
// Make newform 5610.2.a.bq in Magma, downloaded from the LMFDB on 19 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_bq();".
// 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_bq();".
// 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], [-4, 0], [1, 0], [4, 0], [-4, -1], [-4, 1], [-2, 3], [4, -2], [6, -2], [-8, 1], [6, -2], [0, 0], [8, -2], [2, -4], [4, 0], [0, -2], [-6, 0], [0, 0], [-4, 4], [2, 2], [-6, 5], [2, 0], [16, -1], [8, -1], [2, -6], [-2, -4], [-2, 4], [8, 0], [-6, -3], [4, 1], [10, -2], [2, -6], [10, 0], [4, 5], [10, -4], [0, 6], [4, 2], [-12, 1], [10, 1], [6, 4], [-4, 4], [-10, 2], [-12, -1], [4, 3], [12, -5], [-6, -2], [6, 3], [-16, 6], [6, -7], [12, 0], [-2, 3], [-12, 3], [14, 0], [-10, -1], [6, -4], [2, -5], [-20, 2], [10, 3], [-12, 0], [20, 4], [2, -1], [6, 3], [20, -3], [-2, 4], [8, -4], [-16, 6], [22, 1], [20, -2], [10, 4], [12, -10], [0, 0], [2, 2], [16, -2], [8, -6], [18, -5], [-2, -6], [36, -1], [-30, -2], [-6, 5], [14, -2], [28, -2], [8, -1], [10, 5], [2, 8], [14, -4], [8, 0], [12, -8], [-26, 3], [-2, -4], [28, 4], [36, 0], [10, -10], [16, -2], [2, 0], [-4, -3], [-18, 6], [12, -2], [-18, 5], [20, 4], [6, 0], [4, 0], [-2, 12], [-8, 9], [14, 8], [34, 3], [10, 8], [8, 0], [-12, 2], [-14, 12], [4, -6], [20, -6], [-26, -3], [8, -1], [-14, 8], [26, -1], [12, -1], [22, 2], [-10, 6], [-28, -2], [28, -8], [8, 0], [6, 0], [-16, -2], [-8, 6], [-4, 3], [-4, 4], [16, 6], [2, -14], [10, -3], [-18, 1], [-2, -11], [-2, 8], [-20, -4], [4, -4], [8, -4], [14, 8], [4, -4], [-24, -1], [10, 0], [-20, 13], [-2, 8], [20, -10], [6, -3], [2, 5], [4, -13], [-14, 10], [-22, 1], [-22, 3], [-12, 4], [22, -10], [36, -5], [0, -12], [38, -3], [22, 1], [6, -12], [-8, -2], [-28, -2], [-34, 8], [-14, 12], [8, 12], [2, -4], [-16, 3], [10, 1], [14, 3], [-14, -9], [-26, -5], [-28, -7], [-42, 2], [-6, -6], [-26, 5], [-32, 0], [-22, -4], [-12, 0], [-34, 0], [-58, -2], [12, 1], [-20, -5], [-24, 4], [-20, 2], [-6, -8], [2, 6], [-8, 10], [2, 9], [-36, 0], [26, 1], [-6, 14], [6, -3], [16, -4], [20, -1], [32, -13], [20, -1], [-10, 6], [-42, -4], [-48, 6], [-10, 3], [8, 9], [-36, 10], [-44, -6], [-22, 17], [10, 6], [24, 2], [-16, 8], [-32, -2], [-12, -4], [-30, 6], [-12, 9], [18, -2], [-36, 9], [38, -10], [52, -1], [2, 14], [-34, 2], [-14, -12], [-58, -3], [8, -16], [4, -10], [48, -3], [-8, 24], [38, -2], [34, -12], [20, -8], [0, -13], [-10, -16], [16, -2], [-58, 4], [-32, -10], [10, 5], [-12, 23], [2, 7], [18, 3], [36, -16], [40, -11], [22, -10], [8, -4], [52, 1], [62, 2], [20, -8], [-4, -2], [0, -13], [34, 4], [-10, 25], [-66, 3], [-10, 0], [-44, 5], [56, -6], [12, -21], [20, -4], [-54, -6], [30, 6], [26, 12], [2, -19], [28, 1], [-8, 14], [20, -18], [-18, -9], [-46, 0], [12, 0], [28, -2], [14, -12], [-14, 3], [-14, -14], [-12, -11], [24, 11], [-6, 14], [12, 6], [-34, 17], [40, 7], [28, -18], [6, 0], [-10, 23], [46, 5], [-24, -4], [0, 12], [-20, 23], [-6, 16], [-20, -16], [-20, 19], [34, 6], [-18, 14], [24, 3], [-20, 2], [-30, 6], [34, 12], [-4, 26], [-36, -2], [-22, -1], [-2, -4], [-60, 8], [22, 10], [-52, -10], [-26, -15], [24, 6], [32, -18], [24, 12], [36, -6], [8, 7], [18, -19], [-22, 3], [-16, 6], [-28, -12], [-12, 20], [10, -12], [0, -9], [8, -6], [22, -23], [64, 2], [-46, -4], [-22, -16], [-2, -14], [28, -20], [-40, 7], [18, 14], [24, 12], [-14, 22], [-2, 8], [54, -11], [28, 5], [26, -17], [34, -2], [-16, 29], [-62, 0], [6, 9], [-56, -4], [16, -23], [-50, 10], [60, 0], [-42, 0], [10, -1], [-40, 6], [60, -12], [4, 16], [20, 8], [6, 6], [6, 13], [18, -6], [-70, 0], [6, -12], [10, -12], [52, 11], [-36, -14], [56, 4], [4, 22], [4, -10], [92, 0], [14, 11], [-38, 11], [-44, 12], [-30, -16], [64, 3], [-8, -12], [-8, 22], [-22, 16], [-46, 13], [-16, -20], [-62, 3], [-10, 4], [-30, -11], [-78, 4], [-30, 18], [-16, 26], [28, -16], [-10, -18], [-28, 6], [46, 0], [34, 8], [-56, -2], [-68, 8], [46, -4], [-6, 8], [-62, -3], [0, 0], [36, 0], [-54, -9], [22, -23], [-18, 24], [18, -24], [-26, 10], [34, 15], [-60, -4], [6, -9], [-28, 15], [-52, 11], [-16, 30], [14, 5], [-44, 2], [-6, 10], [-62, 2], [-26, 17], [-24, 14], [92, 2], [2, 14], [-6, -10], [-18, -10], [-2, 0], [-60, 6], [-70, -6], [-22, -6], [-54, 20], [14, -18], [6, -10], [34, 0], [52, -10], [54, -8], [26, 0], [-20, 14], [-36, 27], [14, -8], [-30, 8], [48, 4], [44, -7], [-2, 17], [14, 0], [-66, 10], [-2, 14], [-30, -6], [-8, 16], [32, 14], [-26, 0], [48, 8], [-36, 0], [58, -8], [60, 6], [60, 0], [-14, 16], [-4, 29], [22, 1]];
    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_bq();".
// 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_bq(: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_bq();".
// 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_bq( : 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![4, 1]>,<19,R![-4, 1]>,<23,R![12, 9, 1]>],Snew);
    return Vf;
end function;