
// Make newform 1890.2.t.a in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1890_t();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1890_t_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_1890_2_t_a();".
// 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_1890_2_t_a();".
// 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, 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_1890_t();"
function MakeCharacter_1890_t()
    N := 1890;
    order := 6;
    char_gens := [1541, 757, 1081];
    v := [5, 6, 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_1890_t_Hecke();"
function MakeCharacter_1890_t_Hecke(Kf)
    N := 1890;
    order := 6;
    char_gens := [1541, 757, 1081];
    char_values := [[0, 0, 1, 0], [1, 0, 0, 0], [1, 0, -1, 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 := [[0, 1, 0, -1], [0, 0, 0, 0], [1, 0, 0, 0], [-1, 0, 3, 0], [-1, 0, 2, 3], [4, 0, -2, 0], [0, 0, 0, 0], [1, -3, 1, 0], [2, 0, -4, 6], [2, 3, 2, 0], [-3, -3, -3, 0], [-1, -3, 1, 6], [0, 0, 9, 0], [2, -3, -2, 6], [0, 0, 9, 0], [-2, 9, 1, -9], [3, -3, -3, 6], [8, 6, -4, -6], [4, 0, -4, 0], [1, 0, -2, -3], [8, 0, -4, 0], [0, -9, 1, -9], [-6, -3, 6, 6], [-6, 6, 6, -12], [-2, 12, -2, 0], [6, 6, 0, -3], [7, 0, -14, 0], [-4, -9, -4, 0], [0, 3, 2, 3], [18, -3, -9, 3], [-1, -12, 0, 6], [6, -12, 0, 6], [-1, 0, 2, -9], [22, 3, -11, -3], [0, 0, 0, 12], [1, -6, 0, 3], [-5, -9, -5, 0], [-2, 6, 2, -12], [0, -6, 0, -6], [0, -3, -3, -3], [-18, -9, 9, 9], [-6, 0, 12, 3], [-4, 0, 2, 0], [-19, 3, 19, -6], [5, 0, -10, -15], [8, 0, -4, 0], [0, 9, 5, 9], [1, 12, 1, 0], [6, 12, 0, -6], [-4, 0, 8, -21], [1, -3, 1, 0], [-14, 9, 7, -9], [-5, 0, 10, -12], [18, 0, 0, 0], [-18, -12, 0, 6], [-1, 0, 2, 12], [0, 0, -18, 0], [9, -9, 9, 0], [-19, -6, 0, 3], [-5, -6, -5, 0], [-8, -3, -8, 0], [0, 6, 6, 6], [12, 0, -24, -3], [21, -3, -21, 6], [-6, -15, 3, 15], [4, -24, -2, 24], [0, 0, 8, 0], [0, 0, -10, 0], [-14, 3, -14, 0], [6, 6, 6, 0], [-15, 18, 0, -9], [-3, 21, -3, 0], [-3, 0, 6, 24], [5, 6, 0, -3], [-20, 12, 0, -6], [-9, -12, 0, 6], [16, 0, -32, 3], [-9, -15, -9, 0], [-9, 0, 18, -12], [-13, 12, -13, 0], [0, -6, 0, -6], [4, 15, -4, -30], [24, 18, -12, -18], [0, 0, 0, 18], [10, 21, -5, -21], [0, -9, 0, 9], [-9, 0, 18, -6], [0, -6, 8, -6], [0, 3, 0, -6], [0, -6, 13, -6], [-12, 3, 12, -6], [0, 12, 0, -6], [0, 0, -34, 0], [14, 15, -7, -15], [-11, -6, 0, 3], [-3, 12, 0, -6], [6, -18, 0, 9], [0, 0, -21, 0], [0, 33, 0, 0], [-8, 0, 8, 0], [10, -9, -10, 18], [-24, -12, 12, 12], [24, -6, -24, 12], [0, 6, 0, -6], [-32, 6, 32, -12], [0, 24, 0, -24], [-36, 3, 36, -6], [6, 12, -6, -24], [-6, 6, -6, 0], [8, 18, 8, 0], [7, 0, -14, 6], [0, 12, -14, 12], [-28, 6, 14, -6], [6, 0, -12, -6], [2, 12, 0, -6], [-12, 0, 24, -24], [32, -9, -16, 9], [0, 6, -15, 6], [8, 0, -16, -36], [17, 9, 17, 0], [-8, 27, -8, 0], [-1, 15, 1, -30], [0, -15, 3, -15], [4, -39, -2, 39], [8, -12, -4, 12], [-10, 0, 20, 27], [0, 12, 26, 12], [0, -24, 0, 48], [-14, -24, -14, 0], [11, 0, -22, -15], [0, -18, -2, -18], [-10, 36, 5, -36], [-32, 12, 0, -6], [7, -18, 0, 9], [33, 0, 0, 0], [18, 0, -9, 0], [0, 0, 18, 0], [-14, 15, -14, 0], [0, 18, -12, 18], [-26, -30, 13, 30], [-2, 0, 4, -48], [-32, -9, 16, 9], [-41, 6, 41, -12], [20, 0, -40, 3], [16, 21, -8, -21], [-9, 3, 9, -6], [12, -24, 12, 0], [33, -18, 0, 9], [-3, 0, 6, -21], [-10, -12, -10, 0], [5, 54, 0, -27], [6, 12, 0, -6], [-8, 42, 0, -21], [39, 12, 0, -6], [28, -6, 0, 3], [-5, -3, -5, 0], [11, -9, -11, 18], [0, 24, 9, 24], [-9, 0, 18, 27], [0, 15, 0, -30], [48, -18, -24, 18], [-1, 0, 2, -3], [0, 0, 16, 0], [6, -12, -6, 24], [4, -12, 4, 0], [21, 12, 0, -6], [0, -9, 29, -9], [-13, 0, 26, 9], [-4, 48, 0, -24], [-24, 0, 0, 0], [-19, 0, 38, -3], [40, -24, -20, 24], [16, 0, -32, -6], [-10, -36, 0, 18], [4, 54, 4, 0], [0, 6, 27, 6], [25, 15, -25, -30], [-40, -18, 20, 18], [-10, 0, 20, 18], [-12, 27, 6, -27], [0, 12, 25, 12], [24, 6, -24, -12], [0, -18, -14, -18], [15, 3, -15, -6], [13, 30, 13, 0], [-6, -24, 0, 12], [0, -6, 56, -6], [12, 0, -24, -21], [-31, -36, 0, 18], [0, 9, 3, 9], [2, 12, 2, 0], [-18, -15, -18, 0], [11, 21, -11, -42], [0, 24, 24, 24], [-32, -33, 16, 33], [9, 15, -9, -30], [47, 6, -47, -12], [0, 0, -26, 0], [-3, 3, 3, -6], [9, -6, -9, 12], [6, -9, 6, 0], [-11, 39, -11, 0], [-3, 0, 6, 33], [15, 0, -30, 18], [48, 0, 0, 0], [0, 21, -9, 21], [-4, 6, -4, 0], [-2, 0, 4, -63], [20, -30, 20, 0], [7, -51, 7, 0], [19, -15, -19, 30], [0, 21, -24, 21], [26, 0, -26, 0], [0, 3, -30, 3], [15, -21, -15, 42], [-14, 18, 7, -18], [5, -30, -5, 60], [-21, -48, 0, 24], [27, 12, 27, 0], [-58, 21, 29, -21], [22, 0, 0, 0], [2, 39, -1, -39], [-14, 0, 28, 12], [5, -30, -5, 60], [0, -9, 48, -9], [-58, -9, 58, 18], [0, -9, 15, -9], [50, -3, -25, 3], [12, 6, -6, -6], [-30, 15, 15, -15], [14, 30, -14, -60], [-60, -6, 30, 6], [0, 18, 14, 18], [-25, 24, -25, 0], [20, -33, 20, 0], [0, -36, 0, 18], [8, 0, -16, -30], [-16, 30, -16, 0], [-26, 33, 13, -33], [-30, 24, 0, -12], [-8, 0, 16, 21], [-15, -51, -15, 0], [1, -60, 1, 0], [-10, 15, 10, -30], [0, 21, -33, 21], [0, 6, 18, 6], [6, 0, -12, 42], [3, 9, -3, -18], [4, -30, -4, 60], [-9, 0, 18, 18], [0, 0, -40, 0], [-27, -12, 27, 24], [-9, 18, -9, 0], [2, -48, 2, 0], [-36, 0, 0, 0], [-6, 54, -6, 0], [0, 3, 4, 3], [-8, 0, 16, -3], [-24, -48, 0, 24], [9, -21, 9, 0], [-64, 12, 0, -6], [-8, -42, -8, 0], [-16, -36, -16, 0], [-15, 0, 30, -45], [-12, 24, 12, -48], [-32, -12, 16, 12], [2, 0, -4, -42], [9, -6, -9, 12], [0, -3, 46, -3], [6, 30, 6, 0], [-2, 0, 4, 45], [0, -9, 32, -9], [-1, 0, 2, 3], [4, 0, 0, 0], [-58, 3, 29, -3], [-13, 12, -13, 0], [6, 0, -12, 21], [-20, -18, 0, 9], [8, -12, -8, 24], [0, 9, -33, 9], [-15, -18, 15, 36], [31, 15, -31, -30], [-12, 12, 12, -24], [-26, -33, -26, 0], [-8, 0, 16, -42], [6, 48, 6, 0], [0, -42, 2, -42], [12, 48, -6, -48], [-8, 0, 16, 0], [27, -48, 0, 24], [-2, 0, 4, -21], [0, 9, -12, 9], [-17, 0, 34, 51], [0, -9, -3, -9], [-26, 0, 26, 0], [6, 9, -6, -18], [-72, 6, 36, -6], [51, -9, -51, 18], [3, 9, -3, -18], [-28, -30, -28, 0], [-3, 0, 6, -33], [30, 0, 30, 0], [0, 27, 1, 27]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1890_t_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1890_2_t_a();".
// 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_1890_2_t_a(:prec:=4)
    chi := MakeCharacter_1890_t();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_1890_2_t_a();".
// 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_1890_2_t_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1890_t();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![36, 0, 24, 0, 1]>],Snew);
    return Vf;
end function;