
// Make newform 1122.2.a.m in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1122_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_1122_2_a_m();".
// 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_1122_2_a_m();".
// 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_1122_a();"
function MakeCharacter_1122_a()
    N := 1122;
    order := 1;
    char_gens := [749, 409, 1057];
    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_1122_a_Hecke(Kf)
    return MakeCharacter_1122_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 := [[1], [1], [2], [4], [-1], [-2], [1], [-4], [-4], [2], [-4], [-6], [-6], [-4], [0], [6], [4], [-6], [-4], [4], [10], [4], [12], [-6], [-6], [14], [16], [4], [18], [-6], [-8], [-4], [-6], [-12], [6], [-16], [-2], [-12], [-12], [18], [20], [26], [0], [-22], [18], [-20], [-4], [-24], [-12], [-10], [18], [24], [26], [-20], [2], [-16], [10], [-8], [10], [10], [20], [6], [4], [12], [-14], [-30], [-20], [2], [12], [14], [18], [-8], [-20], [-10], [28], [16], [-18], [-14], [34], [-6], [4], [14], [-4], [2], [20], [28], [2], [10], [6], [-40], [-12], [-12], [4], [-20], [-20], [-12], [6], [26], [4], [26], [28], [-42], [-36], [42], [-20], [34], [-4], [18], [-16], [10], [36], [30], [-14], [-36], [0], [34], [36], [8], [-6], [12], [-18], [26], [50], [-4], [44], [-26], [-30], [-36], [-8], [30], [-20], [4], [20], [-10], [-54], [2], [-18], [44], [-2], [42], [-4], [50], [-44], [-44], [14], [-44], [26], [-54], [-20], [-8], [2], [-6], [44], [12], [20], [-20], [-32], [18], [26], [-6], [-36], [-38], [8], [4], [-30], [36], [-12], [-62], [10], [42], [-60], [-26], [-36], [-6], [0], [-46], [28], [58], [56], [-10], [-8], [-4], [10], [-54], [-16], [54], [-38], [52], [2], [-20], [2], [12], [-20], [38], [20], [-14], [-30], [-62], [26], [8], [-54], [-60], [-26], [66], [-28], [-42], [-56], [-44], [18], [-20], [10], [-66], [-20], [44], [12], [50], [32], [-30], [-60], [14], [-2], [-36], [66], [-52], [-12], [-58], [66], [-12], [-48], [60], [22], [52], [16], [-54], [52], [32], [-54], [66], [20], [32], [52], [-20], [8], [-2], [58], [4], [60], [52], [-20], [0], [-42], [-54], [-24], [42], [46], [68], [-38], [44], [10], [74], [4], [76], [66], [-78], [18], [76], [-66], [42], [28], [-42], [10], [-28], [58], [32], [2], [80], [4], [-34], [58], [44], [8], [-28], [-52], [-10], [-28], [24], [-6], [2], [72], [-30], [10], [-44], [-54], [-28], [42], [-30], [40], [54], [76], [-28], [58], [-66], [-52], [36], [28], [-6], [-60], [-30], [0], [14], [76], [-30], [2], [44], [-24], [-38], [-12], [-20], [-46], [-78], [36], [50], [46], [-24], [-6], [-30], [-52], [20], [68], [-30], [-46], [-62], [-84], [12], [-52], [-60], [-42], [-6], [-70], [-64], [-2], [-22], [34], [-48], [46], [-44], [74], [68], [-36], [90], [4], [-62], [6], [4], [-50], [26], [-24], [12], [-14], [56], [66], [-78], [-48], [-36], [60], [-70], [-30], [-80], [34], [-84], [-12], [-4], [-34], [16], [58], [36], [4], [-30], [18], [10], [58], [-54], [68], [42], [-92], [-60], [-32], [6], [92], [-8], [2], [42], [52], [4], [24], [50], [-24], [42], [-44], [14], [58], [18], [16], [10], [6], [60], [62], [-62], [4], [20], [-14], [38], [-28], [-52], [90], [-30], [-20], [68], [-30], [48], [14], [-86], [-20], [-20], [2], [-74], [76], [-30], [-28], [60]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1122_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1122_2_a_m();".
// 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_1122_2_a_m(:prec:=1)
    chi := MakeCharacter_1122_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_1122_2_a_m();".
// 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_1122_2_a_m( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1122_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-2, 1]>,<7,R![-4, 1]>,<13,R![2, 1]>,<19,R![4, 1]>],Snew);
    return Vf;
end function;