
// Make newform 1800.2.a.k 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_k();".
// 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_k();".
// 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], [-1], [-4], [1], [4], [1], [-4], [-4], [-5], [6], [-12], [-5], [8], [-12], [-8], [7], [-13], [-12], [6], [12], [-8], [0], [13], [-12], [4], [12], [-7], [0], [-12], [-12], [8], [0], [-8], [-19], [19], [-23], [-12], [4], [16], [5], [4], [7], [8], [17], [5], [23], [-12], [1], [-12], [-12], [17], [24], [-24], [12], [28], [-12], [-7], [20], [-31], [32], [-11], [16], [-13], [-24], [0], [13], [-8], [-18], [4], [36], [17], [-5], [-29], [24], [12], [7], [12], [25], [-12], [-6], [28], [11], [19], [24], [24], [-26], [12], [36], [-20], [8], [13], [0], [-19], [-36], [-40], [20], [-35], [-11], [8], [16], [20], [4], [25], [11], [12], [12], [-24], [-17], [36], [6], [-36], [-7], [-17], [-24], [-24], [48], [-32], [-12], [-42], [-30], [-28], [-48], [8], [-20], [29], [-4], [47], [18], [-48], [48], [12], [-43], [-48], [-17], [-36], [-7], [24], [36], [53], [-16], [31], [12], [34], [-20], [-49], [24], [-40], [-24], [19], [-24], [-25], [8], [24], [-40], [5], [-52], [-29], [-16], [44], [-48], [36], [-32], [-8], [-48], [-13], [-42], [5], [40], [4], [-5], [-32], [-43], [-7], [-24], [1], [4], [-5], [-53], [11], [0], [-19], [32], [-56], [16], [53], [5], [29], [-4], [41], [36], [35], [-44], [68], [16], [18], [18], [-56], [52], [36], [-44], [-22], [-66], [0], [-20], [-11], [-4], [-24], [-13], [55], [-32], [35], [-48], [8], [-6], [19], [28], [-12], [44], [-7], [25], [32], [-60], [-36], [-5], [4], [64], [-7], [4], [-11], [-72], [-19], [32], [24], [0], [-18], [36], [8], [12], [-4], [-17], [-35], [-65], [16], [-56], [-47], [52], [55], [-8], [66], [68], [-24], [29], [28], [48], [-7], [-48], [12], [-6], [7], [-68], [-6], [71], [64], [64], [-40], [-24], [24], [-40], [71], [16], [6], [-61], [65], [-61], [56], [-49], [-49], [24], [12], [-12], [4], [-41], [-59], [56], [17], [-60], [-36], [-12], [56], [24], [24], [72], [-65], [36], [-1], [-12], [-76], [24], [-6], [0], [59], [-32], [-24], [73], [24], [-59], [36], [38], [68], [-24], [0], [31], [4], [37], [-28], [0], [-35], [-4], [-64], [-37], [36], [-53], [-88], [-18], [-23], [53], [52], [36], [65], [80], [7], [-4], [49], [-48], [-37], [12], [49], [-60], [-61], [12], [-4], [89], [-52], [-64], [-35], [25], [-36], [-24], [-47], [-90], [-76], [-67], [23], [56], [-36], [-4], [44], [-68], [-25], [-40], [76], [24], [48], [7], [48], [-52], [-25], [-48], [-37], [-20], [-72], [60], [77], [60], [-8], [71], [-96], [-43], [-56], [-52], [12], [-12], [24], [-16], [-5], [-54], [48], [92], [-23], [84], [-60], [7], [-8], [-55], [-25], [-12], [-85], [24], [30], [-84], [-11], [-12], [60], [12], [-47], [-60], [1], [-72], [19], [48], [40], [73], [-71], [-96], [-88], [-11], [-36], [12], [24], [-65], [8], [-16], [-6], [-60], [-84], [-60], [107], [52]];
    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_k();".
// 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_k(: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_k();".
// 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_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1800_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![1, 1]>,<11,R![4, 1]>,<13,R![-1, 1]>],Snew);
    return Vf;
end function;