
// Make newform 1800.2.a.f in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1800_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_1800_2_a_f();".
// 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_1800_2_a_f();".
// 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_1800_a();"
function MakeCharacter_1800_a()
    N := 1800;
    order := 1;
    char_gens := [1351, 901, 1001, 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_1800_a_Hecke(Kf)
    return MakeCharacter_1800_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], [-2], [-2], [-4], [-2], [4], [8], [10], [4], [0], [0], [8], [8], [6], [14], [-14], [4], [-12], [-6], [-12], [4], [12], [14], [-6], [14], [-12], [2], [18], [-18], [18], [2], [12], [-10], [8], [-4], [-4], [0], [-2], [14], [2], [-4], [-22], [2], [8], [20], [22], [0], [-2], [-18], [-12], [-10], [-6], [18], [0], [2], [0], [16], [28], [4], [2], [32], [20], [10], [-30], [-12], [2], [16], [-6], [-26], [-24], [-26], [20], [28], [-24], [-18], [8], [12], [10], [-30], [30], [-16], [10], [-32], [24], [-12], [-10], [30], [-30], [-20], [4], [-22], [18], [20], [-24], [-14], [-20], [-28], [10], [-8], [-26], [8], [20], [4], [-26], [36], [42], [24], [22], [-6], [48], [6], [44], [-20], [-24], [-12], [-24], [-14], [-6], [18], [-30], [26], [-12], [-28], [-22], [14], [40], [-38], [24], [12], [24], [-36], [-32], [36], [-50], [42], [28], [30], [24], [-28], [-14], [2], [-24], [-38], [32], [16], [-18], [-4], [48], [20], [-36], [-44], [56], [-24], [16], [20], [4], [-10], [22], [56], [-54], [-18], [2], [-2], [-24], [8], [-36], [38], [22], [14], [-26], [-40], [34], [20], [60], [52], [-34], [2], [22], [-38], [-6], [16], [50], [16], [-10], [-44], [-20], [-10], [-44], [-14], [24], [20], [14], [-28], [58], [-30], [-36], [-38], [16], [-18], [-44], [4], [-18], [30], [-38], [4], [-40], [-36], [-28], [2], [38], [22], [-24], [-44], [18], [-10], [-52], [-24], [-10], [38], [16], [40], [66], [-12], [22], [-38], [-40], [10], [-10], [-52], [36], [-16], [-44], [60], [0], [-30], [-6], [-2], [-12], [-64], [-20], [-46], [-38], [-74], [20], [-10], [38], [4], [-8], [-24], [64], [-24], [50], [46], [-6], [50], [12], [54], [-54], [14], [-32], [-42], [-56], [-14], [28], [-26], [24], [36], [-70], [-10], [-16], [18], [-4], [-14], [-26], [32], [2], [-22], [42], [-24], [0], [40], [-2], [-64], [-44], [-74], [-54], [-24], [0], [-50], [-36], [-42], [-18], [-64], [6], [-28], [-42], [-74], [-48], [66], [18], [80], [-20], [84], [62], [-12], [70], [-12], [52], [-16], [42], [60], [56], [88], [-26], [-26], [-84], [-46], [40], [-28], [-38], [66], [50], [62], [-42], [-20], [16], [-32], [6], [-82], [38], [52], [92], [-44], [36], [-82], [-18], [10], [78], [-20], [30], [34], [32], [-34], [22], [46], [-16], [12], [54], [52], [-42], [-62], [22], [2], [38], [48], [-2], [-70], [64], [-20], [-32], [16], [-66], [-12], [14], [-18], [70], [38], [-90], [-4], [40], [-6], [0], [-56], [54], [56], [22], [-12], [-2], [-82], [74], [6], [-66], [-84], [32], [-20], [-48], [-24], [-64], [46], [78], [54], [-76], [-16], [-2], [80], [-24], [-52], [-66], [-90], [-6], [38], [66], [-30], [12], [32], [60], [-4], [18], [-106], [-102], [28], [28], [50], [90], [88], [62], [66], [-96], [54], [-52], [32], [-14], [66], [42], [84], [48], [-100], [-16]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1800_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1800_2_a_f();".
// 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_1800_2_a_f(:prec:=1)
    chi := MakeCharacter_1800_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_1800_2_a_f();".
// 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_1800_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1800_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![2, 1]>,<11,R![2, 1]>,<13,R![4, 1]>],Snew);
    return Vf;
end function;