
// Make newform 224.3.r.b in Magma, downloaded from the LMFDB on 21 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_224_r();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_224_r_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_r_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_r_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 := [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_224_r();"
function MakeCharacter_224_r()
    N := 224;
    order := 6;
    char_gens := [127, 197, 129];
    v := [3, 6, 4];
    // 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_r_Hecke();"
function MakeCharacter_224_r_Hecke( : Kf := false)
    N := 224;
    order := 6;
    char_gens := [127, 197, 129];
    char_values := [[-1, 0, 0, 0], [1, 0, 0, 0], [0, 0, -1, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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]];
    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, 1, 0, 0], [0, 0, -1, 0], [0, 0, 0, -7], [0, -17, 0, 0], [24, 0, 0, 0], [-1, 0, 1, 0], [0, 7, 0, -7], [0, -7, 0, 7], [24, 0, 0, 0], [0, -41, 0, 0], [0, 0, 49, 0], [-48, 0, 0, 0], [0, 0, 0, 24], [0, 55, 0, -55], [25, 0, -25, 0], [0, 17, 0, 0], [0, 0, 1, 0], [0, 65, 0, 0], [0, 0, 0, 96], [-95, 0, 95, 0], [0, 41, 0, -41], [0, 0, 0, 72], [0, 0, -95, 0], [144, 0, 0, 0], [-73, 0, 73, 0], [0, 89, 0, -89], [0, -185, 0, 185], [71, 0, -71, 0], [96, 0, 0, 0], [0, 0, 0, -144], [0, 185, 0, -185], [143, 0, -143, 0], [0, 0, 0, -216], [0, 0, 47, 0], [0, 199, 0, 0], [73, 0, -73, 0], [0, -55, 0, 55], [0, 0, 0, 206], [0, 0, -239, 0], [0, -17, 0, 0], [70, 0, 0, 0], [0, 199, 0, -199], [47, 0, -47, 0], [24, 0, 0, 0], [0, -137, 0, 0], [0, 0, 0, 264], [0, 0, 0, 0], [0, 415, 0, 0], [0, 0, 143, 0], [0, 0, -289, 0], [0, 0, 0, 226], [-95, 0, 95, 0], [0, 0, 0, -38], [0, 0, -193, 0], [0, 233, 0, 0], [-455, 0, 455, 0], [0, 425, 0, -425], [-167, 0, 167, 0], [-432, 0, 0, 0], [0, 223, 0, 0], [26, 0, 0, 0], [0, 0, 0, -264], [0, -343, 0, 0], [0, 0, -335, 0], [0, 0, 529, 0], [0, -55, 0, 55], [-240, 0, 0, 0], [0, -31, 0, 0], [-120, 0, 0, 0], [527, 0, -527, 0], [0, 535, 0, -535], [0, 89, 0, 0], [0, 0, -335, 0], [0, 0, 0, 38], [0, -665, 0, 665], [-73, 0, 73, 0], [0, 0, -1, 0], [0, 0, -337, 0], [479, 0, -479, 0], [0, 0, 0, -552], [-216, 0, 0, 0], [0, 439, 0, 0], [-288, 0, 0, 0], [0, -617, 0, 617], [0, 247, 0, -247], [-288, 0, 0, 0], [0, 0, 97, 0], [312, 0, 0, 0], [0, 0, 0, 192], [0, -295, 0, 295], [0, 617, 0, 0], [0, -247, 0, 0], [0, 0, 0, 134], [0, -727, 0, 727], [0, 0, 0, -432], [0, 0, -433, 0], [-479, 0, 479, 0], [0, -905, 0, 905], [0, 0, 191, 0], [0, 0, 0, -374], [-937, 0, 937, 0], [0, -799, 0, 0], [0, 0, 529, 0], [0, 1039, 0, 0], [529, 0, -529, 0], [0, 0, 0, 840], [0, 0, 383, 0], [0, -343, 0, 0], [-624, 0, 0, 0], [0, -137, 0, 137], [71, 0, -71, 0], [384, 0, 0, 0], [0, 593, 0, 0], [0, 0, 0, 384], [-767, 0, 767, 0], [0, 0, 0, 456], [0, 89, 0, 0], [0, 0, -527, 0], [0, 0, 0, -936], [745, 0, -745, 0], [-720, 0, 0, 0], [0, 0, 1151, 0], [0, -737, 0, 0], [0, -1015, 0, 1015], [1226, 0, 0, 0], [0, 0, 625, 0], [0, 137, 0, -137], [0, 0, 0, -960], [0, 0, -479, 0], [0, 511, 0, 0], [0, 0, 0, -528], [0, -809, 0, 809], [-120, 0, 0, 0], [0, 0, -769, 0], [-144, 0, 0, 0], [407, 0, -407, 0], [0, -1135, 0, 0], [312, 0, 0, 0], [1297, 0, -1297, 0], [0, 0, 0, -1128], [0, 0, 383, 0], [0, -55, 0, 0], [0, 0, 0, 696], [-505, 0, 505, 0], [0, 0, 0, -48], [696, 0, 0, 0], [-1199, 0, 1199, 0], [0, 775, 0, -775], [0, -343, 0, 343], [0, 0, 1151, 0], [-866, 0, 0, 0], [0, 0, 0, 648], [0, 679, 0, -679], [0, 353, 0, 0], [0, 0, 0, 144], [0, -1097, 0, 1097], [0, 0, 529, 0], [146, 0, 0, 0], [-743, 0, 743, 0], [0, 521, 0, -521], [1440, 0, 0, 0], [0, 0, 0, 1920], [0, 329, 0, -329], [-721, 0, 721, 0], [0, 631, 0, 0], [0, -727, 0, 0], [25, 0, -25, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_224_r_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_224_3_r_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_r_b(:prec:=4)
    chi := MakeCharacter_224_r();
    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_r_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_r_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_224_r();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![1, 0, -1, 0, 1]>],Snew);
    return Vf;
end function;