
// Make newform 7728.2.a.h 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_h();".
// 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_h();".
// 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], [-6], [-1], [2], [2], [-2], [-10], [8], [6], [-6], [4], [-10], [8], [-12], [-2], [-8], [6], [-8], [-8], [0], [-12], [12], [2], [18], [4], [-12], [-2], [-8], [-10], [4], [10], [-4], [2], [0], [-12], [-2], [0], [6], [-18], [-4], [-12], [6], [10], [-10], [-10], [12], [-20], [6], [2], [-16], [0], [-14], [14], [-10], [26], [-6], [-24], [-2], [12], [-10], [-20], [22], [12], [8], [-18], [0], [-28], [-26], [-20], [0], [10], [20], [-10], [2], [38], [10], [16], [-16], [-10], [28], [-18], [-14], [8], [-40], [-38], [-4], [-8], [-12], [-4], [-24], [-24], [40], [-22], [-2], [28], [46], [38], [14], [-40], [22], [-24], [-46], [-16], [-26], [-38], [26], [-34], [-10], [0], [-18], [46], [18], [-30], [0], [46], [-34], [-22], [4], [-16], [-2], [14], [-6], [-4], [-14], [-44], [-48], [-16], [-38], [-22], [-28], [46], [-18], [30], [-50], [-20], [6], [-4], [36], [-36], [16], [-12], [54], [-48], [36], [-14], [12], [20], [26], [-52], [-8], [-16], [6], [20], [-6], [44], [10], [-48], [54], [14], [32], [44], [20], [-30], [12], [44], [16], [-16], [38], [-58], [40], [12], [-30], [40], [50], [28], [30], [50], [62], [0], [22], [-34], [-26], [-58], [-34], [2], [20], [44], [60], [-24], [-48], [22], [-62], [36], [42], [58], [58], [-20], [68], [2], [68], [40], [60], [10], [0], [-6], [0], [0], [42], [-34], [-52], [8], [26], [-8], [-38], [50], [24], [-46], [40], [-4], [34], [48], [-60], [-12], [4], [26], [-66], [24], [-42], [26], [36], [44], [50], [76], [6], [0], [-20], [18], [38], [-6], [6], [2], [-66], [-32], [16], [26], [-40], [-72], [-12], [48], [44], [-54], [22], [36], [-6], [-8], [72], [28], [-26], [-18], [-2], [46], [48], [80], [26], [-48], [-52], [-8], [40], [62], [42], [84], [22], [18], [-54], [-10], [-48], [0], [-74], [-72], [52], [-62], [22], [-2], [6], [30], [-38], [66], [-10], [86], [42], [48], [-84], [-32], [20], [64], [-30], [88], [-20], [-32], [-6], [-44], [54], [-48], [86], [10], [-22], [-18], [36], [-56], [0], [40], [-28], [-40], [18], [-86], [-10], [72], [16], [-70], [36], [-80], [-64], [-64], [54], [-66], [74], [10], [-60], [-24], [-24], [22], [-40], [74], [-8], [-10], [-18], [-20], [-80], [-30], [-28], [12], [52], [24], [34], [20], [10], [-50], [88], [-10], [-26], [48], [18], [-90], [24], [2], [-2], [-24], [-88], [-20], [-10], [-36], [8], [-38], [-36], [-56], [-24], [-74], [-52], [-2], [-24], [48], [-84], [72], [-82], [62], [42], [-60], [66], [30], [6], [84], [-20], [40], [32], [-58], [-12], [28], [-32], [-48], [-74], [88], [-48], [-36], [34], [8], [-90], [-88], [-56], [4], [54], [-82], [-26], [-102], [32], [-30], [-78], [96], [76], [14], [40], [-40], [86], [16], [-56], [-34], [-78], [48], [-12], [2], [-88], [-4], [74], [56], [78], [24], [24], [-4], [44], [72], [56], [-26], [98], [92]];
    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_h();".
// 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_h(: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_h();".
// 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_h( : 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;