
// Make newform 5520.2.a.q in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5520_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_5520_2_a_q();".
// 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_5520_2_a_q();".
// 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_5520_a();"
function MakeCharacter_5520_a()
    N := 5520;
    order := 1;
    char_gens := [4831, 1381, 1841, 4417, 1201];
    v := [1, 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_5520_a_Hecke(Kf)
    return MakeCharacter_5520_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 := [[0], [1], [-1], [-4], [-4], [6], [-2], [-4], [-1], [-2], [0], [2], [2], [0], [0], [-6], [8], [-2], [16], [4], [10], [0], [-4], [6], [-2], [-10], [-4], [-4], [14], [14], [8], [0], [6], [20], [-6], [-24], [2], [-12], [8], [18], [-16], [-10], [8], [10], [-6], [24], [28], [-16], [28], [14], [22], [28], [18], [12], [6], [-16], [-10], [-24], [-10], [22], [16], [-6], [-12], [4], [-26], [10], [28], [-2], [-28], [30], [-18], [24], [-28], [-6], [36], [-32], [-30], [-2], [-26], [10], [-36], [-18], [0], [22], [8], [-12], [-30], [22], [6], [-8], [12], [-24], [0], [8], [36], [16], [-34], [22], [-16], [30], [20], [34], [4], [-18], [28], [10], [12], [-42], [-12], [42], [-40], [-6], [-42], [28], [8], [-10], [-8], [24], [-6], [-4], [38], [-14], [26], [-36], [-36], [-30], [14], [12], [-20], [26], [44], [24], [40], [26], [18], [-22], [-6], [-8], [-30], [42], [-20], [-18], [8], [-12], [-2], [-16], [38], [-42], [12], [-24], [22], [-50], [20], [48], [16], [24], [-16], [-30], [-58], [18], [-12], [-10], [-24], [-20], [30], [-24], [-32], [-34], [34], [42], [-12], [-18], [-56], [-18], [56], [14], [44], [54], [-20], [-10], [24], [20], [-26], [14], [-8], [50], [-42], [-16], [-22], [-44], [10], [28], [28], [6], [44], [54], [26], [-14], [-66], [24], [42], [-24], [-26], [18], [36], [34], [16], [-12], [50], [44], [-46], [46], [60], [-52], [-60], [-46], [-8], [42], [24], [-6], [-10], [-40], [34], [20], [44], [-10], [6], [60], [-60], [8], [-34], [-20], [-16], [-54], [64], [-72], [10], [10], [24], [-4], [-76], [-28], [24], [-50], [46], [12], [8], [-12], [52], [-40], [50], [70], [-8], [50], [-54], [64], [22], [32], [-30], [50], [52], [12], [-26], [-14], [-42], [-28], [-6], [-50], [-40], [-30], [-34], [48], [-34], [0], [-30], [-16], [-84], [62], [18], [52], [-32], [0], [-24], [62], [-12], [44], [-74], [-6], [8], [-38], [2], [68], [6], [20], [-10], [50], [40], [74], [-56], [-12], [54], [66], [48], [-36], [52], [58], [60], [6], [8], [-26], [-40], [66], [-18], [4], [-56], [10], [72], [20], [-26], [18], [28], [6], [-58], [48], [54], [2], [20], [-4], [-8], [-38], [14], [-54], [56], [-4], [60], [92], [-74], [30], [-70], [28], [-74], [-42], [54], [-16], [-30], [-16], [22], [4], [64], [-54], [36], [10], [78], [28], [-10], [-90], [8], [60], [-10], [56], [2], [18], [72], [20], [76], [-30], [26], [60], [34], [24], [92], [-24], [58], [-88], [-26], [-56], [-24], [86], [-98], [50], [42], [54], [-56], [30], [-20], [96], [88], [-74], [-32], [40], [10], [-14], [-64], [92], [56], [-66], [0], [54], [76], [30], [-66], [-74], [-68], [86], [6], [-80], [2], [-70], [56], [-56], [34], [-38], [-12], [-4], [6], [-74], [4], [-48], [14], [24], [90], [106], [88], [-48], [18], [18], [36], [74], [-100], [52]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5520_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5520_2_a_q();".
// 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_5520_2_a_q(:prec:=1)
    chi := MakeCharacter_5520_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_5520_2_a_q();".
// 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_5520_2_a_q( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5520_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![4, 1]>,<11,R![4, 1]>,<13,R![-6, 1]>,<17,R![2, 1]>,<19,R![4, 1]>],Snew);
    return Vf;
end function;