
// Make newform 1680.2.a.t in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1680_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_1680_2_a_t();".
// 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_1680_2_a_t();".
// 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_1680_a();"
function MakeCharacter_1680_a()
    N := 1680;
    order := 1;
    char_gens := [1471, 421, 1121, 337, 241];
    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_1680_a_Hecke(Kf)
    return MakeCharacter_1680_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], [1], [1], [4], [-2], [2], [4], [0], [-10], [0], [6], [-6], [4], [8], [6], [4], [-10], [-4], [16], [-14], [-8], [4], [10], [10], [-10], [-8], [4], [14], [-6], [-16], [12], [18], [-4], [-2], [-16], [14], [-4], [0], [-18], [-4], [-2], [-24], [2], [6], [-24], [4], [16], [4], [-2], [18], [8], [-14], [20], [2], [0], [-18], [0], [-10], [-22], [-20], [6], [4], [-24], [-30], [14], [12], [18], [36], [-26], [-14], [-32], [-32], [-10], [-20], [-24], [30], [-18], [-30], [-22], [-4], [22], [8], [-38], [8], [-28], [-14], [-6], [-2], [32], [20], [0], [40], [4], [36], [0], [-34], [-6], [-4], [46], [28], [-18], [4], [-22], [-20], [-38], [-36], [34], [-32], [10], [-32], [6], [18], [-20], [16], [-30], [4], [16], [14], [28], [46], [18], [6], [20], [36], [-10], [38], [0], [24], [-34], [4], [16], [-40], [22], [42], [2], [6], [-44], [30], [42], [44], [-2], [24], [52], [-42], [56], [-10], [26], [-52], [8], [-18], [2], [-20], [32], [52], [8], [32], [50], [50], [-18], [12], [18], [-56], [-44], [-54], [48], [-8], [22], [-14], [-10], [4], [-10], [-32], [-22], [16], [10], [12], [30], [24], [22], [48], [-36], [38], [26], [24], [-58], [-34], [36], [42], [48], [26], [52], [-12], [-18], [44], [-38], [18], [-2], [-30], [0], [-58], [-32], [-26], [-46], [36], [30], [16], [-52], [-38], [-52], [-62], [-42], [40], [12], [-24], [-22], [32], [-30], [0], [-34], [6], [40], [66], [32], [-60], [6], [-6], [24], [-72], [20], [46], [20], [56], [-70], [-36], [-72], [66], [6], [20], [-40], [-36], [44], [24], [46], [-46], [-24], [-48], [12], [-4], [56], [62], [18], [48], [-54], [46], [-4], [-10], [-52], [-26], [2], [-32], [-36], [46], [-2], [50], [-28], [6], [42], [-44], [70], [-26], [-52], [82], [24], [-22], [8], [-44], [-66], [74], [12], [24], [-16], [-48], [-18], [-52], [8], [-62], [-74], [40], [18], [38], [36], [-30], [-12], [-18], [14], [48], [22], [36], [-44], [34], [78], [-24], [-36], [60], [34], [-60], [54], [0], [70], [24], [78], [-14], [-20], [-32], [-70], [60], [-72], [58], [18], [68], [-22], [-18], [16], [-78], [-78], [20], [-36], [8], [54], [62], [30], [48], [-28], [-4], [-20], [-50], [-62], [-70], [-64], [54], [50], [-90], [64], [-18], [-36], [30], [-12], [32], [-42], [84], [-70], [-10], [-32], [22], [74], [64], [20], [42], [64], [-58], [-38], [-24], [-60], [20], [-6], [78], [8], [-38], [-20], [92], [72], [-50], [24], [30], [-68], [-8], [-6], [66], [18], [46], [66], [40], [-6], [68], [0], [-24], [-10], [-60], [56], [2], [6], [-28], [68], [-32], [58], [16], [26], [28], [30], [38], [-6], [32], [-6], [70], [56], [-50], [66], [-60], [76], [58], [38], [-44], [-28], [-22], [14], [-24], [8], [-94], [-48], [-10], [-10], [-40], [36], [50], [14], [-36], [-6], [92], [-24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1680_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1680_2_a_t();".
// 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_1680_2_a_t(:prec:=1)
    chi := MakeCharacter_1680_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_1680_2_a_t();".
// 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_1680_2_a_t( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1680_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![-4, 1]>,<13,R![2, 1]>,<17,R![-2, 1]>,<19,R![-4, 1]>],Snew);
    return Vf;
end function;