
// Make newform 3192.2.a.n in Magma, downloaded from the LMFDB on 25 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_n();".
// 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_n();".
// 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], [0], [-1], [6], [4], [0], [-1], [-2], [2], [4], [10], [-6], [0], [0], [-6], [4], [6], [2], [-8], [-6], [-6], [16], [10], [-8], [0], [-4], [16], [-6], [2], [6], [20], [14], [12], [-10], [-14], [10], [16], [-8], [14], [16], [-8], [-10], [-14], [-18], [-24], [-6], [-16], [12], [-10], [-26], [6], [4], [20], [14], [26], [-10], [32], [-10], [18], [-4], [-10], [8], [-4], [-14], [-2], [-10], [-10], [-6], [-14], [8], [-30], [-16], [-18], [-2], [24], [-14], [26], [14], [-32], [-36], [-10], [-8], [-16], [4], [22], [2], [-26], [-16], [8], [-8], [-28], [30], [18], [24], [12], [-26], [-30], [16], [-30], [18], [-10], [4], [-42], [-12], [-38], [-16], [-12], [16], [40], [8], [-34], [30], [-44], [-28], [-30], [-36], [12], [-6], [0], [32], [-2], [10], [20], [20], [-30], [10], [44], [-8], [-22], [-20], [-36], [-42], [-2], [28], [34], [14], [-32], [22], [-46], [-40], [46], [-32], [40], [44], [0], [30], [18], [4], [20], [-22], [48], [16], [0], [-26], [48], [0], [-24], [38], [-18], [-38], [-6], [-52], [-4], [-18], [8], [2], [2], [-30], [48], [-36], [16], [-22], [18], [-40], [12], [-20], [2], [-56], [46], [-20], [-4], [-6], [14], [-54], [-24], [38], [-12], [-50], [16], [4], [-14], [50], [-6], [18], [2], [-54], [54], [-32], [4], [-14], [40], [8], [12], [0], [-32], [56], [-20], [66], [-32], [46], [-32], [-40], [20], [-40], [-28], [-16], [2], [48], [-30], [-46], [64], [6], [-72], [-12], [-42], [-34], [-44], [44], [-38], [-46], [72], [62], [-18], [-4], [-32], [74], [46], [6], [56], [36], [20], [-8], [14], [-2], [-60], [32], [20], [-26], [-14], [2], [28], [-2], [-12], [-4], [-34], [26], [-52], [42], [78], [38], [-60], [34], [-12], [28], [-20], [66], [-36], [34], [-62], [64], [70], [42], [-68], [40], [-40], [-66], [-2], [-46], [32], [64], [-32], [12], [-4], [-76], [-78], [46], [66], [56], [30], [6], [-24], [66], [52], [46], [-36], [52], [32], [-68], [76], [-26], [18], [4], [4], [0], [-82], [-28], [-76], [-26], [34], [48], [18], [82], [18], [86], [-56], [-24], [72], [-26], [78], [-52], [-10], [-30], [42], [-30], [60], [46], [8], [40], [-22], [34], [-58], [-40], [4], [16], [0], [-14], [42], [30], [24], [46], [-34], [-6], [-10], [-42], [0], [-62], [58], [8], [16], [-44], [-70], [-58], [24], [-66], [-30], [-8], [60], [-78], [-48], [74], [-76], [64], [-28], [28], [62], [-20], [-2], [34], [30], [60], [-70], [54], [-88], [42], [12], [78], [-22], [-36], [-12], [82], [2], [36], [-54], [76], [-96], [12], [-18], [72], [-64], [-26], [-6], [42], [-92], [36], [-58], [-52], [14], [-2], [10], [-12], [90], [82], [30], [70], [48], [46], [-50], [72], [0], [-56], [-102], [-24], [8], [-10], [36], [-12], [0], [-48], [-96], [-90], [96], [-78], [-100], [-48], [102], [24], [46], [76], [-40]];
    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_n();".
// 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_n(: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_n();".
// 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_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3192_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![0, 1]>,<11,R![-6, 1]>,<17,R![0, 1]>],Snew);
    return Vf;
end function;