
// Make newform 1800.2.a.j in Magma, downloaded from the LMFDB on 10 March 2026.
// 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_j();".
// 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_j();".
// 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 := 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]];
    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], [4], [-4], [0], [-4], [-2], [-2], [0], [-4], [-2], [6], [-6], [-4], [12], [-10], [-14], [-8], [-8], [16], [2], [-6], [-16], [-6], [-14], [-10], [-6], [16], [6], [12], [8], [4], [-18], [-8], [4], [-2], [18], [-12], [4], [22], [0], [16], [-12], [-8], [4], [-6], [-2], [-6], [-24], [16], [-22], [20], [0], [30], [6], [-16], [12], [-18], [6], [28], [2], [-24], [-8], [12], [28], [16], [22], [2], [0], [8], [-10], [20], [20], [6], [30], [36], [14], [14], [-12], [-2], [-16], [0], [-24], [-6], [-6], [24], [18], [26], [-18], [-32], [-2], [4], [-4], [-34], [-34], [-10], [-10], [-2], [34], [12], [18], [18], [-20], [-32], [38], [-16], [-40], [18], [-10], [20], [-8], [4], [-40], [6], [-18], [2], [36], [36], [14], [16], [-12], [-38], [36], [-22], [-38], [-16], [-18], [-36], [12], [30], [-16], [-52], [38], [-2], [-20], [2], [12], [42], [-36], [18], [2], [-42], [10], [-24], [-28], [-8], [-44], [-42], [-44], [-10], [-50], [18], [-22], [-48], [-24], [-38], [40], [-14], [30], [-24], [-50], [36], [32], [46], [-32], [12], [14], [-36], [-20], [30], [24], [-40], [-16], [26], [12], [58], [-14], [-6], [38], [12], [-12], [24], [6], [-10], [4], [14], [-2], [0], [-32], [-54], [4], [-54], [-18], [8], [10], [44], [-64], [-50], [-18], [16], [28], [-2], [-4], [-4], [-32], [50], [10], [28], [32], [18], [-46], [22], [40], [-62], [22], [-50], [34], [20], [-34], [40], [-30], [-54], [-2], [50], [24], [64], [46], [-60], [-4], [60], [0], [-2], [-26], [42], [-10], [60], [-20], [-72], [2], [44], [34], [-46], [-48], [56], [38], [12], [-60], [-26], [-12], [30], [-30], [-2], [52], [-60], [-10], [10], [4], [-24], [74], [62], [-38], [28], [48], [12], [-50], [70], [6], [-36], [46], [82], [8], [64], [48], [-46], [38], [-30], [66], [28], [-10], [40], [-62], [-26], [-70], [48], [-32], [36], [40], [-70], [26], [-2], [56], [36], [12], [-42], [32], [12], [76], [2], [-56], [44], [-16], [-30], [-52], [-48], [-42], [-14], [24], [-28], [54], [54], [-34], [-2], [-14], [54], [36], [-32], [0], [-86], [4], [-88], [-86], [42], [-8], [-22], [-20], [-90], [10], [44], [-26], [-68], [64], [-14], [-36], [-26], [10], [-64], [-30], [-90], [52], [-40], [62], [40], [-60], [-12], [-2], [-6], [-48], [-92], [-44], [-8], [90], [42], [74], [-8], [-64], [4], [-64], [14], [44], [-42], [74], [12], [-14], [56], [-36], [34], [50], [12], [100], [6], [46], [88], [-76], [4], [96], [32], [-22], [-88], [-30], [-40], [-2], [32], [-52], [78], [48], [-20], [-26], [-54], [-90], [-36], [-4], [-14], [40], [88], [-32], [42], [-36], [50], [-54], [80], [-42], [-104], [-66], [-88], [-28], [46], [-34], [-76], [-80], [-12], [-6], [20], [-56], [50], [96], [-50], [-64], [-82], [-74], [-52], [-102], [12], [-24], [-84], [34], [82], [-52], [-8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1800_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1800_2_a_j();".
// 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_j(: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_j();".
// 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_j( : 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![-4, 1]>,<13,R![4, 1]>],Snew);
    return Vf;
end function;