
// Make newform 5610.2.a.bb in Magma, downloaded from the LMFDB on 07 May 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_bb();".
// 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_bb();".
// 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], [-2], [1], [-4], [1], [-2], [6], [-2], [-8], [-6], [2], [-6], [8], [0], [10], [-6], [-8], [-10], [12], [-6], [4], [-18], [-2], [-10], [-16], [12], [-10], [-4], [0], [20], [-6], [0], [-6], [-4], [-2], [-4], [-16], [18], [-10], [-22], [-12], [-16], [-2], [0], [-8], [-8], [20], [2], [6], [8], [4], [6], [-14], [24], [14], [0], [-2], [-26], [32], [22], [6], [-26], [6], [-6], [-28], [-36], [-12], [8], [-6], [-8], [12], [4], [-12], [36], [0], [-34], [-16], [22], [4], [-10], [16], [-2], [-6], [-18], [20], [-18], [-30], [-40], [14], [-20], [-4], [36], [-12], [-28], [0], [8], [14], [18], [20], [22], [4], [-6], [40], [-2], [-2], [-18], [-44], [32], [14], [4], [-8], [28], [8], [16], [36], [32], [6], [28], [-22], [-16], [-2], [12], [44], [-6], [-26], [34], [-16], [28], [-2], [52], [-20], [26], [6], [26], [-4], [-28], [0], [30], [20], [-38], [40], [-36], [-10], [30], [54], [-6], [-52], [36], [-2], [44], [44], [-16], [-28], [54], [20], [36], [-42], [18], [40], [-18], [-8], [-22], [-30], [-10], [60], [-46], [-8], [18], [-36], [10], [36], [38], [-32], [24], [36], [-22], [40], [-4], [-48], [-52], [-34], [-10], [-16], [0], [2], [24], [-8], [-20], [6], [20], [24], [24], [12], [44], [40], [30], [26], [-64], [50], [-14], [-42], [10], [-2], [0], [0], [16], [-66], [-24], [8], [-60], [-60], [-22], [24], [-30], [-68], [-46], [54], [44], [0], [-50], [-62], [-8], [16], [64], [36], [-18], [28], [20], [-46], [64], [28], [6], [52], [24], [2], [42], [-36], [36], [52], [-26], [64], [18], [-50], [0], [8], [-4], [-58], [-48], [12], [18], [24], [-58], [-18], [-12], [-10], [-52], [-30], [-34], [-6], [-12], [-22], [22], [-24], [-16], [-8], [10], [-72], [-6], [-30], [16], [66], [-36], [62], [-56], [-42], [44], [38], [-64], [-24], [64], [-68], [0], [-4], [-24], [70], [-54], [-40], [18], [-10], [52], [-54], [60], [-62], [30], [-16], [0], [-36], [-2], [-62], [-10], [-34], [8], [4], [-62], [42], [50], [-36], [56], [36], [54], [90], [24], [88], [-26], [-78], [-36], [86], [-82], [32], [34], [-54], [72], [22], [-66], [20], [12], [-60], [-18], [-66], [74], [-46], [-60], [-84], [72], [-54], [-34], [30], [-40], [-66], [-38], [26], [8], [88], [60], [14], [32], [-60], [74], [-50], [-14], [40], [-26], [0], [-38], [-80], [-92], [82], [4], [-46], [2], [60], [-36], [-28], [-10], [-82], [-68], [-40], [42], [20], [-64], [70], [-80], [-66], [0], [-12], [42], [-30], [-18], [-42], [-62], [-62], [-6], [-40], [6], [32], [-78], [-52], [68], [2], [-94], [46], [-24], [-12], [16], [-44], [34], [12], [38], [50], [34], [-16], [-92], [-62], [50], [-22], [2], [-16], [24], [-44], [46], [38], [-48], [50], [-22], [-4], [-52], [12], [96], [72], [-2], [58], [60], [6], [-60], [-36], [-30], [24], [-24]];
    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_bb();".
// 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_bb(: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_bb();".
// 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_bb( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5610_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![2, 1]>,<13,R![4, 1]>,<19,R![2, 1]>,<23,R![-6, 1]>],Snew);
    return Vf;
end function;