
// Make newform 1904.2.a.q in Magma, downloaded from the LMFDB on 01 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1904_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_1904_2_a_q();".
// 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_1904_2_a_q();".
// 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 := [11, 1, -8, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [4, -4, -2, 1], [-8, 3, 3, -1]];
    Rf_basisdens := [1, 1, 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_1904_a();"
function MakeCharacter_1904_a()
    N := 1904;
    order := 1;
    char_gens := [239, 1429, 1361, 785];
    v := [1, 1, 1, 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;

function MakeCharacter_1904_a_Hecke( : Kf := Kf)
    return MakeCharacter_1904_a();
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, 0, 0], [-1, 1, 0, 0], [0, -1, 0, -1], [1, 0, 0, 0], [-1, 0, 1, 0], [-2, 0, -2, 0], [1, 0, 0, 0], [-4, 2, 0, 0], [-2, 0, 0, 2], [1, 0, -1, 2], [-1, -1, 1, 3], [-1, 0, -3, -2], [-2, 1, -3, 0], [-4, -3, 1, 0], [-2, 0, -2, -4], [1, 1, 3, 3], [0, 0, 0, 2], [-5, -3, 2, 0], [-3, -3, 3, 1], [0, -2, 4, 2], [-2, 3, 3, 0], [6, 0, -2, -4], [-2, -2, -2, -4], [-4, 0, -4, -6], [-3, -3, 3, -1], [-4, 6, -4, -2], [-6, 2, 2, 2], [-1, 2, 1, 6], [-15, 2, -3, -2], [2, 4, -2, 2], [1, 3, 3, 3], [-1, 2, -3, 0], [10, -1, 3, 0], [0, 5, -4, -1], [0, 3, 5, 4], [-2, -3, 3, 0], [-4, -2, 2, 4], [5, 2, 1, -4], [4, -1, -1, 0], [1, -3, 4, 4], [2, -5, -3, 0], [-11, -2, 5, 0], [3, 5, -3, -3], [4, -4, -4, -6], [3, -2, 3, -2], [4, -9, -1, 0], [-3, 4, -3, -2], [12, 2, 0, 2], [-8, 5, 4, 3], [-14, 2, 2, 2], [12, -2, -2, -2], [-6, -3, 3, 0], [-1, 5, -3, -5], [-2, 2, 10, 0], [-4, 0, 4, 6], [0, 0, 0, -4], [-5, 2, 7, 4], [-8, 4, -10, -6], [9, -6, 1, -2], [15, -1, -3, -7], [-8, -1, -8, 1], [8, 0, -4, -4], [2, -6, 0, 2], [-21, 1, -7, -3], [-3, 9, 3, 3], [9, 2, -5, -4], [-1, 3, -11, -1], [6, 4, -2, 6], [-7, 6, 1, -4], [-10, -2, -4, 0], [2, 10, -2, 0], [5, 7, -1, 3], [-2, -1, 5, -4], [2, 9, -3, 0], [-1, -8, 5, -4], [2, -2, 2, 6], [-5, 5, -3, -1], [2, -1, 2, 7], [8, -6, 2, -2], [8, -2, -6, -10], [-2, -1, -10, -7], [2, 5, -3, 4], [-12, 6, 2, 0], [6, 4, 0, -4], [2, 1, -13, -4], [-22, -4, 2, -4], [4, 4, -2, -4], [1, -9, -1, -3], [-8, 4, 2, -2], [-2, 1, -5, -4], [0, -8, -6, 0], [20, 1, 5, -4], [0, 10, -2, -4], [-17, 1, -7, -3], [-7, 0, -3, 6], [1, 3, -9, 3], [6, -8, -4, 2], [16, -13, 13, 8], [-10, 4, 12, 4], [-13, 2, 3, -6], [15, -8, -1, -6], [12, -8, -2, -4], [6, 10, -2, 12], [8, 11, -7, 8], [-7, -4, 3, -8], [2, -4, -4, 4], [-8, -8, -6, 4], [-10, 10, -6, -4], [-2, 7, 9, 4], [2, 12, 0, 0], [-5, 1, 5, 9], [-5, 1, 13, 3], [8, 2, -8, -8], [3, -6, 5, 4], [-17, 1, 9, 1], [-16, -2, 6, 6], [-1, -1, -6, -12], [-24, -6, 4, -2], [25, -2, 1, -2], [7, -3, 5, 1], [-10, 2, 16, 4], [-16, 12, 10, 4], [-7, -10, 9, 0], [27, -6, 11, 8], [15, -7, 16, 12], [8, -8, 2, 0], [-9, 2, 9, 12], [12, 3, 3, 0], [-6, 6, 4, 8], [-16, -4, -2, 6], [6, 7, -7, 0], [-20, -10, 0, -6], [2, -6, -6, -2], [16, -15, 11, 8], [-22, -2, 10, 4], [-12, -2, 2, 2], [-16, 0, -12, -4], [11, -10, -7, -12], [8, -10, -4, -2], [-30, -8, 0, -4], [3, -1, -2, -16], [7, -4, 1, -14], [4, 10, 0, 14], [13, -8, -7, -10], [-20, -8, 0, -4], [-6, -4, -10, -8], [19, 2, 3, 0], [11, -1, 1, 5], [-16, -12, 4, -2], [-3, -5, -1, -5], [-11, 6, 5, -2], [5, 15, -9, 1], [15, 7, 9, 3], [3, -5, -3, -9], [-21, 14, -5, -4], [18, 6, 8, -4], [-18, 7, -7, -8], [-3, 3, 11, -3], [-24, -12, 8, -6], [-17, 1, -10, -4], [-19, 4, 3, 4], [15, -7, 5, -1], [-26, 1, -1, 4], [-22, 2, -10, -12], [-11, -13, 7, 1], [39, -5, 9, 7], [4, -8, 2, 10], [-25, 5, -10, 0], [-20, -4, -12, 2], [3, 3, -8, -4], [8, 0, -8, -12], [-10, -4, -10, -12], [4, 14, -6, 8], [1, -9, 7, -7], [0, -12, -4, -12], [1, -15, 15, -1], [-3, 16, 5, 2], [-13, 14, 11, 10], [-38, 0, -2, 4], [-12, 6, 2, 16], [0, 4, 4, 4], [25, -7, 6, 8], [9, 6, -5, -12], [-8, -4, 4, 2], [-15, -1, -5, -17], [16, -8, 18, 18], [-5, 6, -1, 2], [-16, -10, 0, 8], [16, -18, 0, 0], [-6, -12, 6, 8], [16, -1, 9, 4], [25, -4, -5, -12], [-12, 4, -4, 8], [2, -4, -4, -6], [-19, 14, -17, -10], [5, 11, -5, -15], [12, 14, -4, 4], [-1, 10, 7, 22], [-16, -11, -1, 4], [-4, 2, 2, 4], [1, 6, 3, -8], [3, -1, -19, -5], [18, 2, -10, 2], [24, 4, 0, 10], [-28, -4, 6, -4], [-24, 6, 6, 0], [16, 2, 2, -4], [2, 7, 5, 0], [4, -8, 10, 0], [0, -12, 2, 0], [-24, 2, -6, -2], [-6, -12, 0, -6], [-30, 6, -16, -12], [-44, -2, 4, 0], [6, -9, 5, 12], [0, -7, -5, 8], [-4, 4, 8, 12], [-6, -4, 12, 4], [-18, -6, 6, 14], [15, 3, 9, 5], [-1, 9, 5, -9], [10, -17, -3, -16], [-10, -15, -3, -4], [-8, 4, -2, -2], [-4, 18, 2, -2], [-13, 7, 9, 13], [-12, 16, -10, -4], [-2, 2, -10, 14], [-10, -10, -10, 2], [-23, -8, -5, 4], [10, -9, 3, 12], [17, 10, -9, -16], [-64, 0, 0, 0], [-20, 15, 5, 16], [-12, -4, 12, -2], [-14, 4, -22, -20], [16, 0, 2, 8], [11, -6, 7, 14], [21, 0, -9, 0], [12, -14, 8, -6], [33, 0, -1, -8], [12, -12, 14, 4], [8, -2, 8, -6], [-48, 7, 9, 4], [-9, -1, -23, -3], [15, 11, 1, 15], [-6, 1, -5, -4], [9, 7, 4, 12], [9, -19, -1, -3], [-10, -13, 5, -8], [-12, -1, -19, -4], [-41, 3, 9, 1], [-11, 7, 15, 23], [-15, -5, -17, 9], [10, 16, -4, 10], [8, -11, -7, 8], [23, -12, 5, -6], [-12, -1, 24, 1], [39, -3, 14, 8], [28, 6, -6, -6], [28, -8, -14, -6], [35, -15, -15, -11], [16, 17, -4, -3], [-12, 11, 12, 19], [14, -17, -1, -4], [-4, -2, -6, 6], [-35, 5, -13, -5], [-4, 10, -10, -10], [27, 10, 5, -10], [28, -27, 3, 0], [22, 5, 6, -11], [-24, -4, -16, -4], [-12, -10, 10, -14], [-23, 11, 3, -1], [38, -8, 12, 12], [28, -16, 6, 2], [-24, -1, 19, 4], [-3, -3, 3, 19], [-16, 5, -5, 0], [24, 0, 8, 10], [32, 20, -10, 2], [-6, 0, -16, -2], [-14, -4, 10, -4], [-16, 18, 12, 22], [22, -21, 6, 9], [-6, -5, 13, 0], [30, -4, -10, -6], [13, -12, 9, 4], [-14, 12, 8, 6], [22, 0, -4, -16], [33, 4, -1, 10], [18, 17, -14, -1], [-28, 0, 6, 0], [8, 3, -4, 9], [-1, 12, 5, -2], [-27, 1, 4, 20], [6, 16, -8, -22], [2, -2, 12, 12], [-12, -15, 16, 3], [8, -6, -6, -2], [-8, 4, 4, -2], [-18, 12, -4, 4], [10, 2, 10, 6], [-29, -4, 13, 4], [13, 16, 5, 6], [4, 0, 14, -8], [7, 1, 5, 13], [-29, 1, -18, 0], [-27, 23, -13, -5], [4, -13, -3, -24], [45, 3, -13, 3], [-7, 4, -17, 6], [-8, -4, -6, 12], [14, 1, 3, 8], [-6, -9, 1, -8], [-2, 8, 0, 24], [14, -10, -16, -2], [-2, 8, -28, -10], [18, -1, -1, -12], [12, 16, -6, 0], [5, -3, 14, -16], [-38, 12, 18, 18], [-4, 20, -4, 4], [9, 15, 7, 11], [2, -14, 2, -4], [14, 10, -18, -12], [11, -8, 15, 6], [2, -3, 6, 15], [-34, 4, -4, 2], [7, 20, -15, -18], [33, -22, -3, -6], [5, 8, 5, -8], [-44, 19, -5, 0], [18, -4, -18, -22], [29, 6, -7, -4], [-3, 14, 23, 4], [-18, -17, 11, 8], [14, -5, 15, -4], [14, 19, -1, 12], [-8, 4, -18, -10], [2, -7, 1, -20], [33, -13, -21, -3], [1, 22, -15, 8], [-11, -21, 7, -5], [18, 27, -9, -4], [7, 0, -13, -18], [22, 11, 14, 11], [51, 7, -11, 7], [-19, 11, 15, 15], [15, 5, 2, 12], [-22, -8, 0, -12], [46, 12, -10, 10], [-52, -9, -3, 4], [-35, 9, 3, 21], [-51, -9, -5, -19], [4, -16, 16, 18], [6, 4, 18, 4], [43, 15, -2, 0], [22, -18, 24, 10], [-20, 16, -20, -4], [-27, -2, -21, -12], [9, -19, -5, -9], [-3, 7, 23, 19], [-31, -6, -15, -20], [8, 36, -14, 4], [23, -1, -27, -11], [1, -3, -4, 20], [18, -11, -1, -4], [-18, -20, 16, 4], [28, -21, 11, 8], [-1, 27, -2, -4], [-10, 4, 18, 20], [14, 5, 1, 4], [-12, -2, -8, -18], [5, 6, -7, 6], [3, -21, -6, -16], [-8, -20, -6, 10], [14, 0, -4, 20], [12, 12, 0, 2], [-2, -2, -18, -32], [-11, 5, 12, 8], [28, 12, -18, -4], [26, 0, 22, 32], [-40, 30, -14, -10], [45, -21, 28, 16], [-20, -20, 16, -16], [-56, -8, 12, -8], [44, 10, 2, -4], [25, -4, 1, 14], [-4, -8, -8, 0], [-14, 8, -18, -2], [-15, 14, 1, -4], [-19, -1, -5, -13], [-10, -14, 8, -14], [-41, 21, 5, 5], [2, 2, 8, 10], [50, -2, 12, 8], [10, 8, 4, -20], [-33, 10, -1, -8], [-37, -17, 9, -3], [-11, 14, 1, 0], [-67, 9, 11, 7], [-6, -5, -11, -8], [-40, 7, 13, 12], [44, -18, -18, -16], [22, 0, 14, 16], [25, 7, -9, -11], [-14, 9, -3, 4], [-50, 0, -8, 0], [3, -3, 12, -4], [35, 5, -11, 11], [19, 13, 13, 7], [-15, -9, 15, 15], [-13, 22, 5, 2], [6, 17, 13, 20], [11, -13, -20, 0], [-54, -6, 4, -12], [13, 9, -1, -15], [-9, -23, 17, 3], [-22, 18, 8, 20], [66, -1, -5, -20], [-11, 3, -14, 20], [-2, -8, -10, 4], [-56, 0, -2, 16], [6, 12, 0, 24], [8, -2, -6, 20], [-21, 8, -21, -14], [38, -22, 2, -4], [36, -2, 10, -2], [-45, -5, -7, 3]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1904_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1904_2_a_q();".
// 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_1904_2_a_q(:prec:=4)
    chi := MakeCharacter_1904_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 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_1904_2_a_q();".
// 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_1904_2_a_q( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1904_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![4, -14, -5, 3, 1]>,<5,R![4, -22, -15, 1, 1]>],Snew);
    return Vf;
end function;