
// Make newform 224.3.d.b in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_224_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_224_d_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_d_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_d_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 := [4, 0, 56, 0, 53, 0, 14, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 0], [8, 0, 24, 0, 11, 0, 1, 0], [0, 42, 0, 51, 0, 14, 0, 1], [12, 0, 38, 0, 13, 0, 1, 0], [0, -32, 0, -42, 0, -13, 0, -1], [16, 0, 11, 0, 1, 0, 0, 0], [0, -110, 0, -77, 0, -16, 0, -1]];
    Rf_basisdens := [1, 1, 2, 4, 2, 2, 1, 2];
    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_d();"
function MakeCharacter_224_d()
    N := 224;
    order := 2;
    char_gens := [127, 197, 129];
    v := [1, 2, 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_d_Hecke();"
function MakeCharacter_224_d_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], [0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0], [0, 0, 0, -2, 0, -1, 0, 1], [4, 0, 0, 0, 1, 0, 0, 0], [-2, 0, 0, 0, 0, 0, 2, 0], [0, 3, 0, 0, 0, 2, 0, 0], [0, -6, 0, -2, 0, 2, 0, 1], [-10, 0, 0, 0, 2, 0, 0, 0], [0, 6, 0, -4, 0, 0, 0, -2], [-22, 0, 6, 0, 0, 0, -2, 0], [18, 0, 6, 0, 0, 0, 2, 0], [0, 0, 0, 10, 0, 7, 0, 3], [0, 6, 0, 0, 0, -4, 0, 4], [6, 0, 6, 0, -2, 0, -2, 0], [0, 3, 0, -4, 0, -8, 0, 2], [-24, 0, 0, 0, 7, 0, -4, 0], [0, 6, 0, 18, 0, -13, 0, -1], [0, 0, 0, 4, 0, 4, 0, -6], [34, 0, 0, 0, 2, 0, -6, 0], [0, -12, 0, 8, 0, 18, 0, 0], [0, 3, 0, 28, 0, -14, 0, 2], [-10, 0, -6, 0, 0, 0, -4, 0], [66, 0, 6, 0, -6, 0, -4, 0], [-16, 0, 0, 0, -5, 0, 12, 0], [0, -6, 0, 28, 0, -4, 0, -6], [0, 18, 0, -22, 0, 1, 0, 3], [26, 0, 18, 0, 16, 0, 2, 0], [-20, 0, -3, 0, 7, 0, -7, 0], [0, 6, 0, -2, 0, 4, 0, 9], [0, -9, 0, -12, 0, 34, 0, 6], [58, 0, 0, 0, -12, 0, 0, 0], [0, 33, 0, 0, 0, -18, 0, 0], [-30, 0, -12, 0, -16, 0, 12, 0], [0, -30, 0, 10, 0, 2, 0, 3], [84, 0, 0, 0, -11, 0, -4, 0], [0, 0, 0, 10, 0, -7, 0, -5], [0, 42, 0, 8, 0, -36, 0, -4], [-76, 0, 12, 0, -15, 0, 12, 0], [0, -6, 0, 34, 0, -25, 0, -1], [36, 0, -12, 0, 29, 0, 8, 0], [0, -24, 0, -68, 0, 42, 0, -2], [32, 0, 15, 0, 1, 0, -9, 0], [6, 0, -6, 0, 30, 0, 10, 0], [0, -42, 0, 20, 0, 0, 0, -2], [0, 6, 0, 42, 0, 39, 0, -13], [0, 24, 0, 60, 0, -12, 0, -14], [0, 27, 0, -48, 0, -38, 0, -16], [-4, 0, -12, 0, 1, 0, -24, 0], [-94, 0, 0, 0, 4, 0, -12, 0], [0, -18, 0, -30, 0, -28, 0, -9], [18, 0, -18, 0, -2, 0, 0, 0], [0, 21, 0, -36, 0, 2, 0, 10], [50, 0, 24, 0, -20, 0, -4, 0], [0, -48, 0, 36, 0, 32, 0, 2], [-44, 0, 12, 0, -7, 0, 4, 0], [0, 36, 0, 20, 0, -24, 0, -18], [18, 0, -24, 0, -8, 0, -24, 0], [66, 0, -24, 0, 36, 0, -4, 0], [0, 15, 0, 88, 0, -16, 0, 12], [-176, 0, 12, 0, -1, 0, -12, 0], [0, -57, 0, 12, 0, 42, 0, -14], [0, 24, 0, 24, 0, 68, 0, 0], [-230, 0, 18, 0, -4, 0, 14, 0], [-146, 0, 30, 0, 2, 0, 14, 0], [0, 0, 0, -54, 0, -53, 0, -37], [80, 0, -33, 0, -43, 0, 27, 0], [0, -24, 0, -38, 0, 27, 0, -21], [-4, 0, 0, 0, 5, 0, -16, 0], [62, 0, -54, 0, -38, 0, -22, 0], [0, 54, 0, 22, 0, 42, 0, -11], [0, 120, 0, -60, 0, -16, 0, -2], [370, 0, 12, 0, 32, 0, -20, 0], [0, -36, 0, -54, 0, -5, 0, 27], [0, 102, 0, 96, 0, -76, 0, 12], [-262, 0, 6, 0, -8, 0, 30, 0], [-300, 0, 12, 0, -7, 0, 28, 0], [244, 0, 21, 0, -21, 0, -23, 0], [134, 0, -24, 0, -26, 0, 20, 0], [0, 21, 0, -48, 0, 96, 0, 0], [202, 0, 36, 0, -44, 0, 28, 0], [0, -30, 0, -122, 0, 8, 0, 13], [-562, 0, -12, 0, -40, 0, -2, 0], [0, -60, 0, -76, 0, -8, 0, 10], [0, 66, 0, 30, 0, -67, 0, 1], [290, 0, 0, 0, 32, 0, -8, 0], [40, 0, 27, 0, 41, 0, 31, 0], [100, 0, -24, 0, -23, 0, 40, 0], [0, 60, 0, 188, 0, 38, 0, -26], [0, -33, 0, 0, 0, -38, 0, -24], [0, 6, 0, 12, 0, 68, 0, -18], [0, -102, 0, -98, 0, 18, 0, 9], [0, 66, 0, 150, 0, -35, 0, 13], [0, -6, 0, -34, 0, -93, 0, -31], [0, 48, 0, 60, 0, 48, 0, -18], [388, 0, -36, 0, -15, 0, -36, 0], [-346, 0, 36, 0, 18, 0, -12, 0], [0, -57, 0, 236, 0, 64, 0, 26], [42, 0, 0, 0, -76, 0, 56, 0], [0, 84, 0, -190, 0, 13, 0, -17], [-26, 0, 42, 0, -66, 0, -30, 0], [0, 45, 0, 212, 0, -8, 0, -26], [-196, 0, -15, 0, -9, 0, -11, 0], [0, -132, 0, -62, 0, -49, 0, 23], [70, 0, 6, 0, 2, 0, 18, 0], [0, -27, 0, 0, 0, 42, 0, 0], [146, 0, 60, 0, 40, 0, -12, 0], [0, -48, 0, 160, 0, 24, 0, 36], [158, 0, 6, 0, 20, 0, 0, 0], [0, -24, 0, 8, 0, 76, 0, 20], [-114, 0, 84, 0, 14, 0, -12, 0], [56, 0, -93, 0, 17, 0, -17, 0], [0, -105, 0, 72, 0, 20, 0, -12], [0, 132, 0, -200, 0, 28, 0, 24], [-252, 0, -27, 0, -101, 0, -15, 0], [0, -27, 0, -252, 0, 24, 0, 6], [0, 6, 0, 68, 0, 52, 0, 46], [-290, 0, -24, 0, 62, 0, -32, 0], [0, -60, 0, 90, 0, -3, 0, -5], [128, 0, -36, 0, -21, 0, 24, 0], [82, 0, -12, 0, 100, 0, 32, 0], [244, 0, 36, 0, 81, 0, 4, 0], [0, -138, 0, -86, 0, 77, 0, -29], [0, -33, 0, -4, 0, 54, 0, 50], [134, 0, 12, 0, -78, 0, -4, 0], [338, 0, -54, 0, 84, 0, 42, 0], [0, -54, 0, -96, 0, 52, 0, -4], [0, -138, 0, -120, 0, 120, 0, 4], [-464, 0, -60, 0, -37, 0, -40, 0], [0, 0, 0, -246, 0, 25, 0, -37], [0, -186, 0, 270, 0, 54, 0, 33], [0, -6, 0, -18, 0, -52, 0, 1], [-426, 0, -24, 0, -78, 0, 32, 0], [122, 0, -18, 0, 124, 0, 2, 0], [-686, 0, 18, 0, 46, 0, 76, 0], [-520, 0, -48, 0, -17, 0, -12, 0], [0, 15, 0, -72, 0, -58, 0, 36], [580, 0, 36, 0, 97, 0, -20, 0], [168, 0, -45, 0, 25, 0, 55, 0], [0, 123, 0, 200, 0, -64, 0, 20], [-70, 0, 60, 0, 52, 0, 28, 0], [0, 36, 0, -164, 0, -216, 0, 6], [0, 162, 0, -6, 0, 105, 0, 19], [752, 0, 12, 0, 55, 0, -20, 0], [0, -6, 0, -188, 0, -140, 0, -58], [-300, 0, -48, 0, -51, 0, -52, 0], [598, 0, -120, 0, -22, 0, 48, 0], [0, 165, 0, -20, 0, 54, 0, -22], [0, 24, 0, -28, 0, 114, 0, -38], [-86, 0, 54, 0, 0, 0, -74, 0], [6, 0, 66, 0, 2, 0, -42, 0], [0, -138, 0, -2, 0, -53, 0, -7], [0, -210, 0, 120, 0, 92, 0, -12], [0, 102, 0, -2, 0, -99, 0, 17], [0, -30, 0, -222, 0, -84, 0, 15], [0, -156, 0, 332, 0, -44, 0, 22], [254, 0, -96, 0, -108, 0, 70, 0], [370, 0, 144, 0, 14, 0, 46, 0], [-320, 0, -36, 0, -49, 0, -28, 0], [0, -36, 0, 266, 0, 141, 0, -29], [-326, 0, -12, 0, -68, 0, -28, 0], [0, -198, 0, -202, 0, 206, 0, -43], [0, 171, 0, -92, 0, -208, 0, -26], [2, 0, -60, 0, -68, 0, -48, 0], [0, 138, 0, 148, 0, 44, 0, 38], [0, 24, 0, 152, 0, -230, 0, 24], [-144, 0, 84, 0, -113, 0, -12, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_224_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_224_3_d_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_d_b(:prec:=8)
    chi := MakeCharacter_224_d();
    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_d_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_d_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_224_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![1024, 0, 3584, 0, 848, 0, 56, 0, 1]>],Snew);
    return Vf;
end function;