27, 2, -9, 1], [1, 0, -14, -6, -8, 5, 14, -2, 0, 0, 0, 0], [7, -4, -12, 4, -3, 0, 0, 0, -7, -3, -7, -2], [9, -4, 10, 4, 1, 0, 0, 0, -9, 1, 9, 0], [-9, 0, 4, 6, 13, 7, -4, 7, 0, 0, 0, 0], [0, 18, 0, 0, 0, 4, 0, 2, 0, 1, 4, -2], [-16, -15, -4, 15, -4, 0, 0, 0, 16, -4, -8, -10], [-1, 0, -4, -20, 5, -6, 4, -5, 0, 0, 0, 0], [6, 0, -8, -2, -12, 10, 8, 0, 0, 0, 0, 0], [-5, -4, 18, 4, 0, 0, 0, 0, 5, 0, -5, 0], [0, -7, 0, 0, 0, 15, 9, 0, -1, -4, 15, 0], [0, -1, 0, 0, 0, 1, 10, -3, -5, -7, 1, 3], [0, -4, 0, 0, 0, 0, -15, 0, 0, -13, 0, 0], [0, 2, 0, 0, 0, -2, 6, 2, 4, 4, -2, -2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_952_q_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_952_2_q_c();".
// 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_952_2_q_c(:prec:=12)
    chi := MakeCharacter_952_q();
    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_952_2_q_c();".
// 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_952_2_q_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_952_q();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![4, 46, 493, 454, 572, 236, 243, 77, 73, 11, 10, 1, 1]>],Snew);
    return Vf;
end function;