
// Make newform 1680.2.a.e in Magma, downloaded from the LMFDB on 06 September 2025.
// 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_e();".
// 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_e();".
// 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 := 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]];
    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], [-6], [-2], [8], [-4], [6], [-4], [-2], [-2], [12], [0], [2], [4], [6], [4], [-8], [6], [16], [4], [-18], [6], [18], [16], [0], [14], [14], [16], [-4], [6], [8], [-2], [-8], [-22], [-4], [16], [-14], [-12], [14], [-24], [-14], [-6], [-20], [4], [-8], [-4], [-10], [14], [24], [2], [4], [-18], [4], [18], [20], [14], [-6], [28], [-6], [12], [-32], [22], [-22], [-36], [-14], [-32], [-10], [-34], [0], [8], [-10], [36], [0], [-10], [-30], [-38], [18], [20], [6], [16], [-18], [-20], [32], [34], [2], [-22], [0], [28], [-8], [-16], [-4], [-4], [8], [-30], [-42], [-4], [14], [-28], [42], [-12], [-14], [-4], [14], [12], [-18], [32], [34], [-24], [-2], [30], [-8], [8], [-46], [28], [16], [42], [4], [-18], [-6], [42], [48], [-8], [6], [6], [-24], [-8], [-38], [20], [-36], [24], [6], [30], [10], [42], [-12], [2], [-14], [8], [6], [8], [-16], [38], [8], [26], [54], [8], [4], [-42], [-34], [-20], [24], [-28], [-24], [0], [-10], [-2], [-30], [16], [-26], [-16], [-52], [6], [-48], [-8], [-38], [34], [10], [36], [-50], [40], [-22], [60], [54], [36], [6], [8], [30], [8], [-44], [-42], [6], [12], [34], [22], [-28], [-38], [24], [22], [-48], [-36], [-22], [-48], [-42], [-30], [-2], [14], [-32], [30], [4], [58], [-62], [-4], [-30], [12], [40], [66], [8], [-54], [42], [-56], [-4], [-40], [2], [56], [-34], [52], [-6], [38], [-44], [-6], [-16], [-20], [22], [54], [-32], [32], [60], [-34], [24], [-48], [-46], [-28], [-24], [-6], [-6], [-20], [0], [-48], [56], [8], [-50], [-2], [24], [64], [-60], [4], [-44], [-74], [30], [36], [-30], [34], [-36], [-10], [28], [-54], [70], [-64], [-8], [22], [-38], [30], [-40], [30], [38], [-36], [50], [-34], [-44], [14], [-24], [22], [-72], [-48], [-66], [-38], [-68], [24], [-8], [80], [6], [-20], [-24], [-6], [18], [20], [-26], [-34], [68], [-66], [60], [-34], [42], [28], [-54], [-44], [-44], [14], [42], [56], [48], [-52], [50], [-8], [14], [-72], [6], [56], [-50], [74], [-4], [60], [42], [-20], [0], [86], [-70], [64], [-46], [42], [-32], [-18], [50], [52], [92], [12], [26], [-18], [-22], [52], [36], [-12], [76], [-2], [-66], [42], [-32], [86], [-18], [-6], [40], [10], [20], [70], [52], [-72], [-22], [24], [-94], [78], [-16], [22], [6], [-8], [-12], [-18], [76], [6], [30], [-92], [76], [52], [66], [18], [32], [-22], [68], [-88], [4], [-66], [12], [-26], [12], [48], [94], [46], [-10], [-22], [-90], [-16], [86], [16], [-24], [40], [34], [60], [0], [-46], [-6], [-4], [60], [-72], [-22], [-20], [-26], [76], [-66], [-50], [-82], [40], [6], [-102], [20], [-42], [-62], [-36], [-36], [86], [90], [-16], [-92], [26], [-30], [24], [-8], [30], [-72], [94], [42], [-60], [-36], [-74], [82], [-64], [82], [-80], [24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1680_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1680_2_a_e();".
// 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_e(: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_e();".
// 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_e( : 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![6, 1]>,<17,R![2, 1]>,<19,R![-8, 1]>],Snew);
    return Vf;
end function;