
// Make newform 5610.2.a.bx in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5610_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_5610_2_a_bx();".
// 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_5610_2_a_bx();".
// 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 := [-8, -1, 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_5610_a();"
function MakeCharacter_5610_a()
    N := 5610;
    order := 1;
    char_gens := [1871, 3367, 1531, 3301];
    v := [1, 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_5610_a_Hecke(Kf)
    return MakeCharacter_5610_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+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 := 2;
    raw_aps := [[1, 0], [1, 0], [1, 0], [0, 1], [1, 0], [-2, 2], [1, 0], [4, 0], [0, -1], [6, -1], [8, -1], [-10, 0], [-6, -2], [4, 1], [0, -4], [6, -2], [-4, -2], [-2, 0], [4, 0], [8, -2], [-6, 0], [0, -4], [-12, 0], [10, 2], [-6, -1], [6, 0], [0, -1], [4, 3], [-2, -2], [2, -4], [0, 6], [4, -4], [2, -5], [4, 3], [6, -2], [8, 4], [-2, 4], [-4, -1], [8, -6], [-2, 4], [-12, -2], [-18, 1], [-8, 1], [-14, 4], [6, 6], [-8, 0], [-4, -3], [-8, -1], [-4, 3], [-10, 2], [2, -5], [0, 6], [-6, -1], [-20, 4], [10, 1], [-16, 5], [14, 0], [8, -3], [22, 0], [2, 5], [12, 2], [14, 5], [4, 0], [8, -4], [2, 5], [-10, 3], [-12, 9], [2, -8], [-4, 8], [-2, 0], [-6, -5], [-8, -6], [-16, -2], [-10, -4], [28, 0], [16, -4], [6, -12], [-2, 0], [-6, -1], [10, 6], [28, -1], [-10, -2], [-8, 3], [34, -2], [-8, -6], [-12, -3], [26, 1], [10, 0], [14, -4], [16, 0], [4, 8], [-8, 13], [8, -2], [-20, 4], [-12, 4], [-24, 8], [-2, 4], [-6, 8], [-28, 1], [14, -6], [-12, 6], [6, -9], [-12, 0], [-6, 8], [-4, 8], [34, 4], [-12, 11], [-14, 4], [0, -9], [-22, 4], [16, 0], [6, -16], [26, -4], [-4, 6], [8, -6], [-38, 5], [-20, 9], [8, -10], [6, 3], [28, 1], [-26, 6], [2, -6], [-10, 0], [-20, 4], [20, 8], [-2, 4], [-10, 12], [-16, -10], [16, -9], [-18, 6], [4, 2], [-8, 4], [-24, 1], [30, -3], [-14, -9], [2, 4], [22, -2], [-12, -4], [46, -2], [-22, -8], [12, -4], [14, 1], [24, -2], [4, -3], [-2, 8], [8, 2], [-18, -5], [34, -7], [-12, -1], [16, -8], [-26, -5], [-6, 7], [20, -16], [8, -20], [20, -15], [-32, 8], [32, -5], [-6, 1], [10, 0], [-2, 0], [36, -2], [-38, 4], [-24, -2], [-4, -12], [-14, 0], [-32, 7], [-8, 13], [-18, -3], [10, 5], [-18, 3], [20, -9], [14, -2], [-8, 4], [-14, 9], [16, 12], [-22, 4], [-4, 8], [38, -4], [-40, -4], [38, -5], [-8, -1], [-28, -4], [-10, 12], [42, 4], [0, 12], [-10, 16], [-10, 19], [4, -16], [-14, 3], [-48, 0], [-38, 3], [-52, 0], [12, 11], [-10, 7], [-4, -17], [-22, 18], [2, 12], [46, 0], [-6, 11], [32, -5], [-2, 8], [16, 10], [30, -7], [18, -10], [-20, 2], [14, 14], [16, -14], [-28, -8], [26, -10], [-44, 5], [-14, -6], [-34, -3], [-24, 12], [-12, 9], [-40, 0], [-6, -10], [-16, 14], [10, -1], [40, -8], [-34, -4], [30, -5], [8, 12], [18, 6], [48, -2], [20, 8], [-2, 17], [-22, 4], [0, 18], [40, 2], [-4, -18], [-10, 21], [-4, 5], [24, 9], [-14, 13], [-20, -12], [8, 15], [2, 2], [-10, 6], [4, -7], [-8, 16], [-12, -8], [-36, 6], [-32, 3], [14, -16], [-38, -7], [16, -11], [0, -6], [-20, -11], [44, 6], [-40, -3], [-18, 10], [18, -6], [-8, -16], [10, 4], [-26, -13], [12, 7], [-58, 4], [-52, -6], [-2, -9], [-6, 0], [16, -12], [-12, 10], [6, 4], [6, -7], [-30, 6], [-20, 1], [6, -17], [10, -14], [-52, 6], [30, -13], [-42, 7], [4, 2], [26, -8], [56, -3], [-38, -7], [-8, -20], [12, 16], [-26, 9], [42, -8], [-12, -16], [8, -19], [56, 8], [-24, -12], [46, 9], [-36, -2], [-32, -8], [-14, 8], [-26, 8], [40, 6], [-22, -7], [14, -4], [20, 8], [-54, 10], [-4, -22], [38, -9], [46, -16], [-16, 6], [70, -2], [44, -14], [44, -1], [-46, 5], [54, 5], [-16, 6], [-12, 0], [-4, -12], [34, -8], [-60, -3], [-2, 0], [48, -3], [70, -4], [-48, -6], [54, 8], [18, 10], [4, -16], [-16, 21], [-70, 6], [4, 16], [16, 0], [2, 8], [10, 15], [-52, 7], [2, -3], [-50, 6], [40, 13], [10, 8], [26, 13], [4, -12], [-44, 7], [-48, -4], [6, 2], [-18, -8], [22, -5], [-32, -14], [4, 0], [12, -16], [-4, -12], [14, -30], [10, 13], [26, -6], [-16, 18], [-10, -12], [26, 8], [-2, 25], [56, 6], [46, -10], [-12, -6], [86, 0], [-84, 4], [8, 21], [46, 7], [52, 0], [-22, 28], [38, 7], [32, -12], [54, 8], [26, -16], [24, 13], [-20, 0], [-22, 15], [-40, 2], [-2, -9], [10, 0], [32, 4], [12, -10], [36, 12], [-6, 2], [62, 8], [-56, -6], [42, 12], [-28, -2], [-20, 0], [-16, -6], [22, -8], [-32, -1], [30, -10], [20, 16], [-40, 5], [-54, -3], [-14, -8], [58, -16], [-82, -2], [50, 9], [8, -16], [58, 7], [-20, -3], [-16, -5], [32, -10], [-2, -15], [12, 14], [16, -20], [-46, 10], [-18, -15], [-20, -10], [-60, -10], [8, 8], [26, 6], [48, 4], [42, -4], [-4, 6], [38, 18], [46, 6], [-30, -8], [-32, -8], [26, 18], [-26, 16], [8, 6], [-18, 32], [18, 12], [-60, -2], [28, 11], [2, 8], [86, 0], [-20, 16], [28, -13], [-46, 5], [-18, -4], [-16, 0], [-40, -12], [50, 6], [-40, 8], [-34, -4], [-58, 12], [-32, 8], [28, 16], [10, -4], [-18, 0], [-12, 24], [-6, -12], [4, -15], [-32, 19]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5610_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5610_2_a_bx();".
// 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_5610_2_a_bx(:prec:=2)
    chi := MakeCharacter_5610_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_5610_2_a_bx();".
// 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_5610_2_a_bx( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5610_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-8, -1, 1]>,<13,R![-32, 2, 1]>,<19,R![-4, 1]>,<23,R![-8, 1, 1]>],Snew);
    return Vf;
end function;