
// Make newform 4800.2.a.bp in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4800_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_4800_2_a_bp();".
// 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_4800_2_a_bp();".
// 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_4800_a();"
function MakeCharacter_4800_a()
    N := 4800;
    order := 1;
    char_gens := [4351, 901, 1601, 577];
    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_4800_a_Hecke(Kf)
    return MakeCharacter_4800_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], [0], [-3], [-2], [-3], [-6], [7], [-6], [2], [-5], [10], [12], [3], [10], [0], [6], [13], [7], [-4], [6], [-8], [-6], [16], [7], [0], [12], [-16], [-9], [-12], [8], [8], [10], [-4], [22], [1], [-9], [1], [8], [2], [-18], [19], [-18], [-19], [14], [-3], [-9], [-11], [-4], [19], [4], [0], [25], [-28], [-4], [-12], [6], [24], [1], [2], [5], [-2], [5], [2], [-19], [32], [4], [7], [-18], [22], [-26], [36], [11], [7], [29], [-36], [-12], [11], [-8], [5], [24], [-14], [-34], [-3], [19], [-16], [12], [10], [-8], [-8], [26], [-38], [11], [24], [-25], [4], [-22], [-6], [29], [15], [0], [18], [-26], [18], [39], [-11], [-16], [36], [-4], [35], [32], [34], [40], [-5], [-25], [-12], [4], [-12], [26], [20], [-42], [50], [-10], [40], [32], [-42], [19], [30], [5], [-2], [-20], [-20], [8], [13], [-52], [-21], [-12], [37], [-20], [48], [-17], [10], [9], [24], [14], [34], [-25], [4], [32], [-8], [31], [8], [-53], [-42], [52], [18], [41], [-6], [-39], [-46], [6], [8], [-24], [-30], [-58], [4], [-17], [42], [-35], [18], [-10], [-7], [-30], [19], [-3], [-40], [23], [-10], [61], [17], [-47], [-8], [61], [6], [-46], [-30], [-59], [17], [-19], [-22], [-25], [-12], [-15], [14], [18], [2], [-34], [-58], [-2], [-6], [44], [20], [-38], [-30], [12], [14], [-11], [42], [36], [17], [-19], [30], [49], [-20], [6], [-18], [13], [-10], [-12], [-6], [-41], [-47], [-14], [-16], [-28], [9], [18], [6], [-13], [30], [41], [4], [-31], [-26], [44], [72], [26], [8], [-14], [64], [50], [-55], [-29], [-59], [10], [-30], [23], [6], [-55], [-26], [-42], [6], [-12], [-11], [-58], [-12], [31], [-28], [40], [2], [-55], [30], [-22], [15], [-42], [32], [6], [0], [20], [-6], [57], [64], [38], [-41], [23], [-71], [18], [1], [31], [-24], [36], [80], [6], [13], [-59], [70], [15], [-12], [24], [-76], [-50], [4], [-36], [48], [-61], [-36], [-77], [-84], [-42], [28], [-22], [36], [39], [-18], [-28], [83], [-4], [31], [24], [-58], [6], [44], [76], [-65], [34], [33], [-42], [-76], [-1], [10], [24], [-3], [8], [61], [-42], [-62], [-17], [1], [-2], [-36], [31], [6], [23], [6], [47], [60], [-35], [48], [-51], [72], [59], [44], [-54], [-51], [42], [70], [7], [57], [96], [-92], [15], [-26], [-82], [31], [41], [82], [84], [-18], [58], [50], [-65], [30], [78], [-40], [44], [21], [40], [44], [63], [44], [97], [46], [-20], [-32], [77], [-60], [-66], [25], [36], [-5], [26], [-34], [16], [20], [-20], [50], [83], [-10], [-28], [-10], [-67], [4], [-32], [-41], [86], [-49], [43], [24], [57], [-36], [62], [-32], [11], [-48], [72], [-64], [13], [12], [-31], [-92], [-7], [-68], [86], [71], [51], [4], [-66], [-85], [44], [8], [32], [27], [-106], [-30], [-46], [36], [-20], [-32], [-71], [-46]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4800_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4800_2_a_bp();".
// 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_4800_2_a_bp(:prec:=1)
    chi := MakeCharacter_4800_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_4800_2_a_bp();".
// 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_4800_2_a_bp( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4800_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![3, 1]>,<11,R![2, 1]>,<13,R![3, 1]>,<19,R![-7, 1]>,<23,R![6, 1]>],Snew);
    return Vf;
end function;