
// Make newform 1792.2.a.h in Magma, downloaded from the LMFDB on 24 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_h();".
// 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_h();".
// 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], [4], [1], [2], [-4], [2], [-6], [0], [8], [8], [8], [-10], [2], [-8], [0], [-10], [4], [-2], [8], [6], [-8], [-6], [-10], [2], [4], [-16], [-6], [8], [2], [-16], [-6], [6], [-2], [-16], [-16], [-4], [22], [0], [12], [14], [12], [8], [10], [0], [-16], [-2], [0], [-6], [-4], [-18], [16], [2], [-18], [18], [-8], [12], [16], [-16], [-2], [22], [4], [-22], [8], [6], [32], [18], [-30], [18], [-20], [-30], [0], [0], [0], [34], [-24], [-24], [28], [10], [-26], [-14], [0], [32], [2], [24], [-6], [10], [-10], [-20], [-40], [26], [-40], [16], [-6], [-34], [-24], [12], [6], [22], [16], [22], [16], [18], [38], [18], [-14], [-2], [-46], [24], [-10], [48], [-8], [14], [-10], [-8], [26], [34], [0], [-24], [-10], [36], [-6], [12], [26], [10], [-24], [-8], [8], [16], [-28], [22], [-16], [48], [-40], [6], [-14], [36], [-38], [28], [-42], [6], [48], [-8], [42], [-12], [48], [12], [22], [-34], [-24], [8], [-30], [30], [0], [-22], [32], [8], [34], [-10], [52], [38], [-42], [-32], [6], [42], [-32], [-40], [20], [-38], [60], [-38], [12], [24], [46], [-40], [54], [26], [-8], [56], [-44], [48], [18], [8], [22], [-48], [12], [16], [-54], [38], [56], [2], [-38], [-2], [28], [-2], [6], [-30], [56], [66], [-24], [-48], [40], [-44], [-14], [-10], [4], [8], [54], [-66], [-34], [-22], [-60], [-16], [-10], [24], [-26], [-24], [-14], [-40], [-8], [-40], [-40], [-54], [16], [-38], [56], [6], [-16], [-24], [-46], [-24], [18], [-8], [62], [-58], [-48], [66], [16], [-6], [40], [22], [6], [24], [64], [34], [-32], [32], [42], [-30], [56], [-64], [-30], [72], [70], [-12], [-18], [-40], [-50], [36], [70], [0], [22], [-44], [4], [-78], [-14], [48], [-74], [-6], [24], [-36], [-26], [70], [80], [18], [56], [-62], [-16], [-34], [-54], [32], [8], [32], [-28], [70], [-48], [26], [-16], [-32], [-78], [64], [-22], [-2], [54], [-16], [60], [-16], [60], [-34], [-14], [-26], [72], [24], [-26], [-62], [-38], [18], [-76], [32], [48], [32], [-24], [-70], [70], [-24], [-10], [18], [-56], [-62], [18], [18], [54], [-60], [-56], [-42], [-62], [-58], [38], [32], [24], [64], [-72], [24], [18], [74], [70], [-8], [2], [-10], [56], [-16], [6], [-84], [24], [24], [6], [-4], [-30], [0], [20], [-62], [-58], [48], [24], [80], [86], [-72], [-10], [26], [48], [64], [6], [32], [90], [-22], [-82], [12], [64], [38], [78], [-34], [40], [-64], [-72], [-32], [-6], [-96], [-46], [34], [6], [-20], [54], [88], [66], [82], [40], [-40], [-28], [42], [72], [-46], [36], [58], [82], [32], [-10], [56], [6], [-22], [56], [4], [10], [-40], [54], [-4], [-32], [64], [90], [98], [-94], [18], [-24], [-94], [-50], [-50], [-36], [-72], [-48], [82], [-8], [72], [20], [88], [30], [-74], [-12], [-26], [-34], [-66], [-64]];
    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_h();".
// 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_h(: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_h();".
// 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_h( : 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![-4, 1]>,<23,R![0, 1]>],Snew);
    return Vf;
end function;