
// Make newform 441.2.u.a in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_441_u();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_441_u_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_441_2_u_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_441_2_u_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, -1, 1, -1, 1, -1, 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_441_u();"
function MakeCharacter_441_u()
    N := 441;
    order := 7;
    char_gens := [344, 199];
    v := [7, 5];
    // 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_441_u_Hecke();"
function MakeCharacter_441_u_Hecke(Kf)
    N := 441;
    order := 7;
    char_gens := [344, 199];
    char_values := [[1, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 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 := [[1, -1, 0, 0, 1, -1], [0, 0, 0, 0, 0, 0], [-1, 1, 0, 0, 1, 0], [2, 0, 0, -1, 2, -2], [-2, 1, -3, 3, -1, 2], [-2, 0, -5, 5, 0, 2], [-2, 3, 0, 3, -2, 0], [-1, 0, 2, 1, -1, -2], [1, 1, -1, -1, 0, 3], [2, 3, 0, 3, 2, 0], [-2, 0, -1, 0, 0, 1], [4, -6, 4, -6, 4, 0], [0, 0, 0, 2, 4, 2], [0, 3, -2, 2, -2, 3], [5, -4, 1, -1, 4, -5], [8, -4, 4, -8, 0, -6], [0, -1, 1, -7, 1, -1], [-3, 2, -3, 2, -3, 0], [-1, 0, 2, -1, 1, -2], [-1, 3, -3, 1, 0, -6], [-3, 9, -3, 0, -5, 5], [11, 0, 2, -1, 1, -2], [-1, 1, -1, 0, -9, 9], [-8, 9, -8, 0, -9, 9], [-1, 0, -1, 2, -2, 1], [5, -5, 0, 1, 3, 1], [6, -6, 0, -3, -3, -3], [10, -6, 10, 0, 3, -3], [-4, 2, -9, 9, -2, 4], [8, -1, 8, 0, 1, -1], [9, 0, 5, 0, 9, 0], [0, -15, 10, -12, 10, -15], [0, 3, -4, -2, -4, 3], [-6, 6, 0, 15, -14, 15], [0, 2, -3, 3, -2, 0], [7, -3, 3, -7, 0, 1], [0, 6, -1, 4, -1, 6], [0, -7, 7, -3, 7, -7], [0, 9, 4, 9, 0, 0], [-9, 14, -14, 9, 0, 11], [4, -3, 9, -3, 4, 0], [-7, 10, -7, 0, -2, 2], [7, -7, 0, -3, -2, -3], [-10, 10, 0, 0, -10, 0], [-11, 0, -8, 5, -5, 8], [0, 0, 0, -3, 4, -3], [3, -5, 3, 0, -1, 1], [6, -16, 16, -6, 0, -5], [-7, 0, -8, -3, 3, 8], [0, 2, 6, -7, 6, 2], [-6, 2, 6, 2, -6, 0], [0, -4, -6, 12, -6, -4], [1, -3, 3, -1, 0, -9], [0, -7, 9, -3, 9, -7], [-9, -7, 3, -7, -9, 0], [-6, 0, 10, 6, -6, -10], [1, 5, 1, 0, 7, -7], [-13, 12, -12, 13, 0, -2], [-12, 15, -4, 15, -12, 0], [0, -1, -2, 2, 1, 0], [21, -14, 12, -12, 14, -21], [-18, 0, -6, 20, -20, 6], [9, -23, 17, -17, 23, -9], [4, -11, 23, -11, 4, 0], [-15, 0, -13, 0, 0, 13], [-4, 22, -22, 4, 0, 14], [8, 10, 0, 10, 8, 0], [0, -2, 1, 18, 1, -2], [0, -16, 16, 0, 0, -5], [0, -7, -1, -9, -1, -7], [0, 13, -8, 3, -8, 13], [0, 15, -25, 7, -25, 15], [11, -12, 11, 0, 2, -2], [-13, 0, -13, 7, -7, 13], [-6, 6, 6, -6, -6, 6], [5, -4, 5, 0, -4, 4], [-7, 7, 0, -14, 2, -14], [4, -4, 0, -18, 6, -18], [-5, 6, -5, 0, -4, 4], [6, 8, 7, 8, 6, 0], [11, -12, 12, -11, 0, -10], [9, -20, -7, -20, 9, 0], [0, 7, -13, 7, -13, 7], [0, 0, 0, 8, -12, 8], [14, 9, 6, -6, -9, -14], [-14, 0, -12, 12, 0, 14], [8, -8, 0, -13, 5, -13], [0, 8, 6, -21, 6, 8], [6, -5, 27, -5, 6, 0], [-7, -6, 6, 7, 0, 3], [-6, 1, -1, 6, 0, 1], [3, 12, -17, 17, -12, -3], [-6, 6, 0, 12, -15, 12], [-17, 0, 4, 0, 0, -4], [-15, -4, -15, 0, -17, 17], [11, -12, -2, 2, 12, -11], [28, 0, 23, -29, 29, -23], [-7, 0, -32, 15, -15, 32], [0, -5, -10, -4, -10, -5], [27, -25, 9, -9, 25, -27], [10, -10, 0, -28, 17, -28], [26, 0, 5, -11, 11, -5], [23, -26, 23, 0, 22, -22], [-8, 0, -15, 7, -7, 15], [7, -11, 11, -11, 7, 0], [-7, 6, -17, 17, -6, 7], [27, 0, 17, -22, 22, -17], [-4, 4, 0, 11, -20, 11], [5, 13, -5, 5, -13, -5], [4, 6, 9, -9, -6, -4], [5, 0, 15, -4, 4, -15], [5, 6, 5, 0, 18, -18], [-1, 13, -9, 13, -1, 0], [-19, 0, -19, 5, -5, 19], [0, -26, 7, -20, 7, -26], [-24, 22, -22, 24, 0, 18], [0, 17, -7, 8, -7, 17], [0, -8, 24, -17, 24, -8], [0, -7, 2, 21, 2, -7], [14, -22, 22, -14, 0, -17], [18, 3, 18, 0, -1, 1], [-25, 7, -23, 23, -7, 25], [-2, -19, -2, 0, -1, 1], [-15, 15, 0, 13, 2, 13], [-12, 12, 0, 17, 10, 17], [-1, 24, -1, 0, -2, 2], [7, -8, 8, -7, 0, -8], [0, 11, -28, -14, -28, 11], [-11, 11, 0, -4, 14, -4], [7, 20, -4, 4, -20, -7], [14, 11, -11, -14, 0, -20], [-6, 6, 0, -17, -17, -17], [0, 0, -16, 7, -16, 0], [19, -10, 10, -19, 0, 0], [0, 21, -21, 0, 0, 1], [-25, 29, -25, 0, -20, 20], [28, -32, 33, -33, 32, -28], [-18, 18, 0, 14, -10, 14], [-32, 26, -5, 5, -26, 32], [7, -5, 7, 0, 30, -30], [13, 16, -16, -13, 0, -6], [18, -15, -6, -15, 18, 0], [0, -25, 31, -19, 31, -25], [0, 8, 6, -16, 6, 8], [13, -20, 20, -13, 0, -21], [0, -7, 2, -45, 2, -7], [-24, 1, 1, 1, -24, 0], [-2, 15, -2, 0, -13, 13], [16, -11, 11, -16, 0, 4], [-4, 0, 0, 17, -17, 0], [8, -8, 0, 3, -21, 3], [18, 0, 2, -3, 3, -2], [13, 0, -8, 8, -8, 8], [22, -22, 0, -28, 39, -28], [29, -14, 29, 0, 26, -26], [3, -17, 28, -17, 3, 0], [-1, -20, -14, -20, -1, 0], [11, 23, -6, 6, -23, -11], [0, -44, 25, -37, 25, -44], [0, -25, 4, -30, 4, -25], [0, 19, 6, -9, 6, 19], [-2, -9, 9, 2, 0, -5], [-11, 10, -6, 6, -10, 11], [2, -34, 2, 0, 12, -12], [-10, 10, 0, 0, 8, 0], [-22, 22, 0, 21, -22, 21], [7, 26, -10, 10, -26, -7], [15, -38, 9, -38, 15, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_441_u_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_441_2_u_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_441_2_u_a(:prec:=6)
    chi := MakeCharacter_441_u();
    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_441_2_u_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_441_2_u_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_441_u();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, -5, 11, -13, 9, -3, 1]>],Snew);
    return Vf;
end function;