
// Make newform 7728.2.a.t in Magma, downloaded from the LMFDB on 18 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_t();".
// 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_t();".
// 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], [2], [-1], [4], [-4], [-4], [2], [-1], [-6], [-6], [-2], [-10], [-8], [-10], [-6], [12], [-10], [-8], [4], [-2], [8], [6], [0], [-8], [16], [-12], [20], [2], [-6], [-20], [12], [6], [16], [6], [-4], [10], [-12], [18], [16], [-4], [-2], [-16], [6], [-26], [12], [-4], [14], [10], [6], [6], [-20], [12], [-26], [2], [-16], [0], [26], [14], [14], [18], [-6], [0], [30], [-20], [30], [4], [-10], [-28], [-16], [-2], [-16], [-28], [6], [28], [-32], [-6], [28], [-2], [-30], [22], [10], [-32], [-16], [14], [4], [14], [18], [-24], [32], [-22], [-4], [-16], [28], [-28], [-24], [-24], [0], [34], [30], [-44], [14], [22], [-10], [40], [38], [-16], [-30], [0], [22], [2], [-38], [6], [14], [32], [-42], [-10], [-14], [-22], [-40], [-50], [30], [-6], [44], [-8], [30], [-18], [-6], [-20], [34], [-52], [-16], [0], [-6], [26], [-28], [-18], [6], [-18], [30], [20], [-2], [20], [-36], [20], [0], [12], [-10], [-8], [4], [-14], [20], [-20], [-6], [28], [24], [-48], [-10], [4], [10], [-28], [-30], [-24], [-42], [-10], [0], [20], [-52], [-30], [-36], [36], [-8], [48], [-42], [-50], [0], [20], [-22], [24], [-46], [-44], [30], [-14], [-18], [32], [6], [-34], [46], [-26], [30], [-14], [12], [12], [-36], [32], [-40], [6], [-62], [44], [-6], [-54], [18], [-12], [52], [-14], [-60], [24], [4], [10], [-56], [58], [-64], [48], [42], [14], [44], [32], [42], [72], [34], [50], [72], [-46], [40], [68], [-62], [24], [-60], [20], [-52], [-38], [-10], [40], [-10], [34], [20], [12], [-62], [36], [-58], [-56], [4], [-6], [70], [42], [-10], [-38], [14], [16], [-16], [-38], [48], [8], [36], [-48], [4], [-54], [14], [-60], [26], [-56], [48], [52], [6], [30], [6], [-50], [56], [0], [82], [40], [28], [-24], [56], [62], [66], [-68], [-58], [34], [-22], [6], [-16], [32], [70], [-32], [36], [-46], [-42], [22], [-42], [-18], [42], [18], [70], [-10], [42], [-64], [-4], [-72], [44], [-16], [18], [8], [4], [-80], [58], [-52], [54], [0], [54], [-86], [10], [-42], [76], [-72], [64], [-48], [-44], [56], [-78], [2], [-10], [72], [8], [-30], [-76], [-16], [-8], [0], [-42], [30], [82], [-14], [-4], [-24], [0], [22], [-80], [10], [-40], [-26], [54], [-4], [-32], [50], [76], [-60], [-52], [-56], [-30], [92], [-54], [22], [40], [-74], [-10], [-80], [-14], [54], [-8], [66], [30], [8], [-48], [-28], [-74], [60], [-72], [-54], [-60], [-16], [24], [-10], [12], [-18], [-32], [48], [-68], [-64], [62], [-18], [34], [92], [34], [86], [-42], [28], [-76], [-88], [48], [-26], [-44], [-44], [-24], [0], [-58], [88], [-72], [92], [26], [32], [86], [40], [32], [36], [62], [46], [86], [18], [-24], [50], [90], [104], [44], [-98], [-72], [-8], [78], [-88], [40], [-34], [-62], [32], [-36], [2], [-72], [-92], [-54], [-48], [-98], [24], [48], [-60], [-36], [16], [-48], [-26], [34], [68]];
    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_t();".
// 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_t(: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_t();".
// 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_t( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7728_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-2, 1]>,<11,R![-4, 1]>,<13,R![4, 1]>,<17,R![4, 1]>],Snew);
    return Vf;
end function;