
// Make newform 224.3.g.b in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_224_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_224_g_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_224_3_g_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_224_3_g_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 := [256, -64, -32, -8, 24, -2, -2, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 96, -160, 4, -6, 4, -5, -1], [0, -96, 0, 12, 6, -12, -3, 1], [-16, 8, 8, -2, -2, -1, 1, 0], [-128, 0, 0, -4, -14, 0, -1, -1], [-384, 128, 48, 84, -22, -4, -1, 3], [-64, -88, 0, 22, -4, -3, 0, 1], [192, 16, -8, -20, 18, 0, 1, -3]];
    Rf_basisdens := [1, 128, 128, 16, 64, 64, 16, 32];
    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_224_g();"
function MakeCharacter_224_g()
    N := 224;
    order := 2;
    char_gens := [127, 197, 129];
    v := [1, 1, 2];
    // 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_224_g_Hecke();"
function MakeCharacter_224_g_Hecke(Kf)
    N := 224;
    order := 2;
    char_gens := [127, 197, 129];
    char_values := [[-1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 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 := 3;
    raw_aps := [[0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 1, 0, 0, 0, 0], [0, -1, 1, 0, 1, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0], [4, 0, 0, 0, 0, 0, 1, 1], [0, 1, -1, 0, 3, 0, 0, 0], [-10, 0, 0, 4, 0, 0, 0, 0], [-7, 0, 0, 1, 0, 0, 0, 0], [0, 3, -5, 0, 0, -2, 0, 0], [0, 6, 2, 0, -4, -2, 0, 0], [0, 6, 6, 0, 0, 0, 0, 0], [0, -2, -14, 0, 4, 2, 0, 0], [16, 0, 0, 2, 0, 0, -4, -2], [0, 0, 0, 4, 0, 0, 1, -3], [0, 4, 4, 0, -8, 4, 0, 0], [0, 4, -4, 0, -6, -2, 0, 0], [-13, 0, 0, -13, 0, 0, -2, 6], [0, -3, 19, 0, -5, 0, 0, 0], [-38, 0, 0, 6, 0, 0, -1, 3], [0, -8, 0, 0, -16, 0, 0, 0], [-14, 0, 0, 0, 0, 0, 4, 4], [0, 16, -8, 0, -12, 4, 0, 0], [-9, 0, 0, -1, 0, 0, -6, 2], [-64, 0, 0, -6, 0, 0, -4, -6], [8, 0, 0, -6, 0, 0, -8, 2], [0, -23, -25, 0, 7, -4, 0, 0], [0, 2, 10, 0, 16, 0, 0, 0], [-86, 0, 0, -2, 0, 0, 3, 3], [0, 10, 22, 0, 12, -2, 0, 0], [17, 0, 0, -19, 0, 0, 8, 3], [0, -1, 23, 0, 20, 2, 0, 0], [59, 0, 0, 3, 0, 0, 6, -2], [16, 0, 0, -6, 0, 0, 0, 6], [93, 0, 0, 21, 0, 0, -4, 4], [0, 4, 20, 0, -2, 14, 0, 0], [0, -27, 13, 0, 40, 2, 0, 0], [0, 21, 11, 0, 7, -16, 0, 0], [-24, 0, 0, 20, 0, 0, 13, 5], [0, 36, -4, 0, -24, -12, 0, 0], [0, 21, 11, 0, 7, 8, 0, 0], [-26, 0, 0, -14, 0, 0, -5, 7], [0, -15, -33, 0, 35, 4, 0, 0], [0, -16, 56, 0, -20, 4, 0, 0], [25, 0, 0, 37, 0, 0, 0, -5], [0, -4, -28, 0, -22, -10, 0, 0], [0, -10, -34, 0, 32, 24, 0, 0], [62, 0, 0, 10, 0, 0, -5, -1], [0, -18, 30, 0, 40, -12, 0, 0], [29, 0, 0, -11, 0, 0, 4, -4], [0, -13, -19, 0, 1, -24, 0, 0], [56, 0, 0, 50, 0, 0, -8, -2], [0, -11, -19, 0, -36, -2, 0, 0], [96, 0, 0, 18, 0, 0, -8, 2], [131, 0, 0, -37, 0, 0, -26, -18], [-98, 0, 0, 4, 0, 0, 0, 20], [0, -12, 76, 0, 8, -16, 0, 0], [0, 33, -1, 0, -53, 0, 0, 0], [0, -10, -58, 0, 16, 20, 0, 0], [0, -8, 0, 0, -6, 38, 0, 0], [130, 0, 0, -32, 0, 0, 16, 16], [-43, 0, 0, 53, 0, 0, 4, -12], [0, -39, -41, 0, 23, 12, 0, 0], [-85, 0, 0, -29, 0, 0, 14, -2], [0, -4, -12, 0, -48, -32, 0, 0], [-88, 0, 0, -22, 0, 0, 12, 38], [0, 48, -80, 0, -14, -6, 0, 0], [-48, 0, 0, 4, 0, 0, -23, -35], [131, 0, 0, 3, 0, 0, -24, -35], [-76, 0, 0, 8, 0, 0, -11, 9], [0, 55, -87, 0, -11, -20, 0, 0], [64, 0, 0, 90, 0, 0, 16, 6], [0, -41, -33, 0, 64, -18, 0, 0], [0, -10, -42, 0, 8, 44, 0, 0], [0, -52, 12, 0, 78, -18, 0, 0], [-80, 0, 0, 36, 0, 0, 1, 29], [0, 32, 128, 0, 32, 12, 0, 0], [0, -30, 158, 0, 56, 10, 0, 0], [0, -29, -3, 0, 57, 12, 0, 0], [-21, 0, 0, -117, 0, 0, -16, 13], [110, 0, 0, -84, 0, 0, 36, 20], [-195, 0, 0, 13, 0, 0, -4, -44], [0, 24, -32, 0, -22, -18, 0, 0], [0, 27, -45, 0, -4, 42, 0, 0], [66, 0, 0, -56, 0, 0, 0, -36], [0, -14, -6, 0, -8, -68, 0, 0], [-142, 0, 0, 78, 0, 0, -9, -13], [-184, 0, 0, -46, 0, 0, -32, 6], [-15, 0, 0, 93, 0, 0, 24, 35], [0, -57, -71, 0, 29, 68, 0, 0], [0, -40, -64, 0, 68, -12, 0, 0], [-55, 0, 0, -47, 0, 0, 16, 32], [0, 138, 42, 0, -32, 0, 0, 0], [0, -17, 55, 0, -64, 6, 0, 0], [-450, 0, 0, -22, 0, 0, 3, 7], [138, 0, 0, -130, 0, 0, -13, 27], [0, 114, -6, 0, -8, 4, 0, 0], [0, -5, -155, 0, -15, 20, 0, 0], [-290, 0, 0, -4, 0, 0, -20, -36], [159, 0, 0, -17, 0, 0, -54, -22], [0, -80, 8, 0, -106, -22, 0, 0], [332, 0, 0, -136, 0, 0, -31, -15], [0, -44, -180, 0, 62, 10, 0, 0], [-89, 0, 0, 55, 0, 0, -22, -6], [19, 0, 0, 83, 0, 0, -16, 13], [208, 0, 0, -124, 0, 0, 49, 29], [16, 0, 0, 42, 0, 0, -8, -66], [565, 0, 0, -35, 0, 0, 36, 44], [562, 0, 0, 48, 0, 0, 32, -32], [0, 52, -84, 0, -144, -8, 0, 0], [-224, 0, 0, -70, 0, 0, -20, -6], [0, -64, 96, 0, -48, -24, 0, 0], [0, 62, 74, 0, -24, 34, 0, 0], [547, 0, 0, -45, 0, 0, -24, -27], [89, 0, 0, -71, 0, 0, -8, -32], [0, -28, -20, 0, -136, -24, 0, 0], [-113, 0, 0, 55, 0, 0, -16, -47], [-297, 0, 0, 103, 0, 0, 10, -22], [0, 26, 18, 0, -40, -24, 0, 0], [0, 46, -22, 0, -44, -58, 0, 0], [172, 0, 0, 232, 0, 0, 9, 17], [0, 37, -69, 0, -21, 28, 0, 0], [162, 0, 0, 0, 0, 0, 40, 80], [0, 25, 327, 0, -5, -4, 0, 0], [402, 0, 0, 86, 0, 0, 11, 27], [35, 0, 0, -53, 0, 0, -74, -42], [0, 114, -90, 0, -8, 22, 0, 0], [0, 126, -78, 0, -180, 82, 0, 0], [0, -88, -168, 0, 128, 36, 0, 0], [0, -188, -4, 0, 176, 20, 0, 0], [0, -159, 63, 0, 191, -24, 0, 0], [-720, 0, 0, -4, 0, 0, 5, -3], [0, -25, 191, 0, 56, 54, 0, 0], [0, 35, -149, 0, 20, 34, 0, 0], [0, -138, -142, 0, 56, -102, 0, 0], [68, 0, 0, -66, 0, 0, -20, 2], [388, 0, 0, 14, 0, 0, 0, 38], [0, 7, 233, 0, 145, 32, 0, 0], [473, 0, 0, 33, 0, 0, 72, -16], [0, -175, -113, 0, -21, -24, 0, 0], [149, 0, 0, 201, 0, 0, 24, 55], [829, 0, 0, 29, 0, 0, -4, 28], [0, -76, 100, 0, 142, -98, 0, 0], [0, 120, -32, 0, -104, 48, 0, 0], [278, 0, 0, 122, 0, 0, -1, -89], [0, 77, 211, 0, -149, 16, 0, 0], [0, 98, -230, 0, 88, -48, 0, 0], [0, -173, -163, 0, 153, 0, 0, 0], [254, 0, 0, 44, 0, 0, -12, 12], [-361, 0, 0, 15, 0, 0, -38, 10], [0, -244, -44, 0, 92, 44, 0, 0], [0, -26, 90, 0, 32, 38, 0, 0], [588, 0, 0, 78, 0, 0, 0, -54], [738, 0, 0, 134, 0, 0, -21, 51], [0, -32, 168, 0, -264, -12, 0, 0], [130, 0, 0, -146, 0, 0, -57, -45], [0, 129, -103, 0, -100, 110, 0, 0], [0, 16, -248, 0, -96, 8, 0, 0], [-278, 0, 0, -208, 0, 0, 48, 28], [-110, 0, 0, -80, 0, 0, -108, -20], [0, -25, 73, 0, -7, -60, 0, 0], [140, 0, 0, -56, 0, 0, 9, -63], [-874, 0, 0, 44, 0, 0, 32, 4], [0, 43, -349, 0, 88, -10, 0, 0], [-589, 0, 0, 67, 0, 0, 102, 54], [-184, 0, 0, -158, 0, 0, -48, 14], [0, -226, 6, 0, 200, 72, 0, 0], [0, -20, 68, 0, -60, 84, 0, 0], [0, 55, -231, 0, 41, 64, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_224_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_224_3_g_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_224_3_g_b(:prec:=8)
    chi := MakeCharacter_224_g();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    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_224_3_g_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_224_3_g_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_224_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![8, 80, -22, -4, 1]>],Snew);
    return Vf;
end function;