
// Make newform 1792.2.a.b in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1792_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_1792_2_a_b();".
// 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_1792_2_a_b();".
// 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_1792_a();"
function MakeCharacter_1792_a()
    N := 1792;
    order := 1;
    char_gens := [1023, 1541, 1025];
    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_1792_a_Hecke(Kf)
    return MakeCharacter_1792_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], [-2], [-2], [1], [0], [2], [6], [-6], [-4], [0], [8], [8], [6], [-8], [0], [-4], [-6], [-2], [-4], [-16], [-6], [-16], [-14], [-14], [-2], [14], [16], [-4], [16], [-10], [4], [-6], [-10], [-10], [-20], [20], [-6], [16], [-8], [-6], [4], [-6], [-8], [-22], [-12], [16], [28], [-8], [-14], [2], [-6], [-20], [14], [-2], [2], [0], [-30], [-8], [20], [26], [2], [30], [10], [8], [6], [-12], [-8], [-18], [24], [34], [-6], [-20], [-24], [-4], [16], [32], [-24], [-22], [18], [-34], [22], [-12], [-12], [-26], [0], [-12], [-2], [2], [18], [-24], [-38], [-8], [12], [20], [28], [-16], [26], [14], [-14], [-36], [-40], [4], [-26], [10], [0], [-22], [-26], [-14], [24], [34], [0], [16], [6], [10], [8], [18], [22], [-48], [-40], [40], [22], [-18], [-6], [-44], [-22], [-24], [8], [-16], [8], [-10], [32], [12], [4], [-16], [6], [30], [-18], [10], [-30], [-14], [-30], [-28], [-16], [-36], [30], [16], [-30], [-2], [22], [24], [8], [26], [36], [8], [28], [60], [32], [-10], [26], [6], [-24], [22], [-52], [-6], [14], [32], [-16], [-2], [10], [-22], [-44], [58], [0], [-54], [16], [2], [-52], [16], [56], [-34], [36], [30], [64], [42], [60], [-30], [-20], [-54], [22], [-8], [-38], [-28], [-44], [18], [-12], [66], [-50], [16], [42], [-32], [-36], [40], [-58], [-70], [34], [-18], [8], [0], [-18], [14], [34], [-26], [-28], [34], [16], [-26], [24], [14], [8], [72], [-8], [-64], [-66], [60], [-6], [40], [-46], [12], [48], [-48], [24], [30], [8], [-54], [2], [8], [54], [-52], [20], [-72], [56], [18], [48], [-20], [42], [48], [32], [-30], [40], [-32], [-52], [10], [8], [50], [-2], [52], [-40], [-26], [-26], [-18], [28], [48], [22], [22], [46], [62], [-20], [34], [60], [48], [42], [-56], [-14], [-28], [-30], [8], [8], [-44], [46], [74], [-24], [80], [24], [-50], [34], [-36], [46], [84], [-8], [-6], [-84], [-38], [-10], [-38], [-28], [-22], [40], [18], [6], [22], [62], [48], [24], [-48], [40], [-2], [-40], [-50], [-12], [52], [56], [-24], [-18], [64], [-40], [34], [70], [-56], [-38], [-62], [-66], [-42], [14], [-32], [18], [22], [-40], [-70], [-36], [-56], [-28], [-72], [-48], [46], [44], [-30], [64], [-38], [-38], [-8], [20], [-6], [-22], [24], [-8], [24], [34], [-48], [-24], [78], [-82], [-10], [60], [-24], [52], [-46], [72], [-14], [-2], [12], [-4], [42], [52], [12], [-22], [-10], [-38], [-52], [-46], [52], [-10], [0], [60], [40], [28], [10], [76], [10], [30], [78], [-78], [82], [-80], [-94], [-82], [-80], [32], [62], [-60], [56], [62], [86], [92], [-34], [68], [70], [-8], [-66], [-20], [-48], [-66], [46], [-72], [62], [-30], [-80], [100], [-6], [-58], [-42], [-54], [8], [48], [-52], [26], [90], [48], [-72], [-2], [64], [32], [46], [-80], [102], [102], [14], [24], [-6], [-90], [-40]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1792_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1792_2_a_b();".
// 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_1792_2_a_b(:prec:=1)
    chi := MakeCharacter_1792_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_1792_2_a_b();".
// 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_1792_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1792_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2, 1]>,<5,R![2, 1]>,<23,R![4, 1]>],Snew);
    return Vf;
end function;