
// Make newform 1416.1.r.b in Magma, downloaded from the LMFDB on 18 April 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_1416_r();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1416_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_1416_1_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 ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        Kf := CyclotomicField(58);
    end if;
    return [ #coeff eq Kf!0 select 0 else &+[ elt[1]*Kf.1^elt[2] : elt in coeff]  : coeff in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_1416_r();"
function MakeCharacter_1416_r()
    N := 1416;
    order := 58;
    char_gens := [1063, 709, 473, 769];
    v := [58, 29, 29, 6];
    // 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_1416_r_Hecke();"
function MakeCharacter_1416_r_Hecke( : Kf := false)
    N := 1416;
    order := 58;
    char_gens := [1063, 709, 473, 769];
    char_values := [[[1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 11]]];
    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 := 1;
    raw_aps := [[[-1, 20]], [[-1, 14]], [[-1, 16], [1, 21]], [[-1, 9], [-1, 15]], [[1, 3], [1, 11]], [], [], [], [], [[1, 19], [-1, 28]], [[-1, 21], [-1, 25]], [], [], [], [], [[-1, 4], [-1, 6]], [[-1, 24]], [], [], [], [[-1, 9], [1, 26]], [[1, 2], [1, 28]], [[1, 5], [1, 27]], [], [[1, 4], [-1, 19]], [[-1, 8], [-1, 26]], [[1, 0], [-1, 7]], [[-1, 18], [1, 25]], [], [], [[1, 10], [-1, 17]], [[1, 9], [-1, 22]], [], [], [[-1, 8], [1, 9]], [[1, 16], [1, 26]], [], [], [], [[1, 1], [1, 21]], [[-1, 4], [1, 7]], [], [], [[-1, 21], [-1, 23]], [[1, 7], [1, 23]], [[1, 12], [-1, 13]], [], [[1, 8], [-1, 23]], [[1, 1], [-1, 26]], [], [], [], [[1, 14], [-1, 23]], [[-1, 10], [-1, 26]], [], [], [[1, 15], [1, 27]], [[-1, 7], [-1, 25]], [], [], [], [[1, 15], [1, 25]], [], [], [[1, 4], [-1, 5]], [[-1, 2], [1, 23]], [], [[1, 8], [1, 26]], [[1, 3], [1, 21]], [], [], [], [[-1, 11], [1, 20]], [], [], [], [[-1, 0], [1, 3]], [], [], [[1, 2], [1, 20]], [[-1, 0], [-1, 10]], [], [], [[1, 14], [1, 16]], [[-1, 15], [-1, 27]], [[-1, 2], [-1, 16]], [], [[-1, 17], [-1, 19]], [[-1, 2], [-1, 12]], [[1, 6], [-1, 21]], [[1, 17], [-1, 20]], [], [[1, 16], [1, 20]], [[1, 19], [-1, 22]], [], [], [[1, 11], [-1, 14]], [], [], [], [], [[1, 17], [1, 25]], [[1, 27], [-1, 28]], [], [], [[-1, 3], [1, 28]], [[1, 9], [1, 19]], [], [], [[1, 4], [1, 10]], [[1, 2], [-1, 3]], [], [], [], [[1, 4], [-1, 5]], [], [], [], [[1, 9], [1, 13]], [[-1, 22], [-1, 26]], [], [[1, 0], [-1, 3]], [[1, 3], [-1, 14]], [[-1, 0], [-1, 16]], [], [[-1, 0], [-1, 24]], [], [], [[1, 14], [-1, 27]], [], [], [], [[-1, 7], [1, 8]], [], [], [[-1, 1], [1, 10]], [[1, 17], [-1, 22]], [], [[1, 1], [1, 17]], [], [], [[1, 9], [-1, 28]], [[-1, 3], [-1, 25]], [[-1, 6], [1, 23]], [], [], [], [], [], [], [], [], [], [], [], [], [[-1, 3], [-1, 13]], [], [[-1, 1], [-1, 23]], [[-1, 12], [-1, 16]], [[-1, 10], [-1, 18]], [], [[1, 6], [1, 14]], [[-1, 6], [-1, 20]], [], [], [[1, 22], [1, 28]], [], [[-1, 15], [1, 24]], [[-1, 4], [1, 15]], [[1, 1], [-1, 14]], [], [], [[1, 6], [1, 12]], [[-1, 1], [-1, 19]], [], [], [[-1, 2], [1, 27]], [[1, 4], [-1, 25]], [], [[1, 18], [-1, 27]], [[-1, 6], [-1, 12]], [], [], [], [[-1, 24], [-1, 26]], [], [], [[-1, 15], [1, 18]], [], [[-1, 7], [-1, 19]], [[1, 7], [1, 27]], [], [[-1, 14], [1, 15]], [[1, 11], [1, 13]], [], [[1, 24], [1, 28]], [], [], [], [[-1, 0], [1, 19]], [[-1, 9], [1, 24]], [], [[1, 0], [-1, 19]], [[1, 5], [1, 25]], [[1, 9], [-1, 14]], [[-1, 15], [1, 26]], [[1, 13], [1, 23]], [], [], [[1, 2], [-1, 27]], [[1, 5], [1, 23]], [[-1, 11], [1, 26]], [[-1, 2], [1, 25]], [], [[-1, 7], [-1, 13]], [[1, 8], [1, 10]], [], [], [[-1, 4], [1, 11]], [], [[-1, 13], [-1, 21]], [], [[1, 6], [1, 18]], [[1, 21], [-1, 22]], [], [], [], [[1, 22], [1, 24]], [[1, 11], [1, 21]], [], [], [[1, 4], [1, 18]], [], [], [], [[-1, 11], [1, 24]], [[-1, 16], [-1, 22]], [[1, 13], [1, 19]], [], [[-1, 0], [-1, 14]], [], [[1, 4], [-1, 23]], [], [], [], [[-1, 1], [1, 12]], [[-1, 10], [1, 15]], [], [], [], [], [], [[-1, 5], [1, 10]], [[1, 9], [1, 21]], [[1, 15], [1, 27]], [], [], [[1, 1], [-1, 6]], [[1, 16], [-1, 21]], [[1, 20], [-1, 23]], [[1, 11], [1, 25]], [], [], [], [], [[-1, 4], [1, 7]], [], [], [[-1, 6], [1, 19]], [], [], [[1, 12], [1, 22]], [[1, 0], [1, 14]], [[1, 26], [-1, 27]], [[1, 0], [1, 2]], [[-1, 14], [-1, 20]], [], [[1, 20], [1, 26]], [[1, 3], [-1, 6]], [], [[1, 4], [-1, 7]], [], [], [], [], [[-1, 9], [-1, 27]], [[1, 15], [-1, 28]], [[-1, 5], [1, 22]], [], [[1, 5], [-1, 26]], [[-1, 2], [-1, 10]], [], [[-1, 4], [1, 11]], [], [[1, 17], [1, 23]], [[1, 6], [1, 16]], [[-1, 20], [-1, 22]], [[1, 7], [1, 19]], [], [[-1, 9], [1, 22]], [[-1, 0], [1, 27]], [[-1, 5], [-1, 19]]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1416_r_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1416_1_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_1416_1_r_b(:prec:=28)
    chi := MakeCharacter_1416_r();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 1));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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;

