
// Make newform 7728.2.a.i 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_i();".
// 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_i();".
// 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], [0], [6], [-6], [0], [1], [6], [-8], [10], [-6], [8], [0], [2], [-4], [10], [-8], [0], [10], [16], [8], [-6], [-14], [14], [0], [-8], [10], [-6], [8], [20], [-6], [-4], [18], [-8], [-14], [-4], [-8], [14], [-20], [2], [8], [18], [14], [8], [4], [24], [0], [-14], [-22], [16], [2], [0], [2], [8], [14], [-32], [-2], [18], [16], [-14], [12], [32], [26], [-26], [28], [-14], [-12], [30], [-14], [0], [-16], [26], [-32], [16], [10], [22], [-30], [10], [24], [26], [24], [-14], [-8], [36], [18], [2], [-2], [8], [-24], [0], [-16], [20], [-4], [16], [-18], [-30], [40], [22], [36], [34], [0], [-38], [0], [2], [-36], [18], [24], [-38], [-8], [34], [34], [-32], [0], [2], [8], [-32], [-34], [0], [-14], [-14], [-30], [-4], [20], [-30], [-22], [-8], [8], [50], [28], [24], [40], [2], [-22], [-14], [42], [24], [-30], [42], [44], [-42], [40], [8], [-18], [-8], [30], [-38], [4], [16], [22], [-30], [36], [-24], [40], [24], [24], [-46], [50], [26], [28], [-22], [40], [-24], [-30], [-16], [-16], [-10], [-6], [-18], [24], [-34], [8], [26], [-16], [26], [-60], [-42], [32], [-14], [48], [-40], [-2], [-6], [-24], [-54], [-18], [8], [-22], [-64], [-14], [12], [-32], [38], [16], [42], [-14], [-46], [-30], [0], [18], [-32], [22], [-38], [24], [-18], [-64], [-60], [-70], [-12], [-30], [30], [48], [-56], [24], [26], [-64], [34], [24], [-2], [-58], [-32], [2], [-40], [-36], [22], [-6], [-8], [48], [-60], [-26], [-24], [-72], [10], [64], [-8], [-14], [-54], [-68], [-72], [40], [-28], [-48], [-2], [2], [-16], [-48], [-8], [40], [-48], [58], [-30], [-8], [-62], [-42], [-20], [2], [-8], [38], [42], [-56], [-48], [70], [-70], [66], [-8], [18], [10], [40], [-18], [78], [48], [82], [-56], [34], [72], [-36], [-50], [42], [0], [8], [-48], [-48], [34], [36], [-32], [-86], [26], [-40], [-30], [-54], [24], [-6], [-72], [-74], [-38], [-80], [46], [-44], [-28], [74], [-30], [0], [68], [0], [-30], [52], [-6], [56], [-26], [-72], [82], [66], [68], [48], [18], [-52], [40], [-22], [-30], [40], [50], [-42], [0], [-6], [2], [-64], [-84], [48], [-38], [54], [46], [16], [-92], [24], [36], [34], [-14], [10], [-40], [-26], [-70], [-34], [8], [-46], [60], [-2], [-44], [56], [66], [44], [10], [70], [-64], [-14], [-54], [80], [0], [18], [0], [-62], [10], [-56], [-32], [20], [-70], [-42], [-40], [34], [92], [-12], [24], [90], [80], [-34], [-12], [-88], [-54], [-6], [-22], [50], [-22], [16], [82], [96], [-64], [-72], [38], [16], [40], [50], [-42], [-84], [-28], [96], [98], [-88], [-62], [-40], [102], [22], [-14], [64], [18], [46], [48], [58], [34], [-24], [-100], [66], [102], [24], [8], [-6], [30], [-40], [16], [-38], [32], [66], [42], [-64], [-100], [58], [38], [24], [-6], [12], [8], [-86], [-48], [-4], [72], [6], [42], [26], [70], [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_i();".
// 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_i(: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_i();".
// 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_i( : 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![0, 1]>,<13,R![-6, 1]>,<17,R![6, 1]>],Snew);
    return Vf;
end function;