
// Make newform 5520.2.a.u in Magma, downloaded from the LMFDB on 20 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_u();".
// 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_u();".
// 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], [-1], [-4], [0], [5], [0], [-1], [5], [-3], [-5], [3], [4], [-6], [-3], [-9], [10], [7], [-7], [-12], [-8], [1], [16], [-6], [-3], [-16], [-3], [-16], [-15], [-6], [12], [-14], [-5], [10], [-24], [17], [-14], [6], [0], [8], [-24], [-6], [4], [-6], [2], [17], [-2], [-12], [4], [16], [-1], [4], [6], [-22], [23], [-21], [7], [-22], [-16], [1], [11], [32], [-16], [29], [-34], [25], [-22], [32], [-33], [26], [18], [-19], [26], [-16], [15], [10], [8], [6], [-31], [-16], [-12], [12], [15], [-8], [-22], [41], [9], [-22], [-36], [-3], [-14], [6], [-15], [-5], [-37], [-2], [44], [44], [-10], [12], [29], [19], [42], [-18], [-28], [-38], [-6], [12], [-25], [-22], [14], [-17], [-36], [10], [-44], [31], [-12], [16], [36], [-6], [6], [35], [30], [8], [18], [-26], [51], [-47], [-33], [45], [32], [-16], [-23], [35], [30], [18], [7], [-13], [-27], [-55], [-10], [40], [43], [13], [-34], [2], [6], [21], [-6], [-38], [-2], [-6], [30], [-9], [-24], [-40], [-27], [-6], [26], [-32], [-22], [-10], [4], [-33], [9], [41], [40], [28], [54], [-40], [5], [-6], [-5], [-32], [-54], [-44], [5], [45], [24], [-40], [18], [-58], [0], [7], [0], [20], [17], [-31], [-27], [-4], [58], [36], [-13], [-16], [-49], [-48], [-2], [54], [54], [-26], [64], [28], [44], [62], [18], [-14], [-24], [35], [40], [54], [-38], [8], [52], [39], [6], [42], [11], [-33], [64], [-59], [54], [-51], [73], [28], [-61], [42], [-9], [-44], [-19], [4], [0], [22], [65], [11], [12], [-14], [42], [-65], [21], [21], [57], [42], [-50], [-40], [-64], [-10], [-36], [38], [-67], [-18], [-6], [15], [18], [0], [-45], [22], [24], [4], [54], [-11], [43], [67], [65], [-16], [-62], [42], [-50], [1], [72], [5], [-29], [47], [74], [-42], [-28], [-30], [-23], [30], [50], [-46], [-38], [-52], [56], [16], [9], [59], [42], [-8], [-58], [-18], [79], [44], [-22], [-10], [-28], [-28], [-36], [-63], [-2], [-35], [-19], [-64], [26], [32], [6], [-12], [-54], [70], [74], [52], [60], [-14], [56], [-57], [-46], [-21], [-57], [-65], [-22], [14], [69], [18], [38], [9], [-2], [-42], [-36], [-72], [3], [-57], [-48], [83], [-24], [-64], [-92], [-18], [14], [-6], [-73], [22], [-22], [30], [-46], [-41], [-3], [-3], [-64], [-30], [11], [-53], [82], [-75], [-11], [-34], [-34], [12], [-78], [-52], [-22], [-65], [15], [78], [-44], [-14], [-14], [82], [-29], [20], [61], [85], [-18], [84], [4], [20], [79], [-14], [19], [-40], [2], [-38], [-54], [-24], [-32], [-72], [18], [-51], [-58], [-49], [85], [20], [46], [-39], [26], [-72], [-21], [22], [0], [0], [51], [-1], [-94], [-27], [-21], [-91], [55], [-87], [-50], [24], [-76], [58], [72], [-44], [44], [-23], [-62], [-4], [-46], [-71], [56], [80], [70], [-24], [-76], [-46], [-16], [-16], [93], [20], [-43]];
    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_u();".
// 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_u(: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_u();".
// 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_u( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5520_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![1, 1]>,<11,R![4, 1]>,<13,R![0, 1]>,<17,R![-5, 1]>,<19,R![0, 1]>],Snew);
    return Vf;
end function;