
// Make newform 90.2.i.b in Magma, downloaded from the LMFDB on 22 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_90_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_90_i_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_2_i_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_2_i_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, 0, 0, -1, 0, 0, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_i();"
function MakeCharacter_90_i()
    N := 90;
    order := 6;
    char_gens := [11, 37];
    v := [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_90_i_Hecke();"
function MakeCharacter_90_i_Hecke( : Kf := false)
    N := 90;
    order := 6;
    char_gens := [11, 37];
    char_values := [[0, 0, 0, 0, -1, 0, 0, 0], [-1, 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, 1, 0, 0, 0, 0, 0], [0, 0, 1, 1, 0, -1, 0, -1], [0, -1, -1, 0, -1, 2, 1, 0], [0, -2, -2, -1, 0, 1, 0, -1], [1, -2, 0, 1, -1, 1, 0, 1], [0, 1, 0, 2, 0, -2, 0, -1], [0, 2, 0, -2, 0, 2, -1, 4], [3, -1, 0, -1, 0, -1, 0, 2], [0, 2, -2, 4, 0, -4, 2, -2], [-6, 0, 0, 0, 6, 0, 0, 0], [0, 1, 0, -2, -4, -2, 0, 1], [0, 0, 0, 0, 0, 0, 8, 0], [0, 0, 0, 0, 1, 0, 0, 0], [0, 2, 5, 1, 0, -1, 0, 1], [0, -2, 2, -1, 0, 1, 0, -1], [0, -1, 0, 1, 0, -1, -6, -2], [0, 5, 0, -10, 1, -10, 0, 5], [-2, 2, 0, -1, 2, -1, 0, -1], [0, 1, 7, 2, 0, -2, -7, -1], [0, 1, 0, 1, 0, 1, 0, -2], [0, 4, 0, -4, 0, 4, -5, 8], [0, -6, 0, 3, 0, 3, 0, 3], [0, 0, -4, 0, 0, 0, 0, 0], [-8, -2, 0, -2, 0, -2, 0, 4], [0, 0, -13, 0, 0, 0, 0, 0], [8, 0, 0, 0, -8, 0, 0, 0], [0, -5, 2, -10, 0, 10, -2, 5], [0, -3, 0, 3, 0, -3, 9, -6], [8, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, -4, 0, 4, 0, 2], [0, -2, 0, 2, 0, -2, 2, -4], [0, -2, 0, 4, 0, 4, 0, -2], [0, 0, -3, 0, 0, 0, 0, 0], [0, 5, 0, -10, 1, -10, 0, 5], [0, -5, 0, 10, 4, 10, 0, -5], [2, 4, 0, -2, -2, -2, 0, -2], [0, 0, 16, 0, 0, 0, -16, 0], [0, 2, 0, -2, 0, 2, -4, 4], [0, -5, -12, -10, 0, 10, 12, 5], [0, 8, -2, 4, 0, -4, 0, 4], [-4, -2, 0, -2, 0, -2, 0, 4], [8, 1, 0, 1, 0, 1, 0, -2], [-6, 10, 0, -5, 6, -5, 0, -5], [0, 6, 1, 12, 0, -12, -1, -6], [0, 0, 0, 0, 0, 0, -8, 0], [-18, -1, 0, -1, 0, -1, 0, 2], [0, -4, 0, 8, 6, 8, 0, -4], [0, 4, 14, 2, 0, -2, 0, 2], [0, -2, 1, -1, 0, 1, 0, -1], [0, -1, 0, 2, -16, 2, 0, -1], [0, -6, 0, 6, 0, -6, -1, -12], [0, -6, 0, 12, 14, 12, 0, -6], [-1, 0, 0, 0, 1, 0, 0, 0], [9, 1, 0, 1, 0, 1, 0, -2], [0, 2, 15, 4, 0, -4, -15, -2], [0, 2, -10, 1, 0, -1, 0, 1], [6, 9, 0, 9, 0, 9, 0, -18], [4, 8, 0, 8, 0, 8, 0, -16], [0, -16, 10, -8, 0, 8, 0, -8], [-12, 0, 0, 0, 12, 0, 0, 0], [0, 0, -4, 0, 0, 0, 4, 0], [0, 0, -18, 0, 0, 0, 18, 0], [0, -11, 0, 11, 0, -11, 3, -22], [0, 2, 0, -4, -18, -4, 0, 2], [0, 12, -7, 6, 0, -6, 0, 6], [0, 22, 4, 11, 0, -11, 0, 11], [-4, -24, 0, 12, 4, 12, 0, 12], [0, -8, -1, -16, 0, 16, 1, 8], [0, -5, -3, -10, 0, 10, 3, 5], [-8, 16, 0, -8, 8, -8, 0, -8], [0, -16, 13, -8, 0, 8, 0, -8], [-6, -1, 0, -1, 0, -1, 0, 2], [0, 16, -8, 8, 0, -8, 0, 8], [0, -8, -6, -16, 0, 16, 6, 8], [13, 7, 0, 7, 0, 7, 0, -14], [0, -4, 8, -8, 0, 8, -8, 4], [20, 2, 0, -1, -20, -1, 0, -1], [0, 4, 0, -4, 0, 4, 8, 8], [0, 4, 0, -8, -19, -8, 0, 4], [0, 2, 0, -4, 11, -4, 0, 2], [0, -2, 0, 4, -4, 4, 0, -2], [14, 2, 0, -1, -14, -1, 0, -1], [26, -5, 0, -5, 0, -5, 0, 10], [0, 0, 0, 0, 0, 0, -23, 0], [0, -18, 0, 9, 0, 9, 0, 9], [0, -10, 9, -5, 0, 5, 0, -5], [-9, 4, 0, 4, 0, 4, 0, -8], [0, -4, 13, -2, 0, 2, 0, -2], [0, -2, 0, 1, 0, 1, 0, 1], [0, 0, -24, 0, 0, 0, 24, 0], [0, 3, 0, -3, 0, 3, -3, 6], [2, -12, 0, 6, -2, 6, 0, 6], [0, -2, 0, 2, 0, -2, -30, -4], [0, 1, 0, -2, -21, -2, 0, 1], [0, -5, 0, 10, -9, 10, 0, -5], [0, 1, 0, -1, 0, 1, -12, 2], [0, 8, 0, -16, 12, -16, 0, 8], [7, -6, 0, -6, 0, -6, 0, 12], [0, 4, 0, -4, 0, 4, 20, 8], [-20, -8, 0, -8, 0, -8, 0, 16], [0, 10, -5, 5, 0, -5, 0, 5], [0, -11, 0, 11, 0, -11, -12, -22], [0, -7, -19, -14, 0, 14, 19, 7], [-7, 20, 0, -10, 7, -10, 0, -10], [0, -5, 0, 10, -25, 10, 0, -5], [0, 6, 0, -6, 0, 6, -1, 12], [0, -22, -3, -11, 0, 11, 0, -11], [0, -12, 0, 12, 0, -12, -12, -24], [0, 4, 0, -8, 24, -8, 0, 4], [-29, 8, 0, -4, 29, -4, 0, -4], [0, 8, -8, 16, 0, -16, 8, -8], [0, -11, 0, 11, 0, -11, 10, -22], [0, 6, 23, 12, 0, -12, -23, -6], [-31, -2, 0, 1, 31, 1, 0, 1], [-4, 3, 0, 3, 0, 3, 0, -6], [37, 0, 0, 0, -37, 0, 0, 0], [0, 5, -3, 10, 0, -10, 3, -5], [0, -2, 0, 2, 0, -2, 30, -4], [0, 0, 34, 0, 0, 0, -34, 0], [-26, 8, 0, -4, 26, -4, 0, -4], [0, -4, 0, 8, 4, 8, 0, -4], [0, 4, -24, 2, 0, -2, 0, 2], [0, 16, -20, 8, 0, -8, 0, 8], [0, -1, 0, 1, 0, -1, 43, -2], [2, 16, 0, -8, -2, -8, 0, -8], [-10, 12, 0, 12, 0, 12, 0, -24], [30, -6, 0, 3, -30, 3, 0, 3], [-32, 4, 0, 4, 0, 4, 0, -8], [0, -6, 18, -3, 0, 3, 0, -3], [0, 7, -4, 14, 0, -14, 4, -7], [5, -5, 0, -5, 0, -5, 0, 10], [0, -2, 18, -4, 0, 4, -18, 2], [0, 20, 0, -40, 4, -40, 0, 20], [0, 0, 0, 0, 0, 0, -10, 0], [0, -10, 0, 20, 24, 20, 0, -10], [0, 10, 0, -20, 0, -20, 0, 10], [0, 12, 0, -12, 0, 12, -6, 24], [0, 12, -22, 24, 0, -24, 22, -12], [0, -8, -16, -16, 0, 16, 16, 8], [-3, -18, 0, -18, 0, -18, 0, 36], [-5, -5, 0, -5, 0, -5, 0, 10], [20, 18, 0, -9, -20, -9, 0, -9], [0, -8, 12, -16, 0, 16, -12, 8], [0, -4, 0, 4, 0, -4, -4, -8], [24, -1, 0, -1, 0, -1, 0, 2], [28, -10, 0, 5, -28, 5, 0, 5], [0, -14, -8, -7, 0, 7, 0, -7], [0, 24, 24, 12, 0, -12, 0, 12], [0, -15, 0, 30, -1, 30, 0, -15], [0, 1, 0, -1, 0, 1, -24, 2], [0, -7, -38, -14, 0, 14, 38, 7], [-30, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, -1, 0, 1, 9, 2], [0, -3, 12, -6, 0, 6, -12, 3], [0, -30, 3, -15, 0, 15, 0, -15], [-12, 10, 0, -5, 12, -5, 0, -5], [-6, -2, 0, -2, 0, -2, 0, 4], [-36, -16, 0, 8, 36, 8, 0, 8], [0, -8, 0, 8, 0, -8, -20, -16], [0, -4, 0, 8, 40, 8, 0, -4], [0, 10, 9, 5, 0, -5, 0, 5], [0, 4, 0, -4, 0, 4, 41, 8], [0, -6, -14, -12, 0, 12, 14, 6], [6, -12, 0, -12, 0, -12, 0, 24], [0, 2, 33, 4, 0, -4, -33, -2], [0, -24, -8, -12, 0, 12, 0, -12], [4, 0, 0, 0, 0, 0, 0, 0], [0, 22, 6, 11, 0, -11, 0, 11]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_90_i_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_90_2_i_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_2_i_b(:prec:=8)
    chi := MakeCharacter_90_i();
    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_90_2_i_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_2_i_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_90_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![16, 0, -80, 0, 396, 0, -20, 0, 1]>],Snew);
    return Vf;
end function;