
// Make newform 416.3.t.b in Magma, downloaded from the LMFDB on 19 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_416_t();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_416_t_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_416_3_t_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_416_3_t_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];
        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_416_t();"
function MakeCharacter_416_t()
    N := 416;
    order := 4;
    char_gens := [287, 261, 353];
    v := [4, 4, 3];
    // 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_416_t_Hecke();"
function MakeCharacter_416_t_Hecke( : Kf := false)
    N := 416;
    order := 4;
    char_gens := [287, 261, 353];
    char_values := [[1, 0], [1, 0], [0, 1]];
    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], [1, 1], [0, 0], [0, 0], [-12, 5], [0, 30], [0, 0], [0, 0], [-40, 0], [0, 0], [-47, 47], [31, 31], [0, 0], [0, 0], [-90, 0], [0, 0], [22, 0], [0, 0], [0, 0], [-7, 7], [0, 0], [0, 0], [119, -119], [-7, -7], [0, -40], [0, 0], [0, 0], [-151, -151], [224, 0], [0, 0], [0, 0], [17, -17], [0, 0], [89, 89], [0, 0], [264, 0], [0, 0], [0, 0], [0, -104], [0, 0], [0, -38], [0, 0], [-263, 263], [167, 167], [0, 0], [0, 0], [0, 0], [0, 0], [-281, 281], [0, -416], [0, 0], [-329, 329], [0, 0], [0, -64], [0, 0], [520, 0], [0, 0], [0, 504], [-391, 391], [0, 0], [-353, 353], [0, 0], [0, 0], [-624, 0], [-383, -383], [0, 0], [0, 350], [0, 0], [-119, 119], [497, 497], [0, 0], [0, 0], [504, 0], [0, 0], [0, 0], [0, -378], [-553, 553], [359, -359], [-511, -511], [0, 0], [-449, -449], [0, 0], [0, 816], [0, 0], [0, 0], [71, -71], [593, 593], [119, 119], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-239, -239], [880, 0], [0, 0], [-79, 79], [0, 0], [-367, 367], [0, 0], [0, -1040], [0, 0], [623, 623], [0, 0], [-97, 97], [0, 0], [-480, 0], [0, 0], [647, 647], [713, 713], [0, 0], [0, 0], [0, 1218], [0, 0], [0, 0], [-1144, 0], [0, 0], [-889, 889], [0, -770], [1350, 0], [0, 0], [0, 0], [0, 1302], [-919, 919], [0, 0], [0, 0], [-833, -833], [0, 0], [0, 0], [0, 0], [1190, 0], [-799, 799], [-1081, -1081], [-943, -943], [0, 0], [0, 1110], [560, 0], [0, 0], [-271, -271], [0, 0], [0, 0], [0, 1080], [0, 0], [1033, -1033], [0, 464], [0, 0], [0, 0], [1153, 1153], [0, -1600], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-1049, -1049], [430, 0], [-161, -161], [0, 0], [0, 1456], [0, 0], [0, 0], [697, 697], [0, 0], [0, 0], [1850, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_416_t_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_416_3_t_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_416_3_t_b(:prec:=2)
    chi := MakeCharacter_416_t();
    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_416_3_t_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_416_3_t_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_416_t();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![0, 1]>,<5,R![2, -2, 1]>],Snew);
    return Vf;
end function;