
// Make newform 2800.2.a.s in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2800_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_2800_2_a_s();".
// 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_2800_2_a_s();".
// 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_2800_a();"
function MakeCharacter_2800_a()
    N := 2800;
    order := 1;
    char_gens := [351, 2101, 2577, 801];
    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_2800_a_Hecke(Kf)
    return MakeCharacter_2800_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], [0], [0], [1], [0], [-4], [-4], [-4], [8], [2], [8], [8], [6], [8], [8], [0], [4], [-6], [8], [-12], [4], [4], [0], [-10], [12], [-18], [-8], [8], [14], [16], [-8], [4], [8], [-4], [6], [-12], [12], [-24], [24], [-12], [8], [14], [20], [16], [-16], [24], [-24], [-8], [0], [-14], [8], [-20], [-18], [4], [12], [24], [-10], [0], [-16], [6], [32], [-12], [-16], [24], [4], [-16], [16], [8], [24], [-6], [12], [20], [-8], [-32], [8], [24], [6], [-28], [18], [10], [28], [10], [-20], [-28], [-8], [24], [2], [16], [-26], [-8], [16], [0], [-8], [0], [8], [-8], [-22], [-6], [-32], [14], [-8], [16], [-16], [42], [40], [4], [-32], [36], [36], [-10], [-40], [24], [-8], [-4], [36], [-2], [16], [24], [24], [-24], [34], [-8], [-4], [-8], [-28], [2], [-6], [-32], [-8], [-20], [8], [-24], [20], [24], [10], [-2], [-36], [-48], [-28], [-26], [36], [-6], [24], [8], [22], [16], [4], [52], [-36], [-8], [-40], [-18], [-8], [-24], [8], [28], [4], [-30], [-4], [38], [24], [0], [56], [-60], [24], [40], [-4], [-28], [-50], [20], [24], [6], [-28], [-40], [-16], [-42], [-8], [-26], [-24], [-58], [-8], [52], [-40], [36], [8], [18], [0], [16], [10], [-48], [28], [24], [-16], [-54], [8], [20], [14], [-24], [28], [8], [34], [-16], [-12], [30], [60], [52], [24], [40], [6], [4], [56], [-14], [-24], [48], [-40], [38], [-24], [18], [-24], [56], [58], [-56], [-18], [-24], [-64], [-42], [12], [52], [40], [-72], [-8], [36], [28], [6], [-32], [-24], [-14], [0], [-48], [-24], [-24], [20], [8], [-30], [-12], [-72], [-8], [12], [0], [24], [-32], [-30], [40], [26], [60], [-16], [-10], [32], [12], [52], [40], [-72], [34], [76], [-28], [-52], [66], [10], [24], [72], [22], [-72], [-52], [4], [-36], [24], [24], [-14], [-10], [12], [24], [-76], [40], [14], [32], [36], [-32], [-48], [-32], [-30], [66], [0], [24], [76], [16], [-74], [8], [-4], [60], [80], [-68], [-56], [-12], [-24], [32], [-40], [-8], [26], [-60], [-16], [-56], [42], [18], [72], [-24], [38], [12], [-52], [-52], [66], [44], [-88], [-22], [-40], [-64], [-82], [-40], [64], [24], [40], [-62], [-72], [-40], [48], [-88], [32], [30], [-92], [-26], [-40], [-16], [8], [-66], [-4], [72], [-40], [-46], [24], [-72], [-92], [-60], [-16], [2], [-72], [-70], [-4], [-40], [60], [48], [72], [0], [90], [-8], [-56], [-32], [24], [-76], [-88], [54], [0], [28], [24], [6], [8], [32], [-44], [-36], [84], [82], [-28], [-90], [32], [-24], [-48], [4], [24], [20], [-68], [-40], [-72], [50], [84], [-16], [-32], [36], [-24], [-16], [-26], [-32], [-22], [102], [-80], [-24], [-36], [30], [16], [48], [66], [-64], [36], [60], [-24], [8], [-24], [-8], [-42], [-12], [8], [36], [24], [-66], [4], [-88], [100], [-12], [-60], [40], [-54], [84], [32]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2800_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2800_2_a_s();".
// 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_2800_2_a_s(:prec:=1)
    chi := MakeCharacter_2800_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_2800_2_a_s();".
// 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_2800_2_a_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2800_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![0, 1]>,<11,R![0, 1]>,<13,R![4, 1]>],Snew);
    return Vf;
end function;