
// Make newform 5220.2.a.h in Magma, downloaded from the LMFDB on 19 September 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5220_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_5220_2_a_h();".
// 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_5220_2_a_h();".
// 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_5220_a();"
function MakeCharacter_5220_a()
    N := 5220;
    order := 1;
    char_gens := [2611, 4061, 4177, 901];
    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_5220_a_Hecke(Kf)
    return MakeCharacter_5220_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]];
    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], [0], [-1], [2], [4], [2], [8], [-4], [6], [1], [0], [4], [-6], [0], [0], [2], [0], [-6], [2], [8], [-4], [-16], [10], [-6], [8], [18], [-14], [-18], [10], [-4], [8], [12], [8], [4], [-10], [-8], [8], [4], [-2], [-18], [20], [-6], [24], [-4], [14], [-4], [4], [10], [-14], [10], [6], [-12], [10], [12], [-22], [4], [18], [-24], [14], [-6], [22], [-24], [8], [16], [-2], [-12], [12], [-16], [18], [18], [30], [-8], [-28], [2], [12], [-26], [-6], [10], [-6], [-22], [24], [26], [-36], [-8], [-20], [0], [14], [34], [10], [-2], [-24], [-40], [26], [-20], [20], [24], [-14], [6], [-38], [-26], [42], [34], [-12], [22], [-24], [28], [-22], [46], [0], [38], [-48], [-26], [24], [-44], [-4], [26], [-34], [-38], [4], [-36], [-10], [22], [-48], [6], [20], [42], [14], [-12], [12], [-8], [36], [-40], [-24], [44], [10], [46], [-12], [6], [4], [-54], [-48], [-10], [-28], [-24], [-50], [0], [-24], [42], [4], [-18], [-2], [-30], [-38], [24], [0], [24], [-52], [2], [-38], [50], [24], [-10], [36], [-36], [10], [8], [8], [28], [-2], [40], [-24], [62], [-24], [0], [-12], [54], [-4], [42], [-20], [14], [32], [-20], [-54], [-54], [22], [18], [-44], [-8], [-38], [36], [-50], [-48], [28], [22], [-4], [-6], [-10], [-46], [42], [-34], [14], [28], [-32], [46], [68], [-34], [-64], [-50], [-26], [28], [-56], [-42], [36], [12], [-16], [-58], [-34], [70], [-14], [-28], [-30], [-24], [10], [44], [26], [46], [48], [0], [64], [-36], [-4], [32], [-8], [2], [-54], [-12], [-62], [-48], [20], [0], [-40], [-44], [26], [26], [70], [0], [-38], [-52], [32], [-4], [-8], [-6], [4], [-6], [-56], [-44], [-26], [52], [-78], [14], [-36], [-12], [78], [-60], [0], [28], [62], [-66], [72], [-22], [-6], [-34], [-34], [-72], [-68], [-4], [36], [2], [78], [28], [-46], [68], [-10], [-10], [-32], [-48], [-32], [48], [-80], [78], [42], [18], [26], [4], [0], [-30], [32], [2], [48], [-64], [-8], [50], [-32], [48], [36], [-82], [72], [-14], [-76], [2], [82], [-42], [-62], [46], [-46], [-58], [-28], [20], [62], [6], [44], [46], [62], [-20], [6], [62], [-36], [10], [8], [26], [10], [6], [8], [72], [-12], [66], [78], [44], [-50], [-38], [-16], [2], [90], [44], [-42], [60], [10], [64], [48], [40], [-44], [-14], [-58], [-26], [34], [68], [-40], [88], [92], [66], [-86], [54], [68], [-4], [-4], [64], [12], [94], [94], [36], [-12], [-66], [42], [-40], [-74], [68], [48], [-84], [14], [-82], [-46], [-54], [-28], [60], [8], [-82], [16], [-78], [-44], [-48], [-22], [-38], [28], [4], [-32], [46], [-88], [2], [-68], [22], [58], [-16], [76], [-78], [-6], [-20], [-18], [-58], [-72], [-36], [22], [-30], [-6], [44], [68], [50], [-16], [-34], [48], [-72], [6], [-12], [-24], [-36], [42], [42], [6], [6], [-28], [-40]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5220_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5220_2_a_h();".
// 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_5220_2_a_h(:prec:=1)
    chi := MakeCharacter_5220_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_5220_2_a_h();".
// 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_5220_2_a_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5220_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-2, 1]>,<11,R![-4, 1]>,<19,R![4, 1]>],Snew);
    return Vf;
end function;