
// Make newform 6840.2.a.b in Magma, downloaded from the LMFDB on 13 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_6840_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_6840_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_6840_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_6840_a();"
function MakeCharacter_6840_a()
    N := 6840;
    order := 1;
    char_gens := [1711, 3421, 5321, 2737, 6481];
    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_6840_a_Hecke( : Kf := Kf)
    return MakeCharacter_6840_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], [0], [-1], [-2], [2], [-4], [2], [-1], [0], [4], [8], [-8], [12], [-6], [0], [2], [0], [2], [-8], [-8], [6], [8], [8], [4], [-12], [-10], [-4], [-12], [-14], [-6], [0], [6], [-2], [0], [-10], [0], [-14], [6], [8], [-26], [0], [10], [-10], [16], [22], [4], [-4], [-20], [20], [-6], [6], [-6], [-22], [-22], [-6], [-28], [-20], [4], [-22], [16], [14], [6], [16], [14], [-14], [-14], [4], [-20], [16], [-26], [-18], [-34], [-6], [-36], [4], [-8], [-30], [2], [8], [22], [18], [14], [-4], [24], [0], [12], [-4], [-22], [-2], [34], [-12], [-18], [4], [14], [24], [28], [-40], [0], [-28], [14], [-32], [18], [-28], [12], [40], [18], [4], [-34], [-28], [30], [-36], [10], [-22], [40], [0], [4], [-34], [-12], [14], [20], [-22], [28], [42], [44], [-48], [-2], [-22], [-6], [10], [-42], [-44], [40], [32], [22], [-30], [-14], [-30], [-52], [6], [30], [4], [-18], [2], [36], [2], [-24], [38], [-2], [-28], [16], [-44], [-6], [46], [8], [-20], [0], [32], [-42], [-34], [-60], [24], [10], [-6], [28], [14], [40], [40], [-14], [-6], [-14], [-12], [-42], [38], [10], [40], [-10], [-8], [-18], [-12], [-50], [-34], [52], [52], [18], [40], [30], [-40], [-44], [6], [-30], [-16], [-28], [-44], [60], [12], [18], [-2], [6], [2], [0], [20], [32], [-20], [-70], [22], [30], [-24], [52], [-6], [12], [18], [34], [-22], [-36], [60], [-42], [26], [0], [-48], [54], [62], [-8], [68], [22], [-60], [-10], [-6], [60], [56], [62], [-2], [20], [-16], [36], [26], [32], [30], [-6], [30], [28], [-12], [68], [-26], [10], [30], [18], [6], [-24], [-20], [-12], [-2], [14], [-36], [-26], [-42], [34], [10], [-68], [58], [-10], [-44], [36], [6], [-52], [-46], [-44], [28], [14], [52], [-34], [38], [-16], [-18], [44], [76], [74], [-56], [-30], [70], [-82], [-24], [-40], [-48], [-50], [-22], [46], [26], [-22], [-32], [-12], [-2], [0], [54], [-60], [-44], [18], [-72], [-10], [80], [-30], [46], [42], [24], [-4], [40], [68], [-60], [-10], [-50], [6], [-56], [26], [24], [-64], [-60], [-70], [-2], [72], [34], [-6], [68], [70], [-84], [8], [58], [42], [-36], [-12], [-32], [-90], [-18], [42], [-44], [-8], [-40], [-48], [-10], [-66], [30], [-10], [-84], [54], [-80], [72], [2], [-28], [2], [-8], [28], [2], [-20], [0], [-6], [-20], [10], [-18], [2], [-30], [18], [72], [-6], [-6], [-48], [20], [38], [-24], [18], [-52], [70], [-26], [-52], [-36], [-72], [40], [58], [96], [50], [-22], [-82], [44], [96], [26], [-14], [30], [60], [-32], [-12], [34], [-26], [-24], [-10], [58], [-14], [6], [60], [0], [-56], [80], [100], [82], [-34], [-94], [-40], [-6], [-12], [-64], [-22], [-44], [-64], [10], [-4], [-58], [-52], [-72], [-58], [102], [-12], [48], [-74], [64], [56], [84], [-64], [-56], [40], [86], [-84], [-18], [88], [-94]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6840_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6840_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_6840_2_a_b(:prec:=1)
    chi := MakeCharacter_6840_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_6840_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_6840_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6840_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![2, 1]>,<11,R![-2, 1]>,<13,R![4, 1]>],Snew);
    return Vf;
end function;