
// Make newform 4248.2.a.b in Magma, downloaded from the LMFDB on 11 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4248_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_4248_2_a_b();".
// 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_4248_2_a_b();".
// 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_4248_a();"
function MakeCharacter_4248_a()
    N := 4248;
    order := 1;
    char_gens := [1063, 2125, 3305, 3601];
    v := [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_4248_a_Hecke(Kf)
    return MakeCharacter_4248_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]];
    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], [0], [-2], [0], [4], [0], [0], [-4], [0], [-6], [-4], [8], [12], [-6], [-4], [2], [1], [8], [-14], [10], [-10], [8], [12], [-14], [-14], [14], [16], [0], [-8], [-14], [-16], [-20], [-12], [-4], [-14], [-16], [-12], [-8], [-18], [-18], [12], [-10], [8], [14], [-6], [-8], [-22], [24], [12], [-4], [14], [-22], [-26], [12], [-12], [-2], [2], [8], [18], [-8], [2], [-6], [16], [18], [-10], [-14], [0], [-26], [-4], [24], [10], [34], [28], [-2], [-16], [6], [22], [-28], [-6], [26], [36], [32], [-12], [-14], [-16], [20], [0], [34], [-2], [-4], [4], [-14], [-8], [8], [4], [-16], [2], [-4], [-44], [-16], [20], [-18], [20], [-42], [10], [-14], [-28], [-4], [-30], [26], [24], [-16], [-12], [16], [16], [-24], [44], [14], [6], [20], [38], [6], [14], [36], [26], [-30], [-10], [0], [-16], [-38], [34], [-34], [-24], [22], [-40], [30], [-18], [-32], [42], [-18], [18], [42], [-32], [-20], [-10], [44], [-6], [42], [-10], [-56], [-38], [-42], [12], [-52], [4], [6], [28], [-42], [-14], [18], [12], [8], [0], [12], [2], [-24], [48], [-38], [58], [-6], [20], [52], [18], [54], [64], [44], [-40], [6], [-8], [62], [8], [20], [40], [0], [-46], [-10], [-48], [62], [-46], [-60], [-26], [-28], [10], [-18], [60], [-6], [22], [12], [-18], [44], [54], [32], [14], [54], [40], [54], [-56], [44], [-46], [-50], [-42], [30], [-48], [-44], [38], [-2], [40], [64], [-12], [-26], [-24], [-12], [6], [-8], [-20], [40], [40], [-28], [48], [4], [-64], [-6], [68], [-18], [26], [46], [-46], [42], [4], [18], [16], [-26], [-16], [-30], [36], [42], [20], [-20], [0], [-22], [-58], [30], [-52], [-74], [10], [-68], [-34], [50], [26], [-54], [60], [-60], [10], [10], [48], [10], [54], [-18], [4], [-26], [8], [16], [-18], [8], [-22], [8], [-68], [22], [10], [-36], [66], [20], [-24], [-68], [-6], [-12], [-30], [54], [16], [0], [-6], [0], [-44], [-84], [10], [30], [48], [-14], [-36], [-54], [54], [26], [40], [-36], [-28], [-38], [60], [16], [-16], [-68], [-6], [18], [16], [-74], [62], [-34], [60], [-54], [-18], [76], [-64], [-18], [42], [-64], [56], [-22], [50], [-40], [-28], [-90], [24], [74], [28], [60], [12], [12], [50], [-62], [-42], [-80], [-58], [-50], [26], [80], [66], [60], [52], [-44], [-72], [42], [86], [26], [66], [-92], [-86], [34], [-60], [-56], [72], [78], [-32], [48], [78], [68], [-72], [-94], [30], [8], [-2], [36], [-50], [-84], [-78], [16], [42], [76], [-44], [10], [30], [-34], [-66], [46], [-88], [38], [16], [-60], [80], [22], [-16], [-92], [-66], [-90], [60], [-26], [72], [-86], [-56], [20], [76], [-90], [-70], [-90], [-32], [-32], [18], [-68], [-52], [-100], [74], [68], [-26], [38], [-68], [64], [50], [-30], [36], [-8], [-42], [-18], [26], [38], [-18], [-40], [58], [-38], [68], [48], [-4], [-38]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4248_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4248_2_a_b();".
// 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_4248_2_a_b(:prec:=1)
    chi := MakeCharacter_4248_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(2999) - 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_4248_2_a_b();".
// 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_4248_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4248_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![2, 1]>,<7,R![0, 1]>,<11,R![-4, 1]>],Snew);
    return Vf;
end function;