
// Make newform 2112.2.a.z in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2112_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_2112_2_a_z();".
// 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_2112_2_a_z();".
// 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_2112_a();"
function MakeCharacter_2112_a()
    N := 2112;
    order := 1;
    char_gens := [2047, 133, 1409, 1729];
    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_2112_a_Hecke(Kf)
    return MakeCharacter_2112_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], [2], [0], [-1], [6], [2], [-4], [4], [-2], [4], [2], [10], [-4], [4], [-6], [-12], [6], [-4], [4], [10], [16], [4], [10], [2], [6], [-20], [-4], [-10], [-14], [-8], [12], [10], [-4], [6], [-8], [18], [-20], [0], [6], [-4], [2], [12], [-22], [-26], [-4], [-4], [-20], [28], [2], [18], [-8], [10], [12], [18], [24], [18], [-16], [-10], [-14], [-28], [-18], [20], [12], [-22], [-6], [-20], [2], [12], [-34], [-30], [24], [4], [-26], [-28], [-12], [-30], [34], [-30], [-14], [4], [-22], [24], [2], [32], [-36], [-30], [2], [-2], [36], [-20], [-8], [-12], [-36], [4], [8], [-14], [-6], [-36], [-10], [-20], [-2], [-4], [-6], [12], [-30], [20], [26], [-12], [42], [8], [-2], [-22], [20], [-20], [-46], [-28], [-28], [34], [-44], [26], [26], [14], [44], [4], [6], [-30], [44], [36], [22], [12], [32], [-12], [-22], [10], [-22], [26], [52], [-6], [-6], [-52], [6], [-52], [36], [-46], [44], [46], [-6], [4], [28], [46], [-30], [4], [-16], [44], [-28], [32], [34], [-6], [6], [20], [-6], [0], [60], [34], [-36], [52], [-26], [10], [-6], [-12], [-14], [-48], [26], [36], [26], [60], [42], [-56], [30], [-44], [20], [42], [-46], [-60], [10], [-2], [44], [-30], [48], [50], [36], [-28], [58], [36], [42], [-30], [-22], [42], [8], [-66], [64], [34], [50], [28], [-54], [-28], [-44], [-30], [60], [10], [10], [-12], [-52], [24], [58], [32], [-30], [12], [-30], [-10], [48], [18], [-20], [28], [6], [42], [-12], [-64], [28], [34], [36], [32], [18], [-60], [-40], [-62], [-26], [-4], [76], [-36], [-28], [12], [18], [74], [-40], [12], [-60], [-20], [8], [-2], [10], [-36], [-70], [-26], [-4], [58], [52], [-30], [-14], [64], [-68], [46], [14], [18], [28], [18], [42], [4], [-18], [34], [76], [-54], [32], [-30], [76], [-12], [-10], [-38], [36], [-24], [-36], [40], [-74], [4], [-28], [2], [-50], [20], [-22], [42], [12], [-62], [-84], [30], [-66], [20], [10], [44], [68], [-14], [-66], [24], [-12], [28], [-14], [36], [-30], [-20], [38], [-40], [2], [18], [12], [8], [-86], [-12], [-72], [-46], [42], [-28], [74], [-42], [-36], [42], [34], [-12], [68], [40], [46], [46], [-6], [-48], [4], [36], [52], [-14], [-22], [26], [-88], [26], [58], [62], [-28], [-14], [20], [2], [-68], [0], [-54], [-44], [-22], [66], [-8], [-50], [-30], [36], [-20], [-62], [68], [14], [-78], [36], [36], [-20], [58], [6], [56], [-14], [36], [20], [48], [-10], [-24], [-62], [92], [-72], [34], [18], [-30], [-22], [-70], [24], [42], [-92], [36], [-28], [74], [36], [-12], [-78], [74], [-28], [-68], [52], [2], [-16], [42], [68], [78], [38], [18], [64], [42], [-98], [-64], [2], [-78], [20], [12], [82], [38], [-76], [-20], [-38], [74], [72], [-28], [82], [-40], [66], [46], [-76], [-12], [42], [-46], [-52], [-30], [4], [32]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2112_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2112_2_a_z();".
// 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_2112_2_a_z(:prec:=1)
    chi := MakeCharacter_2112_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_2112_2_a_z();".
// 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_2112_2_a_z( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2112_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-2, 1]>,<7,R![0, 1]>,<13,R![-6, 1]>,<17,R![-2, 1]>,<19,R![4, 1]>,<23,R![-4, 1]>],Snew);
    return Vf;
end function;