
// Make newform 1152.2.a.q in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1152_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_1152_2_a_q();".
// 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_1152_2_a_q();".
// 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_1152_a();"
function MakeCharacter_1152_a()
    N := 1152;
    order := 1;
    char_gens := [127, 901, 641];
    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_1152_a_Hecke(Kf)
    return MakeCharacter_1152_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], [2], [2], [4], [2], [4], [-4], [-8], [6], [-6], [-2], [12], [12], [-8], [6], [8], [-10], [-8], [0], [2], [-14], [12], [-8], [-2], [6], [14], [-8], [18], [8], [-2], [0], [4], [8], [-6], [10], [-18], [-4], [-8], [-6], [-24], [2], [8], [-2], [-26], [-10], [8], [-18], [-12], [-14], [-24], [24], [-14], [-12], [0], [0], [-26], [14], [18], [0], [0], [22], [24], [-16], [22], [-18], [-16], [10], [12], [-26], [24], [32], [10], [22], [-20], [-24], [-14], [30], [12], [-26], [20], [-14], [-8], [-30], [6], [12], [-28], [6], [34], [-2], [4], [0], [-38], [32], [32], [-16], [-2], [-4], [-4], [-38], [4], [2], [36], [-4], [-40], [-10], [32], [24], [-24], [-18], [14], [22], [16], [16], [-38], [36], [-12], [-24], [18], [-48], [22], [-14], [26], [36], [-36], [-10], [2], [48], [18], [-6], [24], [-40], [26], [2], [4], [14], [6], [20], [42], [-12], [-12], [-2], [26], [-8], [-14], [0], [-10], [-36], [4], [-8], [-10], [32], [-20], [48], [20], [24], [-6], [-36], [-38], [-14], [-8], [-20], [22], [-28], [12], [-16], [58], [-18], [-34], [-14], [-48], [30], [40], [38], [10], [-48], [48], [26], [38], [-10], [-14], [12], [-6], [-32], [-24], [-30], [26], [-64], [10], [16], [46], [24], [16], [-22], [16], [32], [-58], [10], [32], [0], [2], [-10], [58], [-38], [12], [54], [-50], [4], [-40], [-44], [62], [58], [70], [-8], [48], [-38], [-6], [4], [-24], [-30], [-22], [-18], [-36], [6], [28], [-10], [56], [-16], [22], [-32], [-46], [20], [14], [-32], [68], [-32], [14], [26], [-12], [-72], [36], [-16], [34], [-2], [64], [64], [26], [24], [-32], [0], [66], [40], [-56], [22], [46], [56], [-70], [-72], [30], [-6], [-10], [8], [26], [62], [12], [-48], [10], [72], [4], [74], [-58], [56], [-26], [50], [-18], [62], [-4], [30], [30], [72], [-16], [-58], [24], [22], [52], [-32], [-14], [-30], [-14], [40], [34], [-24], [-24], [72], [-26], [-30], [22], [70], [-36], [4], [-10], [42], [-6], [-48], [-68], [-30], [4], [-86], [48], [70], [40], [54], [-12], [28], [-16], [-42], [32], [64], [-6], [-40], [32], [-58], [6], [-70], [-36], [-34], [60], [44], [32], [58], [-14], [-78], [62], [24], [-56], [60], [-34], [-40], [18], [38], [-26], [-52], [30], [-26], [-18], [-76], [74], [-48], [-48], [-2], [52], [-10], [6], [30], [-10], [-16], [72], [0], [-60], [48], [-50], [-24], [-24], [-4], [-8], [-82], [14], [-62], [-22], [28], [76], [0], [-90], [-62], [10], [20], [-24], [-14], [-36], [58], [58], [-12], [26], [-48], [4], [24], [10], [82], [56], [-56], [46], [-30], [16], [76], [8], [-74], [22], [76], [12], [42], [-18], [60], [6], [-36], [78], [90], [-94], [12], [-52], [20], [54], [-30], [4], [92], [-46], [14], [40], [-46], [60], [96], [58], [6], [32], [-12], [102], [78], [-92], [-72], [-12], [-24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1152_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1152_2_a_q();".
// 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_1152_2_a_q(:prec:=1)
    chi := MakeCharacter_1152_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_1152_2_a_q();".
// 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_1152_2_a_q( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1152_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-2, 1]>,<7,R![-2, 1]>,<11,R![-4, 1]>,<13,R![-2, 1]>],Snew);
    return Vf;
end function;