
// Make newform 3192.2.a.k in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3192_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_3192_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_3192_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_3192_a();"
function MakeCharacter_3192_a()
    N := 3192;
    order := 1;
    char_gens := [799, 1597, 2129, 913, 1009];
    v := [1, 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_3192_a_Hecke(Kf)
    return MakeCharacter_3192_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], [-4], [-1], [4], [2], [-4], [-1], [0], [4], [-8], [-2], [-6], [12], [-10], [0], [-12], [-2], [-8], [6], [14], [-8], [6], [2], [2], [-12], [-8], [-14], [-14], [-16], [-4], [-22], [-6], [8], [-22], [4], [-2], [4], [-12], [6], [-10], [-2], [8], [-14], [-10], [20], [-12], [16], [-24], [-14], [6], [28], [26], [26], [6], [-12], [18], [-8], [10], [16], [8], [-34], [20], [6], [26], [-12], [24], [-2], [32], [-26], [-36], [24], [36], [-34], [-32], [4], [18], [-2], [28], [6], [30], [2], [-10], [2], [16], [-24], [20], [-26], [20], [-24], [-14], [-6], [-8], [16], [28], [6], [18], [-10], [4], [-18], [-4], [-46], [-24], [24], [-4], [18], [-38], [-24], [-6], [-38], [40], [-14], [2], [8], [8], [-12], [-16], [-18], [6], [6], [2], [2], [2], [22], [-28], [-10], [26], [-14], [-8], [10], [12], [22], [12], [-34], [-32], [-6], [6], [-20], [30], [-22], [-12], [2], [-56], [-6], [46], [-24], [-2], [-6], [44], [2], [-2], [36], [-12], [20], [-36], [-46], [-40], [-4], [-38], [18], [12], [-44], [8], [28], [-44], [-56], [56], [62], [-2], [-36], [-30], [10], [44], [54], [-16], [-24], [-28], [-2], [-8], [-26], [32], [12], [26], [58], [24], [36], [-14], [36], [14], [50], [30], [52], [16], [-14], [-44], [-66], [-26], [54], [28], [-22], [40], [8], [-34], [-50], [10], [-20], [4], [34], [-18], [-4], [46], [-4], [48], [-48], [40], [-50], [0], [70], [-6], [18], [34], [-40], [64], [-40], [-12], [-10], [-2], [42], [-64], [-24], [-10], [-4], [32], [0], [-44], [66], [38], [14], [-24], [52], [54], [44], [-4], [-66], [54], [34], [12], [8], [20], [48], [-18], [-48], [36], [54], [24], [12], [-58], [28], [26], [-46], [-56], [-70], [58], [-46], [-8], [-36], [-72], [-56], [20], [18], [-82], [-4], [46], [-24], [-62], [36], [56], [-26], [10], [58], [84], [80], [22], [-74], [-76], [44], [-62], [-36], [4], [22], [-38], [-10], [24], [-20], [14], [-16], [48], [60], [-24], [-16], [-70], [12], [-8], [-54], [20], [30], [34], [18], [12], [50], [-26], [58], [12], [-32], [64], [58], [-6], [62], [-66], [18], [36], [-2], [-78], [-36], [14], [-22], [-52], [-28], [-18], [14], [62], [20], [-4], [-26], [28], [26], [78], [-14], [30], [-88], [-66], [-62], [-82], [32], [12], [54], [10], [-84], [-40], [84], [-28], [-42], [-50], [48], [-90], [54], [30], [-66], [78], [14], [86], [-48], [-26], [74], [72], [-62], [-72], [44], [-46], [20], [-76], [-36], [-8], [0], [-74], [60], [88], [74], [24], [-62], [-70], [-6], [8], [82], [76], [60], [-40], [34], [68], [52], [6], [6], [36], [28], [-90], [-2], [88], [-62], [72], [-78], [34], [-14], [-72], [30], [-94], [12], [70], [84], [-60], [-42], [-14], [-6], [-6], [20], [-86], [-48], [-10], [8], [44], [-24], [-4], [38], [-36], [20], [70], [-18], [6], [62], [-40], [22]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3192_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3192_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_3192_2_a_k(:prec:=1)
    chi := MakeCharacter_3192_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_3192_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_3192_2_a_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3192_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![4, 1]>,<11,R![-4, 1]>,<17,R![4, 1]>],Snew);
    return Vf;
end function;