
// Make newform 5520.2.a.f in Magma, downloaded from the LMFDB on 25 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_f();".
// 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_f();".
// 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], [0], [4], [-6], [-6], [-4], [-1], [-6], [8], [6], [10], [-4], [8], [-14], [0], [10], [-4], [-8], [2], [12], [16], [-2], [-14], [18], [16], [-8], [2], [-14], [-12], [-8], [2], [-12], [10], [-16], [6], [-4], [16], [18], [0], [-6], [8], [10], [26], [12], [20], [20], [-8], [-14], [22], [8], [2], [12], [6], [32], [10], [8], [-30], [30], [-4], [-14], [-12], [8], [10], [18], [20], [-14], [28], [-34], [6], [8], [32], [-18], [-4], [-16], [-6], [2], [6], [10], [12], [-38], [-8], [2], [24], [-12], [10], [10], [2], [20], [-8], [-24], [-20], [0], [-12], [24], [-6], [-42], [20], [-34], [-20], [18], [8], [-18], [-20], [34], [4], [6], [40], [10], [28], [14], [18], [-20], [-20], [-2], [44], [-24], [-14], [36], [34], [-22], [-6], [-44], [28], [18], [10], [40], [-8], [6], [12], [24], [28], [54], [10], [-6], [2], [-36], [18], [-30], [-20], [-46], [-20], [8], [-18], [48], [-22], [-42], [-4], [-24], [2], [6], [-44], [-8], [4], [-8], [-12], [-6], [10], [-14], [20], [-22], [-28], [-12], [2], [24], [0], [-22], [2], [-22], [36], [-42], [32], [-38], [-16], [6], [12], [58], [24], [26], [-28], [-28], [-14], [-42], [-32], [42], [26], [-36], [42], [-24], [-38], [12], [-20], [50], [0], [-30], [-46], [14], [-22], [16], [-14], [48], [26], [10], [52], [-62], [20], [-4], [2], [-4], [-6], [-6], [-56], [-24], [-40], [2], [68], [66], [32], [2], [38], [-68], [-22], [-16], [36], [-26], [-6], [32], [-32], [32], [66], [36], [-60], [58], [-4], [24], [34], [42], [24], [-16], [-72], [12], [-4], [-74], [-18], [-16], [4], [68], [4], [-48], [-50], [-58], [16], [-62], [50], [0], [-22], [-44], [42], [34], [40], [24], [-66], [54], [54], [76], [18], [-50], [28], [26], [38], [52], [26], [-68], [-38], [36], [-28], [70], [58], [-60], [-40], [-28], [0], [10], [-20], [32], [-46], [-54], [-72], [42], [-46], [48], [-2], [-52], [-54], [66], [20], [26], [48], [84], [42], [66], [-20], [-76], [-76], [82], [-84], [-22], [-24], [50], [-24], [10], [-50], [4], [24], [-30], [8], [24], [-46], [66], [20], [-6], [26], [28], [82], [-14], [-4], [-52], [-72], [-6], [-42], [34], [40], [-52], [12], [68], [-22], [-86], [-70], [-40], [2], [-30], [-30], [36], [-22], [-8], [70], [-84], [-48], [-30], [20], [58], [-70], [80], [34], [-2], [-24], [36], [62], [24], [38], [98], [32], [-84], [-4], [74], [-14], [72], [-30], [32], [20], [32], [42], [68], [2], [64], [-24], [-14], [-66], [26], [66], [-14], [-4], [46], [-28], [-24], [-64], [18], [-52], [40], [-22], [-70], [0], [-92], [-16], [10], [68], [30], [-68], [-6], [62], [-2], [32], [-54], [18], [56], [46], [-30], [68], [-24], [34], [10], [24], [-60], [-38], [26], [-48], [84], [-6], [-72], [-30], [-66], [-24], [24], [-94], [66], [16], [26], [100], [24]];
    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_f();".
// 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_f(: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_f();".
// 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_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5520_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![0, 1]>,<11,R![-4, 1]>,<13,R![6, 1]>,<17,R![6, 1]>,<19,R![4, 1]>],Snew);
    return Vf;
end function;