
// Make newform 5202.2.a.j in Magma, downloaded from the LMFDB on 27 December 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5202_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_5202_2_a_j();".
// 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_5202_2_a_j();".
// 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_5202_a();"
function MakeCharacter_5202_a()
    N := 5202;
    order := 1;
    char_gens := [2891, 4627];
    v := [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_5202_a_Hecke(Kf)
    return MakeCharacter_5202_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 := [[1], [0], [0], [-2], [0], [2], [0], [-4], [-6], [0], [10], [-8], [6], [-4], [-12], [-6], [12], [-8], [-4], [6], [-2], [10], [-12], [18], [-14], [-6], [-4], [0], [-20], [-6], [8], [0], [6], [-8], [6], [8], [-10], [-20], [18], [0], [12], [-8], [12], [-14], [-12], [-2], [-8], [-28], [12], [14], [-30], [12], [10], [-12], [-18], [-12], [24], [-16], [4], [-30], [16], [-18], [20], [18], [-14], [12], [20], [22], [-12], [26], [18], [0], [10], [-22], [28], [12], [30], [16], [-18], [38], [-36], [-22], [-30], [14], [-26], [36], [6], [-10], [-30], [20], [-12], [6], [10], [36], [-32], [-6], [6], [6], [20], [16], [-8], [42], [12], [6], [-20], [2], [12], [-18], [0], [22], [10], [-34], [18], [28], [-4], [30], [4], [-36], [-24], [-36], [-22], [34], [-12], [-12], [-32], [42], [-20], [6], [8], [26], [-4], [42], [-2], [-34], [54], [-22], [6], [-32], [-6], [6], [4], [-36], [34], [48], [-10], [-18], [28], [6], [-28], [-48], [4], [-54], [20], [-18], [40], [6], [-16], [-18], [-10], [12], [24], [-54], [-4], [-12], [30], [6], [-2], [-8], [-38], [60], [36], [-22], [30], [38], [8], [42], [4], [60], [20], [14], [44], [-24], [-32], [54], [-12], [-6], [-56], [-28], [34], [-54], [-50], [36], [-4], [6], [36], [42], [-26], [64], [-54], [0], [-36], [10], [-10], [-46], [-60], [-18], [44], [-60], [-30], [-28], [-14], [-66], [46], [36], [-18], [10], [-28], [-42], [54], [-6], [-10], [58], [18], [10], [60], [-22], [-6], [42], [56], [12], [74], [-20], [44], [-6], [-4], [24], [-14], [12], [0], [-48], [12], [20], [32], [38], [42], [-30], [-14], [-12], [-52], [0], [-58], [-54], [-36], [46], [-18], [60], [16], [4], [-60], [26], [-62], [-12], [40], [-44], [-42], [-28], [-30], [-6], [-20], [6], [4], [-76], [-70], [-4], [14], [20], [-12], [50], [-58], [-60], [-60], [-14], [6], [26], [-32], [-24], [34], [-12], [8], [18], [24], [-60], [-42], [-72], [64], [-24], [-28], [18], [84], [20], [-58], [30], [-26], [-60], [52], [34], [-36], [-32], [72], [62], [-54], [0], [-42], [-76], [48], [-10], [-36], [90], [-74], [-30], [-44], [-14], [-42], [44], [-66], [-70], [-56], [-8], [-18], [-36], [-44], [84], [-26], [-12], [-92], [-24], [-70], [-6], [-26], [-52], [-46], [54], [-12], [44], [-54], [-84], [-56], [-4], [-6], [60], [-4], [10], [78], [10], [-58], [-90], [-48], [-60], [-18], [36], [64], [42], [84], [60], [-52], [-10], [0], [-16], [58], [-12], [64], [42], [-6], [32], [-68], [84], [-18], [-46], [18], [26], [72], [-6], [22], [18], [-44], [6], [80], [62], [52], [-24], [-2], [-60], [-84], [-28], [60], [-86], [80], [-54], [-80], [-102], [-56], [-66], [-4], [54], [-42], [-14], [-34], [-18], [-4], [-36], [-26], [6], [-36], [-80], [62], [-60], [-30], [34], [-18], [12], [-54], [-68], [30], [36], [-14], [-42], [-24], [-54], [-52], [-66]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5202_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5202_2_a_j();".
// 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_5202_2_a_j(:prec:=1)
    chi := MakeCharacter_5202_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_5202_2_a_j();".
// 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_5202_2_a_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5202_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![0, 1]>,<7,R![2, 1]>,<23,R![6, 1]>,<47,R![12, 1]>],Snew);
    return Vf;
end function;