
// Make newform 4410.2.a.bi in Magma, downloaded from the LMFDB on 11 February 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_4410_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_4410_2_a_bi();".
// 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_4410_2_a_bi();".
// 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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_4410_a();"
function MakeCharacter_4410_a()
    N := 4410;
    order := 1;
    char_gens := [3431, 2647, 1081];
    v := [1, 1, 1];
    // 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;

function MakeCharacter_4410_a_Hecke( : Kf := Kf)
    return MakeCharacter_4410_a();
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 := 2;
    raw_aps := [[1], [0], [1], [0], [0], [-2], [-6], [-8], [0], [-6], [4], [-10], [-6], [-4], [0], [6], [-12], [10], [-4], [-12], [10], [8], [12], [-6], [10], [6], [-8], [-12], [14], [-6], [8], [-12], [-6], [16], [-6], [8], [22], [20], [0], [6], [24], [10], [-12], [2], [-18], [-20], [-4], [-8], [12], [-14], [-30], [12], [-26], [-12], [-30], [24], [-18], [-20], [-10], [-18], [4], [30], [4], [24], [-14], [-18], [-28], [2], [12], [10], [-6], [12], [-8], [-34], [-4], [24], [-6], [-2], [30], [-26], [-12], [38], [12], [10], [28], [12], [-18], [2], [-18], [-40], [-36], [0], [32], [0], [20], [0], [30], [-6], [28], [-34], [44], [-18], [36], [6], [-4], [10], [12], [-6], [-36], [-2], [-8], [38], [18], [-32], [-16], [-42], [4], [0], [6], [-48], [34], [26], [-18], [-36], [16], [-6], [-10], [-24], [-32], [-2], [-28], [24], [8], [-34], [-6], [-26], [-18], [4], [-42], [30], [16], [18], [32], [36], [34], [0], [-26], [-6], [40], [24], [14], [-30], [44], [24], [-28], [-12], [32], [-30], [-38], [-18], [36], [-30], [8], [36], [-6], [24], [8], [-26], [-46], [30], [0], [-14], [36], [26], [-20], [-6], [-28], [42], [16], [-14], [32], [12], [62], [18], [0], [6], [-34], [4], [26], [24], [10], [-12], [20], [-18], [-12], [66], [50], [-10], [18], [-48], [42], [-44], [-50], [-26], [-12], [6], [-20], [-36], [30], [40], [-22], [-42], [8], [36], [0], [-2], [-40], [18], [-24], [6], [-10], [-20], [-42], [-16], [-60], [-10], [-54], [-60], [40], [0], [-58], [-56], [-16], [30], [-68], [48], [-50], [30], [-48], [-48], [12], [16], [-8], [-34], [-30], [-48], [40], [-60], [68], [24], [14], [-30], [48], [-26], [6], [-24], [-10], [4], [78], [34], [-40], [36], [-62], [70], [-54], [-32], [42], [-54], [68], [-18], [-14], [20], [-14], [-40], [58], [16], [-12], [-82], [74], [60], [-48], [-16], [-24], [10], [76], [-60], [26], [-42], [-20], [-30], [42], [-36], [18], [60], [86], [-66], [-20], [-42], [-12], [-44], [58], [6], [-16], [-12], [-52], [50], [36], [10], [36], [-34], [-48], [-78], [-66], [-4], [0], [46], [-60], [-60], [10], [78], [-32], [-46], [78], [-16], [-6], [-26], [68], [4], [24], [-42], [-82], [6], [-68], [-60], [68], [60], [-34], [-30], [-2], [-56], [-34], [-30], [6], [8], [6], [0], [-62], [-28], [24], [-18], [40], [2], [-54], [64], [-58], [66], [24], [-84], [-6], [24], [-82], [-6], [0], [24], [52], [-46], [-66], [56], [-70], [24], [88], [-48], [90], [4], [14], [36], [-60], [-38], [-30], [-38], [-66], [18], [-16], [18], [40], [0], [-40], [22], [20], [-48], [-46], [54], [-72], [52], [-12], [-22], [-20], [-6], [44], [66], [-14], [-102], [32], [-6], [102], [4], [62], [-66], [-68], [-84], [82], [102], [60], [44], [-22], [-18], [60], [-32], [-6], [-24], [66], [-26], [-96], [84], [34], [6], [12], [54], [-56], [48]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4410_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4410_2_a_bi();".
// 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_4410_2_a_bi(:prec:=1)
    chi := MakeCharacter_4410_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 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_4410_2_a_bi();".
// 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_4410_2_a_bi( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4410_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![0, 1]>,<13,R![2, 1]>,<17,R![6, 1]>,<19,R![8, 1]>,<29,R![6, 1]>,<31,R![-4, 1]>],Snew);
    return Vf;
end function;