
// Make newform 5520.2.a.r 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_r();".
// 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_r();".
// 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], [2], [0], [2], [0], [-1], [-4], [0], [10], [6], [-2], [-12], [6], [-12], [-14], [-2], [2], [6], [-8], [-8], [-8], [0], [-12], [-4], [12], [14], [6], [-6], [0], [-2], [-20], [-14], [12], [-22], [16], [0], [-6], [-4], [-6], [0], [-14], [-6], [-16], [-4], [-2], [-12], [-26], [10], [-22], [-14], [-18], [2], [32], [24], [-8], [8], [32], [10], [14], [-4], [2], [20], [-10], [28], [-4], [-4], [-30], [-34], [0], [-4], [-22], [-4], [24], [22], [32], [-12], [-10], [26], [-6], [-24], [24], [-20], [20], [-34], [-24], [20], [-18], [-24], [40], [-18], [24], [28], [8], [28], [32], [-22], [-22], [-12], [26], [-8], [-12], [36], [-10], [4], [-30], [30], [-10], [14], [14], [-14], [24], [16], [-8], [-14], [-12], [-26], [18], [6], [-18], [-22], [36], [20], [6], [22], [-18], [-8], [-30], [-36], [-16], [-40], [22], [2], [6], [6], [34], [50], [-30], [-52], [-16], [10], [-44], [10], [20], [56], [-42], [-36], [-36], [-56], [40], [48], [36], [-42], [12], [32], [30], [48], [26], [52], [14], [-58], [-50], [18], [-24], [-52], [40], [-38], [18], [-10], [-34], [48], [16], [28], [36], [4], [-64], [48], [-18], [-34], [-42], [56], [-30], [-56], [54], [8], [-10], [-10], [30], [-22], [28], [36], [-28], [32], [14], [30], [58], [-42], [36], [46], [-44], [28], [50], [14], [42], [-8], [36], [-22], [52], [30], [-20], [44], [64], [-6], [30], [-54], [-70], [-48], [-50], [22], [-72], [-6], [-32], [-20], [-10], [6], [30], [-68], [20], [-44], [60], [40], [-10], [-22], [-24], [-50], [6], [4], [30], [72], [-12], [18], [-62], [-46], [-10], [-46], [18], [-40], [32], [30], [-48], [-24], [-18], [54], [24], [22], [-30], [-26], [42], [76], [4], [-74], [-58], [14], [-44], [-42], [-56], [-26], [54], [14], [34], [-40], [-64], [-18], [74], [36], [-62], [-18], [62], [-16], [-8], [-16], [-46], [-8], [66], [20], [54], [4], [-10], [18], [28], [-10], [-34], [-52], [-70], [-40], [66], [84], [52], [4], [74], [56], [-28], [56], [78], [-12], [78], [4], [-52], [-44], [66], [40], [-28], [48], [26], [24], [-18], [64], [2], [-16], [84], [72], [74], [-30], [-62], [-48], [-84], [48], [42], [54], [-18], [68], [36], [-40], [-20], [-78], [-46], [-66], [8], [-44], [50], [-60], [-16], [58], [-12], [78], [56], [-84], [-22], [76], [82], [12], [16], [-22], [-10], [84], [-6], [-94], [80], [-50], [30], [-72], [70], [-20], [-26], [-74], [4], [-34], [-8], [-20], [-24], [30], [-32], [-52], [-8], [-44], [4], [-16], [14], [-50], [6], [-90], [-26], [-60], [-36], [60], [8], [-10], [-8], [-34], [46], [-24], [20], [-24], [-12], [40], [-12], [36], [-12], [62], [-46], [24], [-42], [-28], [28], [-18], [-86], [-6], [-100], [-14], [-6], [-104], [-64], [-92], [40], [50], [-70], [-98], [56], [-82], [-86], [-48], [-52], [26], [-46], [-52], [-6], [20], [-58]];
    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_r();".
// 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_r(: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_r();".
// 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_r( : 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![-2, 1]>,<13,R![0, 1]>,<17,R![-2, 1]>,<19,R![0, 1]>],Snew);
    return Vf;
end function;