
// Make newform 6720.2.a.s in Magma, downloaded from the LMFDB on 28 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6720_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_6720_2_a_s();".
// 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_6720_2_a_s();".
// 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_6720_a();"
function MakeCharacter_6720_a()
    N := 6720;
    order := 1;
    char_gens := [1471, 3781, 4481, 5377, 1921];
    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_6720_a_Hecke(Kf)
    return MakeCharacter_6720_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], [0], [-6], [-2], [-4], [4], [-6], [0], [-6], [-2], [4], [8], [2], [12], [-6], [4], [-12], [10], [-8], [12], [14], [2], [14], [16], [8], [-14], [-6], [8], [-4], [-6], [12], [-6], [-24], [2], [4], [16], [6], [16], [10], [-4], [18], [2], [0], [12], [0], [-12], [-6], [-22], [-12], [-22], [-4], [6], [-12], [30], [-16], [26], [2], [-4], [-34], [-28], [16], [34], [10], [20], [34], [0], [26], [-18], [20], [16], [10], [20], [32], [-30], [34], [-6], [34], [4], [-6], [28], [10], [16], [0], [2], [-38], [-26], [40], [-28], [24], [-24], [-8], [-20], [24], [-18], [-26], [36], [34], [20], [26], [36], [34], [-4], [-38], [12], [-2], [-28], [26], [-48], [-38], [10], [28], [16], [-6], [-12], [0], [-30], [40], [2], [2], [-18], [0], [20], [26], [26], [0], [-16], [10], [-28], [-36], [0], [42], [14], [-30], [14], [-36], [-10], [42], [12], [-38], [32], [8], [34], [-16], [26], [-34], [-12], [-20], [50], [14], [-4], [-48], [20], [-20], [40], [6], [2], [-34], [32], [-30], [32], [-28], [-62], [24], [-16], [-54], [2], [-18], [-16], [10], [-28], [10], [-40], [6], [-36], [-30], [0], [58], [-8], [-20], [26], [6], [60], [46], [-14], [-20], [10], [48], [-14], [24], [-52], [54], [-32], [-10], [34], [2], [54], [-32], [-62], [-40], [50], [-22], [4], [-42], [-40], [32], [42], [-28], [2], [-26], [40], [52], [-64], [58], [16], [-50], [4], [66], [-38], [56], [-30], [32], [-52], [26], [30], [36], [-32], [-72], [-14], [28], [32], [-38], [12], [-48], [10], [-30], [8], [-56], [-24], [-68], [24], [-14], [-26], [48], [-8], [-12], [-36], [44], [50], [-18], [-68], [-46], [38], [24], [74], [-4], [-18], [2], [-8], [-8], [-70], [34], [62], [20], [34], [-74], [-4], [-38], [10], [12], [10], [-32], [18], [-48], [72], [-30], [-6], [84], [0], [-32], [-8], [2], [-28], [12], [-30], [-62], [40], [-42], [2], [12], [-6], [-4], [50], [-26], [-56], [6], [-84], [-68], [82], [-62], [40], [-48], [60], [50], [64], [18], [52], [-38], [80], [66], [26], [44], [68], [34], [-4], [-12], [18], [90], [28], [-6], [54], [48], [82], [90], [-76], [76], [-12], [-38], [50], [-14], [64], [-44], [92], [-68], [-78], [6], [-14], [-72], [-22], [-6], [-10], [56], [-46], [8], [-38], [-76], [16], [-18], [-68], [-86], [-30], [-64], [10], [70], [48], [52], [58], [76], [-38], [78], [84], [-32], [-4], [-22], [78], [80], [74], [0], [-44], [4], [-22], [-40], [-46], [12], [-60], [-6], [-66], [26], [-42], [34], [72], [-54], [4], [56], [80], [18], [-4], [56], [-30], [-58], [-64], [68], [-92], [-54], [-56], [86], [28], [50], [-70], [-54], [-24], [62], [6], [64], [-94], [-30], [-4], [-12], [90], [90], [-64], [-4], [-22], [14], [-12], [-16], [-18], [0], [-38], [-70], [36], [52], [34], [6], [104], [66], [92], [0], [82], [-24], [76], [48], [2], [-58], [26], [-38], [108]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6720_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6720_2_a_s();".
// 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_6720_2_a_s(:prec:=1)
    chi := MakeCharacter_6720_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(3067) - 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_6720_2_a_s();".
// 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_6720_2_a_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6720_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![0, 1]>,<13,R![6, 1]>,<17,R![2, 1]>,<19,R![4, 1]>,<23,R![-4, 1]>,<29,R![6, 1]>,<31,R![0, 1]>],Snew);
    return Vf;
end function;