
// Make newform 6800.2.a.k in Magma, downloaded from the LMFDB on 07 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_6800_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_6800_2_a_k();".
// 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_6800_2_a_k();".
// 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_6800_a();"
function MakeCharacter_6800_a()
    N := 6800;
    order := 1;
    char_gens := [5951, 1701, 2177, 1601];
    v := [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_6800_a_Hecke( : Kf := Kf)
    return MakeCharacter_6800_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], [0], [5], [-4], [3], [1], [2], [8], [0], [5], [-12], [-10], [4], [2], [-1], [0], [2], [-8], [-5], [4], [17], [16], [-6], [16], [3], [-10], [-11], [4], [12], [-16], [9], [-3], [-5], [-7], [-2], [9], [-11], [16], [-10], [22], [2], [-10], [16], [-18], [8], [-1], [-10], [9], [13], [-8], [0], [10], [8], [3], [8], [-14], [26], [18], [17], [24], [6], [4], [-31], [-2], [-30], [18], [8], [-25], [-17], [27], [-2], [35], [-9], [25], [22], [30], [28], [-10], [19], [-20], [35], [11], [30], [19], [30], [-36], [3], [-30], [4], [-26], [12], [12], [-12], [-27], [-8], [33], [18], [36], [-24], [-3], [-9], [38], [34], [-35], [-34], [-16], [-23], [12], [8], [17], [-27], [-4], [36], [12], [28], [-31], [-12], [20], [6], [39], [-30], [36], [25], [11], [-2], [-22], [37], [-46], [31], [-34], [1], [24], [-10], [-31], [-23], [-31], [7], [-21], [-30], [7], [-42], [-31], [12], [13], [-21], [-30], [-52], [2], [-32], [0], [38], [-34], [9], [28], [37], [-4], [16], [1], [-30], [-36], [-39], [-4], [-32], [-27], [15], [9], [28], [40], [18], [20], [2], [-17], [-31], [-2], [8], [-36], [-60], [14], [1], [-56], [55], [-48], [-19], [6], [31], [-12], [26], [8], [-8], [-14], [12], [-22], [45], [59], [-12], [-48], [40], [6], [-62], [40], [-28], [-66], [35], [-10], [-35], [-50], [6], [-12], [-32], [16], [-70], [-13], [6], [51], [-32], [12], [39], [-17], [3], [-43], [8], [13], [41], [6], [18], [-44], [0], [0], [19], [-65], [11], [-30], [23], [-6], [46], [26], [-24], [3], [30], [39], [-12], [-54], [-50], [30], [-56], [44], [16], [32], [52], [-31], [-36], [-38], [14], [-3], [-22], [74], [52], [34], [-1], [-8], [-6], [-12], [38], [4], [-78], [-54], [-18], [-20], [-26], [-42], [-26], [-62], [-44], [58], [-10], [-22], [-33], [13], [-42], [-36], [23], [-51], [-49], [-64], [-2], [50], [78], [60], [-9], [0], [-40], [-6], [15], [-46], [58], [54], [18], [-81], [-46], [41], [-18], [25], [-63], [76], [-32], [-12], [74], [-54], [1], [-69], [-86], [-56], [36], [30], [25], [40], [-16], [28], [-45], [-28], [56], [-30], [-12], [30], [37], [7], [11], [-21], [-62], [0], [-14], [59], [70], [-52], [-15], [-5], [-4], [-72], [54], [-21], [-14], [72], [-8], [34], [52], [-70], [-92], [-57], [-18], [-18], [-82], [-39], [-31], [57], [-54], [4], [91], [-18], [42], [-12], [12], [-66], [-11], [2], [-10], [-54], [-56], [-8], [2], [-40], [35], [11], [-46], [-58], [39], [63], [-66], [-99], [23], [0], [-3], [53], [-32], [-93], [-24], [66], [53], [-7], [-42], [-8], [36], [-4], [38], [42], [-40], [-76], [-78], [-76], [55], [-68], [34], [-50], [-48], [42], [32], [37], [42], [-38], [-5], [14], [-34], [72], [84], [-101], [74], [19], [-37], [-104], [-6], [-49], [-36], [43], [34], [94], [50], [51], [-30], [-66], [12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6800_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6800_2_a_k();".
// 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_6800_2_a_k(:prec:=1)
    chi := MakeCharacter_6800_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_6800_2_a_k();".
// 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_6800_2_a_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6800_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, 1]>,<7,R![-5, 1]>,<11,R![4, 1]>,<13,R![-3, 1]>],Snew);
    return Vf;
end function;