
// Make newform 5610.2.a.bg in Magma, downloaded from the LMFDB on 24 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_bg();".
// 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_bg();".
// 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_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], [1], [-1], [-1], [-1], [-4], [1], [-4], [9], [9], [-1], [2], [12], [-1], [0], [0], [6], [2], [8], [-6], [2], [-4], [12], [0], [17], [6], [-1], [3], [20], [-6], [2], [0], [-21], [5], [12], [20], [2], [-1], [-18], [-6], [-18], [11], [-9], [2], [-12], [-16], [-13], [23], [3], [-28], [-3], [6], [-25], [-12], [-3], [-3], [-18], [11], [26], [15], [14], [-9], [20], [0], [-1], [3], [-7], [14], [12], [-22], [3], [18], [-22], [14], [-16], [0], [30], [2], [15], [32], [-39], [8], [27], [8], [26], [3], [-3], [-10], [-18], [32], [12], [-3], [-34], [24], [-40], [24], [-18], [-18], [-25], [-28], [-10], [33], [-36], [-6], [-4], [26], [-27], [18], [9], [26], [-40], [-46], [30], [-10], [26], [33], [-7], [-6], [3], [-15], [-16], [-16], [-42], [36], [8], [-18], [-22], [-6], [-49], [32], [-22], [24], [-31], [47], [9], [-10], [24], [-28], [-24], [-42], [-16], [27], [14], [-27], [38], [-18], [-13], [-45], [-1], [36], [-37], [-9], [20], [36], [-55], [0], [29], [-27], [-34], [6], [-6], [18], [-22], [24], [30], [33], [29], [-7], [-31], [3], [-57], [-16], [60], [29], [8], [-6], [44], [18], [-28], [-37], [-1], [-36], [38], [30], [36], [-6], [11], [-28], [-13], [48], [29], [12], [-37], [15], [-33], [-24], [-22], [50], [-3], [51], [-18], [2], [-61], [32], [-42], [-36], [-34], [-48], [-36], [-19], [56], [9], [8], [-57], [-48], [8], [38], [-39], [24], [-42], [-13], [32], [-48], [74], [0], [-19], [18], [54], [14], [-30], [23], [-37], [-25], [-9], [-28], [39], [-28], [24], [15], [0], [-60], [-22], [-13], [-34], [27], [-27], [62], [-21], [50], [-27], [20], [-36], [-72], [74], [69], [15], [26], [-46], [-57], [38], [-52], [-6], [38], [29], [-24], [-31], [39], [-48], [14], [45], [65], [14], [74], [-61], [77], [20], [84], [-19], [38], [-12], [-51], [-16], [-12], [5], [14], [0], [2], [-6], [14], [3], [-30], [-60], [12], [-54], [71], [78], [-82], [24], [-6], [-7], [89], [63], [-70], [-36], [-40], [2], [3], [-82], [-45], [38], [-6], [-90], [-12], [8], [-51], [20], [48], [24], [74], [-39], [41], [-1], [24], [-43], [18], [23], [56], [-1], [-72], [-36], [-22], [27], [-22], [-60], [68], [-60], [20], [39], [32], [74], [74], [54], [27], [-82], [0], [6], [-34], [-28], [21], [63], [-52], [38], [-33], [-52], [-70], [6], [27], [-72], [-15], [-42], [23], [18], [-72], [-54], [68], [68], [-6], [-34], [-70], [-42], [-52], [54], [54], [41], [92], [-36], [69], [65], [-30], [38], [48], [69], [80], [-63], [83], [21], [50], [-13], [-34], [84], [-28], [-27], [54], [-10], [0], [44], [-52], [-90], [-22], [-12], [-4], [-42], [44], [48], [30], [74], [-34], [-90], [-94], [-75], [-70], [78], [12], [-67], [17], [-42], [0], [-16], [-12], [-48], [-18], [-34], [48], [12], [14], [-54], [-48], [78], [65], [-93]];
    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_bg();".
// 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_bg(:prec:=1)
    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_bg();".
// 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_bg( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5610_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![1, 1]>,<13,R![4, 1]>,<19,R![4, 1]>,<23,R![-9, 1]>],Snew);
    return Vf;
end function;