
// Make newform 7728.2.a.k in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7728_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_7728_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_7728_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_7728_a();"
function MakeCharacter_7728_a()
    N := 7728;
    order := 1;
    char_gens := [4831, 5797, 5153, 6625, 6721];
    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_7728_a_Hecke(Kf)
    return MakeCharacter_7728_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], [4], [-1], [-3], [-2], [4], [7], [1], [10], [2], [-10], [7], [4], [-3], [-9], [-9], [-1], [0], [-8], [4], [-14], [6], [6], [-2], [3], [13], [20], [-8], [10], [17], [-3], [3], [16], [-11], [-19], [-13], [17], [9], [22], [-12], [-6], [-5], [-19], [18], [11], [-15], [-14], [-14], [23], [12], [12], [25], [30], [-5], [29], [-30], [8], [3], [-2], [28], [16], [20], [11], [-17], [2], [-23], [34], [-26], [-10], [22], [24], [-13], [18], [-14], [18], [-6], [34], [-15], [-14], [-10], [-20], [-7], [-11], [8], [0], [6], [14], [30], [21], [-6], [-36], [28], [-6], [20], [-28], [-29], [18], [-1], [-9], [-16], [2], [12], [-35], [-28], [2], [9], [-6], [-10], [40], [-20], [-22], [-34], [28], [46], [13], [-35], [-28], [-36], [-8], [-1], [-15], [-28], [8], [8], [-33], [-46], [-8], [17], [34], [-20], [45], [4], [32], [-15], [-26], [-44], [13], [-26], [26], [38], [8], [19], [-19], [26], [-42], [-44], [-13], [44], [-6], [-21], [2], [40], [-16], [24], [8], [26], [46], [-59], [-12], [-24], [35], [28], [-42], [47], [18], [-19], [-28], [26], [-57], [-3], [-2], [45], [-28], [-42], [-14], [39], [-58], [11], [6], [-1], [-34], [21], [3], [9], [14], [13], [-12], [33], [-48], [-26], [-36], [-8], [-45], [-3], [12], [66], [-26], [48], [1], [-45], [56], [62], [-1], [-50], [53], [40], [18], [60], [-2], [-43], [-15], [-14], [-42], [-32], [14], [-33], [-50], [40], [-42], [-17], [31], [-20], [-4], [-11], [-19], [26], [-20], [-64], [-32], [-23], [-20], [44], [52], [9], [28], [9], [-14], [-72], [8], [7], [-8], [-4], [15], [11], [-49], [-26], [-42], [10], [3], [-72], [-20], [-21], [55], [-14], [-50], [-28], [-43], [-15], [-50], [4], [-43], [-6], [11], [73], [-1], [-5], [18], [-48], [40], [-32], [2], [34], [-46], [68], [26], [-16], [55], [-56], [-60], [-67], [24], [42], [-42], [-62], [-6], [-54], [63], [-8], [14], [-71], [-46], [58], [-36], [-62], [16], [4], [-27], [15], [50], [34], [62], [-38], [-24], [6], [-45], [60], [-39], [81], [66], [27], [54], [-9], [-32], [-55], [53], [-5], [20], [-87], [-12], [-47], [50], [-37], [55], [75], [-63], [4], [20], [-75], [-79], [58], [-12], [-4], [21], [16], [-37], [-54], [74], [42], [-64], [34], [-47], [65], [34], [15], [10], [-68], [59], [-6], [62], [-22], [67], [-64], [-95], [-18], [-21], [-70], [-42], [14], [86], [28], [-87], [44], [55], [40], [87], [65], [26], [-50], [-52], [12], [-10], [17], [-63], [70], [28], [-69], [-39], [92], [8], [48], [6], [-5], [16], [84], [-89], [35], [20], [-20], [-80], [-44], [-29], [-30], [-32], [-65], [-54], [-89], [-42], [30], [66], [40], [-84], [50], [46], [41], [-46], [-44], [-66], [56], [12], [-50], [14], [33], [-28], [78], [-50], [90], [-80], [-56], [-20], [6], [-35], [-42], [-28], [-72], [-85], [-33], [-106], [-18], [-33], [-21], [-55], [63], [-78], [-104], [60], [-87], [41], [-79]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7728_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7728_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_7728_2_a_k(:prec:=1)
    chi := MakeCharacter_7728_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_7728_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_7728_2_a_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7728_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-4, 1]>,<11,R![3, 1]>,<13,R![2, 1]>,<17,R![-4, 1]>],Snew);
    return Vf;
end function;