
// Make newform 5520.2.a.c in Magma, downloaded from the LMFDB on 19 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_c();".
// 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_c();".
// 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], [-3], [6], [-2], [3], [6], [1], [-9], [3], [3], [-3], [0], [-4], [-9], [3], [-8], [-3], [9], [6], [4], [-3], [-8], [18], [11], [-16], [5], [-2], [-1], [4], [20], [-18], [9], [24], [-4], [17], [14], [12], [-4], [8], [-12], [20], [-12], [6], [14], [-5], [2], [-12], [20], [20], [-1], [-26], [18], [12], [-21], [21], [5], [-10], [14], [-5], [5], [-6], [24], [-31], [-12], [-9], [2], [8], [31], [-4], [-20], [-17], [22], [-8], [27], [34], [32], [-22], [-35], [-30], [10], [0], [19], [-24], [24], [-33], [29], [38], [-2], [29], [-8], [4], [33], [-27], [-1], [-6], [-18], [-4], [-2], [-24], [-9], [-13], [10], [16], [-8], [18], [-12], [24], [3], [40], [-2], [45], [40], [16], [42], [-31], [-18], [14], [-30], [18], [-8], [33], [-36], [-12], [-44], [4], [23], [7], [43], [-5], [8], [26], [13], [49], [-28], [38], [49], [33], [11], [19], [-38], [4], [23], [17], [-54], [-10], [-18], [-1], [46], [-34], [-40], [24], [42], [5], [-52], [-24], [15], [2], [48], [-12], [-22], [-32], [20], [-23], [-31], [55], [8], [-48], [-4], [48], [-39], [58], [-37], [0], [40], [-28], [-37], [19], [-22], [-34], [24], [-46], [-18], [-21], [-38], [50], [35], [1], [-51], [-22], [-6], [-22], [-11], [16], [-55], [38], [-10], [62], [-14], [0], [52], [-50], [-26], [-4], [-46], [-16], [-42], [37], [4], [-54], [6], [32], [-36], [-53], [-48], [-50], [45], [11], [-10], [-23], [14], [-1], [-1], [46], [-5], [38], [55], [-44], [-27], [-30], [70], [38], [23], [13], [68], [-64], [-22], [-57], [57], [53], [11], [-14], [-74], [60], [60], [8], [-38], [-58], [-63], [54], [18], [43], [14], [-66], [15], [12], [80], [-18], [2], [-49], [47], [-41], [-3], [0], [4], [0], [10], [-69], [-24], [-19], [37], [59], [46], [-72], [-14], [-48], [25], [-60], [60], [-14], [-36], [-48], [-24], [-22], [-55], [3], [50], [16], [50], [-30], [11], [-40], [40], [48], [-60], [-50], [-30], [57], [-22], [29], [-9], [84], [-8], [52], [-32], [30], [2], [-8], [64], [0], [42], [-52], [64], [-57], [-68], [75], [23], [-29], [-50], [16], [89], [-14], [-74], [-57], [-44], [-40], [50], [-80], [13], [-49], [-78], [37], [58], [52], [68], [-34], [90], [2], [-91], [2], [58], [-6], [-12], [1], [-71], [-31], [-38], [18], [57], [1], [-24], [63], [-45], [0], [94], [-34], [42], [-34], [58], [43], [-7], [62], [-24], [78], [10], [48], [-31], [4], [-51], [35], [38], [-12], [76], [78], [-41], [30], [-81], [76], [2], [38], [62], [-44], [-22], [-16], [-66], [-89], [72], [93], [-7], [10], [-22], [37], [-56], [-66], [55], [50], [-56], [-32], [65], [-69], [54], [-81], [69], [-49], [49], [-91], [-70], [4], [12], [-104], [98], [-92], [-52], [-43], [-78], [88], [76], [67], [-60], [-30], [38], [-26], [-84], [88], [12], [52], [-49], [20], [5]];
    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_c();".
// 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_c(: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_c();".
// 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_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5520_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![3, 1]>,<11,R![-6, 1]>,<13,R![2, 1]>,<17,R![-3, 1]>,<19,R![-6, 1]>],Snew);
    return Vf;
end function;