
// Make newform 7728.2.a.q in Magma, downloaded from the LMFDB on 25 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_q();".
// 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_q();".
// 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], [0], [1], [2], [-6], [-6], [6], [-1], [2], [-8], [8], [6], [-10], [8], [-4], [-8], [0], [-14], [0], [-2], [8], [-14], [-10], [-2], [-6], [20], [-18], [-4], [-2], [0], [-8], [2], [-8], [20], [-16], [-12], [24], [-16], [14], [0], [0], [-20], [10], [-14], [-16], [12], [8], [6], [16], [-6], [8], [-18], [-30], [-18], [24], [-2], [-16], [10], [30], [2], [24], [12], [0], [-10], [-22], [20], [-22], [16], [-2], [-14], [-20], [0], [-16], [2], [-16], [0], [2], [10], [-22], [10], [8], [4], [-2], [8], [-24], [-10], [14], [30], [-8], [-38], [-24], [-16], [-24], [40], [-28], [6], [-38], [-34], [42], [-28], [24], [6], [-22], [34], [-26], [-40], [30], [-24], [30], [8], [-12], [-2], [18], [-4], [10], [10], [32], [-34], [6], [32], [14], [-24], [-20], [48], [24], [40], [-8], [-8], [-4], [0], [-32], [-36], [28], [-6], [-2], [-36], [26], [44], [26], [40], [46], [16], [6], [-18], [32], [50], [10], [28], [8], [34], [-26], [-20], [0], [22], [4], [-48], [30], [-22], [0], [24], [-14], [16], [54], [2], [-60], [16], [-58], [38], [-46], [-18], [14], [-32], [-30], [-40], [-18], [-56], [54], [44], [-12], [32], [2], [-46], [-38], [12], [-4], [22], [-6], [-10], [-16], [-50], [-44], [6], [-14], [-18], [-2], [50], [40], [30], [8], [-48], [-40], [22], [42], [30], [54], [16], [-68], [6], [-20], [-50], [-6], [20], [46], [-48], [62], [-8], [34], [8], [-34], [46], [4], [-30], [12], [-12], [70], [38], [56], [-20], [64], [-22], [-62], [68], [-58], [-38], [-20], [-10], [32], [28], [0], [74], [52], [64], [-26], [-54], [40], [-56], [-30], [70], [40], [20], [22], [0], [58], [-54], [-72], [-4], [78], [78], [-22], [68], [-26], [-66], [-24], [34], [-22], [48], [-2], [38], [-38], [62], [-58], [-14], [4], [66], [-56], [-60], [-18], [74], [-42], [-16], [44], [12], [0], [20], [24], [46], [72], [-48], [-34], [12], [-78], [54], [30], [74], [48], [-24], [-70], [-84], [8], [46], [-48], [64], [44], [-70], [2], [-48], [-64], [24], [-50], [-8], [-36], [10], [36], [-36], [-46], [-36], [-24], [66], [-50], [-46], [38], [42], [-8], [50], [-50], [-74], [72], [0], [40], [34], [-42], [16], [24], [78], [-12], [-64], [30], [-54], [-4], [-10], [34], [66], [-8], [-48], [24], [34], [8], [52], [56], [-44], [38], [-46], [-64], [60], [26], [8], [-22], [2], [16], [28], [54], [56], [42], [-72], [86], [-6], [-16], [62], [-80], [20], [96], [-72], [-44], [-50], [-16], [60], [66], [-46], [-78], [52], [-14], [-80], [-18], [-66], [72], [-96], [-18], [10], [44], [74], [-58], [76], [-8], [4], [-26], [-60], [42], [14], [86], [10], [-18], [-24], [-6], [-22], [-16], [-4], [-98], [50], [-36], [-98], [62], [6], [-78], [-22], [-58], [0], [-64], [42], [-36], [-88], [28], [-24], [8], [-22], [42], [10], [42], [48], [-88], [-2], [62], [68], [-12], [-54], [-38], [70], [22], [52]];
    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_q();".
// 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_q(: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_q();".
// 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_q( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7728_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![0, 1]>,<11,R![-2, 1]>,<13,R![6, 1]>,<17,R![6, 1]>],Snew);
    return Vf;
end function;