
// Make newform 180.6.d.b in Magma, downloaded from the LMFDB on 06 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_180_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_180_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_180_6_d_b();".
// 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_180_6_d_b();".
// 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 := [8, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [-2, 4]];
    Rf_basisdens := [1, 1];
    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_180_d();"
function MakeCharacter_180_d()
    N := 180;
    order := 2;
    char_gens := [91, 101, 37];
    v := [2, 2, 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_180_d_Hecke();"
function MakeCharacter_180_d_Hecke( : Kf := false)
    N := 180;
    order := 2;
    char_gens := [91, 101, 37];
    char_values := [[1, 0], [1, 0], [-1, 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 := 6;
    raw_aps := [[0, 0], [0, 0], [5, 5], [0, -11], [100, 0], [0, 66], [0, -88], [2244, 0], [0, 307], [-7854, 0], [-2144, 0], [0, 934], [7414, 0], [0, 1595], [0, 847], [0, 2178], [25972, 0], [-3058, 0], [0, 5279], [-37608, 0], [0, 2156], [-79728, 0], [0, -1463], [-826, 0], [0, -3376], [143594, 0], [0, -10043], [0, -8283], [106238, 0], [0, -10152], [0, -4631], [89100, 0], [0, 3444], [134684, 0], [-248006, 0], [313720, 0], [0, -22054], [0, 35695], [0, -17061], [0, -7282], [142108, 0], [250790, 0], [209472, 0], [0, -32032], [0, -7766], [-749208, 0], [287364, 0], [0, 106745], [0, 87857], [-506474, 0], [0, -139612], [-374704, 0], [843634, 0], [1720500, 0], [0, 139392], [0, 36619], [-1827102, 0], [-616880, 0], [0, -164978], [1220934, 0], [0, -61853], [0, 76890], [0, 98439], [2124648, 0], [0, -26476], [0, -227338], [-1170212, 0], [0, -167816], [0, 146685], [-2006290, 0], [0, 162416], [4506744, 0], [0, -271919], [0, 104478], [-832052, 0], [0, 257687], [-311926, 0], [0, 265474], [-2770, 0], [-1979846, 0], [-5101204, 0], [-2432234, 0], [-918896, 0], [0, -184504], [676632, 0], [0, -227283], [-5176194, 0], [0, 279532], [2649570, 0], [0, 232737], [0, -594615], [6893216, 0], [0, 507717], [-5883900, 0], [6720804, 0], [0, -42141], [-294414, 0], [7100246, 0], [0, -535733], [-2723666, 0], [0, -828113], [0, -307890], [0, 518969], [-3892570, 0], [-5062772, 0], [0, 296416], [0, 464277], [0, -1423224], [16614600, 0], [7882490, 0], [0, -70279], [0, -216634], [0, 867812], [18003580, 0], [-4800808, 0], [-14495042, 0], [0, -1638273], [0, -866229], [0, 1731454], [9705916, 0], [-4283962, 0], [0, -1172688], [0, -756646], [0, -1473659], [-11213852, 0], [-20470670, 0], [81654, 0], [-8610064, 0], [0, 1056813], [0, -353606], [23060180, 0], [0, 155067], [-2580304, 0], [0, -517154], [-14049882, 0], [5598978, 0], [0, -569910], [0, 1559767], [0, 724646], [19455414, 0], [-2852036, 0], [5484314, 0], [0, 590589], [0, -376915], [-27888882, 0], [12775192, 0], [0, 482350], [0, -1451076], [-33464276, 0], [0, 2261127], [0, -2212078], [11056398, 0], [0, 3364647], [0, 2661219], [0, 174603], [-12806352, 0], [35840376, 0], [-8477922, 0], [0, -1441924], [42467458, 0], [0, -2504391], [0, 4291628], [0, 4055381], [27090084, 0], [0, -3903096], [0, 589139], [17294176, 0], [0, -2012538]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_180_d_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_180_6_d_b();".
// 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_180_6_d_b(:prec:=2)
    chi := MakeCharacter_180_d();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 6));
    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_180_6_d_b();".
// 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_180_6_d_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_180_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<7,R![15004, 0, 1]>,<11,R![-100, 1]>],Snew);
    return Vf;
end function;