
// Make newform 2112.2.a.f 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_f();".
// 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_f();".
// 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], [4], [1], [2], [2], [-4], [0], [6], [4], [2], [-6], [4], [-8], [6], [-12], [10], [-4], [0], [10], [12], [-12], [10], [18], [14], [4], [4], [18], [-14], [-12], [12], [2], [12], [6], [20], [-22], [20], [-8], [14], [12], [-22], [-16], [18], [-2], [4], [4], [12], [20], [-22], [18], [-16], [-30], [-20], [-30], [-8], [-2], [-28], [10], [18], [12], [-2], [-28], [8], [10], [14], [-12], [-14], [-20], [10], [18], [16], [-28], [18], [12], [32], [-2], [10], [26], [26], [-12], [-6], [-24], [-14], [-4], [-36], [-22], [10], [30], [28], [20], [24], [-4], [4], [-20], [16], [-34], [18], [-12], [-30], [-4], [14], [-28], [-6], [-4], [18], [-12], [-6], [16], [-22], [-12], [34], [-6], [4], [12], [10], [-28], [-48], [38], [-20], [-46], [-30], [-42], [-20], [36], [-26], [10], [24], [12], [-14], [-44], [-8], [-28], [26], [42], [18], [6], [-4], [6], [-6], [-44], [-18], [-20], [-12], [2], [0], [2], [-38], [-20], [32], [26], [2], [52], [24], [28], [48], [20], [42], [10], [-18], [12], [-6], [-36], [-36], [58], [24], [-44], [2], [-14], [14], [4], [-22], [56], [26], [-28], [-54], [-20], [14], [28], [-22], [-44], [20], [-54], [-46], [32], [-18], [18], [-52], [58], [-48], [2], [20], [-20], [-26], [12], [42], [2], [18], [-54], [-40], [-2], [-44], [-22], [34], [-52], [-18], [-4], [-12], [-62], [44], [34], [14], [36], [52], [0], [-22], [-68], [-46], [40], [6], [-6], [12], [-54], [-4], [68], [2], [-70], [40], [28], [-60], [18], [28], [68], [18], [-20], [-8], [34], [-26], [-68], [48], [36], [28], [-4], [-38], [42], [16], [-28], [-20], [-20], [0], [-62], [42], [32], [-22], [38], [-68], [-22], [-20], [54], [-6], [4], [-20], [-38], [-70], [-38], [-20], [-58], [-38], [36], [70], [42], [44], [-70], [76], [34], [-28], [28], [-22], [-70], [-20], [-40], [20], [16], [2], [68], [32], [82], [-58], [-52], [-86], [54], [-36], [2], [4], [-54], [-50], [-84], [-26], [-36], [28], [-70], [-18], [-4], [-4], [-84], [34], [-36], [-30], [80], [34], [0], [54], [66], [-44], [8], [-6], [44], [-72], [-46], [-54], [-28], [-54], [-42], [4], [-70], [-62], [-12], [44], [-48], [6], [50], [54], [68], [76], [60], [12], [10], [42], [58], [4], [82], [-78], [-66], [52], [-34], [-60], [10], [-44], [8], [22], [60], [26], [78], [-28], [-62], [-30], [0], [68], [2], [-40], [50], [50], [-80], [-36], [76], [90], [70], [20], [26], [68], [92], [48], [-34], [-28], [-78], [20], [-16], [66], [50], [42], [-10], [82], [52], [42], [-68], [-72], [-28], [42], [-20], [-72], [2], [-34], [-76], [-92], [72], [58], [-44], [-30], [-36], [-42], [10], [10], [-28], [18], [-66], [20], [82], [-30], [68], [100], [34], [54], [-84], [44], [42], [-50], [16], [28], [42], [0], [102], [18], [80], [-36], [-70], [94], [36], [66], [-52], [-16]];
    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_f();".
// 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_f(: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_f();".
// 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_f( : 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![-4, 1]>,<13,R![-2, 1]>,<17,R![-2, 1]>,<19,R![4, 1]>,<23,R![0, 1]>],Snew);
    return Vf;
end function;