
// Make newform 930.2.n.d in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_930_n();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_930_n_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_930_2_n_d();".
// 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_930_2_n_d();".
// 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 := [400, 1000, 1340, 1170, 1931, 823, 263, 1, 60, -9, 13, -2, 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], [60463522673300, -14600309174840, -33360822706280, -38781999340004, -22850390835465, 216123116006, -1902177059364, -495858189935, -111496966669, -178529954095, 17625053431, -17849017864], [-15010929677480, 148807987112405, -23111477672266, 20833220372194, 3863106684695, 10498141078366, -3751671881315, 2950126373569, -1139133730021, 512829808605, -106837550849, 23361760035], [-205909846055800, -315224424729940, -302595918088550, -382232489020961, -127545198552457, -48748704399099, -637596034947, -9624980905700, 626624859189, -1782233600297, 214983815436, -146654331477], [53049709415200, 212990793099520, -257444750971091, 278674930944664, 134261867661482, 72430523469356, -17907584046427, 19051303779369, -6329673984486, 4037834562557, -820941472449, 259180521843], [-663036463432360, -943654494647340, -862551212637626, -1258952186163283, -496750299137235, -166627772868235, -2588067986473, -36724101470656, 4727597224217, -7789757733805, 1137874382010, -604635226733], [103672208737200, 206130812427800, 134311106170100, 560685961527401, 221802656734169, 79043701815307, -4266046224647, 18166764568270, -3500472468789, 3997049287899, -668487778598, 302580428763], [-228519186690980, 87053037871525, -359469623547681, 381784730133684, 168010412077976, 115791065278799, -34700982692066, 30636118502149, -10496150942691, 6108799410656, -1246157058016, 377873811249], [-797530905591380, -583577757140760, -1493427761388610, -1801907179914651, -848523389812191, -156547132073865, -47642147786361, -32876644690520, -1669598790561, -8730178441851, 1070796378400, -811947191771], [1778835879579520, 3110122517771760, 578163383466308, 3005207185931801, 1468271773696657, 604977651733493, -91084996607083, 156459753762298, -42245615072839, 32819025256117, -6036163855492, 2196856365657], [-1579157825751160, -1085086519421180, -226203310184226, -3452889698168529, -1830786108310665, -567823275403729, 42032352335873, -145106089008796, 35569652132971, -33979177074555, 6234274597186, -2487542474027]];
    Rf_basisdens := [1, 1, 107519805038390, 107519805038390, 215039610076780, 215039610076780, 430079220153560, 215039610076780, 107519805038390, 215039610076780, 430079220153560, 430079220153560];
    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_930_n();"
function MakeCharacter_930_n()
    N := 930;
    order := 5;
    char_gens := [311, 187, 871];
    v := [5, 5, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_930_n_Hecke();"
function MakeCharacter_930_n_Hecke(Kf)
    N := 930;
    order := 5;
    char_gens := [311, 187, 871];
    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], [0, 0, 0, 0, 0, 0, 1, 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 := [[-1, 0, 0, -1, 0, 1, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0], [1, 0, 0, 1, 0, -1, 0, 0, 1, 0, 0, 0], [-1, 2, 2, 0, 2, -1, -1, 2, 1, -1, 1, 1], [0, 0, 1, 2, 0, -2, 0, 1, 0, 1, -1, -1], [-1, -1, -1, -1, 0, 1, -1, 1, 0, 0, -1, 0], [0, 0, -1, -1, -1, 1, -1, -1, 0, 2, -2, -1], [0, 1, 1, 0, 0, 0, 0, -1, -1, 1, -1, 0], [1, 0, -1, 0, 0, 1, 0, -1, 2, -2, 1, 1], [-2, 0, 0, 3, 0, 0, 3, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 1, -1, 1, -1, -1, 0], [4, -1, -1, 4, 0, -5, 5, -1, 1, -1, -1, 0], [-2, 3, 2, 0, 2, -1, -2, 3, 1, 0, 1, 0], [0, 0, 3, 0, 1, 0, 0, 3, 0, 2, -2, -2], [0, 1, 1, 0, 1, -2, 0, 1, -1, 0, -1, 0], [1, -2, -1, -1, -2, 0, -1, 0, 1, -3, 0, 1], [2, -1, 2, 0, -1, 0, 0, 0, -2, 2, 0, -2], [0, 0, 3, 2, 1, -2, 2, 3, 0, -2, 2, 2], [4, -3, 0, 7, -1, -4, 0, -1, 1, 0, 0, 0], [0, 0, -2, 1, -1, -1, -3, -2, 0, 0, 0, 2], [-3, 0, 0, -3, 0, 0, 0, 3, -1, 1, -2, 0], [-2, 3, 0, 1, 0, 2, 0, 0, 1, 0, 0, 0], [-3, 2, 0, -3, 0, 3, 0, 0, 2, 0, 3, 3], [0, 0, 2, 2, 2, -2, 5, 2, 0, -1, 1, 1], [-7, 2, 2, -7, 0, 4, -4, 1, 3, -3, 2, 0], [0, 0, 2, -5, -3, 5, 0, 2, 0, 1, -1, -1], [-3, 0, 2, 0, 2, -4, -3, 0, 0, -1, 0, 1], [5, 2, 0, 7, 2, -5, 0, 2, 2, 0, 1, 1], [0, -2, -3, 0, -3, 7, 0, -2, 1, 3, 1, -3], [-4, 2, 0, 3, 3, 4, 0, 3, 2, 0, -1, -1], [-2, -1, -1, 0, -1, 0, -2, -1, 4, -2, 4, 2], [2, -1, 3, 0, 3, -9, 2, -1, -1, -1, -1, 1], [2, -2, -3, 2, -2, 0, 2, 0, -1, 3, 0, -1], [3, -3, 0, 11, -1, -3, 0, -1, -4, 0, -3, -3], [-6, -3, -3, -6, 0, 5, -5, -3, 0, 0, 5, 0], [0, 0, 1, 6, -2, -6, 4, 1, 0, -1, 1, -2], [4, -3, -3, 4, 0, -7, 7, -3, -2, 2, 2, 0], [-3, -1, -1, 0, -1, 2, -3, -1, 0, -1, 0, 1], [-6, 3, 0, -5, 0, 6, 0, 0, 1, 0, -3, -3], [-6, 3, 3, -3, 3, 0, -3, 0, -1, -2, 0, -1], [-1, 2, 4, -2, 2, 0, -2, 0, 0, 3, 0, 0], [-4, 5, 0, -1, 0, 4, 0, 0, 3, 0, 4, 4], [0, -3, 0, -4, -6, 0, 0, -6, 2, 0, 2, 2], [-7, -4, -3, 0, -3, 8, -7, -4, -7, 7, -7, -7], [-2, 4, -1, -2, 4, 0, -2, 0, 0, -2, 0, 0], [7, 0, -4, 14, 0, 0, 14, 0, -2, 3, 0, -2], [-5, -2, -2, -5, 0, 4, -4, -6, -5, 5, 0, 0], [18, 0, 0, 18, 0, -14, 14, 3, 0, 0, 2, 0], [2, 4, 2, 0, 2, 0, 2, 4, 2, 1, 2, -1], [0, 0, 1, -1, 0, 1, 14, 1, 0, -3, 3, 3], [3, -1, 0, -14, 2, -3, 0, 2, 1, 0, 0, 0], [-5, 1, -2, 0, -2, 7, -5, 1, 1, 1, 1, -1], [0, 0, 1, -4, 5, 4, -3, 1, 0, -3, 3, -1], [6, -1, -6, 0, -6, -2, 6, -1, -1, 2, -1, -2], [0, 0, 0, 0, 0, -4, 4, 6, 1, -1, 4, 0], [0, 0, -2, -8, 5, 8, -2, -2, 0, -1, 1, -3], [7, -3, -3, 7, 0, 5, -5, 2, -1, 1, 1, 0], [1, 0, 0, 1, 0, 0, 0, 6, 1, -1, 0, 0], [1, -9, 0, 1, -1, -1, 0, -1, -2, 0, -4, -4], [0, 0, -3, -5, 2, 5, 3, -3, 0, -1, 1, 0], [2, 7, 5, 0, 5, -5, 2, 7, -2, 3, -2, -3], [0, -5, -1, -5, -5, 0, -5, 0, -1, 2, 0, -1], [1, -3, 3, 0, 3, 0, 1, -3, 0, -3, 0, 3], [-9, -1, 0, 1, 3, 9, 0, 3, -2, 0, -1, -1], [5, 3, 3, 5, 0, -15, 15, 0, -2, 2, 1, 0], [-5, -3, 0, 0, -6, 5, 0, -6, 1, 0, 3, 3], [-2, 1, -8, 0, 1, 0, 0, 0, 2, -4, 0, 2], [0, 0, -6, 0, -4, 0, -1, -6, 0, 2, -2, 5], [3, 0, 0, 3, 0, -16, 16, -2, 4, -4, -1, 0], [2, -6, -1, 0, -1, -5, 2, -6, -1, -1, -1, 1], [-15, -1, 2, -12, -1, 0, -12, 0, 0, 1, 0, 0], [-3, -2, -2, -3, -2, 0, -3, 0, 0, -2, 0, 0], [-11, 8, 0, -6, 7, 11, 0, 7, 1, 0, -1, -1], [-12, -6, 0, -12, -3, 12, 0, -3, -7, 0, -3, -3], [0, 0, 5, -2, 3, 2, 8, 5, 0, -4, 4, -2], [20, -3, 2, 11, -3, 0, 11, 0, 0, -1, 0, 0], [3, 6, 6, 3, 0, 4, -4, 9, 1, -1, 4, 0], [1, -1, -1, 20, -1, 0, 20, 0, 3, -7, 0, 3], [-1, 6, 3, 0, 3, -5, -1, 6, -5, -1, -5, 1], [-7, 8, 4, 0, 4, 7, -7, 8, 3, -8, 3, 8], [-2, -8, 2, 0, 2, -12, -2, -8, 2, -4, 2, 4], [-10, -1, 1, -7, -1, 0, -7, 0, 3, 0, 0, 3], [-2, 0, -1, -13, 0, 0, -13, 0, -4, 4, 0, -4], [0, 0, -3, -6, -3, 6, -4, -3, 0, 2, -2, 2], [-7, -7, 0, 0, 0, -11, -7, -7, -1, -2, -1, 2], [0, 0, -1, -9, -1, 9, 2, -1, 0, 1, -1, 5], [2, 7, 0, 13, 4, -2, 0, 4, 3, 0, 0, 0], [18, -2, -2, 18, 0, -6, 6, -2, 0, 0, -2, 0], [7, -1, -1, 7, 0, -5, 5, -2, 2, -2, -9, 0], [0, 0, -5, -11, 0, 11, 12, -5, 0, -1, 1, -3], [0, 0, -7, 4, 4, -4, 12, -7, 0, 4, -4, -6], [12, 3, 5, 4, 3, 0, 4, 0, 1, 1, 0, 1], [-2, 1, 2, 0, 2, -12, -2, 1, -2, 6, -2, -6], [5, 9, 0, 18, 8, -5, 0, 8, 1, 0, 0, 0], [-8, -7, -9, 0, -9, 8, -8, -7, -3, -2, -3, 2], [8, -8, -2, 9, -8, 0, 9, 0, 1, 2, 0, 1], [5, 0, 0, -2, -1, -5, 0, -1, 1, 0, -2, -2], [0, 0, 3, 5, 7, -5, 20, 3, 0, -1, 1, 1], [1, 6, 0, 0, -1, -1, 0, -1, 7, 0, 6, 6], [11, 0, -2, -4, 0, 0, -4, 0, 2, -11, 0, 2], [11, 0, -5, -8, 0, 0, -8, 0, -2, -3, 0, -2], [-13, 2, 0, -1, 1, 13, 0, 1, -4, 0, -3, -3], [-8, -3, 1, 0, 1, 17, -8, -3, 3, -7, 3, 7], [4, 1, 1, 4, 0, -13, 13, 4, -1, 1, 5, 0], [18, -3, -3, 18, 0, 4, -4, 0, -4, 4, -8, 0], [-6, 8, 0, -9, 5, 6, 0, 5, 7, 0, -1, -1], [-4, -4, -4, -4, 0, 12, -12, 4, 2, -2, 6, 0], [10, 0, 0, 10, 0, -7, 7, 1, -6, 6, -2, 0], [5, -5, 0, 0, 0, -7, 5, -5, -3, 2, -3, -2], [-9, 8, 0, -11, 8, 9, 0, 8, 3, 0, 2, 2], [8, 0, -1, 0, -1, -26, 8, 0, 0, -1, 0, 1], [5, 7, -1, 13, 7, 0, 13, 0, 1, -1, 0, 1], [-7, -8, 0, 0, -2, 7, 0, -2, 1, 0, -2, -2], [-18, -1, -1, -18, 0, 1, -1, -10, -1, 1, -7, 0], [0, 0, -1, -2, -12, 2, 6, -1, 0, 1, -1, -4], [2, -1, 0, 18, 7, -2, 0, 7, -2, 0, 1, 1], [-18, 2, 2, -18, 0, 7, -7, -1, -1, 1, 1, 0], [0, 0, 0, 4, -5, -4, -5, 0, 0, -1, 1, 4], [2, -3, -3, 2, 0, 16, -16, 4, 2, -2, -2, 0], [0, 0, -1, 7, -6, -7, 8, -1, 0, 1, -1, 7], [-6, -5, 2, -6, -5, 0, -6, 0, 0, -4, 0, 0], [11, -6, 0, 18, -6, 0, 18, 0, 2, -5, 0, 2], [0, 0, 7, 3, 2, -3, 15, 7, 0, -3, 3, 2], [-7, 3, 3, -7, 0, 4, -4, 6, 2, -2, 3, 0], [-15, 4, 0, -25, 1, 15, 0, 1, 6, 0, 5, 5], [12, 10, 3, 5, 10, 0, 5, 0, -5, 2, 0, -5], [0, 0, -2, -5, -5, 5, 0, -2, 0, 11, -11, -5], [-8, 4, 0, 13, 4, 8, 0, 4, -2, 0, 3, 3], [7, -8, -10, -5, -8, 0, -5, 0, -1, -3, 0, -1], [16, 3, 4, -7, 3, 0, -7, 0, 0, 4, 0, 0], [9, -8, 0, 2, -8, -9, 0, -8, 2, 0, 0, 0], [-7, -12, -7, 0, -7, -8, -7, -12, 4, -5, 4, 5], [0, 0, -5, 0, 0, 0, 14, -5, 0, -6, 6, 6], [18, 0, 6, 13, 0, 0, 13, 0, 4, 2, 0, 4], [16, -5, -5, 16, 0, -3, 3, -5, 7, -7, 9, 0], [-3, 11, 11, -3, 0, 16, -16, 11, 2, -2, 1, 0], [0, 0, 3, 10, 1, -10, 29, 3, 0, 2, -2, -3], [10, -3, 1, 0, 1, 7, 10, -3, 1, 3, 1, -3], [9, 1, 2, -13, 1, 0, -13, 0, 5, -11, 0, 5], [5, 7, 4, 0, 4, -28, 5, 7, -4, 12, -4, -12], [0, 0, 0, 11, 4, -11, 11, 0, 0, 1, -1, -10], [-1, 2, 2, -1, 0, -5, 5, 0, 4, -4, -9, 0], [0, 0, -3, 9, -7, -9, -6, -3, 0, -3, 3, -5], [-16, -2, -2, -16, 0, 4, -4, -6, -2, 2, 14, 0], [16, -3, -4, 0, -4, 6, 16, -3, 1, -6, 1, 6], [-11, -9, 0, 6, -3, 11, 0, -3, -5, 0, -6, -6], [0, 0, -3, 2, 14, -2, 4, -3, 0, 8, -8, -4], [2, -3, -7, 5, -3, 0, 5, 0, 3, 6, 0, 3], [0, 0, 4, -25, 3, 25, -15, 4, 0, 3, -3, -4], [-4, 12, 2, 0, 2, 6, -4, 12, -4, 10, -4, -10], [3, 1, 5, 0, 5, -19, 3, 1, 7, -3, 7, 3], [-12, 9, 9, -12, 0, 13, -13, 0, 2, -2, -8, 0], [0, 0, -2, 21, 7, -21, 16, -2, 0, 0, 0, 6], [7, 1, 1, 7, 0, -14, 14, 3, 5, -5, 2, 0], [11, -11, 0, 15, -7, -11, 0, -7, 0, 0, 2, 2], [15, -8, -7, 20, -8, 0, 20, 0, -8, 3, 0, -8], [-6, -6, 0, -13, -7, 6, 0, -7, 1, 0, -6, -6], [11, 4, 0, 12, 2, -11, 0, 2, 4, 0, 2, 2], [0, 0, -1, 3, 3, -3, 2, -1, 0, 9, -9, -7], [0, 0, 10, -10, -6, 10, -27, 10, 0, 3, -3, -9], [7, 6, 4, 31, 6, 0, 31, 0, -1, 1, 0, -1], [3, -4, -4, 3, 0, 18, -18, -3, -2, 2, 5, 0], [-7, 17, 13, 0, 13, 11, -7, 17, 8, -10, 8, 10], [0, 0, -1, 3, -3, -3, 19, -1, 0, -6, 6, -2], [11, 1, -10, -1, 1, 0, -1, 0, -1, -3, 0, -1], [16, 5, -1, 5, 5, 0, 5, 0, 8, -1, 0, 8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_930_n_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_930_2_n_d();".
// 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_930_2_n_d(:prec:=12)
    chi := MakeCharacter_930_n();
    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_930_2_n_d();".
// 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_930_2_n_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_930_n();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![400, -1200, 2960, -3850, 3421, -2116, 1417, -543, 190, -17, -3, 1, 1]>],Snew);
    return Vf;
end function;