
// Make newform 2112.2.a.k in Magma, downloaded from the LMFDB on 23 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_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_2112_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_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], [-2], [-1], [2], [0], [2], [0], [8], [-4], [6], [-4], [6], [8], [6], [-4], [6], [4], [8], [-10], [14], [-8], [2], [14], [0], [-8], [-8], [2], [14], [14], [20], [-2], [10], [12], [-10], [2], [16], [12], [-8], [-20], [10], [-24], [18], [8], [12], [18], [16], [0], [6], [-12], [-12], [14], [-28], [18], [-4], [-26], [10], [-22], [-12], [22], [4], [34], [-8], [-2], [-6], [-16], [-18], [-8], [-26], [-6], [8], [-8], [22], [-32], [-24], [-14], [-34], [-18], [-18], [-12], [26], [-20], [-34], [2], [4], [-30], [-18], [-12], [40], [-28], [-4], [-4], [40], [28], [-36], [22], [-10], [6], [10], [-26], [-20], [36], [-32], [22], [34], [44], [12], [24], [-22], [-10], [-2], [-2], [-40], [20], [34], [-16], [-32], [6], [36], [-38], [10], [0], [12], [-40], [44], [38], [-24], [28], [18], [14], [40], [20], [10], [-24], [10], [-10], [22], [-22], [-24], [-38], [0], [-48], [-8], [38], [0], [14], [16], [40], [-24], [-42], [6], [-4], [0], [-48], [-48], [10], [-10], [26], [-48], [12], [48], [-58], [-4], [38], [16], [-4], [-26], [38], [14], [-4], [46], [-24], [46], [-32], [-10], [-10], [-18], [-10], [14], [-12], [-36], [58], [64], [56], [-58], [2], [24], [-26], [-52], [10], [16], [12], [18], [-4], [-14], [58], [50], [32], [24], [60], [30], [-6], [-58], [-12], [42], [24], [-20], [-24], [-20], [-30], [30], [36], [-28], [32], [62], [-38], [-12], [-48], [6], [58], [6], [-66], [32], [-68], [-50], [6], [0], [-38], [0], [-10], [-14], [-2], [72], [-52], [40], [-54], [36], [-12], [16], [-36], [-10], [-68], [18], [-44], [0], [-52], [60], [54], [24], [10], [44], [72], [-74], [20], [0], [46], [-54], [-22], [2], [-22], [28], [-70], [78], [-54], [44], [-66], [-74], [34], [-16], [-2], [0], [50], [50], [-50], [-28], [4], [78], [-14], [-20], [68], [-76], [72], [-2], [-10], [-8], [-18], [-48], [-20], [-28], [2], [-52], [-52], [-4], [-66], [-60], [20], [-2], [-88], [10], [66], [28], [-74], [36], [20], [-50], [-28], [18], [80], [54], [-84], [-22], [-12], [-68], [68], [-34], [-44], [48], [26], [8], [-62], [26], [36], [4], [44], [82], [-12], [20], [-36], [32], [-62], [50], [14], [-72], [30], [-68], [-2], [36], [-38], [62], [22], [66], [-20], [16], [54], [-32], [10], [-32], [-64], [-42], [-22], [90], [-10], [54], [-50], [-48], [-40], [32], [24], [-40], [-42], [28], [-24], [0], [92], [-70], [-12], [-34], [-6], [52], [16], [-72], [0], [82], [34], [12], [-100], [-14], [-48], [-18], [-22], [-78], [46], [48], [78], [-24], [44], [62], [-82], [24], [-34], [-42], [-60], [28], [-64], [14], [6], [-78], [-40], [-24], [6], [102], [-42], [42], [76], [30], [54], [20], [-4], [20], [-62], [84], [-4], [-50], [-26], [10], [-60], [-64], [18], [-76], [6], [38], [0], [40], [58], [46], [-36], [20], [8], [36]];
    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_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_2112_2_a_k(: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_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_2112_2_a_k( : 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![2, 1]>,<13,R![-2, 1]>,<17,R![0, 1]>,<19,R![-2, 1]>,<23,R![0, 1]>],Snew);
    return Vf;
end function;