
// Make newform 1600.2.a.h in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1600_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_1600_2_a_h();".
// 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_1600_2_a_h();".
// 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_1600_a();"
function MakeCharacter_1600_a()
    N := 1600;
    order := 1;
    char_gens := [1151, 901, 577];
    v := [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_1600_a_Hecke(Kf)
    return MakeCharacter_1600_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], [-2], [5], [0], [-5], [5], [6], [-4], [-10], [-10], [5], [-4], [-8], [-10], [0], [10], [-3], [0], [5], [-10], [1], [-9], [-10], [-2], [-16], [3], [-10], [-15], [18], [-20], [-5], [5], [20], [10], [10], [21], [8], [0], [-15], [-18], [10], [-5], [-10], [-20], [-5], [-4], [-12], [4], [10], [20], [-15], [-5], [10], [14], [0], [-10], [20], [-10], [29], [-30], [7], [-10], [30], [20], [25], [5], [27], [14], [-30], [-10], [12], [10], [15], [-14], [10], [20], [-27], [5], [-15], [20], [-10], [-5], [-20], [-39], [25], [15], [12], [4], [12], [-30], [2], [-20], [20], [4], [6], [-3], [-1], [-8], [23], [-40], [24], [-15], [-20], [5], [-7], [-15], [30], [-5], [-8], [-10], [-30], [20], [30], [-30], [-36], [12], [-30], [-25], [20], [-10], [-40], [31], [-15], [0], [-44], [10], [28], [0], [-20], [6], [20], [40], [37], [-51], [-20], [-52], [-30], [-6], [-20], [-10], [36], [-23], [40], [0], [10], [-45], [5], [-4], [-20], [-10], [-1], [28], [52], [-20], [10], [30], [5], [-8], [32], [-25], [-8], [45], [15], [-46], [-10], [-60], [-19], [20], [15], [-22], [30], [-35], [-60], [39], [35], [38], [16], [-10], [12], [65], [10], [-15], [-24], [44], [30], [-21], [25], [-60], [65], [-59], [35], [-30], [3], [-10], [35], [10], [45], [34], [54], [20], [40], [11], [20], [10], [-10], [44], [-34], [0], [30], [-70], [44], [23], [-10], [25], [-22], [13], [18], [30], [50], [40], [-31], [-6], [-57], [30], [-15], [20], [-48], [15], [-50], [15], [50], [-65], [-16], [18], [-35], [60], [-45], [60], [-49], [-25], [26], [36], [5], [-40], [-22], [-60], [-65], [-36], [0], [27], [2], [75], [-60], [55], [-28], [-63], [30], [75], [-14], [53], [-76], [-40], [35], [-15], [-6], [22], [31], [30], [8], [17], [-55], [60], [-50], [-4], [-33], [-16], [-50], [25], [-84], [-30], [32], [12], [-27], [-40], [-30], [60], [-80], [50], [-30], [-12], [45], [-45], [20], [-30], [50], [-20], [-60], [17], [-65], [-30], [40], [-21], [-5], [-25], [12], [46], [-50], [10], [66], [-26], [-7], [-16], [-12], [-34], [-35], [80], [-55], [-5], [20], [-45], [40], [34], [30], [10], [35], [-44], [-18], [40], [80], [-50], [30], [51], [25], [48], [20], [-50], [-25], [78], [-40], [15], [-30], [-40], [70], [20], [-80], [-43], [-60], [10], [55], [-10], [30], [64], [-66], [-75], [-40], [25], [50], [36], [-50], [67], [-78], [60], [-7], [85], [-60], [64], [45], [60], [-35], [-36], [60], [40], [50], [45], [-30], [-75], [-15], [-50], [78], [15], [-28], [-15], [40], [6], [-40], [-80], [56], [-38], [-5], [60], [-5], [-63], [-90], [-85], [-60], [-71], [5], [-90], [-60], [75], [32], [-25], [40], [60], [10], [2], [36], [75], [-55], [0], [9], [-60], [25], [62], [90], [58], [-25], [-54], [-20], [50], [-78], [-65], [-45], [60], [64], [-85], [65], [90]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1600_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1600_2_a_h();".
// 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_1600_2_a_h(:prec:=1)
    chi := MakeCharacter_1600_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_1600_2_a_h();".
// 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_1600_2_a_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1600_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, 1]>,<7,R![2, 1]>,<11,R![-5, 1]>,<13,R![0, 1]>],Snew);
    return Vf;
end function;