
// Make newform 2394.2.a.o in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2394_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_2394_2_a_o();".
// 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_2394_2_a_o();".
// 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_2394_a();"
function MakeCharacter_2394_a()
    N := 2394;
    order := 1;
    char_gens := [533, 1711, 1009];
    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_2394_a_Hecke(Kf)
    return MakeCharacter_2394_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], [0], [4], [1], [-2], [-2], [0], [1], [-2], [6], [4], [2], [10], [4], [-6], [-2], [-4], [2], [-12], [0], [-14], [8], [-10], [18], [-18], [0], [0], [0], [-10], [-2], [4], [18], [12], [-12], [-4], [-8], [-14], [-4], [-16], [14], [0], [-10], [-22], [18], [8], [8], [-20], [16], [0], [-6], [-8], [-14], [-10], [30], [-10], [-30], [14], [16], [-14], [26], [-28], [6], [-8], [18], [22], [10], [0], [-2], [-18], [-2], [24], [2], [0], [26], [-24], [12], [-24], [-18], [-10], [38], [26], [-26], [-24], [26], [-32], [-10], [10], [18], [-16], [-32], [6], [-18], [-4], [-10], [12], [-26], [14], [-14], [20], [-18], [8], [28], [-24], [-38], [36], [14], [-42], [36], [-4], [42], [4], [-6], [-20], [-20], [16], [30], [-12], [18], [-36], [44], [26], [-46], [-14], [48], [4], [12], [38], [26], [16], [-2], [4], [8], [-24], [10], [0], [-18], [14], [52], [46], [0], [24], [-36], [32], [48], [-10], [12], [14], [-22], [52], [-32], [2], [-24], [-20], [48], [-28], [36], [-24], [-36], [-38], [14], [6], [2], [-48], [16], [14], [-24], [-4], [-22], [14], [-32], [0], [26], [54], [6], [-12], [12], [-60], [60], [-44], [-42], [-56], [-16], [14], [22], [46], [0], [-2], [36], [-58], [10], [30], [-58], [32], [58], [-66], [26], [34], [62], [4], [-26], [46], [-20], [-58], [-26], [-30], [-8], [0], [0], [-36], [40], [70], [28], [64], [12], [36], [18], [48], [42], [24], [-32], [22], [-64], [54], [56], [24], [-70], [-42], [-40], [28], [6], [-14], [-40], [-20], [-26], [4], [-14], [10], [24], [-2], [-52], [-48], [68], [48], [-70], [-2], [46], [64], [48], [16], [30], [50], [-4], [38], [-10], [-36], [30], [-42], [-64], [2], [62], [32], [-28], [6], [-54], [52], [20], [30], [68], [-40], [-24], [-62], [12], [-74], [56], [-2], [64], [82], [10], [66], [6], [-36], [56], [-42], [-30], [-60], [-58], [30], [2], [56], [54], [-12], [38], [-54], [-36], [10], [-4], [32], [4], [-40], [-60], [-66], [-18], [-80], [-24], [-36], [-10], [-20], [-58], [62], [86], [54], [-24], [22], [-32], [-18], [-26], [-22], [36], [-14], [20], [44], [58], [90], [-64], [-28], [30], [40], [12], [-8], [-44], [82], [30], [56], [-54], [-4], [-50], [2], [42], [-10], [48], [58], [-68], [-54], [68], [-66], [-60], [82], [-28], [0], [-48], [-68], [38], [56], [36], [82], [-78], [30], [-18], [36], [96], [-18], [-80], [-48], [-96], [52], [-6], [48], [36], [-62], [-30], [48], [-18], [54], [40], [-34], [-80], [54], [66], [-36], [-2], [-30], [-16], [24], [88], [44], [12], [-48], [-86], [-76], [0], [-58], [-54], [46], [44], [48], [-66], [76], [30], [-68], [-24], [-86], [-24], [40], [102], [-90], [-80], [14], [-54], [64], [-54], [74], [-32], [-68], [28], [-38], [8], [36], [76], [56], [-76], [-66], [-70], [102], [-16], [50], [-66], [-40], [-48], [68], [-14]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2394_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2394_2_a_o();".
// 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_2394_2_a_o(:prec:=1)
    chi := MakeCharacter_2394_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_2394_2_a_o();".
// 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_2394_2_a_o( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2394_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-4, 1]>,<11,R![2, 1]>,<13,R![2, 1]>,<17,R![0, 1]>],Snew);
    return Vf;
end function;