
// Make newform 90.4.c.b in Magma, downloaded from the LMFDB on 12 March 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_90_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_90_c_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_90_4_c_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_90_4_c_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 := [1, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [0, 2]];
    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_90_c();"
function MakeCharacter_90_c()
    N := 90;
    order := 2;
    char_gens := [11, 37];
    v := [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_90_c_Hecke();"
function MakeCharacter_90_c_Hecke( : Kf := false)
    N := 90;
    order := 2;
    char_gens := [11, 37];
    char_values := [[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 := 4;
    raw_aps := [[0, 1], [0, 0], [5, -5], [0, -13], [28, 0], [0, 6], [0, -32], [60, 0], [0, 29], [90, 0], [-128, 0], [0, -118], [-242, 0], [0, 181], [0, 113], [0, 54], [-20, 0], [542, 0], [0, 217], [1128, 0], [0, 316], [720, 0], [0, 239], [-490, 0], [0, -728], [578, 0], [0, 731], [0, 483], [-370, 0], [0, 264], [0, 767], [-12, 0], [0, -612], [-3100, 0], [250, 0], [2152, 0], [0, 262], [0, -1759], [0, -267], [0, -2126], [2500, 0], [-2578, 0], [768, 0], [0, -1304], [0, 2558], [3480, 0], [3132, 0], [0, 31], [0, -2657], [190, 0], [0, 1204], [-5680, 0], [-278, 0], [-3252, 0], [0, 768], [0, 2429], [2610, 0], [-5168, 0], [0, 962], [-3042, 0], [0, -859], [0, -1146], [0, -2703], [5688, 0], [0, 3676], [0, -1742], [-7868, 0], [0, -328], [0, -2877], [3110, 0], [0, 3904], [-9240, 0], [0, 1607], [0, -174], [-4940, 0], [0, -3071], [3050, 0], [0, -2698], [-14482, 0], [1090, 0], [-7180, 0], [-8138, 0], [208, 0], [0, 6496], [-1080, 0], [0, 4539], [14310, 0], [0, 1172], [-11382, 0], [0, 8031], [0, 8583], [7520, 0], [0, 5907], [-14052, 0], [-7620, 0], [0, 909], [17850, 0], [19238, 0], [0, -3139], [-9818, 0], [0, 6257], [0, 5298], [0, -7001], [-7330, 0], [5812, 0], [0, -8368], [0, -3717], [0, -12936], [-3720, 0], [-12958, 0], [0, 3607], [0, -2414], [0, 13828], [21220, 0], [17672, 0], [-7322, 0], [0, -4119], [0, 3213], [0, 2954], [-26780, 0], [-24538, 0], [0, -14424], [0, -13442], [0, -7141], [-3428, 0], [-26942, 0], [1950, 0], [12080, 0], [0, -8613], [0, -394], [2060, 0], [0, 1629], [-4528, 0], [0, -9118], [18678, 0], [-27390, 0], [0, -4626], [0, -2863], [0, 13618], [10950, 0], [-8828, 0], [16058, 0], [0, 20931], [0, -6077], [15390, 0], [-4280, 0], [0, 7226], [0, -11292], [26740, 0], [0, 249], [0, 6622], [-40842, 0], [0, -6039], [0, -9147], [0, -11283], [6768, 0], [-22200, 0], [-6330, 0], [0, 9772], [9898, 0], [0, 20703], [0, -12716], [0, -6053], [-7812, 0], [0, 6288], [0, -2171], [26272, 0], [0, -22398]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_90_c_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_90_4_c_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_90_4_c_b(:prec:=2)
    chi := MakeCharacter_90_c();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_90_4_c_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_90_4_c_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_90_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<7,R![676, 0, 1]>],Snew);
    return Vf;
end function;